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ELEMENTARY TEXT-BOOK
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ALG^RA
AN ELEMENTAEY TEXT-BOOK
FOR THE
HIGHER CLASSES OF SECONDARY SCHOOLS
AND FOR COLLEGES
G. CHRYSTAL, MA., LL.D.
HONORARY FELLOW OF CORPUS CHRISTl OOLLEQE, CAMBRIDGE;
PROFESSOR or MAXUKMATICS IN THE UNIVEUSITV Off ElUNBURQH
PART II.
SECOND EDITION
LONDON
ADAM AND CHARLES BLACK
1906
First Edition published November, 1889.
Second Edition published March, 1900 ; reprinted July, 1906.
PEEFACE TO THE SECOND EDITION
OF PART II.
The present edition of this volume has been carefully
revised and corrected throughout. The principal alterations
will be found in the Theory of Series; which has been
developed a little in some places, with a view to rendering
it more useful to students proceeding to study the Theory
of Functions. In the interest of the same class of readers,
I have added to the chapter on limits a sketch of the
modern theory of irrational quantity, one of the most
important parts of the purely Arithmetical Theory of
Algebraic Quantity, which forms, as the fashion of mathe-
matical thinking now runs, the most widely accepted basis
for the great structure of Pure Analysis reared by the
masters of our science.
I am indebted for proof-reading and for useful criticism
to my friends Prof. G. A. Gibson and Mr. C. Tweedie, B.Sc.
It is but right, however, to add that the careful and
intelligent readers of the Pitt Press have rendered the
work of correcting the proofs of this volume more of a
sinecure than it often is when mathematical works are
in question.
G. CHRYSTAL.
Edinburgh, 3rd March, 1900.
PEEFACE TO FIRST EDITION.
The delay in the appearance of this volume finds an apology
partly in circumstances of a private character, partly in
public engagements that could not be declined, but most of
all in the growth of the work itself as it progressed in my
hands. I have not, as some one prophesied, reached ten
volumes ; but the present concluding volume is somewhat
larger and has cost me infinitely more trouble than I
expected.
The main object of Part II. is to deal as thoroughly as
possible with those parts of Algebra which form, to use
Euler's title, an Introductio in Analysin Infinitorum. A
practice has sprung up of late (encouraged by demands for
premature knowledge in certain examinations) of hurrying
young students into the manipulation of the machinery of
the Differential and Integral Calculus before they have
grasped the preliminary notions of a Limit and of an
Infinite Series, on which all the meaning and all the uses
of the Infinitesimal Calculus are based. Besides being to
a large extent an educational sham, this course is a sin
against the spirit of mathematical progress. The methods
of the Differential and Integral Calculus which were once
an outwork in the progress of pure mathematics threatened
for a time to become its grave. Mathematicians had fallen
PREFACE Vll
into a habit of covering their inability to solve many-
particular problems by a vague wave of the hand towards
some generality, like Taylor's Theorem, which was sup-
posed to give "an account of all such things," subject only
to the awkwardness of practical inapplicability. Much
has happened to remove this danger and to reduce dfdx
and jdx to their proper place as servants of the pure
mathematician. In particular, the brilliant progress on the
continent of Function-Theory in the hands of Cauchy,
Riemann, Weierstrass, and their followers has opened for us
a prospect in which the symbolism of the Differential and
Integral Calculus is but a minor object. For the proper
understanding of this important branch of modern mathe-
matics a firm grasp of the Doctrine of Limits and of the
Convergence and Continuity of an Infinite Series is of much
greater moment than familiarity with the symbols in which
these ideas may be clothed. It is hoped that the chapters
on Inequalities, Limits, and Convergence of Series will help
to give the student all that is required both for entering
on the study of the Theory of Functions and for rapidly
acquiring intelligent command of the Infinitesimal Calculus.
In the chapters in question, I have avoided trenching on
the ground already occupied by standard treatises: the
subjects taken up, although they are all important, are
either not treated at all or else treated very perfunctorily
in other English text-books.
Chapters xxix. and xxx. may be regarded as an
elementary illustration of the application of the modern
Theory of Functions. They are intended to pave the way
Vlll PREFACE
for the study of the recent works of continental mathe-
maticians on the same subject. Incidentally they contain
all that is usually given in English works under the title of
Analytical Trigonometry. If any one should be scandalised
at this traversing of the boundaries of English examination
subjects, I must ask him to recollect that the boundaries in
question were never traced in accordance with the principles
of modem science, and sometimes break the canon of
common sense. One of the results of the old arrangement
has been that treatises on Trigonometry, which is a geometri-
cal application of Algebra, have been gradually growing into
fragments more or less extensive of Algebra itself: so that
Algebra has been disorganised to the detriment of Trigono-
metry ; and a consecutive theory of the elementary functions
has been impossible. The timid way, oscillating between ill-
founded trust and unreasonable fear, in which functions of a
complex variable have been treated in some of these manuals
is a little discreditable to our intellectual culture. Some
expounders of the theory of the exponential function of an
imaginary argument seem even to have forgotten the obvious
truism that one can prove no property of a function which
has not been defined. I have concluded chapter xxx. with
a careful discussion of the Reversion of Series and of the
Expansion in Power-Series of an Algebraic Function —
subjects which have never been fully treated before in an
English text-book, although we have in Frost's Curve Tracing
an admirable collection of examples of their use.
The other innovations call for little explanation, as they
^ipi inerely at greater completeness on the old lines. In
PREFACE IX
the chapter on Probability, for instance, I have omitted
certain matter of doubtful soundness and of questionable
utility; and filled its place by what I hope will prove a
useful exposition of the principles of actuarial calculation.
I may here give a word of advice to young students
reading my second volume. The matter is arranged to
facilitate reference and to secure brevity and logical
sequence; but it by no means follows that the volume
should be read straight through at a first reading. Such
an attempt would probably sicken the reader both of
the author and of the subject. Every mathematical book
that is worth anything must be read "backwards and
forwards," if I may use the expression. I would modify the
advice of a great French mathematician* and say, "Go on,
but often return to strengthen your faith." When you come
on a hard or dreary passage, pass it over ; and come back to
it after you have seen its importance or found the need for
it further on. To facilitate this skimming process, I have
given, after the table of contents, a suggestion for the course
of a first reading.
The index of proper names at the end of the work will
show at a glance the main sources from which I have drawn
my materials for Part II. Wherever I have, consciously
borrowed the actual words or the ideas of another writer
I have given a reference. There are, however, several
works to which I am more indebted than appears in the
bond. Among these I may mention, besides Cauchy's
♦ "Allez en avant, et la I'oi vous viendia."
X PREFACE
Analyse Algdhrique, Serret's Aigihre Supirieure, and Schlo-
milch's Algebraische Analysis, which have become classical,
the more recent work of Stolz, to which I owe many indica-
tions of the sources of original information — a kind of help
that cannot be acknowledged in footnotes.
I am under personal obligations for useful criticism, for
proof-reading, and for help in working exercises, to my
assistant, Mr. R. E. Allardice, to Mr. G. A. Gibson, to
Mr. A. Y. Fraser, and to my present or former pupils —
Messrs. B. B. P. Brandford, J. W. Butters, J. Crockett,
J. GOODWILLIE, C. TWEEDIE.
In taking leave of this work, which has occupied most
of the spare time of five somewhat busy years, I may be
allowed to express the hope that it will do a little in a
cause that I have much at heart, namely, the advancement
of mathematical learning among English-speaking students
of the rising generation. It is for them that I have worked,
remembering the scarcity of aids when I was myself a
student; and it is in their profit that I shall look for my
reward.
G. CHRYSTAL.
Edinbubqh, let November 1889.
CONTENTS.
The principal technical terms are printed in italics in tJie
following table.
CHAPTER XXIII.
PERMUTATIONS AND COMBINATIONS.
PAGE
Definition of r-permutation and r-combination .... 1
Methods of Demonstration 2
Permutations 2-6
Number of r-permutations of n letters 2
Kramp's Notation for Factorial-w (n!) 4
Linear and Circular Permutations 4
Number of r-permutations with repetition 4
Permutations of letters having groups alike .... 5
Examples 0
Combinations 6-12
Combinations from Sets 6
Number of r-combinations of n letters 6
Various properties of ^O^ — Vandermonde's Theorem . . 8-9
Combinations when certain letters are alike .... 10
Combinations with repetition 10
Properties of „Hy — Number of r-ary Products .... 12
Exercises 1 12
Binomial and Multinomial Theorems 14-18
Examples 16
Exercises II. . 18
Examples of the application of the Law of Distribution . . 21-22
Distributions and Derangements 22-25
Distribution Problem 22
Derangement Problem 24
Subfactorial n (n\) defined 25
Theory of Substitutions 25-32
Notation for Substitutions 26
Order and Group .......... 27
Cyclic Substitutions and Transpositions 27
62
xu
CONTENTS
Cycles of a Substitution 27
Decomposition into Transpositions 28
Odd and even Substitutions ....... 29
Exercises III 32
Exercises lY 33
CHAPTER XXIV.
GENERAL THEORY OF INEQUALITIES.
Definition of Algebraic Inequality 35
Elementary Theorems 36
Examples 38
Derived Theorems 41-50
A Mean-Theorem for Fractions 41
{xP-l)lpx{x'i-l)lq 42
mj;"»-i(x-l)^(a;"'-l)^m(a;-l) 43
jna»»-i(a-6)^a'"-b'»^m6'"-i(a-i^) 45
Inequality of Arithmetic and Geometric Means ... 46
Spa^/Sp > < (2pa/2^)'» 48
Exercises V 50
Applications to Maxima- and Minima-Theorems .... 52-64
Fundamental Theorem 52
Eeciprocity Theorem 53
Ten Theorems deduced 53-59
Grillet's Method 69
Method of Increments 61
Purkiss's Theorem 61
Exercises VI 63
CHAPTER XXV.
LIMITS.
Definition of a Limiting Value and Corollaries
Enumeration of Elementary Indeterminate Forms
Extension of Fundamental Operations to Limiting Values
Limit of a Sum ....
Limit of a Product
Limit of a Quotient .
Limit of a Function of Limits .
Limiting forms for Rational Functions
Forms 0/0 and oo/oo .
Fundamental Algebraic Limit L (x"' - l)/{x - 1) when x = ]
Examples— Li"" (x -t- Ij/^x, LVxlVx, when a; = Qo, &o.
66
69
69
70
70
71
71
72
72
74
76
CONTENTS XHl
PAOE
Exponential Limits 77-81
X((l + l/x)* when x = co, Napierian Base 77
L (l + x)^, L {1+ylxf, L {l + xyjV'', L (a^-l)/a; . . 79
j6=0 x=«> K=0 a;=0
Exponential and Logarithmic Inequalities .... 80-81
Euler's Constant 81
General Limit Theorems 82
L{/(x) }*<*)= {L/(x)}^'''W 82
L{f{x + l)-f{x)}=Lf{x)lx^\henx = x 83
Z/(x + l)//(a;) = L{/(x)}V^ when x = oo 84
Exponential Limits Resumed 85
L a'lx, L log^xlx, L xlogg^x 85
Examples — L x'^jnl, L m(wi-l) . . . {m-n + l)lnl . . 86
n=co n=ao
L a;*=l 87
a;=+0
General Theorem regarding the form 0" 88
Cases where O^ + l 88
Forms 00° and 1°° 89
Trigonometrical Limits 89
Fundamental Inequalities 90
Lsinxlx, Ltanxjx, when x = 0 91
L
(sin-/-), ifcos-j, L (tan -/-), when x^ao . 91
Limit of the Sum of an Infinite Number of Infinitely Small Terms 92
i(ir + 2'-+. . . + ?i'-)/;i'-+i 92
Dirichlet's Theorem 94
Geometrical Applications 95
Notion of a Limit in General, Abstract Theory of Irrational Numbers 97-109
The Rational OnefSld 99
Dedekind's Theory of Sections 99
Systematic Eepresentation of a Section 101
Cantor's Convergent Sequence 103
Null Sequence 105
Arithmeticity of Irrational Onefold 105
General Definition of a Limit 107
Condition for Existence of a Limit 109
Exercises VII. . 110
CHAPTER XXVI.
CONVERGENCE OF INFINITE SERIES AND OF INFINITE PRODUCTS.
Definition of the terms Convergent, Divergent, Oscillating, Non-
Convergent 114
Necessary and Sufficient Conditions for Convergency . . . 115
Residue and Fariial Residue 117
XIV CONTENTS
PAGE
Four Elementary Comparison Theorems 118
Ratio of Convergence 120
Absolutely Convergent and Semi-Convergent Series . . . 120
Special Tests of Convergency for Series of Positive Terms . . 120-132
Lw„V»<>l 121
i'«n+i/"n<>l • • •. ■'^^^
Examples — Integro-Geometric, Logarithmic, Exponential, Bi-
nomial Series 122-123
Cauchy's Condensation Test 123
Logarithmic Criteria, first form 125
Logarithmic Scale of Convergency 128
Logarithmic Criteria, second form 129
Examples — Hypergeometric and Binomial Series . . . 130-132
Historical Note 132
Semi-Convergent Series 133-137
Example of Direct Discussion 134
M1-M2 + W3- 135
Trigonometrical Series 135
Abel's Inequality 136
Convergence of a Series of Complex Terms .... 137
Necessary and Sufficient Condition for Convergency . . 138
Convergence of the Series of Moduli sufficient .... 138
Examples — Exponential and Logarithmic Series, &6. . . 138
Application of the Fundamental Laws of Algebra to Infinite Series 139-143
Law of Association 139
Law of Commutation 140
Addition of Infinite Series 141
Law of Distribution 142
Theorem of Cauchy and Mertens . . . • . . . 142
Uniformity and Non-Uniformity in the Convergence of Series whose
terms are functions of a variable 143-148
Uniform and Non-Uniform Convergence 144
Continuity of the sum of a Uniformly Converging Series . 146
Du Bois-Reymond's Theorem 148
Special Discussion of the Power Series 148-157
Condition for Absolute and Uniform Convergency of Power Series 149
Circle and Radius of Convergence 149
Cauchy's Rules for the Radius of Convergence .... 150
Behaviour of Power Series on the Circle of Convergence . . 151
Abel's Theorems regarding Continuity at the Circle of Convergence 152
Principle of Indeterminate Coefficients 156
Infinite Products 157-168
Convergent, Divergent, and Oscillating Products . . . 158
Discussion by means of 2log(l-F«J 158
Criteria from 2(/„ 159
Independent Criteria , , . . 160
CONTENTS XV
PAOB
Convergence of Complex Products 160
General Properties of Infinite Products 161
Estimation of the Residue of a Series or Product . . . 168
Convergence of Double Series 171-182
Four ways of Summation 172
Double Series of Positive Terms 174
Cauchy's Test for Absolute Convergeney 177
Examples of Exceptional Cases 179
Imaginary Double Series 181
General Theorem regarding Double Power-Series . . . 182
Exercises VIII 182
CHAPTER XXVII.
BINOMIAL AND MULTINOMIAL SERIES FOR ANY INDEX.
Binomial Series 186-199
Determination of Coefficients, validity being assumed . . 186
Euler's Proof 188
Addition Theorem for the Binomial Series .... 189
Examples 192
Ultimate Sign of the Terms 193
Integro-Binomial Series 194
Examples 196
Exercises IX 199
Series deduced by Expansion of Rational Functions . . . 200-210
Expression of a" + (3" and (a"+i - |3"+i)/(a - j3) in terms of afi and
a + /3, and connected series 201
Sum and Number of r-ary Products 205
Examples 208
Exercises X 210
Multinomial Series 213-215
Numerical Approximation by Binomial Series .... 215-219
Numerically Greatest Term 216
Limits for the Residue 217
Exercises XI 219
CHAPTER XXVIII.
EXPONENTIAL AND LOGARITHMIC SERIES.
Exponential Series 221-228
Determination of Coefficients, validity being assumed . . 221
Deduction from Binomial Theorem 222
Calculation of e 224
Cauchy's Proof 226
Addition Theorem for the Exponential Series .... 227
XVI
CONTENTS
PAGE
Bernoulli's Numbers 228-233
Expansions of x/(l-c-'^), x(e^ + e-^)l{e'' - e-^), &c. . .229,232
Bernoulli's Expression f or F + 2'' + . . . + n'' . . . . 233
Summations by means of Exponential Theorem .... 233-236
Integro-Exponential Series 233
Examples 234
Exercises XII. 236
Logarithmic Series 237-251
Expansion of log (1 + a;) 238
Derived Expansions 239
Calculation of log 2, log 3, &c 241
Factor Method for calculating Logarithms .... 243
First Difference of log x 245
Summations by Logarithmic Series 245-250
20(n)x"/(n + a)(n + 6) 246
Examples — Certain Semi-Convergent Series, &c. . . . 248
Inequality and Limit Theorems 250
Exercises XIII 251
CHAPTER XXIX.
SUMMATION OF THE FUNDAMENTAL POWER-SERIES FOR COMPLEX
VALUES OF THE VARIABLE.
Preliminary Matter
Definition and Properties of the Circular Functions
Evenness, Oddness, Periodicity .
Graphs of the Circular Functions
Addition Formulce for the Circular Functions
Inverse Circular Functions .
Multiple-valuedness
Principal and other Branches
Inversion of w = 2" and wP=z'^
Circulo-Spiral Graphs for w — z^
Multiplicity and Continuity of ^^to
Rieviann's Surface ....
Principal and other Branches of ^w .
Circulo-Spiral Graphs for w'^=z*
Principal and other Values of gjio^ .
Exercises XIV.
Geometric and Integro-Geometric Series
Sr"cos(a-J-?i^), c&c
FormulsB connected with Demoivre's Theorem and the Binomial
Theorem for an Integral Index
Generalisation of the Addition Theorems for the Circular Functions
Expansions of cos nO, sin ndlsin 0, &c., in powers of sin d or cos 6
Expression of cos'^^sin"^ in the form SapCosj?^ or lap sin p0
254-272
254-262
255
256
258
259
260
260
262-271
264
265
265
267
269
270
271
272
273
274-279
275
276
277
CONTENTS
XVll
Exercises XV
Expansion of cos 0 and sin 6 in powers of 0
Exercises XVI.
Binomial Theorem for a Complex Variable
Most general case of all (Abel) .
Exponential and Logarithmic Series for a Complex Variable
Definition of Exp z . . . .
Exp (a; + 7/?) = e^ (cos 7/ + i sin y)
Graphic Discussion of ic = Expz
Imaginary Period of Exp^
Log i<; = log 1 10 I + 1 amp ty
Principal and other Branches of liogw
Definition of Exp^z ....
Addition Theorem for Log z
Expansion of (Log(l + z)
Generalisation of the Circular Functions
General Definitions of Cos z, Sin z, &c.
Euler's Exponential FormuljB for Cosz and Sin 2
Properties of the Generalised Circular Functions
Introduction of the Hyperbolic Functions
Expressions for the Hyperbolic Functions
Graphs of the Hyperbolic Functions .
Inverse Hyperbolic Functions
Properties of the General Hyperbolic Functions
Inequality and Limit Theorems .
Geometrical Analogies between the Circular and
Functions
Gudermanidan Function ....
Historical Note
Exercises XVII
Graphical Discussion of the Generalised Circular
Cos [x + yi)
Sin (a; + 2/1)
T&n(x + yi)
Graphs oi f{x + yi) and Iff {x + yi)
General Theorem regarding Orthomorphosis
Exercises XVIII
Special Applications to the Circular Functions
Series derived from the Binomial Theorem .
Series for cosm^ and sinm</) (m not integral)
Expansion of sin~i x, Quadrature of the Circle
Examples — Series from Abel, Ac.
Series derived from the Exponential Series .
Series derived from the Logarithmic Series .
Sin^-isin2^ + isin3e-. . . = ^5
Eemarkable Discontinuity of this last Series
Functions
the
Hyperbolic
PAGE
279
280-283
284
285-288
287
288-297
288
290
290
292
293
293
294
295
296
207-313
297
298
299
300-313
300
301
303
303-307
307
308
311
312
313
316-325
316
319
320
322
323
325
226-334
327
327
329
330
331
831
332
332
XVIU CONTENTS
PAGE
Series for tan-^x, Gregory's Quadrature of the Circle . . 333
Note on the Arithmetical Quadrature of the Circle . . . 333
Exercises XIX., XX 334, 335
CHAPTER XXX.
GENERAL THEOREMS REGARDING THE EXPANSION OF FUNCTIONS
IN INFINITE FORMS.
Expansion in Infinite Series 337-344
Expansion of a Fimction of a Function 337
Expansion of an Infinite Product in the form of an Infinite Series 337
Examples — Theorems of Euler and Cauchy .... 339
Expansion of Sech x and Sec x 341
Euler's Numbers 342
Expansion of Tanh x, x Coth x, Cosech x ; Tan x, x Cot x, Cosec x 343
Exercises XXI 344
Expression of Certain Functions as Infinite Products . . . 346-357
General Theorem regarding the Limit of an Infinite Product . 346
Products for sinhjpu, sinh u ; sin pd, sin 6 .... 348
Wallis's Theorem 351
Products for coshpu, coshu; cospd, cos 5 .... 351
Products for cos 0 + sin 0 cot 0, cos <^ - sin ^ tan 0,1 + cosec 0 sin ^ 354
Product for cos 0 - cos 0 355
Remark regarding a Certain Fallacy 356
Exercises XXII 357
Expansion of Circular and Hyperbolic Functions in an Infinite
Series of Partial Fractions 359-362
Expressions for tan 0, 0 cot 0, 0 cosec 0, secO . . . . 360
Expressions for tanh u, u coth u, u cosech u, sech u . . . 362
Expressions for the Numbers of Bernoulli and Euler . . . 362-367
Series for S^ 363
Product for B^ 364
Certain Properties of B^ 364
Eadii of Convergence of the Power-Series for tan 0, 6 cot 0, 0 cosec 0 364
Series and Product for E^ 365
Certain Properties of £^ 366
Radius of Convergence of the Power- Series for sec 0 . . 366
Sums of Certain Series involving Powers of Integers . . 367
Power-Series for log sin 0, &c. 367
Stirling's Theorem 368
Exercises XXIII 372
Reversion of Series — Expansion of an Algebraic Function . . 373-396
General Expansion-Theorem regarding S (m, w)x"*!/"=0 . . 374
Reversion of Series 378
Branch Point 378
CONTENTS XIX
PAGE
Expansion of the various Branches of an Algebraic Function . 379
Irreducibility , Ordinary and Singular Points, Multiple Points,
Zero Points, Poles, Zeros, and Infinities of an Algebraic
Function 380
Expansion at an Ordinary Point 382
Expansion at a Multiple Point 383
Cycles at a Branch Point 386
Neicton's Parallelogram, Degree-Points, Effective Group of Degree-
Points 386
All the Branches of an Algebraic Function expansible . . 389
Algebraic Zeros and Infinities and their Order .... 392
Method of Successive Approximation 392
Historical Note 396
Exercises XXIV 397
CHAPTER XXXI.
SUMMATION AND TRANSFORMATION OP SERIES IN GENERAL.
The Method of Finite Differences 398-409
Difference Notation 398
Two Fundamental Difference Theorems 401
Sammation by Differences 402
Examples — Factorial Series, S sin (a + ?2/3) 40b
n
Expression of 2m„ in terms of the Differences of j/j . . 405
1
Montmort's Theorem 407
Euler's Theorem 408
Exercises XXV 409
Recurring Series 411-415
Scale of Relation . 411
Generating Function 412
To find the General Term 413
Solution of Linear Difference Equation with Constant Coefficients 414
Summation of Eecurring Series 414
Exercises XXVI 415
Simpson's Method for Summation by taking every kth term of a
Series whose sum is known 416-418
Miscellaneous Methods 418-420
Use of Partial Fractions 418
Euler's Identity 419
Exercises XXVII ,,,... 420
XX
CONTENTS
a Terminating
CHAPTER XXXII.
SIMPLE CONTINUED FRACTIONS.
Nature and Origin of Continued Fractions
Terminating, Non-Terminating, Becurring or Periodic, Simple
Continued Fractions ....
Component Fractions and Partial Quotients
Every Number convertible into a S.G.F. .
Every Commensurable Number convertible into
S.C.F
Conversion of a Surd into a S.G.F. .
Exercises XXVIII
Properties of the Convergents to a S.C.F.
Complete Quotients and Convergents .
Pecurrence-Formiilm for Convergents .
Properties of p.^ and q^ . . . .
Fundamental Properties of the Convergents
Approximation to S.C.F
Condition that pjqn be a Convergent to arj
Arithmetical Utility of S.C.FF. .
Convergence of S.C.F. .....
Exercises XXIX
Closest Rational Approximation of Given Complex
Closeness of Approximation of pjq^ •
Principal and Intermediate Convergents
Historical Note
Examples — Calendar, Eclipses, &c.
Exercises XXX
ity
PAGE
423-429
423
423, 424
424
426
428
430
431-441
431
432
433
435
437
439
440
441
442
444-451
445
446
448
449
451
CHAPTER XXXIII.
ON RECURRING CONTINUED FRACTIONS.
Every Simple Quadratic Surd Number convertible into a Recurring
S.C.F 453-458
Recurrence-Formulae for P„ and Q^ 454
Expressions for P„ and Q„ 455
Cycles of P„, Q^, «„ 457
Every Recurring S.C.F. equal to a Simple Quadratic Surd Number 458-460
On the S.C.F. which represents sJ{ClB) 460-469
Acyclic Quotient of JNJM 462
Cycle of the Partial Quotients of ^NjBI 463
Cycles of the Rational Dividends and Divisors of ^N/U . . 464
Tests for the Middle of the Cycles 467
Examples — Rapid Calculation of High Convergents, &c. . . 468
Exercises XXXI 469
CONTENTS
XXI
PAGE
Applications to the Solution of Diophantine Problems . . . 473-488
ax-by=c 474
ax + bij~c 475
ax + by + cz = d, a'x + b'i/ + c'z = d' 477
Solutions of a;2-Cj/2= ± if and a;2-<7r/'-^=±l .... 478
General Solution ot x^-Cy-^J=H y/hen H<sjC ... 479
General FormulaB for the Groups of Solutions of x^- Cy^= =3=1
and a;2-C</2=±iI 480
Lagrange's Eeduction of x^-Cy'^==i=H when H>^C . . 482
Eemaining Cases of the Binomial Equation .... 486
General Equation of the Second Degree 486
Exercises XXXII 489
CHAPTER XXXIV.
GENERAL CONTINUKD FRACTIONS,
Fundamental Formulse ....
Meaning of G.C.F
G.C.FF. of First and Second Class .
Properties of the Convergents
Continuants
Continuant Notation — Simple Continuant
Functional Nature of a Continuant
Euler's Construction . . . .
Euler's Continuant-Theorem
Henry Smith's Proof of Fermat's Theorem that a Prime of the
form 4\ + l is the Sum of Two Integral Squares
Every Continuant reducible to a Simple Continuant
C.F. in terms of Continuants
Equivalent Continued Fractions
Beduction of G.C.F. to a form having Unit Numerators
Exercises XXXIII
Convergence of Infinite C.FF.
Convergence, Divergence, Oscillation of C.F.
Partial Criterion for C.F. of First Class .
Complete Criterion for C.F. of First Class
Partial Criterion for C.F. of Second Class
Incommensurability of certain C.FF
Legendre's Propositions ......
Conversion of Series and Continued Products into C.FF.
Euler's Transformation of a Series into an equivalent C.F,
Examples— Brouncker's Quadrature of the Circle, Ac.
C.F. equivalent to a given Continued Product .
Lambert's Transformation of an Infinite Series into an Infinite
C.F
491-494
492
492
492
494-502
494, 495
495
496
498
499
500
601
501
502
602
505
505
506
507
510
512-514
512
514-524
514
516
617
617
XXll CONTENTS
PAGE
Example — after Legendre 620
C.FF. for tanx and tanhx 522
Incommensurability of v and e 523
Gauss's Conversion of the Hypergeometric Series into a C.F. . 523
Exercises XXXIV 525
CHAPTER XXXV.
GENERAL PROPERTIES OF INTEGRAL NUMBERS.
Numbers which are congruent with respect to a given Modulus . 528-534
Modulus and Congruence 528
Feriodicity of Integers 529
Examples of Properties deduced from Periodicity — Integrality of
x{x + l) . . . {x+p-l)lpl, Pythagorean Problem, &c. . . 529
Property of an Integral Function 532
Test for Divisibility of /(a;) 532
f(x) represents an Infinity of Composites 532
Difference Test of Divisibility 533
Exercises XXXV 534
On the Divisors of a given Integer 536-546
Limit for the Least Factor of 2^ 536
Sum and Number of the Divisors of a Composite . . . 537
Examples — Perfect Number, &c. 538
Number of Integers < N and prime to iV^, <p (N) . . . 539
Euler's Theorems regarding <I>{N) 540
Gauss's Theorem ^(p(dn)-N 542
Properties of m! 543-546
Exercises XXXVI " 546
On the Eesidues of a Series of Integers in Arithmetical Progression 547-554
Periodicity of the Besidues of an A.P 648, 549
Format's Theorem 660
Historical Note 650
Euler's Generalisation of Fermat's Theorem .... 651
Wilson's Theorem 552
Historical Note 553
Theorem of Lagrange including the Theorems of Fermat and
Wilson 553
Exercises XXXVII 654
Partition of Numbers 555-564
Notation for the Number of Partitions 556
Expansions and Partitions 556
Euler's Table for P(n|*|}>g) 658
Partition Problems solvable by means of Euler's Table . . 659-561
Constructive Theory of Partitions 661-564
CONTENTS
XXIU
Graph of a Partition, Regular GrapJis, Conjugate Partitions .
Franklin's Proof that (1- a;) (l-a;2)(l.-a;3)...^ S (_)Pa;i(-V±i')
Exercises XXXVIII "T" . . .
PAGE
5G2
563
564
CHAPTER XXXVL
PROBABILITY, OR THE THEORY OP AVERAGES.
Fundamental Notions, Event, Universe, Series, &c.
Definition of Probability or Chance, and Eemarks thereon
Corollaries on the Definition
Odds on or against an Event
Direct Calculation of Probabilities
Elementary Examples .
Use of the Law of Distribution
Examples — Demoivre's Problem, &c. .
Addition and Multiplication of Probabilities
Addition Rule for Mutually Exclusive Events
Multiplication Rule for Mutually Independent Events
Examples
General Theorems regarding the Probability of Compound Events
Probability that an Event happen on exactly r out of n occasions
More General ' Theorem of a Similar Kind . . . .
Probability that an Event happen on at least r out of n occasions
Pascal's Problem
Some Generalisations of the Foregoing Problems
The Recurrence Method for calculating Probabilities
"Duration of Play"
Evaluation of Probabilities involving Factorials of Large Numbers
Exercises XXXIX
Mathematical Measure of an Expectation
Value of an Expectation
Addition of Expectations
Life Contingencies
Mortality Table
Examples of the Use of a Mortality Table . . . .
Annuity Problems, Notation, and Terminology, Average Ac-
counting
Calculation of Life Insurance Premium . . . . .
Recurrence- Method for calculating Annuities . . . .
Columnar or Commutation Method
Bemarks, General and Bibliographical
Exercises XL
RESULTS OF EXERCISES
INDEX OF PROPER NAMES FOR PARTS I. AND II.
560
567
569
670
571-575
571
573
574
575-581
575
576
577-581
581-586
581
582
583
584
585
586
587
589
590
593-595
594
594
595-604
596
597
598-601
602
603
603
605
605
609
614
SUGGESTION FOE THE COUESE OF A FIEST HEADING
OF PAET II.
Chap, xxin., §§ 1-15. Chap, xxxvi., §§ 1-4. Chap, xxiv., §§1-9.
Chap. XXV. Chap, xxvi., §§ 1-5, 12-19, 32-35. Chap, xxvii. Chap, xxviii.,
§§ 1-5, 8-15. Chap, xxix., §§ 1-19, 23-81. Chap. xxxi. Chap, xxxii.
Chap, xxxiii., §§ 10-14. Chap. xxxv. Chap, xxxvi., §§ 5-22.
CHAPTER XXIII.
Permutations and Combinations.
§ 1.] We have already seen the importance of the enume-
ration of combinations in the elementary theory of integral
functions. It was found, for example, that the problem of finding
the coefficients in the expansion of a binomial is identical with
the problem of enumerating the combinations of a certain
number of things taken 1, 2, 3, &c,, at a time. Besides its
theoretical use, the theory of permutations and combinations
has important practical applications ; for example, to economic
statistics, to the calculus of probabilities, to fire and life assur-
ance, and to the theory of voting.
Beginners usually find the subject somewhat difficult. This
arises in part from the fineness of the distinctions between the
diff'erent problems, distinctions which are not always easy to
express clearly in ordinary language. Close attention should
therefore be paid to the terminology we are now to introduce.
§ 2.] For our present purpose we may represent individual
things by letters.
By an r-permutation of n letters we mean r of those letters
aiTanged in a certain order, say in a straight line. An w-permu-
tation, which means all the letters in a certain order, is sometimes
called a permutation simply.
Example. The 2-permutations of the three letters a, b, c are be, cb;
ac, ca; ab, ba. The permutations of the three letters are abc, acb; bac, bca;
cab, cba.
By an r-comhination of n letters we mean r of those letters
considered without reference to order.
Example. The 2-combinations of a, b, c are be, ac, ab.
C. II. 1
2 MODES OF PROOF OH. XXIII
Unless the contrary is stated, the same letter is not supposed
to occur more than once in each combination or permutation.
In other words, if the n letters were printed on n separate
counters each permutation or combination could be actually
selected and set down before our eyes.
Another point to be attended to is that in some problems
certain sets of the given letters may be all alike or indifferent ;
that is to say, it may be supposed that no alteration in any
permutation or combination is produced by interchanging these
letters.
§ 3.] The fundamental part of every demonstration of a
theorem in the theory of permutations and combinations is an
enumeration. It is necessary that this enumeration be systematic
-and exhaustive. If possible it should also be simplex, that is,
each permutation or combination should occur only once ; but it
may be multiplex, provided the degree of multiplicity be ascer-
tained (see § 8, below).
Along with the enumeration there often occurs the process
of reasoning step by step, called mathematical induction.
The results of the law of distribution, as applied both to
closed functions and to infinite series, are often used (after the
manner of chap, iv., §§5, 11, and exercise vi. 30) to lighten the
labour of enumeration.
All these methods of proof will be found illustrated below.
We have called attention to them here, in order that the student
may know what tools are at his disposal.
PERMUTATIONS.
§ 4.] The number of r-permutations of n letters (nPr) ^
n{n-l){n-2) . . . (n-r+l).
1st Proof — Suppose that we have r blank spaces, the problem
is to find in how many difterent ways we can fill these with n
letters all different.
We can fill the first blank in n different ways, namely, by
putting into it any one of the n letters. Having put any one
letter into the first blank, we have «- 1 to choose from in filling
§§ 2-4 r-PERMUTATIONS 3
the second blank. Hence we can fill the second blank in n- 1
different ways for each way we can fill the first. Hence we can
fill the first two in n{n- 1) ways.
When any two particular letters have been put into the first
two blanks, there are n — 2 left to choose from in filling the third.
Hence we can fill the first three blanks in n{n- 1) times (w — 2)
ways.
Reasoning in this way, we see that we can fill the r blanks in
n{n—l){n-2) . . . (w-r+l)ways.
Hence nPr = n{n-l) . . . (n-r+l).
2nd Proof. — We may enumerate, exhaustively and without
repetition, the nPr r-permutations as follows : —
1st. All those in which the first letter ai stands first ;
2nd. All those in which «2 stands first : and so on.
There are as many permutations in which «! stands first as
there are (r— l)-permutation3 of the remaining w— 1 letters, that
is, there are n-iPr-i permutations in the first class. The same
is true of each of the other n classes.
Hence nPr = nn-iPr-i •
Now this relation is true for any positive integral values of
n and r, so long, of course, as r ^ n. Hence we may write
successively
n-i r = Tlfi-il'^r-it
n-lPr-i = {n— l)n-2Pr-2t
n-r-i
If now we multiply all these equations together, and observe
that all the P's cancel each other except „P^ and n-r+iPi, and
observe further that the value of n~r+iP*i is obviously n-r+l,
we see that
„Pr = n{n-l) . . . {n-r+2){7i-r+l) (1).
The second proof is not so simple as the first, but it illustrates
a kind of reasoning which is very useful in questions regarding
permutations and combinations.
1—2
4 LINEAR AND CIRCULAR PERMUTATIONS CH. XXIII
Cor. 1. The number of different ways in which a set of n
letters can be arranged in linear order is
n{n-l) . . . 3.2.1,
that is, the product of the first n integral numbers.
This follows at once from (1), for the number required is the
number of ^^-permutations of the n letters. Putting r = w in (1)^
we have
,,Pn^n{n-l) ... 2.1 (2).
The product of the first n consecutive integers may be re-
garded as a function of the integral variable n. It is called
factm-ial-n^ and is denoted by n\*.
Cor. 2. ^Pr = n\l{n-r)\.
For nPr = n{n-\) . . . {n-r + \),
n{n-l) . . . (n-r+l)(n-r) . . . 2.1
{n-r) ... 2.1
nl
Cor. 3. The number of ways of arranging n letters in circular
oi'der is (w-1)!, or {n-l)lj2, according as clock-order and
counter-clock-order are or are not distinguished.
Since the circular order merely, and not actual position, is
in question, we may select any one letter and keep it fixed. We
have thus as many different arrangements as there are (n - 1)-
permutations of the remaining n—1 letters, that is (w— 1)!.
If, however, the letters written in any circular order clock-
wise be not distinguished from the letters written in the same
order counter-clock-wise, it is clear that each arrangement will
be counted twice over. Hence the number in this case is
(w-l)!/2.
§ 5.] When each of the n letters may be repeated, the number
of r-permutations is if.
* This is Kramp's notation. Formerly In^was used in English works, but
this is now being abandoned on account of the difficulty in printing the |_.
The value of 11 is of course 1. Strictly speaking, 0! has no meaning. It is
convenieijt, however, to use it, with the understanding that its value is 1 ; by
so doing we avoid the exceptional treatment of initial terms in many series.
§§ 4-6 CASE WHERE LETTERS ARE ALIKE 5
Suppose that we have r blanks before us. We may fill the
first in n ways ; the second also in n ways, since there is now no
restriction on the choice of the letter. Hence the first two may
be filled in n x n, that is, n' ways. With each of these »* ways
of filling the first two blanks we may combine any one of the n
ways of filling the third ; hence we may fill the first three blanks
in n^ X n, that is, n^ ways, and so on. Hence we can fill the r
blanks in n^ ways.
§ 6.] The number of permutations of n letters of which a
group of a are all alike, a group of P all alike, a group of y all
alike, &c., is ,
w!/a!y3!y! . . .
Let us suppose that a; denotes the number in question. If
we take any one of the a; permutations and keep all the rest of
the letters fixed in their places, but make the a letters unlike
and permutate them in every possible way among themselves,
we shall derive a! permutations in which the a letters are all
unlike. Hence the effect of making the a letters unlike is to
derive xal permutations from the a; permutations.
If we now make all the /? letters unlike, we derive xalftl
permutations from the xa\.
Hence, if we make all the letters unlike, we derive xa\fi]y] . . .
permutations. But these must be exactly all possible permuta-
tions of n letters all unlike, that is, we must have
a;a!/3!y! . . . —n\.
Hence a; = w!/a!^!y! . . .
Cor. The number of ways in which n things can he put into
r pigeon-holes, so that a shall go into the first, p into the second,
y into the third, and so on, is
n\ja\p\y\ . . .
N.B. — The m'der of the pigeon-holes is fixed, and must be at-
tended to, but the oi'der of the things inside the holes is indifferent.
Putting the things into the holes is evidently the same as
allowing them to stand in a line and affixing to them labels
marked with the names of the holes. There will thus be a
6 EXAMPLES CH. XXIII
labels each marked 1, (3 each marked 2, y each marked 3, and
so on.
The problem is now to find in how many ways n labels, a of
which are alike, )8 alike, y alike, &c., can be distributed among
n things standing in a given order. The number in question is
w!/a!)8!y! . . ., by the above proposition.
Example 1. In arranging the crew of an eight-oared boat the captain has
four men that can row only on the stroke-side and four that can row only on
the bow-side. In how many different ways can he arrange his boat — 1st,
when the stroke is not fixed ; 2nd, when the stroke is fixed ?
In the first case, the captain may arrange his stroke-side in as many
ways as there are 4-permutations of 4 things, that is, in 4! ways, and he
may arrange the bow-side in just as many ways. Since the arrangements of
the two sides are independent, he has, therefore, 4! x 4! ( = 576) different
ways of arranging the whole crew.
In the second case, since stroke is fixed, there are only 3! ways of
arranging the stroke-side. Hence, in this case, there are 3! x 4! ( = 144)
different ways of arranging the crew.
Example 2. Find the number of permutations that can be made with the
letters of the word transalpine.
The letters are traannslpie, there being two sets, each containing
two like letters. The number required is therefore (by § 6) ll!/2!2! =
11. 10. 9. 8. 7. 6. 5. 8. 2 = 9979200.
Example 3. In how many different ways can n different beads be
formed into a bracelet?
Since merely turning the bracelet over changes a clock-arrangement of the
stones into the corresponding counter-clock-arrangement, it follows, by § 4,
that the number required is (n - l)!/2.
COMBINATIONS.
§ 7.] The number of ways in which s things can he selected hy
taking one out of a set ofn^ , oiie out of a set of n^, &c., is Wi«2 • • • w«.
The first thing can be selected in Wi ways ; the second in n^
ways; and so on. Hence, since the selection of each of ihQ
things does not depend in any way on the selection of the others,
the number of ways in which the s things can be selected is
»1 X ^ X . . . X w,.
§ 8.] The number of r-comhinations of n letters (nCr) is
n{n-l) . . . (w-r + l)/1.2 . . . r.
§§ 6-8 ?^-COMB [NATIONS 7
1st Proof. — We may enumerate the combinations as follows : —
1st. All those that contain the letter «i ;
2nd. „ „ „ a.,;
nth. „ „ „ an.
In each of these classes there is the same number of
combinations ; namely, as many combinations as there are
{r - l)-combinations of w - 1 letters ; for we obviously form all
the r-combinations in which a^ occurs by forming all possible
(r — l)-combinations of aaj «3> • • •, «» and adding ai to each
of them.
This enumeration, though exhaustive, is not simplex ; for
each r-combination will be counted once for every letter it
contains, that is, r times. Hence
TnCr = rin-iCr-i (l).
This relation holds for all values of n and r, so long as r:^n.
Hence we have successively —
p —- r
C — ^~ ^ (^
ft_ll^,._l — ^ n-'V^r-'it
r — L
If we multiply these r - 1 equations together, and observe that
the (7's cancel, except nCr and n-r+i^i, and that the value of
„_r+iCi is obviously n-r+1, we have
p _ n{n-l) . . . (n-r+l) . .
"^^ ~ 1.2 ... r ^^^•
2nd Proof. — Since every r-combination of n letters, if permu-
tated in every possible way, would give r! r-permutations, and
all the r-permutations of the n letters can be got once and only
$ PROPERTIES OF „(7^ CH. XXIII
once by dealing in this way with all the r-combinations, it follows
that JJ.,r\ = nPr- Hence
„a- = «i^r/r! = w(w-l) . . . («-r+l)/1.2 . . . r.
Cor. 1. If we multiply both numerator and denominator of
the expression for „C,. by {ii -r){n-r-l) . . . 2.1, we deduce
nGr = nl/rl{n-r)l (3).
vOr. Z, n^r — ti^n—r'
This follows at once from (3). It may also be proved by
enumeration ; for it is obvious that for every r-combination of
the n things we select we leave behind an (n — r)-combinatiou ;
there are, therefore, just as many of the latter as of the former.
Cor. 3. n^r = n-lC^r + n-lGr-l (4).
This can be proved by using the expressions for „Cr, n-i^r,
n-iCr-i, and the remark is important, because it shows that the
property holds for functions of n having the form (2) irrespective
of any restriction on the value of n.
The theorem (when n is a positive integer) also follows at
once by classifying the r-combinations of w letters aj, ag, • • • , «»
into, 1st, those that contain «i, „-iC,-i in number, and, 2nd,
those that do not contain ai, n-iCr iw number.
Cor. 4. n-\Gs + n-iC!s + n-zCs"^ ' • • + s^s = nW+l \p)-
Since the order of letters in any combination is indifferent,
we may arrange them in alphabetical order, and enumerate the
(s + l)-combinations of n letters by counting, 1st, those in
which Ui stands first ; 2nd, those in which a^ stands first, &c.
This enumeration is clearly both exhaustive and simplex ; and
we observe that ai cannot occur in any of the combinations of
the 2nd class, neither a^ nor a^ in any of the 3rd class, and so on.
Hence the number of combinations in the 1st class is n-\Gs ', in
the 2nd, n-aC'* ; in the 3rd, n-zGs ; and so on. Thus the theorem
follows.
Cor. 5.
If we divide p + q letters into two groups of p and q re-
spectively, the p^qCg s-combinations of the p + q letters may be
classified exhaustively and simplexly as follows : —
§ 8 vandermonde's theorem 9
1st. All the s-combinations of the j9 letters. The number of
these is pCg.
2nd. All the combinations found by taking every one of
the {s — l)-combinations of the p things with every one of the
1 -combinations of the q things. The number of these is
3rd. All the combinations found by taking every one of
the {s — 2)-combinations of the p things with every one of the
2-combinations of the q things. The number of these is
And so on. Thus the theorem follows.
It should be noticed that Cor. 4 and Cor. 5 furnish proposi-
tions in the summation of series. For example, we may write
Cor. 5 thus —
p{p- 1) . . . (^-.9 + 1) ^pjj>-})
1.2 ... s 1.2
pip -I)
1.2
^p q(q-l)
1 1.2
. (p-s+2) I
. is-l) -1
. ip-s + 3) q(q-l)
. {s-2) ' 1.2
• ' ig-s + 2)
. . (5-1)
(7).
1.2 ... g
^ {p + q)ip + q-l) . . • ip + q-s+l)
1.2 ... s
It is obvious that (7) is an algebraical identity which could
be proved by actually transforming the left-hand side into the
right (see chap, v., § 16). If we take this view, it is clear that
the only restriction upon p, q, s is that s shall be a positive integer.
Thus generalised, (7) becomes of importance in the establishment
of the Binomial Theorem for fractional and negative indices.
Cor. 6. If we multiply both sides of (7) by 1 . 2 ... 5, and
denote p(p — l) . . . (p-s+ 1) by jt?,, we deduce
(P + q)s =Ps + sCiPs-iqi + fys-iq-L + . • ■ + 5'» (8),
which is often called Vandermonde's theorem, although the result
was known before Vandermonde's day.
10 p LETTERS ALIKE CH. XXIII
§ 9,] To find the number of r-combinations of p + q letters
p of which are alike.
1st. With the q unhke letters we can form gC,. r-com-
binations,
2nd. Taking one of the p letters, and r - 1 of the q, we can
form qCr-\ r-combinations.
3rd. Taking two of the p, and r - 2 of the q, we can form
gCv_2 r-combinations; and so on, till at last we take r of the
p (supposing p > r), and form one r-combination.
We thus find for the number required
qCr + qCr-\ + qCr-'i + . . . + qC^ + 1
"^' lr!(^-r)!"^(r-l)!(^-r+l)!'*'" * "^ 1! (g- - 1)! "^ ^ J '
Cor. The number of r-permutations of p + q things p of which
are alike is
^'^' \r\{q-r)\ "^ l\{r-l)l(q -r + 1)! "*" 2!(r- 2)!(g'-r + 2)! '*'
1_ 1 ]
• • ''^ {r -ly.V.iq- 1)1 '^ rlqlj'
For, with the qCr combinations of the 1st class above we can form
qCrrl permutations ;
With the gCr-i combinations of the 2nd class, qCr-i r! per-
mutations ;
With the qCr-2 combinations of the 3rd class (in each of
which two letters are alike), g(7r_2r!/2! permutations: and
so on.
Hence the whole number of permutations is
qCrr\+qCr.,r\/V. + qCr-,rll2l + . . .+qC,rll{r-l)\ + l,
whence the result follows.
A similar process will give the number of r-combinations,
or of r-permutations, when we have more than one group of
like letters ; but the general formula is very complicated.
§ 10.] The number of r-combinations of n letters (nffr), when
each letter may be repeated any number of times up to r, is
n{n + l){n + 2) . . . (w + r- 1)/1 . 2. 3 . . . r (1).
§§ 9, 10 COMBINATIONS WITH REPETITION 11
In the first place, we remark that the number of (r + 1 )-com-
binations, in each of which the letter ai occurs at least once, is
the same as the number of r-combinations not subject to this
restriction. This is obvious if we reflect that every (r + 1)-
combination of the kind described leaves an r-combination when
tti is removed, and, conversely, every r-combination of the n
letters gives, when ai is added to it, an (r + l)-combination of
the kind described.
It follows, then, that if we add to each of the r-combinations
of the theorem all the n letters, we get all the {n + r)-corabinations
of the n letters, in each of which each letter appears at least
once, and not more than r+ 1 times. We may therefore
enumerate the latter instead of the former.
This new problem may be reduced to a question of permuta-
tions as follows. Instead of writing down all the repeated letters,
we may write down each letter once, and write after it the letter
s (initial of same) as often as the letter is repeated. Thus, we
write asssbsscs . . . instead of aaaahhhcc . . . With this notation
there will occur in each of the {n + r)-combi nations the n letters
ai, a^, . . ., ttn along with r s's. The problem now is to find
in how many ways we can arrange these n + r letters. It must
be remembered that there is no meaning in the occurrence of s at
the beginning of the series ; hence, since the order of the letters
fli, ttj, • • ., «»i is indifferent, we may fix ai in the first place.
We have now to consider the different arrangements of the n-1
letters a^, a^, . . ., a„ along with r s's. In so doing we must
observe that nothing depends on the order of ^a, «3, • • •, <*n
inter se ; so that in counting the permutations they must be
regarded as all alike. We have, therefore, to find the number of
permutations of w - 1 + r things, w - 1 of which are alike, and r
of which are alike. Hence we have
(n + r-l)l
" *■ (w-l)!r! ^^^'
_n{n+ I) . . . (n + r-l)
12 THEOREMS REGARDING nHr CH. XXllI
L/Or. 1. nttr ~ re+r-iW»
This follows at once from (2).
Lor. 2. nJ^ir — n-\'Jr + n'Jr-l-
For the r-combiuations consist, 1st, of those in which a^ occurs
at least once, the number of which we have seen to be nHr-i ;
2nd, of those in which aj does not occur at all, the number of
which is n-\Hr.
Cor. 3. Jfr = n-Jir + n-JIr-l + n-JIr-2 + . . . + n-1^1 + 1-
This follows from the consideration that we may classify the
r-combinations into
1st. Those in which a^ does not occur at all, n-iHr in
number ;
2nd. Those in which ai occurs once, n-iffr-i in number ;
3rd. Those in which «! occurs twice, n-iHr-i in number :
and so on.
Cor. 4. The number of different r-ary products that can he
made with n different letters is n{n+\) . . . (w + r - 1)/1 . 2 . . . r ;
and the number of terms in a complete integral function of the rth
degree in n variables is (n + I) (n + 2) . . . (w + r)/l . 2 . . . r.
The first part of the corollary is of course obvious. The
second follows from the consideration that the complete in-
tegral function is the sum of all possible terms of the degrees
0, 1, 2, . . ., r respectively. Hence the number of its terms is
1 + „//i + „//2 + . . . + »//,-.
But, by Cor. 3, tliis sum is n+iffr-
We have thus obtained a general solution of the problems suggested in
chap. IV., §§ 17, 19. As a verification, if we put n=2, we have for the
number of terms in the general integral function of the rth degree in two
variables 3.4 .. . (r + 2)/1.2 , . . r, which reduces to (r + 1) (r + 2)j2, in
agreement with our former result.
Exercises I.
Combinations and Permutations.
(1.) How many different numbers can be made with the digits
11122333450?
(2.) How many different permutations can be made of the letters of the
sentence Ut tensio sic vis ?
§ 10 EXERCISES I 13
(3.) How many different numbers of 4 digits can be formed with 0123456?
(4.) How many odd numbers can be formed with the digits 3694?
(5.) If 2„<^„-i/2n-2C»= 1^2/35, find n.
(6.) If m = „C2, show that ^C2 = 3„+iC4.
(7.) In any set of n letters, if the number of r-permutations which con-
tain a be equal to the number of those that do not contain a, prove that the
same holds of r-combinations.
(8.) In how many ways can the major pieces of a set of chess-men be
arranged in a line on the board?
If the pawns be included, in how many ways can the pieces be arranged
in two lines ?
(9.) Out of 13 men, in how many ways may a guard of 6 be formed in line,
the order of the men to be attended to ?
(10.) In how many ways can 12 men be selected out of 17 — Ist, if there be
no restriction on the choice ; 2nd, if 2 particular men be always included ;
3rd, if 2 particular men never be chosen together ?
(11.) In how many ways can a bracelet be made by stringing together 5
like pearls, 6 like rubies, and 7 like diamonds ?
How many different settings of 3 stones for a ring could be selected
from the above?
What modification of the solution of the first part of the above problem
is necessary. when two, or all three, of the given numbers are even ?
(12.) In how many ways can an eight-oared boat be manned out of 31
men, 10 of whom can row on the stroke-side only, 12 on the bow-side only,
and the rest on either side ?
(13.) In a regiment there are 10 captains, 20 lieutenants, 30 sergeants,
and 60 corporals. In how many ways can a party be selected, consisting of
2 captains, 5 lieutenants, 10 sergeants, and 20 corporals ?
(14.) Three persons have 4 coats, 5 vests, and 6 hats between them ; in
how many different ways can they dress ?
(15.) A man has 12 relations, 7 ladies and 5 gentlemen ; his wife has 12
relations, 5 ladies and 7 gentlemen. In how many ways can they invite a
dinner party of 6 ladies and 6 gentlemen so that there may be 6 of the man's
relations and 6 of the wife's ?
(16.) In how many ways can 7 ladies and 7 gentlemen be seated at a
round table so that no 2 ladies sit together?
(17.) At a dinner- table the host and hostess sit opposite each other. In
how many ways can 2n guests be arranged so that 2 particular guests do
not sit together?
(18.) In how many ways can a team of 6 horses be selected out of a stud
of 16, so that there shall always be 3 out of the 6 ABCA'B'C, but never AA',
BB', or CC together ?
(19.) With 9 consonants an,d 7 vowels, how many words can be made,
each containing 4 consonants and 3 vowels — 1st, when there is no restriction
on the arrangement of the letters ; 2nd, when two consonants are never
allowed to come together?
(20.) In how many ways can 52 cards, all different, be dealt into 4 equal
14 BINOMIAL THEOREM CH. XXIII
hands, the order of the hands, but not of the cards in the hands, to be
attended to?
In ho'.v many cases will 13 particular cards fall in one hand ?
(21.) In how many ways can a set of 12 black and 12 white draught-men
be placed on the black squares of a draught-board ?
(22.) In how many ways can a set of chess-men be placed on a chess-board?
(23.) How many 3-combinations and how many S-permutations can be
made with the letters of parabola?
(24.) With an unlimited number of red, white, blue, and black balls at
disposal, in how many ways can a bagful of 10 be selected ?
In how many of these selections will all the colours be represented ?
(25.) In an election under the cumulative system there were p candidates
for q seats ; (1) in how many ways can an elector give his votes ; (2) if there
be r voters, how many different states of the poll are there?
If there be 15 candidates and 10 seats, and a voter give one minute to the
consideration of each way of giving his vote, how long would it take him to
make up his mind how to vote ?
BINOMIAL AND MULTINOMIAL THEOREMS.
§ 11.] It has already been shown, in chap, iv., § 11, that
{a + b)''--a'' + r,Cia''-^b + . . . -t- ^O-a""'"^'' + • • .+b\
where „Ci, nOi, . . ., nOr . . . denote the numbers of 1-, 2-,
. . ., /--combinations of n things. Using the expressions just
found for nCi, JJ^, &c., we now have
(a + hf = a" + wa"-^ h + ^-^^ dJ'-^h'' + . . .
+ -i /-t: -^ ^a" '^b'^+. . . +b'' (1).
1 . 2 . . . r ^ ^
This is the Binomial Theorem as Newton discovered it, proved,
of course, as yet for positive integral indices only.
§ 12.] We may establish the Binomial Theorem by a some-
what different process of reasoning, which has the advantage of
being applicable to the expansion of an integral power of any
multinomial.
Consider
{ai + a2+. . . + amY (2).
We have to distribute the product of n factors, namely,
(ffli + aa + . . . + am){ai + a.2 + . . . + a^) • • • («i + ^2 + • • • +a,n) (3) ;
§§ 10-12 MULTINOMIAL THEOREM 15
and the problem is to find the coefficient of any given term, say
a^'^^a^'^ . . . am"'- (4),
where of course a-^ + a2+ . . . + a,„ = n. In other words, we have to
find how often the partial product (4) occurs in the distribution
of (3).
We may write out (4) in a variety of ways, such as
aiaia2a2a2«3«4«4 • • • (5),
there being always a^ ai's, a^ Uz's, &c.
Written as in (5) we may regard the partial product as
formed by taking «! from the 1st and 2nd brackets in (3) ; aj
from the 3rd, 4th, and 5th ; as from the 6th ; and so on. It
appears, therefore, that the partial product (4) will occur just as
often as we can make different permutations of the n letters, such
as (5). Now, since a^ of the letters are all alike, a^ all alike, &c.,
the number of difierent permutations is, by § 6, nlja^la^l . . . a„!.
Hence we have
(ai + 052 + . . .+am)" = 2-r— j — ■,«i»-a2"^ . . . a„> (6):
a^Iaa! ...aj.
wherein «!, a2> • • • «m assume all positive integral values con-
sistent with the relation
ai + 02 + . . . + a^ = n (7).
This is the Multinomial Theorem for a positive integral index.
The Binomial Theorem is merely the particular case where
m-2. We then have, since 01 + 03 = n, and therefore oj = w - a^,
(ai + a2r-=S^^,^;i^^^,ai».«2— ,
= 5 n(n-l).. (n-a,^l) ^^^_ ^^„_
ail
which agrees with (1).
Cor. To find the coefficient of x^ in the expansion of
{h, + hx + . . . + bmx'^-^f (8)
we have simply to pick out all the terms which contain a?*". The
general term is
16 EXAMPLES CH. XXIII
Hence we have to take all the terms which are such that
a2 + 2a3 + . . .+(m-l)a^ = r (9).
The coefficient of x^ in the expansion of (8) is therefore
w!
^-i^'Ja
Urn >«
(10),
where a-^,a.^, . . ., a^ have all positive integral values subject
to the restrictions (7) and (9).
Example 1. The coefficient of a%^ in the expansion of (a + fe + c + d)" is
3!2!0!0!
Example 2. To find the coefficient of a;' in (1 + 2x + x^Y.
Here we must have oj + aj + 03 = 4,
02 + 203 = 5.
Hence 01 = 03 — 1, o, = 5 — 203.
Since Oj and 03 must both be positive, the only two admissible values of 03
are 1 and 2. We have therefore the following table of values : —
«1
a..
«3
0
1
3
1
1
The required coefficient is therefore
4! 4!
Jll — 102311 , ^•- 112112 = 56
0!3!1! 1!1!2!
The correctness of the result may be easily verified in the present case ;
for (l + 2a; + xY=(l + a;)8, the coefficient of a;^ in which is gCj^SG.
Example 3. To find the greatest coefficient, or coefficients, in the
expansion of {a^ + a.^+. . .+a^)'".
This amounts to determining x,y,z,... so that nl/xl ylzl . . . shall be a
maximum, where x + y + z+ . . .=n. This, again, amounts to determining
x,y, z, , . . so that
u = xlylzl ... (1)
shall be a minimum, subject to the condition
x + y + z+. . .=n (2).
Let us first consider the case where there are only two variables, x and y.
We obtain all possible values of x\yl by giving y successively the values
0, 1, 2, . . ., n, X taking in consequence the values n,n-l,n-2, . . ., 0. The
consecutive value to xly\ is (x-l)\{y + l)l, and the ratio of the latter
to the former is {y + l)/x ; that is (since x + y-n), {n + 1- x)/x, that is,
p
§12 MAXIMUM COEFFICIENT 17
{n + l)lx - 1. This ratio is less than unity so long as (;H- l)/x<2, that is, so
long as x>(w+l)/2. Until x falls below this value the terms in the series
above mentioned will decrease ; and after x falls below this limit they will
begin to increase.
If n be odd, =2A; + 1 say, then (n + l)/2 = fc + l. Hence, if we make
x = fc + l, the ratio (n + l)/x-l = l, and two consecutive values of x\y\, viz.
(k + 1)1 k\ and k\ (k + 1)1 , are equal and less than any of the others.
If n be even, —2k say, then (« + 1)/2 = ft + ^. Hence, if we make x = A-,
we obtain a single term of the series, viz. klkl, which is less than any of
the others.
Eetuming now to the general case, we see that, if u be a minimum for all
values of x,y, z, . . . subject to the restriction (2), it will also be a minimum
for values such that x and y alone are variable, z, . . . being all constant.
In other words, the values of x and y for which x\y\z\ ... is a minimum
must be such as render x\y\ b. minimum. Hence, by what has just been
proved, x and y must either be equal or differ only by unity. The like
follows for every pair of the variables x,y,z, ... Let us therefore suppose
that 2? of these are each equal to | ; then the remaining m-p must each be
equal to $ + 1. Further, let q be the quotient and r the remainder when n is
divided by m; so that n=viq + r. We thus have
p^ + {m-p){^ + l)~mq + r.
Hence m^-\-{m-p) = mq + r\
so that ^ + {m-p)lni = q + rlm.
Now (m-p)lm and r/m are proper fractions ; hence we must have
^ = q, m-p = r.
It follows, therefore, that r of the variables are each equal to q + 1, and
the rest are each equal to q. The maximum coefficient is therefore
n\l{q\r-'-{(q + l)l}r;
that is, nlKqlyiq + iy (3).
This coefficient is, of course, common to all terms of the type
«l'«2« • • • am-r««m-r+l«+' • • • ««.«+'•
As a special case, consider (Oi + a^ + a^)*. Here 4 = 3x1 + 1; q = l,r = l.
Hence the terms that have the greatest coefficient are those of the type
a^a^a^, and the coefficient in question is 4!/(lI)32i = 12, This is right; for
we find by distributing that
(fli + aj + a^Y = Stti* + iZa^a^ + QlUi^H^ + 12'Lai^a^a3 .
Example 4. Show that
^ n \+x «(n-l) l + 2x n(ra-l)(ra-2) l + 3x
1 1 + nx ~lT2~ (iTnxp 17173 (T+ni)» + •••-"•
(Wolstenholvie.)
The left-hand side may be written
- _ n 1 w (n - 1) 1 n h. - 1) (n - 2)
+
1 1 + na; 1.2 (\ + nxf 1.2.3 (l + nxf
n X n(n-\) 2x n (n - 1) (n - 2) 3x
"ll + na: 1.2 (l + jiip" 1.2.3 (1 + nx)*"'"
18 MOtERTIES OP nCr CH. XXIII
_ , n 1 n (n-1) 1 _ n(n-l)(n-2) 1
~ 11 + nx^ 1.2 (l + nx)2 1.2.3 (l + «x)3
nx f (n-1) 1 (n-l)(n-2) 1 1
~l + nx\ 1 (1 + nx)"^ 1.2 (l + nx)'^ • • -p
I l + nx] l + nx\ l + nx\ '
_ \ nx )"■ nx i vx \"'~^
~ [1+11x1 l + nx\l + vx] *
_ j nx \^ j i)x I "
~ |l + «a:) (1 + 7(J-J '
= 0.
13.] The Binomial Theorem can be used in its turn to
establish identities in the theory of combinations ; as the two
following examples will show : —
Example 1. We have
l=^(T+x-xY
= (l + xY-^CiX(l + xY-''+^C2x'^{l + xY--^- . . . {-Y^.c^x'-.
On the right-hand side of this identity the coefficient of every power of x
must vanish. Hence, s being any positive integer less than r, we have
rC, X 1 - r-iC,_i XrC, + r- 2C._2 x^C^-. .. + {- )»-V_H-iCi x ^C,.^ + ( - )VC, = 0
Example 2. To find the sum of the squares of the binomial coefficients.
We have (l+x)2'* = (l + x)"x (a;-|-l)»
= {l + nGiX + „C^''+ . . . +„C7„.T")
x(x» + „CiX»-i+„C2.T'»-2+ . . . +„C„).
If we imagine the product on the right to be distributed, we see that the
coefficient of x™ is 12 + „Ci2-t-„C'2-+ . . . +„C„2 ; the coefficient of x" on the
left is 2„C„. Hence
l=' + nCi' + nC/+ . . . +„C„2 = 2„C„ = 2nI/H!«!.
Since
2nI = 2/i(27i-l)(2rt-2) . . . 4 . 3 .2 . 1:=2™. 1 . 2 . . . jixl.B . . . (2;j-l),
wehave V + nCi' + nCi'+ ■ • • +„C„2=2™.1.3 . . . (27i-l)/n!.
A great variety of results can be obtained by the above process of equating
coefficients in identities derived from the binomial theorem ; some specimens
are given among the exercises below.
Exercises II.
(1.) Find the third term in the expansion of (2 + 3x)'".
(2.) Find the coefficient of x'' in the expansion of (1 + x + x-) (1 -x)".
(3.) Find the term which is independent of x in the expansion of
(x + l/x)=».
§§ 12, 13 EXERCISES II 19
(4.) Find the coefficient of a-"*" in the expansion of (x - l/.r)2»».
(5.) Find the ratio of the coefficients of x-" in (l + x)^" and (l + .r)-".
(6.) Find the middle term in the expansion of (2 + |x)".
(7.) The product of the coefficients in (l+x)»+i : the product of the
coeflBcients in (l + x)"=(7i + l)" : n\.
(8.) The coefficient of a;'- in { (r - 2) x^ + nx - r} (x + 1)" is n „Cr-2-
(9.) If I denote the integral part and F the proper fractional part of
(3 + ^5)", and if p denote the rational part and o- the irrational part of the
same, show that
I=2{3" + „C2 3»-2.5 + „C^3«-^52+ . . .}-!,
F=l-(3-V5)",
(10.) If (^2 + l)2'»+i = I+i^, where F is a positive proper fraction and I ia
integral, show that F(I+F) = 1.
(11.) Find the integral parts of (2^3 + 3)2"', and of (2^/3 + 3)2»'+i,
(12.) Show that the greatest term in the expansion of (a + 3;)" is the
(r + l)th, where r is the integral part of (n + l)/(a/x + 1).
Exemplify with (2 + 3)J« and with (2 + 1)9.
(13.) Find the condition that the greatest term in (a + a:)" shall have the
greatest coefficient. Find the limits for x in order that this may be so
in (l + x)ioo.
(14.) If the^jth term be the greatest in (a + x)"*, and the qth the greatest
in (a + x)", then either the (p + q)th. or the {p + q- l)th or the (p + 2-2)th is
the greatest in (a + a;)'"+".
(15.) Sum the series , .
•£i+2»^2^3 4!''+ . . . +n~f-'^-.
(16.) Sum the series
l + 2„Ci + 3„C2 + 4„a,+ . . .
(17.) If jPr denote the coefficient of x^ in (l + x)", prove the following
relations : —
1°. Pi-^P2 + ^P:i- ' • • +n(-l)"-'i^„ = 0.
(-1)"-! n
2°. hPi-lP2 + • ■ • + —rr- Pn=—,-T-
iri. jx-j jt + 1 ■^" 11+1
3.1+2 + 3 + • • • +„^i - ,^^.i •
(18.) lipr have the same meaning as in last question, show that
(-l)»-i ,11 1
Pi-\P-2 + lP,- ■ • • +— „-:P" = 1 + 2 + 3+ • • • +«•
(19.) Show that
^C^xl+r-iGs-iX r(^l + r-2Cs-2^r(^2+ • ■ ■ +r-m^l X r^^s-l + 1 >< r<7« = r^82'.
(20.) Show that
• (l-nC-2 + nCi- ■ ■ ■ r- + LCi-,fi,+ . . . )'^-l + „C^i + „C2+ . . .
2—2
20 EXERCISES II CH. XXIII
(21.) Show that
lx„C2 + „CiX„C3+ . . . +„C„_2X„C„ = (2«)!/(n + 2)I(n-2)l.
(22.) Showthatl-n^4.("J^)y-(-^"-^3V"-^y4- . . . =0 if n
be odd, and = ( - l)'"2(n + 2) (n + 4) . . . 2n/2.4 . . . n if n be even.
(23.) Show that
l.n(n + l)+jj(n-l)n+-5-2J— («-2)(n-l)+ -^ gj^^ -'(n-3)(n-2)
+ . . . =2(2/i + l)!/(n + 2)!(«-l)!.
(24.) If Wr stand for «'■ + l/x*", show that
TV+i + r+lCi«r-l + r+lC2"r-3+ • • • =«! (l«r + rCiMr-2 + rC2 Wr-4+ • • • )•
(25.) If a^ denote the coefficient of x'' in (l + x)2('»-'")(l -.t)-'", show that
<^o-n^i<'i + n^2<'^2~ ... =0 for all valucs of p except p — n, in which case
the right-hand side of the equation is 4".
(26.) Show that
x + 1 x + 2 ' ' ' x + n X (x + l) . . . (x + n)'
(27.) Findthe coefficient of x*" in (l + x + x2-t- . . . )^.
(28.) Find the coefficient of x^^ in (1 + x^ + x« + x^)*.
(29.) Find the coefficient of x» in (1 + X + 2x2 + 3x3+ . . . )\
(30.) If Oo, ttj, , . ., ttjii he the coefficients of the powers of x in
(l + 2x + 2x')", show that <'o''2>» ~ "i^'an-i + • • • +^2nflo — ^ ^^ " ^^ odd,
= 2"n!/{(4n)!}2 if n be even.
(31.) If ttj. be the coefficient of x*" in (I4 x + x2+ . . . +xP)", show that
a^ - „Ci a^i + ^Cj a^_2 - ... =0, unless n be a multiple of p + 1. What
does the equation become in the latter case ?
(32.) Find the coefficient of x" in (1 + 2x + 3x2 + 4^3^12.
(33.) Write out the expansion of (a + 6 + c + df.
(34.) Show that
^1''2* . . . nfc^ 1 (?t(ri + l)]P
rl»l . . . fil~ p\\ 2 f '
where r, «, . . ., A; have all values between 0 and p, both inclusive, subject
to the restriction r + 8+ . . . +k =p.
(35.) If „J/y have the meaning of § 10 above, prove that
2°. l-„(7iX,,Hi + „CjX„H2-„C3X„H3+ . . . +(-l)\(7„„Jf„=0.
(36.) IfXr=a;(x + l) . . . (x + r-1), show that
(x + r/)r = Xr + ,.CiXr_i2/i + ,C2X,._2y2+ . . . +yr'
(37.) Find the largest coefficient in the expansion of (a + 6 + c + d+ e)^.
§§ 13-15 LAW OF DISTRIBUTION USED 21
EXAMPLES OF THE APPLICATION OF THE LAW OP
DISTRIBUTION.
§ 14.] If we haver sets, consisting of Ux^n^, . . ., n^ different
letters respectively, the whole number of different ways of making
combinations by taking 1, 2, 3, . . . up to r of the letters at a
time, but never more than one from each set, is
(wi + 1) {no, + 1) . . . (Wr + 1) - 1.
Consider the product
(1 + cfi + &i + . . . 7?i letters)
X (1 + rt2 + ^2 + • • • ^2 letters)
y. {1 -v ttr + br + . . . nr letters).
In the distributed product there will occur every possible com-
bination of the letters taken 1, 2, 3, . . ., r at a time, with the
terra 1 in addition. If we replace each letter by unity, each
term in the distributed product will become unity, and the sum
of these terms will exceed the whole number of combinations by
unity. Hence the number required is
(1 + Wi) (1 + W2) . . . (1 + ih) - 1
= SWj + 2Wi7?2 4- . . . + Wi?i3 . . . Wr.
This result might have been obtained by repeated use of § 7.
§ 15.] If we have r sets of counters, marked with the following
numbers —
°-ii Hit ' ' -> "ii
02, /3o, . . ., /fo,
a J. , p ,. , t . • , Kry
the number of counters not being necessarily the same for each set,
and the inscribed numbers not necessarily all different, then the
number of different ways in which r counters can be drawn, one
from each set, so that the sum of the inscribed numbers shall be n,
is the coefficient of x^ in the distribution of the product
22 DISTRIBUTION PROBLEM CH, XXIII
{af^^ + iT^i + . . . + x"^)
X {x^ + x^'- + . . . + X"').
This theorem is an obvious result of the principles laid down
in chap. iv.
Cor. 1. If in the first set there he a^ counters marked with
the number a^, h marked with Pi, &c., in the second a^ marked
with ttj, ^o marked with P^, <^c-, the number of ways in which r
counters can be drawn so that the sum of the numbers on them is
n, is the coefficient of x^ in the distribution of
(«ia^i + hiX^' + . . . + kiX"^)
X. {a^^ + b.^^-2 + . . . + Z'._wr«2)
X (a,^"r + bi^^r + . . . + k^^r).
Cor. 2. In a box there are a counters marked a, b marked /3,
&c. A counter is drawn r times, and each time replaced. TJie
number of ways in which the sum of the drawings can amount to
n is the coefficient ofx'^ in the distribution of
{ax°- + bx^ +...)'".
DISTRIBUTIONS AND DERANGEMENTS.
§ 16.] The variety of problems that arise in connection with
the subject of the present chapter is endless, and it would be
difficult within the limits of a text-book to indicate all the
methods that have been used in solving such of these problems
as mathematicians have already discussed. The following have
been selected as types of problems which are not, very readily at
least, reducible to the elementary cases above discussed.*
§ 17.] To find the number of ways in which n different letters
can be distributed among r pigeon-holes, attention being paid to
the order of the pigeon-holes, but not to the wder of the letters in
any one pigeon-hole, and no hole to contain less than one letter.
Let Dr denote the number in question.
* For further information see Whitworth's Choice and Chance.
§§ 15-17 DISTRIBUTION PROBLEM 23
If we leave s specified holes vacant and distribute the letters
among the remaining r-s holes under the conditions of the
question, we should thus get Dr-s distributions. Hence, if rCs
have its usual meaning, the number of distributions when s of
the holes are blank is rCgDr-a-
Again, the whole number of distributions when none, one,
two, &c., of the holes may be blank is evidently r", for we can
distribute the n letters separately among the r holes in r" ways.
Hence
Dr + rC,Dr-i + rC^Dr-..+ . . . + ,C,_i A = r" (A).
The equation (A) contains the solution of our problem, for, by
putting r = 2, /• = 3, &c., successively, we could calculate D^, D^,
&c., and Di is known, being simply 1.
We can, however, deduce an expression for Dr in terms of n
and r, as follows. Writing r - 1 in place of r we have
Dr-, + r-.0,Dr-,+ . . . ^ r~^C,-,D, = {r - l)^ (B).
From (A) and (B), by subtraction, remembering (§ 8, Cor. 3)
that
we derive
Dr + )— iW Dr-i + r-iPi Dr-2 + . . . + r^^Cf-i Di
= r''-{r-\Y (1).
From (1), putting r- 1 in place of r, we derive
X'r-i + r- 2^1 -^r-2 "t" • • • "^ (•-2^r-2 -^1
= {r-lY-{r-2f {!').
From (1) and (1'), by subtraction, we derive
X/r + r- 2^1 -^r-l "*■ r-2^2 -^1 — 2+ • • • + r-2^r-2 -^2
= r"-2(r-l)"+(r-2)" (2).
Treating now (2) exactly as we treated (1) we derive
Dr + r-S^i Dr~\ + r-3^2 -^r-2 + . . . 4- r_sCv-3 D3
= r''-3(r-l)'' + 3(r-2)»-(r-3)" (3).
The law of formation of the right-hand side is obvious, the
coefficients being formed by the addition rule peculiar to the
binomial coefficients (see chap, iv., § 14). We shall therefore
finally obtain
24 DERANGEMENTS CH. XXIII
= r^.^(r-ir + '^^^{r-2r-. . . (-^^^1" (4).
Cor. If the order of the pigeon-Jwles he indifferent, the numb&r of
distributions is Dr/rl. In other words, the number of partitions of
n different letters into r lots, no vacant lots being allowed, is Dr/rl
We shall discuss the closely-allied problem to find the
number of r-partitions of n — that is, to find the number of
ways in which n letters, all alike, may be distributed among
r pigeon-holes, the order of the holes being indifferent, and no
hole to be empty — when we take up the Theory of the Partition
of Numbers.
§ 18.] Given a series of n letters, to find in how many ways
the order may be deranged so that no one out of r assigned letters
shall occupy its original position.
Let n^r denote the number in question.
The number of diff"erent derangements in which the r assigned
letters do all occupy their original places is {n-r)\. Hence the
number of derangements in which the r assigned letters do not
all occupy their original places is nl-{n-r)l Now, this last
number is made up of —
1st. The number of derangements in which no one of the r
letters occupies its original place ; that is, „A^.
2nd. The number of derangements in which any one of the r
letters occupies its original place, and no one of the remaining
r-1 does so; that is, rCm-Ar-i-
3rd. The number of derangements in which any two of
the r letters occupy their original places, and no one of the
remaining r-2 does so; that is, rO-in-Ar-z- And so on.
Hence
n\-{n-r)l = ,Ar + rOin~Ar-l+r02n~Ar-2+ • • .
+ r6>_i n-r+Al (A).
If we write in this equation w - 1 for n, and r-1 for r, and
subtract the new equation thus derived from (A), we deduce
n\-{n-l)\=n^r + r-lCin-Ar-l + r-l0.n-Ar-ii+ • • •
+ ,_iC/y_j J^_r+Al \1).
§§17-19 SUBFACTORIAL n 25
We can now treat this equation exactly as we treated
equation (1) of § 16. We thus deduce
n^r = nl-[in-l)l + '^^^{n-2)\-. . . (-nn-r)\ (2).
If we remember that {n — r)l, above, stands for the number
of derangements in which the r letters all occupy their original
positions, we see that, when r = n, {n — r)l must be replaced by 1,
Hence
Cor. T/is number of derangements of a series of n letters in
which no one of the original n occupies its original position is
The expression (3) may be written
n( . . . (4(3(2 (1-1) + 1)-1) + 1) . . .-(-i)») + (-l)».
Hence it may be formed as follows: — Set down 1, subtract 1 ;
multiply by 2 and add 1 ; multiply by 3 and subtract 1 ; and
so on. The function thus formed is of considerable importance
in the present branch of mathematics, and has been called by
Whitworth suhfactorial n. He denotes it by \\ii. A more con-
venient notation would be n] .
SUBSTITUTIONS.
§ 19.] Hitherto we have merely counted the permutations
of a group of letters. If we direct our attention to the actual
permutations, and in particular to the process by which these
permutations are derived from each other, we are led to an order
of ideas which forms the foundation of that important branch of
modern algebra which is called the Theory of Substitutions.
Consider any two permutations, becda, beads, of the five letters
a, b, c, d, e. The latter is derived from the former by replacing
a hy e, b by b, c hy a, d by d, e by c. This process may be
represented by the operator (77); and we may write
febadc\ ,7,7
I 7 7 I becda - bcade :
\abcdej
26 THE SUBSTITUTION OPERATOR CH. XXIII
or, omitting the letters that are unaltered, and thus reducing the
operator to its simplest form,
( ] hecda = heads.
\acej
The operator ( ) , and the operation which it effects, are called
a Substitution ; and the operator is often denoted by a single
capital letter, S, T, &c.
Since the number of different permutations of a group of n
letters is n\, it is obvious that the number of different substitu-
tions is also 7i\, if we include among them the identical substi-
tution ill " j ' (denoted by S'^ or by 1), in which no letter
is altered.
We may effect two substitutions in succession upon the same
permutation, and represent the result by writing the two symbols
representing the substitutions before the permutation in order
from right to left. Thus, if >S^ = {^^^\ , T = (^^\ ,
STaebcd = ecabd.
We may also effect the same substitution twice or three times
over, and denote SS by ;S^^, SSS by S^, &c. Thus, S being as
before,
S'^aebcd - Sceabd = becad.
It should be observed that the multiplication of substitution
symbols is not in general commutative. For example, S and T
being as above, STaebcd = ecabd, but TSaebcd = caebd. If, when
reduced to their simplest form, the symbols S and T have no
letter in common, they are obviously commutative. This con-
dition, although sufficient, is not necessary ; for we have
(dcab\ (badc\ , , „ (badc\ (dcab\ , ,
\ I. j) ( 7 J aocde = cdOae = ( , , ( , , abcde.
\abcdj \abcdj \abcdj \abcdj
§ 20.] Since the number of permutations of n letters is
limited, it is obvious that if we repeat the same substitution, >S',
sufficiently often we shall ultimately reproduce the permutation
that we started with. The smallest number, /a, of repetitions
for which this happens is called the order of the substitution S,
§§ 19-22 ORDER AND GROUP 27
Hence we have S'^ = l, and >S^^'^ = 1, where p is any positive
integer.
We may define a negative index in the theory of substitu-
tions by means of the equation S'"^ = S^'^''^, n being the order of
8, and p such that j9/a > q. From this definition we see that
S!iS-'^ = S^S^"-^ = S^'' = 1. In other words, S'^ and S''^ are inverse
to each other; in particuUir, if
^^ _ /dahc\ 1 o-i _ (ahcd\ _ /hcda\
\abcdj' \dabcj \abcdj'
A set of substitutions which are such that the product of
any number of them is always one of the set is called a group ;
and the number of distinct substitutions in the group is called
t/ie order of the group. The number of letters operated on is
called the degree of the group.
It is obvious from what has been shown that all the powers
of a single substitution, >S^, form a group whose order is the
order of S.
§21.] A substitution such as i i/i f)> where each letter
is replaced by the one that follows it, and the last by the first, is
called a Cyclic Substitutmi, and is usually denoted by the symbol
(abcdef).*
The cyclic substitution (a), consisting of one letter, is an
identical substitution; it may be held to mean that a passes into
itself.
The cyclic substitution of two letters (ab), or what is the
same thing (ba), is spoken of as a Transposition.
The effect of a cyclic substitution may be represented by
writing the n letters at equal intervals round the circumference
of a circle, and shifting each through Ijnih. of the circumference.
Thus, or otherwise, it is obvious that the order of a cyclic sub-
stitution is equal to the number of the letters which it involves.
§ 22.] Every substitution either is cyclic or is the product of a
number of independent cyclic substitutions {cycles).
Consider, for example, the substitution
* Or, of course, by (bcdefa), (cdefai), &c.
28 CYCLES CH. XXIII
o _ fhfdcgaeh\
~ \abcdefgh)''
This replaces ahyh, h by/, /by a; these together constitute
the cyclic substitution {ahf). Next, c is replaced by d, and d by
c; this is equivalent to the cycle {cd). Again, e is replaced by
g, and ghj e; this gives the cycle {eg). Finally, h is unaltered.
Hence we have the following decomposition of the substitution
S into cycles —
8={ahf){cd){eg){h).
The decomposition is obviously unique; and the reasoning
by which we have arrived at it is perfectly general. It should
be noticed that, since the cycles are independent, that is, have
no letters in common, they are commutative, and it is indifferent
in what order we write them.
§ 23.] Every cyclic substitution of n letters can he decomposed
into the product ofn — 1 transpositions.
For example,, we have (abed) = (ab){bc)(cd) ; and the process
is general.
Cor. Every substitution can be decomposed into n-r transpo-
sitions, where n is the number of letters which it displaces, and r
the number of its proper cycles.
= (ab)(bf){cd)(eg).
This decomposition into transpositions is not unique, as will
be seen presently, but the above gives the minimum number.
§ 24.] The following properties of a product of two trans-
positions are of fundamental importance.
I. The product of two transpositions which ham two letters
in common is an identical substitution.
This is obvious from the meaning of {ah).
II. In the product of two transpositions, TT', which have a
letter in comnfion, T' may he placed first, provided we replace the
common letter in T by the other letter in T.
§§ 22-25 DECOMPOSITION INTO TKANSPOSITIONS 29
For we have {ah){hc) = Q^) , {hc){ac) = (^^^) ,
therefore {ab){bc) = {hc){ac).
Cor. 1. {ef){af) = {ae){ef).
Cor. 2. {ae){af) = {af){ef).
Ill, If two transpositions, T and T', have no letter in common,
they are commutative.
This is a mere particular case of a remark already made
regarding two independent substitutions.
§ 25.] The decomposition of a given substitution into transpo-
sitions is not unique.
For we can always introduce a pair of factors {ab){ab), and
then commutate one or both of them with the others, in accord-
ance with the rules of § 24.
In this way we always increase the number of transpositions
by an even number. In fact, we can prove the following im-
portant theorem —
The number of the transpositions which represent a given sub-
stitution is always odd or always even.
We may prove this by reducing the product of transpositions
to a standard form as follows —
Select any one of the letters involved, say a ; take the last
transposition, T, on the right that involves a, and proceed to
commutate this transposition successively with those to the left
of it. So long as we come across transpositions that have no
letter in common with T, neither T nor the others are affected.
If we come to one that has a letter in common with T which is
not a, we see (§ 24, 11. , Cor. 1) that the a in jT remains, the other
letter being altered, and the transposition passed over remains
unaltered. If we come to a transposition that has a, and a only,
in common with T, by § 24, II., Cor. 2, T passes to the left un-
altered, and the transposition passed over loses its a. Lastly, if
we come to a transposition that has both a and its other letter
in common with T, then both it and T may be removed. If
this last happen, we must now take that remaining transposition
containing a which is farthest to the right, and proceed aa
before.
80 DECOMPOSITION INTO TRANSPOSITIONS CH. XXIII
The result of this process, so far as a is concerned, will be,
either that all the transpositions containing a will have dis-
appeared, or that some even number (including 0) will have done
so, and one only, say {ah), will remain on the extreme left.
Consider now b. If among the remaining factors b does not
occur, then we have obtained a cycle {ab) of the substitution ;
and we now proceed to consider some other letter.
If, however, b does occur again, we take the factor farthest
to the right in which it occurs, and cominutate as before ; the
result being, either that all the transpositions (even in number)
containing b disappear, or that an even number of them do, and
we are left with, say {be), in the second place. We now deal
with c in like manner ; and obtain in the third place, say {cd).
This goes on until all the letters are exhausted, or until we
come to a letter, say /, that disappears from the factors not yet
finally arranged. We thus arrive at a product {ab}{bc){cd){de){ef)
on the left.
Now {ab){bc)(cd){de){ef) ^ (^'^'^^^)
= (abcdef).
We have, in fact, arrived at one of the independent cycles of
the substitution. If we now take any other letter that occurs in
one of the remaining substitutions on the right, we shall in like
manner arrive at the cycle to which it belongs, after losing an
even number, if any, of the transpositions ; and so on, until all
the letters are exhausted, and all the cycles arrived at. Since
the whole number of transpositions lost is even, the truth of the
theorem is now obvious ; and our proof furnishes a method for
reducing to the minimum number of transpositions.
It appears, therefore, that we may divide all the substitutions
of a set of n letters into two classes — namely, eve?i substitutions,
which are equivalent to an even number of transpositions, and
odd substitutions, wliich are equivalent to an odd number of
transpositions.
Cor. 1. If n be the number of letters altered by a substitution, r
the number of its cycles, and 2s an arbitrary even integer, the number
offactoi's in an equivalent product of transpositions is n-r+ 2s.
§§ 25-27 EVEN AND ODD SUBSTITUTIONS 31
Cor, 2, The number of thd even is equal to the number of the
odd substitutions of a set of n letters.
For any one transposition, applied in succession to all the
different odd substitutions, will give as many even substitutions,
all different. Hence there are at least as many even as there
are odd substitutions. In like manner we see that there are at
least as many odd as there are even. Hence the number of the
even is equal to the number of the odd substitutions.
Cor. 3. A cyclic substitution is even or odd according as the
number of the letters which it involves is odd or even.
For example, {abc) = {ab) (be) is even.
Cor. 4. The product of any number of substitutions is even or
odd according as the number of odd factors is even or odd. In
particular, ajiy power whatever of an even substitution, and any
even power of any substitution whatever, form even substitutions.
Cor. 5. All the even substitutions of a set of n letters form a
group whose order is n\/2.
§ 26.] If we select arbitrarily any one, say P, of the nl per-
mutations of a set of n letters, and call it an even permutation,
then we can divide all the nl permutations into two classes —
1st, w!/2 even permutations, derived by applying to P the 7i!/2
even substitutions ; 2nd, w!/2 odd permutations, derived by
applying to P all the 72!/2 odd substitutions.
The student who is familiar with the theory of determinants
will observe that the above is precisely the classification of the
permutations of the indices (or umbra?) which is adopted in
defining the signs of the terms in a determinant.
It is farther obvious, from the definitions given in chap, iv.,
§ 20, that symmetric functions of a set of n variables are un-
altered in value by any substitution whatever of the variables ; or,
as the phrase is, they are said to " admit any substitution what-
ever." Alternating functions, on the otlier hand, admit only even
substitutions of their variables, the result of any odd substitution
being to alter their sign without otherwise affecting their value.
§ 27.] The limits of the present work will not permit us to
enter farther into the Theory of Substitutions, or to discuss its
applications to the Theory of Equations. The reader who desires
82 EXERCISES III CH. XXIII
to pursue this subject farther will find iuformation in the follow-
ing works : Serret, Cours d'Alg^bre Superieure (Paris, 1879) ;
Jordan, Traite des Substitutions (Paris, 1870) ; Netto, Suhstitu-
tionen-theorie (Leipzig, 1882) ; Burnside, Theory of Groups
(Cambridge, 1897).
Exercises III.
(1.) There are 10 counters in a box marked 1, 2, . . . , 10 respectively.
Three drawings are made, the counter drawn being replaced each time. In
how many ways can the sum of the numbers drawn amount — 1st, to 9
exactly; 2nd, to 9 at least?
(2.) Out of the integers 1, 2, 3, . . .,10 how many pairs can be selected
so that their sum shall be even ?
(3.) How many different throws can be made with n dice?
(4.) In how many ways can 5 black, 5 white, 5 blue balls be equally
distributed among three bags, the order of the bags to be attended to?
(5.) A selection of c things is to be made partly from a group of a, the
rest from a group of b. Prove that the number of ways in which such a set
can be made will never be greater than when the number of things taken
from the group of a is next less than (a + 1) (c + l)/(a + 6 + 2),
(6.) In how many ways can p + 's and n - 's be placed in a row so that no
two - 's come together ?
(7.) In the Morse signalling system how many signals can be made
without exceeding 5 movements ?
(8.) In how many ways can 3 pairs of subscribers be set to talk in a
telephone exchange having n subscribers ?
(9.) There are 3 colours, and m balls of each. In how many ways can
they be arranged in 3 bags each containing m, the order of the bags to
be attended to ?
(10.) If of ^ + g' + r things p be alike, q alike, and r different, the total
number of combinations will be (p + 1) (g + 1) 2^ - 1.
(11.) In how many ways can 2n things be divided into n pairs?
(12.) The number of combinations of 8« things {n of which are alike),
taken n at a time, is the coefficient of x" in (1 + x)-"/(l - x).
(18.) ^boat clubs have a, 6, c, 1, 1, . . ., 1 boats each. In how many
ways can the boats be arranged subject to the restriction that the 1st boat of
any club is to be always above its 2nd, its 2nd always above its 3rd, &c. ?
(14.) If there be p things of one sort, q of another, r of another, Ac. , the
number of combinations of the p + q + rJr . , . things, taken k at a time, is
the coefficient of x* in (l-a;'^i)(l-a;3+i) . . . /(I -x) (1 -x) . . .
(15.) In how many ways can an arrangement of n things in a row be
deranged so that — 1st, each thing is moved one place ; 2nd, no thing more
than one place ?
(16.) Given n things arranged in succession, the number of sets of 3
§ 27 EXERCISES III 33
which can be formed under the condition that no set shall contain two things
which were formerly contiguous is {n -2){n- 3) {n - 4), the order inside the
sets to be attended to.
(17.) In how many ways can m white and n black balls be arranged in a
row so that there shall be 2r- 1 contacts between white and black balls?
(18.) In how many ways can an examiner give 30 marks to 8 questions
without giving less than 2 to any one question?
*(19.) The number of ways in which n letters can be arranged in r pigeon-
holes, the order of the holes and of the letters in each hole to be attended to
and empty holes admitted, is r(r + l) (r + 2) . . . (r+n-1).
(20.) The same as last, no empty holes being admitted, n!(n-l)!/(n-r)I
(r-1)!.
(21.) The same as last, the order of the holes not being attended to,
hI (n- 1)1/(71 -r)Ir!(r-l)!.
(22.) The number of ways in which n letters, all alike, can be distributed
into r pigeon-holes, the order of the holes to be attended to, empty holes to
be excluded, is „_i(7^_i.
(23.) Same as last, empty holes being admitted, n+r-i^r-i'
(24.) Same as last, no hole to contain less than q letters, „_i_^(g_i|Cy_i.
(25.) The number of ways of deranging a row of n letters so that no letter
may be followed by the letter which originally followed it is 7i] + (« - l)i .
(26.) The number of ways of deranging m + n terms so that m are dis-
placed and n not displaced is (m + n)Imj/wi!n!.
(27.) The number of ways in which r different things can be distributed
among n+p persons so that certain n of those persons may each have one at
least is
Sr={n+pY-n(n+p-lY+^^-^(n+p-2y-. . .
Hence prove that
Si = Sf2=. . .=-S„_i = 0, S„=n!, S„+i = (^|+i>)(n + l)l.
{Wolsten?iolme.)
(28.) Fifteen school-girls walk out arranged in threes. How many times
can they go out so that no two are twice together? (See Cayley's Works, vol.
I., p. 481.)
Exercises IV.
Topological.
(1.) The number of sides of a complete n-point is i|n(ra-l), and the
number of vertices of a complete n-side is the same,
(2.) The number of triangles that can be formed with 2n Unes of lengths
1, 2, . . ., 2nisn(ra-l)(4n-5)/6.
(3.) There are n points in a plane, no three of which are collinear, How
* Exercises 19-25 are solved in Whitworth's Choice and Chance ; q.v.
0. n. 3
34 EXERCISES IV CH. XXIII
many closed r-sided figures can be formed by joining the points by straight
lines?
(4.) If m points in one straight line be joined to n points in another in
every possible way, show that, exclusive of the m+n given points, there are
mn {m - 1) (n - 1)/2 points of intersection.
(5.) On three straight lines, A, B, C, are taken Z, m, n points respectively,
no one of which is a point of intersection. Show that the number of triangles
which can be formed by taking three of the Z + m + n points is |(7n+n) (n+Z)
(Z + m) - mn -nl- Im.
(6.) There are n points in a plane, no three of which are coUinear and no
four concyclic. Through every two of the points is drawn a straight line and
through every three a circle. Assuming each straight line to cut each circle
in two distinct points, find the number of the intersections of straight lines
with circles.
(7.) In a convex polygon of n sides the number of exterior intersections of
diagonals is ^^^2^1 (71 - 3) (n - 4) (n - 5), and the number of interior intersections
i8^^n(n-l)(7i-2)(n-3).
(8.) There are n points in space, no three of which are collinear, and no
four coplanar. A plane is drawn through every three. Find, 1st, the num-
ber of distinct lines of intersections of these planes; 2nd, the number of these
lines of intersection which pass through one of the given n points ; 3rd, the
number of distinct points of intersection exclusive of the original n points.
(9.) Out of n straight lines 1,2, . . .,n inches long respectively, four can be
chosen to form a pericyclic quadrilateral in { 2n (n - 2) (2n - 5) - 3 + 3 ( - l)"}/48
ways.
(10.) Show that n straight lines, no two of which are parallel and no three
concurrent, divide a plane into \{rfi->i-n + 2) regions. Hence, or otherwise,
show that n planes through the centre of a sphere, no three of which are
coaxial, divide its surface into n^-n + 2 regions.
(11.) Show that two pencils of straight lines lying in the same plane, one
containing m the other n, divide the plane into mn + 2m + 2ra - 1 regions, it
being supposed that no two of the lines are parallel or coincident.
(12.) If any number of closed curves be drawn in a plane each cutting all
the others, and if ??,. be the number of points through which r curves pass,
the number of distinct closed areas formed by the plexus i?
l + n3 + 2?i3+. . .+r«r+i+. . .
CHAPTER XXIV.
General Theory of Inequalities.
Maxima and Minima.
§ 1.] The subject of the present chapter is of importance in
many branches of algebra. We have already met with special
cases of inequalities in the theory of Ratio and in the discussion
of the Variation of Quadratic Functions of a single variable ; and
much of what follows is essential as a foundation for the theory
of Limits, and for the closely allied theory of Infinite Series. In
fact, the theory of inequalities forms the best introduction to the
theory of infinite series, and, for that reason, ought to be set as
much as possible on an independent basis.
§ 2.] We are here concerned with real algebraical quantity
merely. As we have already explained, no comparison of com-
plex numbers as to relative magnitude in the ordinary sense can
be made, because any such number is expressed in terms of two
absolutely heterogeneous units. Strictly speaking, there is a
similar difficulty in comparing real algebraical quantities which
have not the same sign ; but this difficulty is met (see chap.
XIII. , § 1) by an extension of the notion of inequality. It will
be remembered that a is defined to be algebraically greater or
less than h according as the reduced value of a - 6 is positive
or negative. An immediate consequence of this definition is
that a positive quantity increases algebraically as it increases
numerically, but a negative quantity decreases algebraically as
it increases numerically. The neglect of this consideration is a
fruitful source of mistakes in the theory of inequalities.
§ 3.] From one point of view the theory of inequalities runs
3—2
86 ELEMENTARY THEOREMS CH, XXIV
parallel to the theory of conditional equations. In fact, the
approximate numerical solution of equations depends, as we have
seen, on the establishment of a series of inequalities*.
The following theorems will bring out the analogies between
the two theories, and at the same time indicate the nature of
the restrictions that arise owing to the fact that the two sides of
an inequality cannot, like the two sides of an equation, be inter-
changed without altering its nature. For the sake of brevity,
we shall, for the most part, write the inequalities so that the
greater quantity is on the left, and the sign > alone appears.
The modifications necessary when the other sign appears are in
all cases obvious.
I. jyP>Q, Q>B, R>S, then P>S.
Proof.— {P -Q)+(Q-B) + (B-S) = P-S, hence, sinceP - Q,
Q-B, E-S aie all positive, P-S is positive, that is, P>S.
II. IfP>Q,thenP±B>Q±B.
For (P ±B) - (Q±B) = P - Q ; hence the sign of the former
quantity is the same as the sign of the latter.
Cor. 1. JfP+Q>B + S,then
P+Q-B>S, -B-S>-P-Q, -P-Q<-B-S.
It thus appears that we may transfer a term from one side of
an inequality to another, provided we change its sign; and we
may change the signs of all the terms on both sides of an inequality,
provided we reverse the symbol of inequality.
Cor. 2. Every inequality may be reduced to one or other of
the forms P>0 or P<0.
In other words, every problem of inequality may be reduced
to the determination of the sign of a certain quantity
III. JfP,>Q„ P,>Q,, . . ., P„>Q„,
thm Pi + P,+ . . . +P„> Qi + Q,+ ...+$„;
for {P, + P,+ . . . +Pn)-iQ^+Q,+ . . . +Qr,)
^{P^-Q^) + {P.-Q.)+. . . +(Pn-Qn),
whence the theorem follows.
It should be noticed that it does not follow that, if Pi>Q,
P^> Q„ then P^ - P,>Q, - Q„
* See, for example, the proof that every equation has a root.
§ 3 ELEMENTARY THEOREMS 37
IV. If P> Q, then PE>QE, and P/B > Q/E, provided E
be positive; but PE<QE, P/E<Q/E, if E be negative.
For (P-Q)E and (P - Q)/E have both the same sign as
P—Q HE be positive, and both the opposite sign if E be
negative.
Cor. 1. If P> QE, and E>S,then P> QS, provided Q be
positive.
Cor. 2. Every fractional inequality can be integralised.
For example, if P/Q>E/S, then, provided QS be positive,
we have, after multiplying by Q8, PS> QE ; but, if QS be
negative, PS<QE.
If there be any doubt about the sign of QS, then we may
multiply by Q^S^, which is certainly positive, and Ave have
QPS'>Q'ES.
V. IfPi>Qi,P2>Q2, . . . , Pn>Qn, ctnd all the quantities
be positive, then
P,P, . . . Pn> Q,Q, . . . Q,.
For P,P,P, . . . P^>Q,P,P, . . . P„,
since Pi > Qi and P2P3 . . . P„ is positive ;
>Q.Q.P. . . . P„, •
since P2>Q2 and Q1P3 . . , P,j is positive ; and so on. Hence,
finally, we have
P,P, . . . P„> Q,Q, . . . Q,.
Cor. 1. If P>Q, and both be positive, then P'>Q'',n being
any positive integer.
Cor. 2. If P>Q, and both be positive, then P^"'>Q^"', n
being any positive integer, and the real positive value of the nth
root being taken on both sides.
For, if P'^"^ Q^"", then, since both are real and positive,
{pvnY^{Qvn)n^ by Cor. 1 ; that is, P? Q, which contradicts our
hypothesis.
Cor. 3. If P>Q, both being positive, and n be any positive
quantity, then P-"<Q-", where, if the indices are fractional,
there is the usual understanding as to the root to be taken.
Eemark. — The necessity for the restrictions regarding the
38 EXAMPLES CH. XXIV
sign of the members of the inequalities in the present theorem
will appear if we consider that, although — 2 > - 3, and - 3 > - 4,
yet it is not true that ( - 2) ( - 3)>( - 3) ( - 4).
These restrictions might be removed in certain cases ; for
example, it follows from - 3 > - 4 that ( - 3)^>( - 4)^ in other
words, that - 27 > - 64 : but the importance of such particular
cases does not justify their statement at length.
Cor. 4. An inequality may be rationalised if due attention he
paid to the above-mejitioned restrictions regarding sign.
§ 4.] By means of the theorems just stated and the help of
the fundamental principle that the product of two real quantities
is positive or negative according as these quantities have the
same or opposite sign, and, in particular, that the square of any
real quantity is positive, we can solve a great many questions
regarding inequalities.
The following are some examples of the direct investigation
of inequalities ; the first four are chosen to illustrate the paral-
lelism and mutual connection between inequalities and equa-
tions : —
Example 1. Under what circumstances is
(3x - l)l(x - 2) + (2x - 3)l(x - 5) > or < 5?
1st. Let us suppose that x does not lie between 2 and 5, and is not equal
to either of these values. Then (x -2)(x- 5) is positive, and we may multiply
by this factor without reversing the signs of inequality.
Hence F=(3a;- l)/(x-2) + (2x-3)/(a;-5)><5,
according as
(3a; -l){x-5) + {2x - 3) (x - 2) >< 5 (x - 2) (a; - 5),
according as Sx^* - 23x + 11 >< 5x2 - 3ox + 50,
according as 12x5> <39,
according as x><3J.
Under our present supposition, x cannot have the value 3J ; but we con-
clude from the above that if x>5, F>5, and if x<2, F<5.
2nd. Suppose 2<x<5. In this case (x-2)(x-5) is negative, and we
must reverse all the signs of inequality after multiplying by it.
We therefore infer that if 2<x<3J, F<5, and if 8J<x<5, then
J'<5.
The student should observe that, as x varies from - oo to + oo , the sign of
the inequaUty is thrice reversed, namely, when x=2, when x = 3^, and when
x = 5; the first and last reversals occur because F changes sign by passage
through an infinite value; the second reversal occurs because F passes
§§ 3, 4 EXAMPLES 39
through the value 5. The student should draw the graph of the func-
tion F.*
Example 2. Under what circumstances is
i?'=(3a;-4)/(x-2)><l?
Multiplying by the positive quantity (x - 2f, we have
(3x-4)/{a;-2)><l,
according as (3a; - 4) (x - 2) > < (x - 2)^,
according as { (3x - 4) - (a; - 2) } (a; - 2) >< 0,
according as 2 (x - 1) (x - 2) > < 0.
Hence F>\, if x<.l or >2;
F<1, if l<a;<2.
Example 3. Under what circumstances is x* + 25x > < Sx^ + 26 ?
x3 + 25x><8x2-(-26,
according as x' - Sx^ + 25x - 26 > < 0,
according as (x-2){x2-6x + 13)> <0,
according as (x-2){(x-3)* + 4}> <0.
Now (x -3)2 + 4 is positive for all real values of x ; hence
x» + 25x><8x2 + 26,
according as x><2.
Example 4. If the positive values of the square roots be taken in all
^(2x + l)+V(x-l)><V(3x)?
Owing to the restriction as to sign, we may square without danger of
reversing the inequality. Hence
V(2x + 1) + ^{x - 1) > < V(3x),
according as 2x + l + x-l + 2;^{(2x + l) (x-l)}> <3x,
according as 2^{(2x + l)(x-l)}> <0.
Now, provided x is such that the value of ^y{(2x + l) (a;-!)} is real, that is,
provided x>l,
2^{(2x + l)(x-l)}>0,
therefore »y(2x + 1) + ^(x - 1) >^(3x), if x > 1.
Negative values of x less than -\ would also make ;^{(2x + l) (x-1)}
real ; but such values would make ;^(2x + l), J{;x-1), and sj{dx) imaginary,
and, in that case, the original inequality would be meaningless.
Example 5. li x, y, z . . . be n real quantities (n - 1) Sx^ <t 2Sx?/.
Since all the quantities are real, S (x - y)^ <t 0.
Hence, since x will appear once along with each of the remaining n - 1
letters, and the same is true of ?/, z, . . ., we have
(»i-l)2x2-22x!/<tO,
that is, (n - 1) Sx^ <t 22xi/.
* The graphical study of inequalities involving only one variable will be
found to be a good exercise.
40 EXAMPLES CU. XXIV
In the case where x = y = z= . . . . we have Sx* = na;^ 2S.Ty = 2„C2.t'^
= n(n-l)x2, 80 that the inequality just becomes an equality.
When n=2, we have the theorem
x^ + j/^ <t 2xy ;
or, if we put x=^a, y—sjh, a and b being real and positive,
a theorem already established, of which the preceding may be regarded as a
generalisation. A more important generalisation of another kind will be
given presently.
Example 6. Iix,y,z, . . . be 7i real positive quantities, and ^J and g any
two real quantities having the same sign, then
xP+<j + yP-t-9 <t xfy'i + x'iyP,
n2xP+9<2xPSx9.
We have seen that x^-yv and afl-yi will both have the same sign as
x-y, or both opposite signs, according as p and q are both positive or both
negative. Hence, in either case, (xp - yf>) {sfl - 2/«) has the positive sign.
Therefore
(xP - y'P) (x9 - 1/9) <i 0,
whence xp+9 + j/P+"9 <f xP(/9 + x'j/P.
If we write down the JJ^ inequalities like the last, obtained by taking
every possible pair of the n quantities x, y, z, . . . , and add, we obtain the
following result —
(n-l)2xP+9<t2xPj/9.
If we now add 2xp+« to both sides, we deduce
?i2xP+9<t2xP2x9.
N.B. — If ^ and q have opposite signs, then
n2xP+9>2xP2x9.
These theorems contain a good many others as particular cases. For
example, if we put q= -p,vfe deduce
2xP2x-P<t:n2,
which, when n=3, p=l, gives
(x + 1/ + 2) (1/x + 1/y + 1/z) <t 9 ;
whence (x + y + z)(yz + zx + xy)<t9xyz;
and so on.
Example 7. If x, y, z be real and not all equal, then 2x5x3x^2,
according as 2x><0.
For 2x» - 3x2/2 = 2x (2x2 -2x?/),
= |2x2(x-j/)2.
Hence the theorem, since 2 (x - y)"^ is essentially positive.
Example 8. To show that
1 1.3 .. . (271-1) V(n+1)
V(2k + 1)^ 2.4 ... 2» '^ 2n + l '
where n is any positive integer.
§§ 4, 5 EXAMPLES 41
From the inequality a + 6>2^(a6) we deduce
(2«-l) + (2n + l)>2V{(2ra-l)(2n + l)};
whence (2rt-l)/2w<V{(2n-l)/(2n + l)} (1);
similarly (2re - 3)/2 {n-l)<J{(2n- 3)/(2n - 1)} (2) ;
5/2. 3 < ^{5/7} (»-2);
3/2.2<V{3/5} - (rt-1);
l/2.1<^{l/3} (n).
Multiplying these inequalities together, we get
1.3.5 ... (2n-l) 1 .^.
2.4.6 . . . (2n) J(2n + 1) ^ ''
Again, n+{n+l)>2^{n(n + l)},
that is, 2n + l>2^{7i{n+l)}.
Hence we have the following inequalities —
{2n + l)/2n>v/{(n + l)/n} (1)',
(2n-l)/2(n-l)>v/{n/(n-l)} (2)',
■ ' * 7/2.3>^{4/3} (n-2)',
5/2.2>V{3/2} . (n-1)',
8/2.1>V{2/l} (n)'.
Multiplying these n inequalities together, we get
1.3.5 . . . (271+1) ,. ^ ,,
Hence 1 -3.5 . . ■ (2n-l) ^^/(n + l) ^^^
■ 2.4.6 . . . 2n 2n + l ^ '
(A) and (B) together establish the theorem in question.
Since J(n + l)l(2n + l)>^(n + l)l(2n + 2)>ll2J(n+l), we may state the
above theorem more succinctly thus,
1 1.3 .. . (2n-l) 1
^(2n + l)^ 2.4. . . 2n ^ 2J(n + l)'
DERIVED THEOREMS.
§ 5.] We now proceed to prove several theorems regarding
inequality which are important for their own sake, and will be
of use to us in following chapters.
If hi, 1)2, . . .yhnhe all positive, the fraction (^1 + ^2 + . . . + a^)/
(61 + ^2 + • • . + tn) «s not less than the least, and not greater tlmn
the greatest, of the n fractions a^jbi, a^jh^, . . . , cinlK-
Let / be the least, and /' the greatest of the n fractions,
then
aijbi<if, a^/bi^if • • •> ajb^^i^f
42 MEANS AMONG RATIOS CH. XXIV
Hence, since hi,h^, . . ., &„ are all positive,
ai^fbi, a2<^fb2, . . ., an^fbn.
Adding, we have
(ai + «2 + . • • +a7i) <t:/(^ + ^j + - • • + ^n);
whence
(«! + ^2 + . . . + «„)/(&! + &2 + • . . + ^») -^f'
In like manner, it may be shown that
(«! + «2 + . . . + a„)/(6i + &2 + . . . + ^n) ^Z'-
Remark. — This theorem is only one among many of the same
kind*. The reader will find no difficulty in demonstrating the
following : —
Iftti, Ui,. . ., a„, bi, bi, . . .,bn be as before, and k, h, - ■ ■, k
be n positive quantities, then "^liai/^libiis not less than the least,
and not greater than the greatest, among the n fractions ai/bi , a^jb^,
. . . , an/ On .
Ifai, a^, . . ., a„, bi, h, . . .,bn, h, k, • • -Jn be all positive,
then {U^a.'^ltkbrYi'' and [a.a^ . . . ajb.b, . • • bnY'"" are,
each of them, not less than the least, and not greater than the
greatest, among the n fractions ai/bi, a-^jb^, . . ., ajbrn-
Example, to prove that
1 » / (1.3 .. . {2n-l)}
2'^\/ I 2.4 ... 2n J*^'
Since the fractions 1/2, 3/4, . . . (2n - 1)/2« are obviously in ascending
order of magnitude, we have, in the second part of the last of the theorems
just stated,
«/ (1.3 . . , (2n-l)) 2n-l
i<7i'
2.4 ... 271 ( 2ra
Now, (2«- l)/2rt = l - l/27i<l, hence the theorem follows; and it holds, be it
observed, however great n may be.
§ 6.] If X, p,q be all positive, and p and q be integers, then
{a^ -\)lp><{afl- \)lq according as pXq.
Since p and q are positive,
{aP-l)/p><{afl-l)/q,
according as q(af-l)><p{afl- 1),
* See the interesting remarks on Mean Values in Cauchy's Analyse
Alg6brique.
§§5-7 {xP-l)/p>(ai^-l)/q 43
according as
(a;-l){q{af-^ + xP-' + . . . + l)-p{af-' + af^-'' + . . . + 1)}><0.
lip>g, we have
X = {a)-l){q(af-'' + a;P-^+. . . +l)-p{afl-^ + afl-^+ . . . + !)},
= {a;-l){q{xP-' + af-^ + . . . +0/^)- (j)-q){afl-' + afl-^ + . . . + 1)}.
Now, if a;>l,
afi-i + afl-^ + , . . + 1 < qafi-^ ;
X>{a;-l){q{p-q)af^-{p-q)qafl-%
>q{p-q)afl-'{w-lf,
>0.
Again, if a:<l,
-^;5-i + ^-2 + , , , + 1 > qa^-^ ;
but, since ^ - I is now negative, the rest of the above reasoning
remains as before.
Hence, in both cases,
{a:'-l)/p>{^-l)lq.
By the same reasoning, if q>p,
{afl-l)lq>{aP-l)lp,
that is, '\ip<q,
{af-l)/p<{afl-l)/q.
§ 7.] I/a; be positive, and +1, then
m^"-^ (a; - l)>a;'^ - l>m (x - 1),
unless m lie between 0 and + 1, in which case
mx""-^ {x -\)<x'^ -\<m{x -I).
From § 6, we have
{^'-l)><{plq){^'^-\) (1),
according as jpXg, where I is any positive quantity + 1, and
p and q positive integers. In (1) we may put x^i'^ for ^, where as
is any positive quantity + 1 (the real positive value of the gth
root to be taken), and we may put m for pjq, where m is any
positive commensurable quantity. (1) then becomes
x'''-l><m{x-l) (2),
44 inx'^-^(oo—l)^x'"' — l^m(x — l) CH. xxiv
according as m> <1, which is part of the theorem to be
established.
In (2) we may replace a; by 1/^, where a; is any positive
quantity 4=1, and the inequality will still hold.
Hence (l/a;)™- l><w(l/^- 1) (3),
according as mxl.
If we multiply (3) by - af^, we deduce
ic'^-lomaf-^iv-l),
that is, ma;™-^ (w-l)><x'^-l,
according as mxl.
We have thus established the theorem for positive values
of m.
Next, let m = -n where n is any positive commensurable
quantity. Then
a;-»-l><(-w)(a;-l),
according as 1 - a;" >< - nx'^ (^ - 1 ),
according as a;" - 1 <> naf (a; — 1 ),
wa;'*+'-wa;''><a;'*-l.
Add af''^^ - af" to both sides, and we see that
a7-»-l><(-w)(a;-l),
according as
(w + 1) a?" (a; - 1) ><a;"+^ - 1.
Now, since n is positive, w + 1 > 1, therefore, by what we
have already proved,
(w + l)a;"(a;-l)>a;''-''-l.
Hence a;-"-l>(-w)(a;- 1) (4).
In (4) we may write 1/a; for w ; and then we have
(l/a;)--l>(-7i)(l/a;-l).
If we multiply by - a;-", this last inequality becomes
a;-"-l<(-M)a7-''-^(a;-l),
that is, (- n) a;-"-^ (a; - l)>a;-" - 1.
Hence, if m be negative,
ma;'"-^ (a: - 1) >a;"' - 1 > w (a- - 1) J
which completes the demonstration.
§ 7 mx'^-'^ (^-y)< ^"* - 2/"* < my'^-^ (a; - y) 45
Cor. If X and y be any two unequal positive quantities, we
may replace a; in the above theorem by a^/y. On multiplying
throughout by y'^, we thus deduce the following —
J[fa! and y be positive and unequal, then
mx^~^ {x-y)>x^'^ — y^>my^~^{x-y),
unless m lie between 0 and + 1, in which case
mx^~^ {x - y)<x'^ -y^ < my^~'^ {x — y).
We have been careful to state and prove the inequality of
the present section in its most general form because of its great
importance : much of what follows, and many theorems in the
following chapter, are in fact consequences of it*.
Example 1. Show that, if x be positive, (1 + a;)"* always lies between
1 + wix and (l + a;)/{l + (l-wi)x}, provided mx<.l + x.
Suppose, for example, that m is positive and < 1. Then, by the theorem
of the present section,
m (1 + x)"*-i a; < (1 + «)•* - 1 < ma;.
Hence (1 + x)™ < 1 + mx.
Also, (l + x)"'-l>mx(l + a;)'»/(l + a;),
{1 - wia;/(l + x) } (1 + x)»» > 1.
If mx<l + x, l-mx/(l + x) is positive, and we deduce
(l + x)'»>l/{l-wix/(l + x)},
>(l + x)/{l+(l-m)x}.
The other cases may be established in like manner.
Remark, — It should be observed that
(l±x)"»> <l±mx,
according as m does not or does lie between 0 and + 1.
Example 2. Show that, if m^ , Wj . . . , u„ be all positive, then
{1 + Mi)(l+Ma) . . . (1 + M„)>l + Ui + M2+ . . .+u„;
also that, if Uj , u^ ■ • • > ^n ^ ^^ positive and each less than 1, then
(l-«l)(l-"2) • • • (l-"n)>l-Wl-'«2- • • • -"n-
The first part of the theorem is obvious from the identity
(1 + Mi)(l+M2) . . . (1+M„) = 1+SMx + SUiU2 + SUjM2Uj,+ . . . +?<il/2 ...«„.
The latter part may be proved, step by step, thus —
1-Mj = l-Uj.
(1 - Ml) (1 - Wj) = 1 - Ui - 1*2 + MjMs,
>l-til-M2.
* Several mathematical writers have noticed the unity introduced into
the elements of algebraical analysis by the use of this inequality. See
especially Schlomilch's Handbuch der Algebraischen Analysis. The secret of
its power lies in the fact that it contains as a particular case the fundamental
limit theorem upon which depends the differentation of an algebraic function.
The use of the theorem has been considerably extended in the present volume.
46 ARITHMETIC AND GEOMETRIC MEANS CH. XXIV
Hence, since 1 - Uj is positive,
(1 - Ui) (1 - M2) (1 - "3) > (1 - U3) (1 - "1 - "2).
>l-Ui-W2-M3 + Ws(Ui + W2)f
>l-Wj-M2-"3'
And so on.
These inequalities are a generalisation of (1 ± x)" > 1 ± wx {x<l ,and n a
positive integer). They are useful in the theory of infinite products.
§ 8.] The arithmetic mean of n positive quantities is not less
than their geometric mean.
Let us suppose this theorem to hold for n quantities
a, b, c, . . ., k, and let I be one more positive quantity. By
hypothesis,
{a + b + c + . . . + k)ln^{ahc . . . kf'",
that is,
a + h^ c^ . . . + k<^n (abc . . . ky\
Therefore
a + b + c + . . .+k + /<}:% (abc . . . ky"" + 1.
Now,
n (abc . . . ky"" + H{n + 1) {abc . . . H)^/("+'),
provided
n{abc . . . k/l"}"'' + l-^in + 1) {abc . . . H/Z«+^p+'),
<i;{n+l){abc . . .^ ^•/^T("+'),
that is, provided
ne^^ + l<^{n+l)e,
where $''^''+^) = abc . . . A//»
that is, provided
which is true by § 7.
Hence, if our theorem hold for n quantities, it will hold for
n+1. Now we have seen that (a + 6)/2<{:(ai)*, that is, the
theorem holds for 2 quantities ; therefore it holds for 3 ; there-
fore for 4 ; and so on. Hence we have in general
{a + b + c+ . . . + k)/n<^{abc . . . ky'\
It is, of course, obvious that the inequality becomes an
equality when a = b = c=^ . . . =k.
§§ 7, 8 ARITHMETIC AND GEOMETRIC MEANS 47
There is another proof of this theorem so interesting and
fundamental in its character that it deserves mention here*.
Consider the geometric mean {ahc . . . ky\ If a, b, e, . . .
be not all equal, replace the greatest and least of them, say a
and k, by {a+k)l2; then, since {{a + k)/2}^>ak, the result has
been to increase the geometric mean, while the arithmetic mean
of the n quantities {a + k)/2, b, c, . . ., (a + k)/2 is evidently the
same as the arithmetic mean oi a, b, c, . . . , k. If the new set
of n quantities be not all equal, replace the greatest and least as
before ; and so on.
By repeating this process sufficiently often, we can make all
the quantities as nearly equal as we please ; and then the
geometric mean becomes equal to the arithmetic mean.
But, since the latter has remained unaltered throughout, and
the former has been increased at each step, it follows that the
first geometric mean, namely, (abc . . . ky'\ is less than the
arithmetic mean, namely, (a + b + c+ . . . + k)/n.
As an illustration of this reasoning, we have (1 . 3 . 5 . 9)^*
<(5 . 3 . 5. 5)i<(5 .4.4. 5)i<(4-5 . 4-5. 4-5 . 4-5)i<4-5<(l + 3
+ 5 + 9)/4.
Cor. Ifa,b,...,kben positive quantities, and p, q, . . .,tbe
n positive commensurable quantities, then
^a + qb + . . -^tk X;.v/(p+,+. . .+o
p + q+ . . . +t ^^ • • • /
It is obvious that we are only concerned with the ratios
p : q : . . . : t. Hence we may replace p, q, . . ., t hy positive
integral numbers proportional to them. It is, therefore, suffi-
cient to prove the theorem on the hypothesis that p, q, . . -, t
are positive integers. It then becomes a mere particular case of
the theorem of the present paragraph, namely, that the arithmetic
mean oi p + q + . . . + t positive quantities, p of which are equal
to a, g' to ft, . . . , ^ to ^, is not less than their geometric mean.
• See also the ingenious proof of the theorem given by Cauchy {Analyse
Algebrique, p. 457), who seems to have been the first to state the theorem in
its most general form.
48 lpa'^/lp^(lpa/lp)'^ CH. xxiv
Example 1. Show that, ii a, b, . . ., khen positive quantities,
V a+b+ . . . +k J
/a + b+ . . .+ky+^- • •+*
The first part of the proposition follows from the above corollary by taking
p = a, q = b, , . ., k = c.
The second inequality is obviously equivalent to
WJ Wb) •••[^) ^^'
which again is equivalent to
\npa J ynpbj \npk J
where p is a positive integer which may be so chosen that pa, pb, . . ., pk are
all positive integers. We shall therefore lose no generality by supposing
a, 6, c, . . ., A to be positive integers.
Consider now a positive quantities each equal to 'S.ajna, b positive quantities
each equal to "Lbjnb, &c. The geometric mean of these is not greater than
their arithmetic mean. Hence
V^^Vf— Y /SayiVS" a (T^alna) + b CLafnb) +. . .+Tc (Zaink)
\\naj \nb J ' ' ' \nk J j a + h+ . . . +k
(sy(sy- •■(!)'-•
Example 2. Prove that 1.3. . . (2n - 1) <?i".
We have {1 + 3+ . . . +(2n-l)}/n>{1.3 . . . (2n-l)}'/'»,
thatis, n2/w>{1.3 . . , (2n-l)}V'».
Hence n«> 1.3 . . . (2n-l).
§ 9.] If a,h, . . ., k be n positive quantities, and p, q, . . .,t
he n positive quantities, thsn
pa"^ + qlf+. . . + tk'^ .^ /pa + qb + . . . + tJcV^ /.v
p + q+...+t ^^\ p + q+ . . . -\-t ) ^'*
according as m does not or does lie between 0 and + 1.
If we denote
Pl(p + q+ ... +0, q/(p + q+' ' '+t), &c.,
^YKh-,- • ', T, and
a/(\a + tib+ . . . + T^), bl{Xa + fjJ)+. . . + rk), &c.,
l>y ^j y> • • • > "^j so that
A. + |A + . . . + T = 1 (2),
\a! + iJ.t/+ . . . +TW==^l (3),
§§8,9 Xa'^jn^iXa/n)"* 49
then, dividing both sides of (1) by
{(pa + qb+ . . . + tk)/(p + q+ . . . + #)}"•,
we have to prove that
\aS^ + iiy^-v. . .+TW"»<^>>1 (4),
according as m does not or does lie between 0 and + 1.
Now, by § 7, if m does not He between 0 and +1, x^-\
■^m{x-\),'ip-\ <^m {y - 1), &c. Therefore, since A, (x, &c., are
positive,
<{;m {2A^ — 2A},
by (2) and (3), that is.
Hence ' 2Aa;™<fl.
In hke manner, we show that, if m hes between 0 and + 1,
Cor. If we make p = q = • . . -t, we have
«'" + &'" + . . .+k'^ /a + b + . . .+kY (r\*
n '^ -ry ^ J \ f i
that is to say, the arithmetical mean of the mth powers of n positive
quantities is not less or not greater than the mth power of their arith-
metical mean, according as m does not or does lie between 0 and + 1.
Remark. — It is obvious that each of the inequalities (1), (4),
(5) becomes an equality if a = & = . . . = A;, if m = 0, or if ;» = 1.
Example. Show that 2Xa;"*, considered as a function of m, increases as m
increases when m>+l, and decreases as m increases when m<-l,
\ IX, V, . . ., X, y, z, . . . being as above.
Ist. Let m>l. We have to show that S\x^+'">2Xa;"*, where r is very
small and positive, that is,
2Xx"*(x'"-l)>0.
Now, ZXx"* {af - 1) > SXx'»rx»-i (x - 1) ,
>rSXx"»+'-i(x-l).
* The earliest notice of this theorem with which we are acquainted is in
Eeynaud and Duhamel's Prohlhnes et Developmens sur Diverses Parties des
Mathimatiques (1823), p. 155. Its surroundings seem to indicate that it
was suggested by Cauchy's theorem of § 8. The original proof rests on a
maximum or minimum theorem, established by means of the Differential
Calculus ; and the elementary proofs hitherto given have usually involved
the use of infinite series.
C. 11. 4
50 EXERCISES V CH. XXIV
Since m>l, 7Ji + r>l, therefore (m + r)x'"+''~^(a;-l)>(m + r) (x-1), that
is, a;'"+'-i(a;-l)>(x-l).
Hence SXx™(a;'--l)>rS\(x-l),
>r{2Xa;-2\),
>0.
Therefore S\a;"'+'- > S\x"*.
2nd. Let m< -1.
I,\x'^ (x'- - 1) < rSXa;"* (x - 1) .
Now (m + l)x'^{x-l)>(m + l){x-l), since m + 1 is negative. Hence,
dividing by the negative quantity m + 1, we have
x'^{x-l)<{x-l).
Hence SXa;™ {x' - 1) < r2X {x-1),
<r(2Xa;-SX),
<0.
Therefore, 2Xa;'"+'- < 2Xx"*.
Exercises V.*
(1.) For what values of xjy is {a + b) xyl(ax + by) j» (aa; + by)l{a + b)?
(2.) If a;, y, z be any real quantities, and x>y>z, then x^y + j/^z + z^x >
xy* + yz* + zx*.
(3.) If X, y, z be any real quantities, then 7:,(y - z) (z-x)>0 and 2j/«/
2a;'' >1.
(4.) If x^ + j/2 + 22 + 2a;?/2 = l, then will all or none of the quantities x, y, z
lie between - 1 and + 1 .
(5.) If X and m be positive integers, show that
a;2m+3 < a; (a; + 1) (2a; + 1) (3x2 + 3^; ^ i)my2 . S*" < (x + l)2m+3_
(6.) (a2/6)i + (62/a)i ^ ai + i<i.
(7.) If Xi,a;2' • • •> a^n 8.11 have the same sign, and 1 + Xj, l + Xj, . . ., l + a;„
be all positive, then
n(l + Xj)>l + 2xi.
(8.) Prove that Qxyz >ll{y-\-z)> 12x3.
(9.) If X, J/, 2, . . ., a, 6, c . . . be two sets, each containing n real
quantities positive or negative, show that
2a22x2<i:(2ax)2;
also that, if all the quantities be positive,
2(x/a)/2x-t2x/2ax;
and, if2x = l, 21/x<tn2.
(10.) If Xj, Xj, . . ., a;„ and also ^i, 2/2' • • •> J/n be positive and in
ascending or in descending order of magnitude, then
2xi2i/i/2xi?/i > 2xj2/2xi . {Laplace.)
* Unless the contrary is stated, all letters in this set of exercises stand
for real positive quantities.
§ 9 EXERCISES V 61
(11.) Iia,b, . . .,lhe in A. P., show that
a^b^ . . . P>aH\
(12.) For what values of x is {x-S)l(x^ + x + l)>{x-i)l{x^-x + l)?
(13.) Find the limits of x and y in order that
c>ax + by>-d,
a>cx +dy>b;
where ad- be =\= 0.
(14.) x^-x'^y + 4:X*!/^-'2x^y^ + ix^y*-xy^ + y^>0, for all real values of
X and y.
(15 . ) Is 10x2 + 5y^ + 13z^> = <8yz + 2xy + 18zx ?
(16.) If ^42-^^2, ihGn^{x'^ + y^) + 'p^(xy)>x-vy.
(17.) lssj{a'' + ab + h'^)-sj{a^-ah + b'')>^<2^{ab)-!
(18.) If X and a be positive, between what limits must x lie in order that
x + a>V{i(a:2 + xa + a2)}+^/{i(x2-xa + a2)}?
(19.) If x<l, then {x + J{x^-\)]^ + {x-^{x"'-\)}^<2.
(20.) If all the three quantities ij {a{b + c-a)}, sj{b{c + a-b)}, sj{c{a^-
b - c)} be real, then the sum of any two is greater than the third.
(21.) If the sum of any two of the three x, y, z be greater than the third,
then f 2x2x2 > Sx' + xyz.
(22.) 21/x>2x8/xV'2*.
(23.) If fr denote the sum of the products r at a time of a, b, c, d (each
positive and < 1) , then f^ + 1^p^ > 2p^ .
(24.) 2x*<tX2/z2x.
(25.) If s = a + b + c+ . . . n terms, then 25/(8 - a) <t n-/(n - 1).
(26.) If wi > 1, X < 1, and 7«x < 1 + x, then 1/(1 =f mx) > (1 ± x)'" > 1 ± mx'.
If m<l, x<l, mx<l + x, then (l + x)/{l±(l-m)x}<(l±x)"'<
1 ± mx.
(27.) If 2"=x" + 7/", then z^> <x"' + 7/"* according as m> <n.
(28.) If X and y be unequal, and x + y<2a, then x"* + ?/"*> 2a"», m being a
positive integer.
(29.) n{(7H-l)'/»-l}<l + l/2+. . .+l/n<n{l-l/(n + l)V» + l/(n + l)}.
(Schlomilch, Zeitschr.f. Math., vol. in. p. 25.)
(30.) IfxiXa . . . x„=j/», n(l + xi)-t(l + y)«.
(31.) If a, 6, . . . , fc be TC positive quantities arranged in ascending order
of magnitude, and if Mr= {2a'-/n}iA-, 2^r={2a>A}'-/n, then
{ab . . . kyi^<Mi<M2< . . .<k,
{ab . . . /<;)'/»<. . .<Ns<N^<:Ni.
(Schlomilch, Zeitschr.f. Math., vol. in. p. 301.)
(32.) Up, q, r be all unequal, and x + 1, then 2px«~'*>2p.
(33.) If n be integral, and x and n each > 1, then
X™ - 1 > 71 (x(»+l)/2 - X ("-l)/'^) .
(34.) Prove for x, y, z that (22^2 - 2x2)2x^ (2x)2*n (2x - 2x)*.
(35.) If 8 = ai + cJa + . . . + «„, then H {sla^ - If' >{n- 1)'.
4—2
52 INEQUALITIES AND TURNING VALUES CH. XXIV
(36.) 3;)i(3m + l)2>4(3mI)iA".
(37.) If s^ be the sum of the nth powers of a^, a^, . . • , «„> ^^^Pm *^^
sum of their products m at a time, then {n - 1)1 s^<i {n - m)\m\p^.
(38.) Ifai>a2>. . . >a„, then
K-«n)"~'>("-l)"~M«l-«2)(«2-«8) • • • K-l-«n)-
Hence, or otherwise, show that {{n-l)\}^>n^~^.
(39.) Which is the greatest of the numbers 4/2, ;^3, ^/4, . . . ?
(40.) If there be n positive quantities Xi, x^, . . . , a;„, each>l, and if
?i. ?2> • • • 1 In be the arithmetic means, or the geometric means, of all but
Xi, all but X.2, . . . , all but a;„, then Jlxi^i>Il^j^i.
(41.) If a, h, c be such that the sum of any two is greater than the third,
and X, y, z such that Sa; is positive, then, if Sa^/x=0, show that xyz is
negative.
(42.) If A = ai + a^+ . . . +(1,^, B = bi + h.2+ . . . +&,i, then 2i{aJA-
h^jB) [a^jb^^ has the same sign as n for all finite values of n.
(Math. Trip., 1870.)
APPLICATIONS TO THE THEORY OF MAXIMA AND MINIMA.
§ 10.] The general nature of the connection between the
theory of maxima and minima and the theory of inequalities
may be illustrated as follows : — Let ^ {x, y, z), f{x, y, z) be any
two functions of w, y, z, and suppose that for all values con-
sistent with the condition
f{x,y,z) = A (1),
we have the inequality
<i>{x,y,z)1^f{x,y,z) (2).
If we can find values of x, y, z, say a, h, c, which satisfy the
equation (1) and at the same time make the inequality (2) an
equality, then </> {a, b, c) is a maximum value of <f> (x, y, z). For,
by hypothesis, ^{a, h, c) = A and «^(a;, y, 'z)1^A\ therefore
^ {x, y, z) cannot, for the values of x, y, z considered, be greater
than A, that is, than ^ {a, b, c).
Again, if we consider all values of x, y, z for which
<f>{x,y,z) = A (1'),
if we have f{x, y, z)<\:<l> (x, y, z)
<^ in
it follows in like manner that, if a, b, c be such that <^(a, b, c)=A,
/{a, b, c)^A, then /(a, b, c) is a minimum value oi/{x, y, z).
§§ 10-12 RECIPROCITY THEOREM 53
The reasoning is, of course, not restricted to the case of three
variables, although for the sake of brevity we have spoken of
only three. The nature of this method for finding turning
values may be described by saying that such values arise from
exceptional or limiting cases of an inequality.
§ 11.] The reader cannot fail to be struck by the reciprocal
character of the two theorems deduced in last section from the
same inequality. The general character of this reciprocity will
be made clear by the following useful general theorem : —
If for all values of x, y, z, consistent with the condition
f{^,y,z)=^,
0 {x, y, z) have a maximum value <f> (a, b,c)=B say (where B depends,
of course, upon A), and if when A increases B also increases, and
vice versa, then f&r all values ofx, y, z, consistent with the condition
<l>{x,y,z)='B,
f(x, y, z) ivill have a minimum value f {a, b, c) = A.
Proof — Let A' <A, then, by hypothesis, when/(ir, y, z) = A',
<j> (x, y, z)1f>B' where B' < B.
Hence, if <fi (x, y, z) = B, f{x, y,z)<^A', for suppose if possible
that /(a?, y, z) = A' <A, then we should have <t> (x, y, z)^B', that
is, since B' <B, ^ (x, y, z) could not be equal to B as required.
Hence, if a, h, c be such that ^(a, h, c) = B and f{a, b, c) = A,
f{a, b, c) is a minimum value oi fix, y, z).
By means of the two general theorems just proved, we can
deduce the solution of a large number of maximum and minimum
problems from the inequalities established in the present chapter.
§ 12.] From the theorem of § 8 we deduce immediately the
two following : —
I. Ifx,y,z, . . . be n positive quantities subject to the condition
%x = k,
then their product Tlx has a maximum value, {JcjnY, when x =
y—. . . =kln.
II. If X, y, z, . . . be n positive quantities subject to the
condition
Hx = k,
54 DEDUCTIONS FROM § 8 CH. XXIV
thsn their sum %x has a minimum value, n¥'"', when x = y-. . .
= F".
The second of these might be deduced from the first by the
reciprocity-theorem.
From the corollary in § 8 we deduce the following : —
III. If X, y, z, . . . he n positive quantities subject to the
condition
%px = k,
where p, q, r, . . . are all positive constants, then '^aP has a
maximum vahie, {k/2p]^^, when x-y-. . . = A;/2/?,
IV. If X, y, z, ... be n positive quantities subject to the
condition
UxP = k,
where p, q, r, . . . are all positive constants, then ^px has a
minimum value, (%>)F^^, when x-y = . . .-k^'^^.
From the last pair we can deduce the following, which are
still more general : —
V. IfX,ix,v,. . ., I, m, n, . . ., p, q, r, . . . be all positive
constants, and x, y, z, . . . be all positive, thsn if
%\x' = k,
Ux^ is a maximum when
IXx^Ip - mixy'^jq^nvz^jr = . . .
VI. And if nxP = k,
SXa;' is a minimum when
l^a^/p - mfjiy'"^lq = nvz^fr = . . .
Proof. — Denote j9//, qfm, r/n, ... by a, /8, y, ... ;
and let W = a$, fiy'" = p-q, vz"" = yC, &c.
So that X - {a^/xy, &c. ; ^ = (al/A)", &c.
We then have in the first case
2a^ = ^ (1),
Uaf^U (a/X)»n^» (2).
Hence, since (a/X)", (^/yu.)^, ... are all constant and all positive,
Hx^ is a maximum when n|" is a maximum. Now, under the
condition (1), n^" is a maximum when ^ = ■>? = . . .^k/^a.
§12 EXAMPLES 55
Hence Tla^ is a maximum when Xa^/a = fiy^jP = . . . , that is,
when lka^\'p = mix.y'^jq = . . .
The maximum vahie of Ux^ is n (a/A)" (^/Sa)-", and the
corresponding vahies of x, y, z, . . . are given by
x = {ak/\:Zay\ . . .
Applying the reciprocity-theorem, we see that, if
n^p = n(a/A)»(>t/2a)2«,
the minimum vahie of ^Xa^ is k, corresponding to
x^{a]cl\taf^ . . .
Whence, putting ^ = n (a/X)" (^•/Sa)^-, we see that, if UxP=j,
the minimum vaUie of %\id is 2a {y/II (a/A.)"}'/^", corresponding
to
x=\a{Jin{ai\Yyriamvi . . .
Cor. If we put l = m = n= . . . =1, p = q = r= . . . =1,
we obtain the following particular cases, which are of frequent
occurrence : —
Jf l,Xx = k, Ux is a maximum when \x = /jii/ = . . . ;
Jf Ux = k, %\x is a minimum when Xx = iiy = . . .
Example 1. The cube is the rectangular parallelepiped of maximum
volume for given surface, and of minimum surface for given volume.
If we denote the lengths of three adjacent edges of a rectangular parallele-
piped by X, y, z, its surface is 2 (yz + zx + xy) and its volume is xyz. If we
put ^=yz, r) = zx, ^=xy, the surface is 2(| + ?; + f) and the volume Ji^rji).
Hence, analytically considered, the problem is to make ^ijf a maximum when
f + 1? + f is given, and to make f + t; + f a minimum when ^ijf is given. This,
by Th. I., is done in either case by making ^ = v = t, that is, yz=zx=xy ;
whence x=y = z.
Example 2. The equilateral triangle has maximum area for given peri-
meter, and minimum perimeter for given area.
The area is A= >Js{s-a){s- b) (s - c). Let x = s-a,y = s-b, z=s-c;
then x + y + z=s ; and the area is Jsxyz. Since, in the first place, s is given,
we have merely to make xyz a maximum subject to the condition x + y-\-z = s.
This leads io x=y = z {hy Th. I.).
Next, let A be given.
Then {x + y + z)xyz=A* (1);
s = A^lxyz (2).
If we put ^=x^yz, 7) = xy'-z, i^=xyz^, we have
» = AWr,i)y* (2').
56 DEDUCTIONS FROM § 9 CH. XXVI
Hence, to make s a minimum when A is given, we have to make ^lyf a
maximum, subject to the condition (!'). This leads to ^ = 7? = f, that is,
x^yz = xy'^z=xyz'^ ; whence x = y — z.
Example 3. To construct a right circular cylinder of given volume and
minimum total surface.
Let X be the radius of the ends, and y the height of the cylinder. The
total surface is Iv (x^-^xy), and the volume is iry^y.
We have, therefore, to make u — x'^-Vxy a minimum, subject to the
condition x^y — c. We have
u=x^ + xy = cly + clx (1);
xV = c (2).
Let l/^ = 2f, 1/2/ = 7?;
then M = c(2^ + 7?) (!');
f'77 = l/4c (2').
We have now to make 2^ + r] (that is, ^ + f + t;) a minimum, subject to the
condition f2^ = constant. This, by Th. II,, leads to ^ = ^ = 7), which gives
2x=y. Hence the height of the cylinder is equal to its diameter.
By the reciprocity-theorem (applied to the problem as originally stated in
terms of x and y), it is obvious that a cylinder of this shape also has maximum
volume for given total surface.
§ 13.] From the inequality of § 9 we infer the following : —
VII. If m do not lie between 0 and + 1, aiid ifp, q,r, . . . he
all constant and positive, then, for all positive values of x, y,z, . . .
such that
Ipx = Jc,
'Xpx^ {m unchanged) has a minimum value when x = y-z = . . .
If m lie between 0 and + 1, instead of a minimum we have a.
maximum.
In stating the reciprocal theorem it is necessary to notice
that, in the inequality, ^px occurs raised to the wth power ; so
that, if m he negative, a maximum of '^px corresponds to a mini-
mum of {'^pxy. Attending to this point, we see that—
VIII. If m> + \, and if p, q, r, . . . be all constant and
positive, then, for all positive values of x,y, z, . . . such that
^px"^ = k {m unchanged),
'^px has a maximum value when x = y = z = . . .
Ifm< + 1, we have a minimum instead of a maximum.
Theorem VIII. might also be deduced from Theorem VII. by
the substitution $ = x"', -q^y^, C = z"', &c. . . .
§§ 12-15 DEDUCTIONS FROM § 9 57
§ 14.] Theorem VII. may be geaeralised by a slight trans-
formation into the following : —
IX. Ifmin do not lie between 0 and + 1, and if p, q,r, . . .,
X, fjL, V, . . . be all constant and positive, then, for all positive
values of x, y, z, . . . such that
2Xa^ -k {n unchanged),
%px^ (m unchanged) has a minimum value when px'^l\x^ =
Ifmjn lie between 0 and + 1, instead of a minimum we have a
maximum.
The transformation in question is as follows : —
Let Xaf = p^, i^f = <^V, ' • . (IX
px^ = p^, qr = <Tt]f, . . , (2).
From the first two equations in (1) and (2) we deduce
^-1 =j9a;"*-"/X, //-I = Va;-^-'"/jt7, &c. Hence, if we take fn = m,
that is, /= m/n, p, a-, . . . will be all constant and obviously all
positive ; we have, in fact,
^^{par--l\f^-'\ v={qr-Vt'y'^-'\ ' ' ' (3),
P = {>//py'^-'\ T = (//gy/tr-i), _ (4).
and we have now to make 2p^ a maximum or minimum, subject
to the condition
^P$ = t
Now, by Th. VII., 2p^ is a minimum or maximum, according
as /does not or does lie between 0 and + 1, when ^ = •»? = . . .
Thus the conditions for a turning value are
which lead at once to
parl\af = qy'^lp.y" = . . .
Cor. A very common case is that where n=l, X = fx. = . . .
= 1.
We then have, subject to the condition 2x = k, Ipaf^, a
minimum or maximum when px^~^ = qy^~'^ = . . ., according as
m does not or does lie between 0 and + 1.
§ 15.] We have hitherto restricted p, q, r, . . . in the in-
58 EXAMPLES CH. XXIV
equality of § 9 to be constant. This is unnecessary ; they may
be functions of the variables, provided they be such that they
remain positive for all positive values of x, y, z.
"We therefore have the following theorem and its reciprocal
(the last omitted for brevity) : —
X. If p, q, r, . . . be functions of x, i/, z, . . . which are
real and positive for all real and positive values of x, y, z, . . .,
then, for all positive values of x, y, z, . . . which satisfy
%px = k,
C^px"^) (2/))"*"^ (m unchanged) has a minimum or maximum value
wJien x = y = . . . , according as m does not or does lie between
0 and + 1.
For example, we may obviously put p='Kx'', q=ny\ • • •
We thus deduce that if m> +1 or <0, then, for all positive values of
x,y,z, . . . consistent with SXa;"+i = A;, (2\a;"'+«) (SXa;»)"»-i is a minimum
■when x=y= . . .
Theorem X. may again be transformed into others in appear-
ance more general, by methods which the student will readily
divine after the illustrations already given.
Also the inequalities of § 8 may be used to deduce maxima
and minima theorems in the same way as those of § 9 were used
in the proof of Theorem X.
Example 1. To find the minimum value of u=x + y + z, subject to the
conditions ajx + bjy + c/z = l,a!>0,2/>0, z>0, a,b,c being positive constants.
Let a'=/)f. y = (T-r/, z=T^f;
ajx = p^, bly = <r7], cIz = t^.
Hence pf~'^ = afja/+^. If we take/= - 1, we therefore get
alx=ija^, by=sjb7), cz^^c^.
The problem now is to make u=2^a$~^ a minimum subject to the con-
dition S^af = l. By Th. VII. this is accomplished by making $ = ?; = f.
Hence f = t; = f = 1/S^a. The minimum value required is therefore
(S,ya)2 ; the corresponding values of x, y, z are »JaI,^a, fJb'Ei^a, iJcZ^^a
respectively.
Example 2. To find a point within a triangle such that the sum of the
mth powers of its distances from the sides shall be a minimum (m>l).
Let a, b, c be the sides, x, y, z the three distances; then we have to make
« = Sa;'" a minimum, subject to the condition Saa5 = 2A, where A is the area
of the triangle.
§§ 15, 16 grillet's method 59
If p^=x^, p^=iax, then /)»»-i = a"», p = a'»/("'-i).
Hence, if we put aa: = a'»/('"-i)^, by = b"'/['^-^)r), c« = c'»/('"-i)f, we have
The solution is therefore given by f = >; = f = 2A/Sa'»/('"-J).
Whence a; = 2AaV("»-i)/Sa'"/('»-i), y = &c., z=&c.
Example 3. Show that, if x^ + y* + z^ = 3, then (x* + y'^ + z^)(x- + y^ + z*)
has a minimum value for all positive values oi x, y, z when x = y = z = l.
This follows from Th. X., if we put nj = 2, p = x^, q = y'\ r=z*, which is
legitimate since x, y, z are all positive.
Example 4. If x,y, z, . . . he n positive quantities, and m do not lie
between 0 and 1, show that the least possible value of (2a;'"~i) (21/a;)'"-i is n"*.
This follows at once from the inequality of § 9, if we put p = llx,
q = lly, . . .
§ 16.] The field of application of some of the foregoing
theorems can be greatly extended by the use of undetermined
multipliers in a manner indicated by Grillet*.
Suppose, for example, it were required to discuss the turning
values of the function
u = {ax+ pY {hx + 3')'" {ex + r)" ( 1 ),
where /, m, n are all positive.
We may write
u ~ {\ax + KpY (jibx + (Mq)"^ (vex + vr)7^V"«'" (2),
where X, /x, v are three arbitrary quantities, which we may sub-
ject to any three conditions we please.
Let the first condition be
l\a + mfib + nvc = 0 (3) ;
then we have
/ {Xax + \p) + m {fibx + fiq) + n (vex + vr)
= IXp + mfiq + nvr = k (4),
where k is an arbitrary positive constant.
This being so, we see by Th. III. that II {\ax + XpY is a
maximum when
Xax + \p = fibx + fJ'-q = vex + vr
= k/^l (5).
* Nouvelle$ Annales de Math., ser. i., tt. 9, 16.
60 EXAMPLES CH. XXIV
The four equations (3) and (5) are not more than sufficient
to exhaust the three conditions on X, fx, v, and to determine a:.
"We can easily determine a) by itself. In fact, from (3) and
(5) we deduce at once
la/ {ax +p) + mb/{bx + g) + nc/{ca; + r) = 0 (6).
This quadratic gives two values for ie, say Xi and w^ ; and the
equations (5) give two corresponding sets of values for X, fi, v,
in terms of k, say X^, fx^, v^ and Xj, fx^, v^.
If, then, ^ifj-i^vi"' be positive, Xi will correspond to a maxi-
mum value of w ; if ^iHi^v^^ be negative, Xi will correspond to
a minimum value of u ; and the like for Wi.
Example 1. To discuss u= (a; + 3)^ (ar - 3).
We have u = i\x + 3\y^ {fix - 3/u)/XV.
Now 2{\x+d\) + {iM-Sfi) = k,
provided 2X + At=0 (1),
6\-3fi=h (2).
Therefore {\x + S\)'^{/jlx-3/jl) will be a maximum, provided
\x + 3\=fMX-3ij. (3).
Hence, by (1),
2/(a; + 3) + l/(x-3) = 0;
which gives x = l. From (2) and (3) we deduce X = fc/12, /t= - A;/6 ; so that
XV is negative.
We therefore conclude that m is a minimum when x = l.
The student should trace the graph of the function u; he will thus find
that it has also a maximum value, corresponding to a; = - 3, of which this
method gives no account.
Example 2. For what values of x and y is
u = {a^x + b^y + c^Y + (a2X + \y + e2f + . . . + (a„a; + &„?/ + c„)2
a minimum?
LetXj.Xo, . . ., X„ be undetermined multipliers. Then we may write
u=SXi2{(a^a;+6jy + Ci)/XiP (1);
and k = SX^^ { {a^x + \y + Ci)/Xi } (2),
where It is an arbitrary positive constant, that is, independent of x and y,
provided
2aiXi=0, S&iXi=0, SCiXi=fc (3).
This being so, by Th. VII., m is a minimum when
{<h^ + \y + e^)l\^(a^ + h^ + c^l\=. . . = ft/SXi2 (4).
The n + 2 equations, (3) and (4), just suffice for the determination of
Xj, Xj, . . ., X„, X, y.
From the first two of (3), and from (4), we deduce
§§ 16, 17 METHOD OF INCREMENTS 61
2ai (a^x + feji/ + Ci) = 0,
26i (ttjX + b^y + Cj) — 0.
Hence the values of x and y corresponding to the minimum value of n are
given by the system
Scij^x + Sajfti?/ + SajCj = 0,
Sajftio; + Sftj^y + SftjCi = 0.
This is the solution of a well-known problem in the Theory of Errors of
Observation.
§ 17.] Method of Increments. — Following the method already
exemplified in the case of a function of one variable, we may
define
I=^(a; + h,y + k, z + l)-<f>{a:,i/, z)
as the increment of <^ (cv, y, z). If, when x = a, y-h, z-c, the
value of / be negative for all small values of h, k, I, then
(f> (a, b, c) is a maximum value of ^ (a:, y, z) ; and if, under like
circumstances, / be positive, ^ {a, b, c) is a minimum value of
«^(«, y, z).
Owing to the greater manifoldness of the variation, the ex-
amination of the sign of the increment when there are more
variables than one is often a matter of considerable difficulty ;
and any general theory of the subject can scarcely be established
without the use of the infinitesimal calculus.
We may, however, illustrate the method by establishing a
case of the following general theorem, which includes some of
those stated above as particular cases.
Purkiss's Theorem*. — If <f){x, y, z, . . .) f{x, y, z, . . .) be
symmetric functions of x, y, z, . . ., and if x, y, z, . . . be
subject to an equation of the form
fix, y,z, . . .)^0 (1),
then <f>(x,y,z, . . .) has in general a turning value when x-y = z
= . . . , provided these conditions be not inconsistent with the
equation (1).
In our proof we shall suppose that there are only three
variables ; and so far as that is concerned it will be obvious that
there is no loss of generality. But we shall also suppose both
• Given with inadequate demonstration in the Oxford, Cambridge, and
Dublin Messenger of Matheviatics, vol. i. (1862).
62 PURKISS'S THEOREM CH. XXIV
<t> (a?, y, z) and f(x, i/, z) to be integral functions, and this sup-
position, although it restricts the generality of the proof, renders
it amenable to elementary treatment.
We remark, in the first place, that the conditions
x = y = z and f{x, y, z) = 0
are in general just sufficient to determine a set of values for x, y, z.
In fact, if the common value of x, y, z be a, then a will be a root
of the equation /(a, a, a) ~ 0.
Consider the functions
I=ff>{a + h, a + k, a + 1)- ^{a, a, a), and /(a + h, a-^k, a + l).
Each of them is evidently a symmetric function of h, k, I, and
can therefore be expanded as an integral function of the
elementary symmetric functions ^h, "^hk, hkl. We observe also
that, since each of the functions vanishes when A = 0, k = 0, 1=0,
there will be no term independent of k, k, I.
Let us now suppose h, k, I to be finite multiples of the same
very small quantity r, say h-ar, k = Pr, l = yr. Then ^h = r%a.
= ru say, %hk = r^^a^ = r^v, hkl = i^w. Expanding as above in-
dicated, and remembering that by the conditions of our problem
/{a + h, a + k, a + l)-0, we have, if we arrange according to
powers of r,
/= Aur + (Bu' + Pv) r" + &c. (1),
0 = Pur + {Qu? + Ev)r' + &c. (2),
where the &c. stands for terms involving r' and higher powers.
From (2) we have
wr = - (Qu^ + Rv) r'/P + &c.,
wV = 0 + &c.,
22a)8r' = - ^^t"" + &c.,
&c. as before including powers of r not under the 3rd.
Hence, substituting in (1) and writing out only such terms
as contain no higher power of r than r^, we have
I={C-AE/P)vr' + &c.,
^-^r'(C-AR/P)'S,a? + &c.
Now (see chap, xv., § 10), by taking r sufficiently small, we
may cause the first term on the right to dominate the sign of /.
§17 EXERCISES VI 63
Hence /will be negative or positive according as {CP-AR)IP
is positive or negative ; that is, <;^ {a, a, a) will be a maximum or
minimum according as {CP - AR)IP is positive or negative.
Example. Discuss the turning values of </> {x, y, z) = xyz + b(yz + zx + xy),
subject to the condition x^ + y^ + z'^= 3a^.
The system
x = y=z, a;2 + r/2 + 22_3a2_o
has the two solutions x = y = z= ^a.
If we take x=y=z= +a, Vfe find, after expanding as above indicated,
1= (a2 + 2ab) ur + {a + b) vr^-i-&c.,
0 = 2aur+{u^-2v)r^.
In this case, therefore, ^ = aH2a6, C=a + b, P=z2a, R= -2 ; a.n6i{CP - AR)I
P=2a + db.
Hence, when x=y = z=+a,(piaa maximum or a minimum according as
2a + Bb is positive or negative.
In like manner, we see that, when x = y = z= -a, 0 is a maximum or a
minimum according as -2a + 3b is positive or negative.
Exercises VI.*
(1.) Find the minimum value of bcx + cay + abz when xyz = abc.
(2.) Find the maximum value of xyz when x^Ja^ + T/^lb'^ + z^jc^— 1.
(3.) If I,x^—c, llilx is a maximum when x : y : z : . . , =1 : m : n : . . .
(4.) Find the turning values of Xa;""» + fiy^^ + vz'^, subject to the condition
px"' + qy^ + rz''=d.
(5.) Find the turning values of aa;^ + 6i/« + m*" when xyz = cP.
(6.) li xyz = a^{x-i-y + z), then yz + zx + xy ia a, minimum when x = j/=2 =
s/3a.
(7.) Find the turning values of (x + l) (y + m) {z + n) where a%Vc*=zd.
(8.) Find the minimum value of ax'^+bjx'^.
(9.) Find the turning values of (3x -2){x- 2f {x - 3)2.
(10.) If ex {b-y) = ay {c-z)=: bz (a-x), find the maximum value of each.
(11.) Find the turning values of a;"*/!/" (m>n), subject to the condition
x-y — c. (Bonnet, Nouv. Ann., ser. i., t. 2.)
(12.) If x^yi + xiy'P — a, then x''+9 + 7/P+9 has a minimum value when x — y =
(a/2)V(P+«) ; and, in general, if ^xPyi=a, Sj;P+« has a minimum value, al{n - 1),
when x=y = z=: . . . ={a/(n-l)n}V(p+9), Discuss specially the case where
p and q have opposite signs.
(13.) If xPyi + x'^y'=c, then x'y^ is a maximum when xv-^jipi - st)=y'-^j
(qt-pu), the denominators, ru-st and qt-pu, being assumed to have the
same sign. (Desboves, Questions cfAlgebre, p. 455. Paris, 1878.)
* Here, unless the contrary is indicated, all letters denote positive
quantities.
64 EXERCISES VI CH. XXIV
(14.) It p>q, and xP+yP = aP, then x^ + y^ is a minimum when x=y =
a/2^/P. State the reciprocal theorem.
(15.) Find the turning values of {ax'^+ln/'^)IJ{a^z^ + bY) ^ben x^+y^= 1.
(16.) If Xj, OTj, . . , , a;„ be each >a, and such that (xj-a) {x^-a) , . .
(x„-a) = fc", the least value of XjXj . . . x,^ is (a + 6)", a and 6 being both
positive.
(17.) If f{m) denote the greatest product that can be formed with n
integers whose sum is m, show that /(m+l)//(m) = l + l/3 where q is the
integral part of mjn.
(18.) ABCD is a rectangle, APQ meets BC in P, and DC produced in Q.
Find the position of APQ when the sum of the areas ABP, PCQ is a
minimum.
(19.) 0 is a given point within a circle, and POQ and. EOS are two per-
pendicular chords. Find the position of the chords when the area of the
quadrilateral PRQS is a maximum or a minimum.
(20.) Two given circles meet orthogonally at A. PAQ meets the circles
in P and Q respectively. Find the position of PAQ when PA . AQ is a
maximum or minimum.
(21.) To inscribe in a given sphere the right circular cone of maximum
volume.
(22.) To circumscribe about a given sphere the right circular cone of
minimum volume.
(23.) Given one of the parallel sides and also the non-paraUel sides of an
isosceles trapezium, to find the fourth side in order that its area may be a
maximum.
(24.) To draw a line through the vertex of a given triangle, such that the
sum of the projections upon it of the two sides which meet in that vertex
shall be a maximum.
CHAPTEE XXY.
Limits.
§ 1.] In laying down the fundamental principles of algebra,
it was necessary, at the very beginning, to admit certain limiting
cases of the operations. Other cases of a similar kind appeared
in the development of the science ; and several of them were
discussed in chap. xv. In most of these cases, however, there
was little difficulty in arriving at an appropriate interpretation ;
others, in which a difficulty did arise, were postponed for future
consideration. In the present chapter we propose to deal
specially with these critical cases of algebraical operation, to
which the generic name of " Indeterminate Forms " has been
given. The subject is one of the highest importance, inasmuch
as it forms the basis of two of the most extensive branches of
modern mathematics — namely, the Differential Calculus and the
Theory of Infinite Series (including from one point of view the
Integral Calculus). It is too much the habit in English courses
to postpone the thorough discussion of indeterminate forms
until the student has mastered the notation of the differential
calculus. This, for several reasons, is a mistake. In the first
place, the definition of a differential coefficient involves the
evaluation of an indeterminate form ; and no one can make
intelligent applications of the differential calculus who is not
familiar beforehand with the notion of a limit. Again, the
methods of the differential calculus for evaluating indeterminate
forms are often less effective than the more elementary methods
which we shall discuss below, and are always more powerful in
combination with them. Moreover the notion of a limiting value
can be applied to functions of an integral variable such as n\ and
to other functions besides, which cannot be differentiated, and
are therefore not amenable to the methods of the Differential
Calculus at all.
c. II. 5
66 MEANING OF A LIMITING VALUE CH. XXV
§ 2.] The characteristic difficulty and the way of meeting it
will be best explained by discussing a simple example. If in
the function (;r^-l)/(a?- 1) we put x = 2, there is no difficulty
in carrying out successively all the operations indicated by the
synthesis of the function ; the case is otherwise if we put x=\,
for we have 1^ - 1 = 0, 1-1=0, so that the last operation in-
dicated is 0/0— a case specially excluded from the fundamental
laws ; not included even under the case a/0 (a 4= 0) already dis-
cussed in chap, xv., § 6. The first impulse of the learner is to
assume that 0/0 = 1, in analogy with «/a = l; but for this he
has no warrant in the laws of algebra.
Strictly speaking, the function {o^-l)l{x-l) has no definite
value when x=l\ that is to say, it has no value that can be
deduced from the principles hitherto laid down. This being so,
and it being obviously desirable to make as general as possible
the law that a function has a definite value corresponding to
every value of its argument, we proceed to define the value of
{a^- \)l{x-\) when x=\. In so doing we are naturally guided
by the principle of continuity, which leads us to define the
value of {x'^-l)l{x-\) when ^=1, so that it shall differ in-
finitely little from values of {x^ —l)l{x -I), corresponding to
values of x that differ infinitely little from 1. Now, so long as
a; =1=1, no matter how little it differs from 1, we can perform the
indicated division; and we have the identity {a^—l)l{x—\) =
a? + 1. The evaluation of a? + 1 presents no difficulty ; and we
now see that for values of x differing infinitely little from 1, the
value of {ic^-l)l{x-l) differs infinitely little from 2. We there-
f(yre define the value of {x^-l)/(x-l) when, x=l to be 2 ; and we
see that its value is 2 in the useful and perfectly intelligible
sense that, bi/ bringing x sufficiently near to 1, we can cause
{o^- i)l{x- 1) to differ from 2 by as little as we please*. The
value of {a? - l)/(x - 1) thus specially defined is spoken of as the
limiting value, or the limit of {p^ - l)l(x - 1) for x=l ; and it is
symbolised by writing
* The reader should observe that the definition of the critical value just
given has another advantage, namely, it enables us to assert the truth of the
identity (a;' - l)/(x - 1) = x + 1 without exception in the case where « = 1.
FORMAL DEFINITION OF A LIMIT 67
where L is the initial of the word "limit." The subscript x=l
may be omitted when the value of the argument for which the
limiting value is to be taken is otherwise sufficiently indicated.
We are thus led to construct the following definition of the
value of a function, so as to cover the cases where the vahie
indicated by its synthesis is indeterminate : —
When, hy causing x to differ sufficiently little from a, we can
make the value of f{x) approach as near as we please to a finite
definite quantity I, then I is said to be the limiting value, or limit,
off{x) when x-a; and we write
Lf{x) = l.
Cor. 1. A function is in general continuous in the neighbour-
hood of a limiting value ; and, therefore, in obtaining that value
we may subject the function to any transformation which is
admissible on the hypothesis that the argument x has any value in
the neighbourhood of the critical value a.
We say "in general," because the statement will not be
strictly true unless the phrase "differ infinitely little from" mean
"differ eit/ier in excess or in defect infinitelj'^ little from." It may
happen that we can only approach the limit from one side ; or
that we obtain two different limiting values according as we in-
crease X up to the critical value, or diminish it down to the critical
valu e. In this last case, the graph of the function in the neighbour-
hood oi x = a would have the peculiarity figured in chap, xv.,
Fig. 5 ; and the function would be discontinuous. The latter
part of the corollary still applies, however, provided the proper
restriction on the variation of x be attended to.
When it is necessary to distinguish the process of taking a
limit by increasing a; up to a from the process of taking a limit
by decreasing x down to a, we may use the symbol L for the
a;=o-0
former, and the symbol L for the latter.
x=a+0
Cor. 2. If L f{x) = I, then f{a + h)-l + d, where d is a
x=a
function of a and h, whose value may be made as small as we
please by sufficiently diminishing h.
5—2
68 CONSEQUENCES OF THE DEFINITION CH. XXV
This is simply a re-statement of the definition of a limit from
another point of view.
Cor. 3. Any ordinary value of a function satisfies the
definition of a limiting value.
For example, Lix"- l)/{w - 1) = (2- - 1)1(2 - 1) = 3. This re-
mark would be superfluous, were it not that attention to the
point enables us to abbreviate demonstrations of limit theorems,
by using the symbol X where there is no peculiarity in the
evaluation of the function to which it is prefixed.
§ 3.] It n)ay happen that the critical value a, instead of
being a definite finite quantity, is merely a quantity greater than
any finite quantity, however great. We symbolise the process
of taking the limit in this case by writing L fix), or L f{x),
a;=+oo a;=-oo
according as the quantity in question is positive or negative.
For example,
L {x + l)lx = L {I + llx) = \.
In this case, we can, strictly speaking, approach the limit from one side
' only ; and the question of continuity on both sides of the limit does not
arise. If, however, we, as it were, join the series of algebraical quantity
-QO...-1...0...+1...+QO through infinity, by considering
+ 00 and - oo as consecutive values ; then we say that / (x) is, or is not, con-
tinuous for the critical value a; = oo , according as I//(.T)and L /(x) have,
a;=a) x=— oo
or have not, the same value. For example, (x + \)lx is continuous for .t = qo ,
for we have L (x + Vjjx = 1 = L (x + l)/j! ; but (x- + l)/a; is not continuous
x=« a:=— «
for a; = 00 .
§ 4.] The value 0 may of course occur as a limiting value ;
for example, L x{x-lfl{x^~l) = Q. It may also happen, even
for a finite value of a, that f{x) can be made greater than any
finite quantity, however great, by bringing x sufficiently near to a.
In this case we write L f{x) = oo . In thus admitting 0 and oo
a:— o
as limiting values, the student must not forget that the general
rules for evaluating limits are, as will be shown presently, sub-
ject in certain cases to exception when these particular limits
occur.
2-6 CLASSIFICATION OF INDETERMINATE FORMS 69
ENUMERATION OF THE ELEMENTARY INDETERMINATE FORMS.
§ 5.] Let u and v be any two functions of x. We have
already seen, in chap, xv., that u + v becomes indeterminate
when u and v are infinite but of opposite sign ; that ?* x -y
becomes indeterminate if one of the factors become zero and
the other infinite ; and that u-^v becomes indeterminate if it
and V become both zero, or both infinite. We thus have
the indeterminate forms — (I.) co — qo, (II.) 0 x go, (III.) 0-^0,
(IV.) 00-00.
It is interesting to observe that all these really reduce to (III.). Take
00-00 for example. Since M + u = (l + r/M)/(l/«), and Ll/w = l/oo =0, this
function will not be really indeterminate unless Lvja— - 1. The evaluation
of the form oo - oo therefore reduces to a consideration of cases {IV. ) and (III.)
at most. Now, since tt-^j; = (l/t))->-(l/M), case (IV.) can be reduced to (III.);
and finally, since uxv — u-i-{\lv), case (II.) can be reduced to (III.).
To exhaust the category of elementary algebraical operations
we have to discuss the critical values of m". This is most simply
done by writing w" = a''*°*''" where a is positive and >1. We
thus see that u" is determinate so long as ^" loga u is determinate.
The only cases where v loga u ceases to be determinate are those
where — (V.) v-0, log^ ^* = + oo , that is v = 0, m = oo ; (VI.) ■» = 0,
logaM = -<», that is ■u = 0, w = 0; (VII.) v = ±qo, logaW=^0,
that is 'y = +Qo, u=l. There thus arise the indeterminate
forms— (V.) 00 «, (VI.) 0«, (VII.) 1*"*.
All these depend on n"^* ; or, if we choose, upon a"/" ; so that it may
be said that there is really only one fundamental case of indetermination,
namely, 0-^0.
EXTENSION OF THE FUNDAMENTAL OPERATIONS TO LIMITING
VALUES.
§ 6.] We now proceed to show that limiting values as above
defined may, under some restrictions, be dealt with in algebraical
* The reader is already aware that 1" gives 1 ; and he may easily convince
himself that 0+", 0"*, oo +°°, qo~* give 0, ±oo, ±oo, 0 respectively, no
matter what their origin.
70 FUNDAMENTAL OPERATIONS WITH LIMITS CH. XXV
operations exactly like ordinary operands. This is established
by means of the following theorems : —
I. The limit of a mm of functions of x is the sum of their limits,
provided the latter does not take the indeterminate form co - go.
Consider the sum f{x)-<ft(a;) + x{it!) for the critical value
«r = a ; and let Lf{x) =/', Z«^ (x) = <f>', Lx{x) = x'- Then, by § 2,
Cor. 2,
f{a>)=f' + a, <i>{x) = <f>' + P, x(^) = x' + y>
where a, p, y can each be made as small as we please by
bringing x sufficiently near to a.
Now, f{x) -<t>ix) + x(^) -f'-^' + X+{a-/3 + y).
But, obviously, a-/3 + y can be made as small as we please by
bringing x sufficiently near to a. Hence
L{fix)-^x) + x{^)}=f'-<l>'+X,
that is, ^If{x)-L*i>{x) + Lx{x) (1).
This reasoning supposes /', ^', x' to be each finite ; but it is
obvious that if one or more of them, all having the same sign,
become infinite, then f '-<(>' + x and L {f{x) - (t>{x) + x(^)} are
both infinite, and the theorem will still be true in the peculiar
sense, at least, that both sides of the equality are infinite. If,
however, some of the infinities have one sign and some the
opposite, f '-<!>' + x' ceases to be interpretable in any definite
sense ; and the proposition becomes meaningless.
II. T/ie limit of a product of functions of x is the product of
their limits, provided the latter does not take tlie indeterminate
form 0 X CO.
Using the same notation as before, we have
f{x) <l>(x) x{^) = (/'+ a)(f + /3)(x'+ y)
= /'</>'x'+ 2a<^'x' + 2a/3x' + a/3y.
Now, provided none of the limits /', ^', x' be infinite, since a, ft,
y can all be made as small as we please by bringing x sufficiently
near to a, the same is true of Sa^'x', SaySx', and aySy. Hence
Lf(x) ^(x) x(^) =/'<^'x' - Lf{x) L<f>(x) Lx(x) (2).
If one or more of the limits/', «/»', x' be infinite, provided none
of the rest be zero, the two sides of (2) will still be equal in the
§§ 6, 7 LF[f{x), 4> {x), ...] = F [Lf{x), L<i> {x), . . .} 71
sense that both are infinite ; but, if there occur at the same time
a zero and an infinite value, then the right-hand side assumes
the indeterminate form 0 x oo ; and the equation (2) ceases to
have any meaning.
III. The limit of the quotient of two functions of x is the
quotient of their limits, provided the latter does not take one of the
indeterminate forms 0/0 or co /go . We have
From this equation, reasoning as above, we see at once that, if
neither /' nor <^' be infinite, and (f>' be not zero,
It is further obvious that if /' = qo , <^' + oo , both sides of (3)
will be infinite ; if ^' = oo , /' 4= oo , both sides will be zero ; and
if <fi' = 0, /' 4= 0, both sides will be infinite. In all these cases,
therefore, the theorem may be asserted in a definite sense. If,
however, we have simultaneously/' = 0, <^' = 0, the right hand of
(3) takes the form 0/0 ; if /' = co , <^' = co , the form go /co ; and
tten the theorem becomes meaningless.
§ 7.] If the reader will compare the demonstrations of last
paragraph with those of § 8, chap, xv., he will see that (except
in the cases where infinities are involved) the conclusions rest
merely on the continuity of the sum, product, and quotient.
This remark immediately suggests the following general theorem,
which includes those of last paragraph as particular cases : —
If F{u, V, w, . . .) he any function ofu,v,w, . . . , which is
determinate, and finite in value, and also continuous when
u^Lf{x), v = Ltf>(x), w = Lx{x), . . .,
then
LF{f(x),cl>(x),x(^\ . . .] = F{Lf(x),L<f>{x),Lx{x), . . .}.
The reader will easily prove this theorem by combining § 2,
Cor. 2, with the definition of a continuous function given in
chap. XV., §§ 5, U.
V2 LIMITS OF RATIONAL FUNCTIONS CH. XXV
The most important case of this proposition which we shall have occasion
to use is that where we have a function of a single function. For example,
L {(x2-l)/(x-l)P-{ L (x^-l)l(x-l)}^=i.
X-l X^'l
L log { (x" - 1)1 (x - 1) } = log { L (.t2 - l)/(x - 1) } = log 2.
ON THE FORMS 0/0 AND X /oo IN CONNECTION WITH
RATIONAL FUNCTIONS.
§ 8.] The form 0/0 will occur with a rational function for
the value a:-0 \i the absolute terms in the numerator and
denominator vanish. The rule for evaluating in this case is to
arrange the terms in the numerator and denominator in order
of ascending degree, divide by the lowest power of x that occurs
in numerator or denominator, and then put ^ = 0. The limit
will be finite, and 4=0, if the lowest terms in numerator and
denominator be of the same degree ; 0 if the term of lowest
degree come from the denominator ; oo if the term of lowest
degree come from the numerator. All this will be best seen
from the following examples ; —
Example 1.
Example 2.
Example 3.
2.r^ + 3.r'' + a;*_ 2j^Bx + x' _2
a:=o '6x^ + x* + a^ ~ x=o^ + x'' + x*~S'
2x» + 3a;Hx»_ 2.T + Sx" + x» _ 0
a;=o 3x'- + X* + X* ~ x^o 3 + x'-* + X* ~ 3 ~~
2x*+x« 2 + x2 2
x=o x« + x» x=^„x^ + x* 0
§ 9.] The form co /oo can arise from a rational function when,
and only when, x= cc. The limit can be found by dividing
numerator and denominator by the highest power of x that
occurs in either. If this highest power occur in both, the limit
is finite ; if it come from the denominator alone, the limit is 0 ;
if from the numerator alone, the limit is co .
Example 1.
J _ 8^+^ ^ T 3/x+l _ Ji^. 1_ _ 1
^'„2x« + x» + 3x^ ,.«2/x2 + l7x + 3~0 + 6 + 3~3*
^7-10
USE OF THE REMAINDER-THEOREM
73
Example 2.
a;--' + 3ar> + 4.r<_ 1/.t* + 3/x^ + 4/.e'^_ 0
r=» 2a; + x^ + 6x'' ~a;=aD 2/x5 + l/ar» + 6 ~6~ '
Example 3.
l/x< + 3/a:3 + 4 _4
a;!* 2.r + 3a;2 + x3 ~^^^2lx^ + 3/x* + 1/^» " 0 ''
J J z;. t: — K-~ — TT — Ij
§ 10.] If the rational function f{iK)l<ii{x) take the form 0/0 for
a finite vahie of x, 4= 0, say for a; = a, then, since f{a) = 0, <f> (a) =^ 0,
it follows from the remainder-theorem that x — a is a common
factor in /(x) and ^ (x). If we transform the function by re-
moving this factor, the result of putting ^ = a in the transformed
function will in general be determinate ; if not, it must be of
the form 0/0, and x — a will again be a common factor, and must
be removed. By proceeding in this way, we shall obviously in
the end arrive at a determinate value, which will be the limit of
f(x)l<f> (x) when x-a.
Example. Evaluate {Sx^ - lOx" + 3x- + 12x - 4)/(.c'» + 2x^ - 22.1-2 + 32a; - 8)
when x = 2. The value is, in the first instance, indeterminate, and of the
form 0/0 ; hence a; - 2 is a common factor. If we divide out this factor, we
find that the value is still of the form 0/0 ; hence we must divide again. We
then have a determinate result. The work may be arranged thus (see chap,
v., § 13) :-
1 + 2 -22 +32 -8
0 + 2 + 8 -28 +8
3-10+ 3+12-4
0+6- 8-10+4
3_ 4- 5+ 2 +0
0+6+4-2
3+2-1
0+ 6+16
+ 0
3+ 81 + 15
1 + 4 -14+ 4
0 + 2+12 - 4
+ 0
1 + 6 - 2,+ 0
0 + 2 +161
_1_+8| + 14
The process of division is to be continued until we have two remainders
which are not both zero. The quotient of these, 15/14 in the present case, is
the limit required.
The evaluation of the limit in the present case may also be
effected by changbig the variable, an artifice which is frequently
of use in the theory of limits. If we put x = a + z, then we have
to evaluate Lf{a + 5;)/<^ {a + z) when z = Q. Since /{a + z) and
^{a + z) are obviously integral functions of z, we can now apply
the rule of § 8. It will save trouble in applying this method if
it be remembered — 1st, that in arranging f{a + z) and <ji{a + z)
according to powers of z we need not calculate the absolute
74 CHANGE OF VARIABLE CH. XXV
terms, since they must, if the form to be evaluated be 0/0, be
zero in each case ; 2nd, that we are only concerned with the
lowest powers of z that occur in the numerator and denominator
respectively.
3a;<- 10x3 + 3a;'' + 12a; -4_ 3(2 + g)<- 10(2 + ^)^ + 3(2 + 2)" + 12(2 + 2) -4
x=2 a;* + 2a;*-22x2 + 32x -8 ~^=o(2 + 2)* + 2(2 + 2;)»- 22(2 + 2)2 + 32(2 + 2) -8
1522 + P23 + &C.
_ I5 + P2 + &C.
~^ol4 + Q2 + &c.'
15
~ 14*
This method is of course at bottom identical with the former ; for, since
z=-x-a, the division by z"^ corresponds to the rejection of the factor (x - a)".
§ 11.] The methods which are applicable to the quotient of
two integral functions apply to the quotient of two algebraic
sums of constant multiples of fractional powers of x. Each of
the two sums might, in fact, be transformed into an integral
function of y by putting x = jf', where d is the L.C.M. of the
denominators of all the fractional indices. It is, however, in
general simpler to operate directly.
Example. Evaluate
1= L, -^ .
If we divide by x', the lowest power of x that occors, we have
, ^x^ + x7 + 3xT^
'— ^ i s— »
*-» l + 2x<r + x5
=?=o.
§ 12.] The following theorem, although partly a special case
under the present head, is of great importance, because it gives
the fundamental limit on which depends the " differentiation " of
algebraic functions : —
If mhe any real commensurable quantity ^ positive or negative
L{x^-l)l{x-l) = m (1).
§§10-12 L(x'^-l)f(x-l) = m 75
First, let w be a positive integer. Then we have
(^'"-l)/(ir-l) = a;"'-i + i2;'"-=' + . . . + a;+l.
Hence
L {a;"'-l)/{x-l) = 1 + 1 + . . . + 1 + 1 (m terms),
= m.
Next, let vw be a positive fraction, say p/q, where p and q are
positive integers. Then the limit to be evaluated is L {x^i'^- 1)/
a:=l
{x — 1)*. If we put X = z^, and observe that to a? = 1 corresponds
z=\, the limit to be evaluated becomes L (z^ - l)l{z^ — 1). This
may be evaluated by removing the common factor z—1; or thus
i(.^-,)/(.-i)=i(^'fi)/(ffi),
=p/q = m.
Finally, suppose m to have any negative value, say - «, where
n is positive. Then
L (^-" - \)l{x - 1) = Z (1 - x'')\x''{x - 1),
--- L{x''-\)\{x-\)x'\
= -{L (;»"- l)/(^- 1)} X L l/x\
Now, by the last two cases, since n is positive, i/(^"-l)/
(a; - 1) = w. Also Z l/x'' = 1. Hence
a;=l
L{x-''-l)/{x-l) = -n;
a;=l
that is, in this case also,
L{x'^-l)l{x-l)^m.
Second Demonstration. — The above theorem might also be deduced at once
from the inequality of chap, xxiv., § 7, as follows : — For all positive values of
X, and all positive or negative values of m, x"* - 1 lies between wta;"'""i (x - 1)
and m (x - 1). Hence (a;"* - l)/(a; - 1) lies between mx^-^ and wt. Now, by
* There is here of course the usual understanding (see chap, x., § 2) as
to the meaning of x '''/«.
76 EXAMPLES CH. XXV
bringing x sufficiently near to 1, mx^~^ can be made to differ as little from m
as we please. The same is therefore true of (a;"' - l)/(u; - 1) ; that is to say,
L(x"»-l)/(a;-l) = »t
for all real values of m.
Example 1. Find the limit of (x" - aP)/(x<' - «») when x=a. We have
L {xv - aP)\(3fl - ai)= L aP-9{(x/a)P- l}/{(a;/a)«- 1},
x—a x=a
where y=xla. Hence we have, by the theorem of the present paragraph
L (xr> - av)j{xfi - ai) = a'P-ipjq.
Example 2. Evaluate log (x^ - 1) - log (x2 - 1) when x = 1.
L{Iog(xi-l)-log(xi-l)} = Llog{(x^-l)/(x4-l)},
= log{L(xi^-l)/(xi-l)}, by §7,
x=a
M''{tl)Ki^)\-
= log3.
Example 3. If Ix, Px, . . . denote logx, log (log x), . . . respectively,
then, when x = cc, LV {x + l)jl''x — l.
In the first place, we have
l{x + l)llx^{l(x + l)-lx + h:}llx,
= l(l + llx)llx + l.
Now, when x^oo, i(l + l/.r) = Zl = 0 and Ix — oa. Hence Li (x + l)/ix=l.
If we assume that LlT{x + VjfVx = 1, we have
i'-+' (x + \)jV+\x= {V^^ (x + 1) - Z'-+ix + i'-+'x}/Z'-+ix,
= l{V{x + l)lVx}ll^^x + l.
Lr+i (X + l)/i'-+Jx = Zl/oo + 1,
= 1;
that is, the theorem holds for r+ 1 if it holds for r. But it holds for r=l, as
we have seen, therefore for r=2, &c. It is obvious that this theorem holds
for any logarithmic base for which ioo = oo .
Example 4. 11 I have the same meaning as before, and X have a similar
meaning for the base a, then
L VxlVx = ll\oga.
x—oo
Let /:i=l/loga. Since 'Kx=fdx, the theorem clearly holds when r=l. It is
therefore sufficient to show that, if it is true for r, it is true for ; +1. Now
X'^'x/Z'^-ix = X (X'-x)/P^-ix,
= mZ(X''x)//h-1x,
=/* {I (X'-x) - r+ix + i-^-ix}//' +'x,
= M{«(X'-x/i'-x)/F+ix+l}.
Hence, if we assume that LVxlirx=n, we have
LX'^-'x/I'+'x ^M { Wa> + 1},
= /*•
§§12,13 L(l + l/.xy = e 77
EXPONENTIAL LIMITS.
§ 13.] The most important theorem in this part of the sub-
ject is the following, on which is founded the differentiation of
exponential functions generally : —
The limit o/(l + llxf when x is increased without limit eit/wr
positively or negatively is a finite number {denoted by e) lying
between 2 and 3.
The following proof is due to Fort*.
We have seen (qhap. xxiv., § 7) that, if a and b be positive
quantities, and m any positive quantity numerically greater
than 1, then
ma"'-'' (a - b)>a'^- b"^ > mb'""-' {a - b) (1).
In this inequality we may put a = {y+ l)/y, b-l, m~y/,v, where
y>x>l. We thus have
\ y J .v'
Hence ( 1 + - ) > 1 + - ,
\ y/ a;
]
y^
that is, (l+^y>(l+_^J (2).
where y>x.
Again, if in (1) we put a = l, b = (y- l)/y, (m, y, x being as
before), we have
X \ 1/ /
Hence (1 — ) >1--,
\ yJ X
and therefore A - -V<[l - i) " (3),
where y>x.
We see from (2) and (3) that, if we give a series of in-
* Zeitschriftfur Mathematik, vii., p. 46 (1862).
78 L{1 + Ijxf = e CH. xxT
creasing positive values to x, the function (1 + IfxY continually
increases, and the function (1 - l/a;)""^' continually decreases.
Moreover, since a?>x^-\, we have
X x+ 1
X- 1 X '
that is, ( 1 - - ) > 1 + - .
\ x/ X
Hence (i _ l)-%(i , 1)' (4).
The values of (1 - l/ar)~^ and (1 + 1/xy cannot, therefore,
pass each other. Hence, when x is increased without limit,
(1 — l/xy must diminish down to a finite limit A, and
(1 + 1/xY must increase up to a finite limit B. The two limits
A and B must be equal, for the difference (1 - l/x)~''-{l + l/xf
may be written {x/{x-l)}''-{{x + l)/x}'' ; and by (1) we have
If a; y f X y (x + iy 1 (x+iy , .
x\x-\) ^\x-\) \ X ) ^x{\-\la^)\ X ) ^^>-
But, since, as has already been shown, {xl{x - \)Y and
{{x + l)lxY remain finite when a; = qo , the upper and lower
limits in (5) approach zero when x is increased without limit ;
the same is therefore true of the middle term of the inequality.
It has therefore been shown that X (1 + IjxY and
L {I- l/x)'" have a common finite limit, which we may denote
by the letter e.
Since (1 + 1/6)" = 2 '521 ... and (1 - 1/6)-" = 2*985 . . .,
e lies between 2 5 and 2 9. A closer approximation might be
obtained by using a larger value of x ; but a better method of
calculating this important constant will be given hereafter, by
which it is found that
e = 27182818285 . . .
The constant e is usually called Napier's Base*; and it is the
logarithmic or exponential base used in most analytical calcula-
tions. In future, when no base is indicated, and mere arith-
* In honour of Napier, and not because he explicitly used this or indeed
any other base.
§13 L(a<'-l)/x = \oga 79
metical computations are not in question, the base of a
logarithmic or exponential function is understood to be ^ ; thus
log a; and expa; are in general understood to mean logga? and
expeW (that is, e') respectively.
Cor. 1. Lil+xY'^'^e.
For L {l + l/zy = e; and, if we put z = l/a;, so that iv = 0
corresponds to z=oo, we have L (l + wy^'' -- e.
Cor. 2. L log„ {(1 + l/wf} = Z log„ {(1 + ;r)v-} = log„ e.
a;=oo x=0
For, since loga^ is a continuous function of y for finite values of
y, we have, by § 7,
L hga {(1 + 1/^)1 = loga { i^ (1 + 1/^)1,
= logae.
The other part of the corollary follows in like manner.
Cor. 3. L{1+ yjxf = X ( 1 + xyf"' = e^.
jc=«) a:=0
If we put \lz = ylx, then to ^= oo corresponds «;= cc ; hence
= {Z(l + l/^)r, by §7,
Cor. 4. L (a* - 1)/^ = log a.
If we put y = a"-\, so that a; = loga(l +?/), and to ^ = 0 corre-
sponds 3/ = 0, we have
X(a--l)/^=Z2//log„(l+7/),
= Xl/log„(l+2/)"^
= i/iog4Z(i+^)n
= l/logae = loga.
It will be an excellent exercise for the student to deduce directly from the
fundamental inequality (1) above, the important result that L (a* - \)lx is
z=0
80 EXPONENTIAL AND LOGARITHMIC INEQUALITIES CH. XXV
finite ; and thence, by transformation, to prove the leading theorem of this
paragraph*.
Cor. 5. If X be any positive quantity,
e">l+a;, log (1 + x) < a- ;
and, if a; be positive and less than 1,
e~^>l-x, -log(l— a?)>;r.
Since e>{l + l/«)", when n may be as great a.s we please,
e^-l>(l + l/w.)"^-l,
> nx {(1 + 1/n) -l}>x, by chap. xxrv. , § 7,
for, however small w, we can by sufficiently increasing n make
nx>l.
Hence 0^>l+w.
It follows at once that log^>log (1 + w), that is, ir>log (1 + a-).
Again, since e<{l- l/w)~" and e~^>(l - 1/w)",
e-^-l>{(w-l)/?i}"*-l,
>na:{(n- l)/w- Ij,
Hence e~'°>\ -x, and therefore 1/(1 -x)>e^.
It follows at once that log {1/(1 - x)], that is, -log(l -x)>x.
Cor. 6t. Iflx,Px, . . . denote log X, log {log x), . . . respect-
ively, x>y>\, and r be any positive integer, then
{x-y)lylyV-y . . . l'y>l'-^'x-l'^'y
>(x-y)/xlxl^x . . . /'■.r.
This may be proved by induction as follows.
By Cor. 5,
Ix -ly = l (x/y) = / { 1 + (a; - 7j)/y} <(x- yl'y,
which proves the first inequality when r = 0.
Assume that it is true for r, i.e. that ,
l''^''x-V^^y<{x-y)lylyl''y . . . Z'*^, then
r^^x-l'-^^y^-l(l^^'x/l'*'y),
= l{l + (l'-*'x-l'-^'y)/l^*hjl
< (I'^^'x - I'-^'yyi^^hj, by Cor. 5.
Hence the induction is complete.
* See Schlomilch, Zeitschrift fiir Mathematik, vol. iii., p. 387 (1858).
+ Malmsten, Grunert's Archiv, viii. (184G).
§ 13 euler's constant 81
Again, we have by Cor. 5,
la;-ly = -l{ylx) = -l{l-{oc-y)lx}>{x-y)lx.
Using this result, and proceeding by induction exactly as before,
we establish the second inequality.
If we put x + l and x for x and y respectively we get the
important particular result
llxlxl^x . . . l'-x>l''^'{x + l)-l'-^'x
>l/(x + l)l{x+l)l^x + l) . . . /'•(.r + 1).
Cor. 7. From the inequality of Cor. 6, combined with the
result of Example 3, § 12, we deduce at once the following im-
portant limits : —
L{l'-{x+l)-l'x} = 0,
L {/'•+' (x + l)- r+M xl.zPx . . . l^x = 1.
X—
Example 1. Show that the limit when n is infinite of 1 + 1/2+ . . .
+ 1/71 - log n is a finite quantity, usually denoted by y, lying between 0 and 1.
(Euler, Comm. Ac. Pet. (1734-5).)
Since, by Cor. 5,
-log(l-l/«)>l/n >log{l + l/n).
We have log {n/(n-l)}>l//i >log{(7i + l)/H},
log{(n-l)/(7i-2)}>l/(n-l)>log{«/(n-l)},
log {3/2} > 1/3 > log {4/3},
log {2/1} > 1/2 > log {3/2},
1 = 1 >log{2/l}.
Hence l+logn>Sl/n>log(n+l).
Therefore 1 > Sl/n - log n > log (1 + 1/;;).
Now, when n = co, log(l + l/H)=0. Thus, for all values of n, however
great, Sl/n - log n lies between 0 and 1.
The important constant 7 was first introduced into analysis by Euler, and
is therefore usually called Euler's Constant. Its value was given by Euler
himself to 16 places, namely, 7= -577215664901532(5). (Seelnst. Calc. Diff.,
chap. VI.)*
* Euler's Constant was calculated to 32 places by Mascheroni in his
Adnotationes ad Eiileri Calculum Integralem. It is therefore sometimes
called Mascheroni's Constant. His calculation, which was erroneous in the
20th place, was verified and corrected by Gauss and Nicolai. See Gauss,
Werke, Bd. in., p. 154. For an interesting historical account of the whole
matter, see Glaisher, Mess. Math., vol. i. (1872).
c. II. 6
82 cauchy's theorems ch. xxv
Example 2. Show that L {1/1 + 1/2+ . . . +l/M}/logre=l.
n— »
This follows at once from the inequality of last example.
From this result, or from Example 1, we see that L {1/1 + 1/2 + ... + 1/n}
= 00 ; and also that L {l/Zc + l/(/c + 1) + . . . + !/«} = oo , where k is any finite
positive integer.
GENERAL THEOREMS.
§ 14.] Before proceeding further with the theory of the limits
of exponential forms, it will be convenient to introduce a few
general theorems, chiefly due to Cauchy. Although these theorems
are not indispensable in an elementary treatment of limits, the
student will find that occasional reference to them will tend to
introduce brevity and coherence into tlie subject.
I. For any critical value of x,L{f{x)\ ={Lf{x)] ', pro-
vided the latter foi'in be not indeterminate.
This is in reality a particular case of the general theorem of
§ 7. The only question that arises is as to the continuity of the
functions of the limits. We may write
... v,*te)_ i>(x)\o^f(x)
Now w = logM is a continuous function of u, so long, at least, as
u lies between + 1 and + so ; and e*"" is a continuous function
of V and w. Hence, so long as 2/<^ {x) and L \ogf{x) are neither
of them infinite, we have
L{f{x)] =Le ,
UMLlogfix)
= e
I4(x)logLf(,x)
-e
Hence L {f[x)f''^ = {Z/(^)j^<-» (i).
An examination of the special cases where either L^ {x) or
L\ogf{x)y or both, become infinite, shows that, so long as
\I^{x)\ does not assume one of the indeterminate forms 0 ,
Qo , 1~ , both sides of (1) become 0, or both qo ; so that the
theorem may be stated as true for all cases where its sense is
determinate.
^ 13, 14 cauchy's theorems 83
II. L{f{x^-\)-f{x)}^Lf{x)lx,promdedL{f{x+l)-f{x)}
K=« x=y> a;==o
he not indeterminate*. (Cauchy's Theorem.)
Since x is ultimately to be made as large as we please, we
may put x-h + n, where A is a number not necessarily an
integer, but as large as we please, and n is an integer as large
as we please.
First, suppose that L {/{x + 1) -/(x)} is not infinite, = k say.
Since L{/{x+ l)—f{x)}=k, we can always choose for h a
definite value, so large that for x = h and all greater values
f{x + 1) -f{x) - ^ is numerically less than a given quantity a, no
matter how small a may be. Hence we have numerically
f{h+\)-f{h)-k<a,
f(k + 2)-f(k+l)-k<a,
f{h + n) -f{h + n-l) -k<a;
and, by addition, /{k + n) —f(h) -nk<na;
that is, / (x) -f{h) -{x — h)k< (x - h) a.
X X \ xj \ X.
A <_ a H .
Since /(^), h, k, and a are, for the present, fixed, it results
that, by making x sufficiently large, we can make f{x)lx—'k
numerically less than a. Now a can be made as small as we
please by properly choosing k ; hence the theorem follows.
Next, suppose that L{f {x + \) -f{x)\ = + <x^ \ then, by
taking h sufficiently large, we can assume that
f{h^l)-f{h)>l,
/(k + 2)-f(h + l)>l,
/(/i + n)-f{h + u~-l)>l,
where / is a definite quantity as large as we please.
* Theorems II. and III. are given by Cauchy in his Analyse Algebrique
(which is Part I. of his Cours d'Analyse de VEcole Royale Polytechnique).
Paris, 1821.
84 CAUCHY'S theorems CH. XXV
Hence f{h + n)-f{h)> nl,
that is f{x) -fih) >{x-h) I.
fix) 7 f{h) hi
Hence ''-^-^ > I +'^-^^ .
a; a; a;
Since /(h), h, I are all definite, we can, by sufficiently in-
creasing a?, render f{h)lx — hljx as small as we please, therefore
f{x)lx>l. Now, by properly choosing h, I can be made as large
as we please ; hence Iif{x)lx = oo .
The case where L{f{x+ \)-f{x)] =- co can be included in
the last by observing that {—/{x+ 1)) - {-/(x)) has in this case
+ Qo for its limiting value.
HI. L f(x + 1)1/ {x) - L {/(a;)}^^ prcmded L/{x + 1)1/ {x)
a;=oo a;=« »=oo
be not indeterminate.
This theorem can be deduced from the last by transformation,
as follows* : —
We have L \^l;{x+\)-xl;{x)] = L"^-^,
a:=«j x=<D X
where ^ (x) is any function such that L {if/ (x + 1) - 1}^ (x)} is not
a=oo
indeterminate. Let no w i/^ (x) = log/(a;) ; so that ij/(x+l)-il/(x) =
log /(x+1)- log / (x) = log {/ (x + 1)1/ {x)] ; and i}/ {x)/x -
{hg/(x)}/x = log {/(x)Y'\ Then we have
£log {^^f^} =Lhg{/{x)Y'\
Hence log { L '^^^ ]=log[L {/(x)n
K. X=ao J \X) } 2=00
provided L/{x-\- 1)1/ {x) be not indeterminate. Hence, finally,
L-^-^^^L{/{x)r.
Cauchy makes the important remark that the demonstrations
of his two theorems evidently apply to functions of an integral
variable such as x\, where only positive integral values of x are
admissible.
* The reader will find it a good exercise to establish this theorem directly
from first principles, as Cauchy does.
^14,15 Ld^jx, L logaos/x, Lx]ogaX 85
For example, we have L (a; + l)!/a;l = L {x + l) = co. Hence L (.rl)V*=Qo,
and consequently L (l/a;!)'/* = 0.
EXPONENTIAL LIMITS RESUMED.
§ 1 5.] If a > 1, then L d^jx = co; L loga^/^ = 0; L x \ogaX = 0.
The first of these follows at once from Cauchy's Theorem
(§ 14, II.) for we have
L (a^+^ - a") = Za^ (a - 1) = 00 .
Hence La^/x = qo .
As the theorem is fundamental, it may be well to give an
independent proof from first principles.
First, we observe that it is sufficient to prove it for integral
values of x alone, for, however large x may be, we can always
put x=/+z where / is a positive proper fraction and z a
positive integer. Then we have
Li — — Jj -? ,
X=ao X Z^tnJ + Z
z=« y + ^ z
f T 1 T <^^
4=00 JlZ + 1 «=„ Z
= ^L"l, (1),
where we have to deal merely with Ldjz, z being a positive
integer.
Let Uz = d/z, then Uz+i/u^ = az/{z + 1) = a/(l + l/z). Now,
since Z «/(l + 1/2;) = a>l, we can always assign an integral
value of z, say z = r, such that, for that and all greater values of z,
Ug+ilug>b, where b>l. We therefore have
tlr+i/Ur>b,
Ur+^/Ur+i > h,
Ug/u.,-i>b,
86 Lx^jn !, LJJn ch. xxv
Hence, by multiplying all these inequalities together, we deduce
Now Urjlf is finite, and, since h>l, h^ can be made as great as
we please by sufficiently increasing z. Hence 7/ w^ = go , on the
supposition that z is always integral. But, since a^ is finite, it
follows at once from (1) that L d^/a;= co, when w is unrestricted.
The latter parts of the theorem follow by transformation.
If we put a" - y, so that a; = log^y, and to a; = go corresponds
y = oc , we have
CO =L d^lx = L yl\ogay-
Hence L \ogay/y = 1/go =0.
If we put a* = l/y, so that a; = - logay, and to ii; = go corre-
sponds y = 0, we have
CO - X a^'/x = - L l/y hgay.
Hence L y hgay = - 1/go = 0.
Example 1. Show that, if a>l and n be positive, then L fl^/x" = Go ;
L log^^x|x^ = 0■, L a;»log„a; = 0.
SC=X X=+0
L a'=lx-"= L {a*/»/a;}",
a=oc ar=ao
= { L (aV«)«/.T}»,
— rr^n—,
for, since a>l and n is positive, we have aV»>l, so that L(a^l^Ylx = <x> and
The two remaining results can be established in like manner, if we put
y = log„ X in the one case, and y= - log„ x in the other.
It should be noticed that if n be negative we see at once that L a'Jx^= co ;
X=<B
L log„x/x»= 00 ; L .i»]og„a;= - oo .
«=■« a:=0
Example 2. If x be any fixed finite quantity, L a;"/Hl=0.
Since n is to be made infinite, and x is finite, we may select some finite
positive integer k such that x<k<n. Then we have
nl (A:-l)l ' k ' k + 1 ' ' ' n'
Now, since x<k, L (.r/fc)"~*+' = 0, hence the theorem.
§§ 15, 16 EXAMPLES 87
Example 3. Lm{m-1) . . . (w- jj + 1)/h! = 0 or oo , according as -?)i>
or < -1.
First, let m> -1, then m + 1 is positive. We can always find a finite
positive integer k such that 7« + l<ft<n. Therefore we may write
w(m-l). . . (m-n + l)_, ^+1 r ( I JI!^\ ( \ '"±1^
-K ) m^k-iy- j^ J\ 'k + lj • ' '
11
= (-)»-fc+\„Ct_iP,say.
Now
logl/P=-log(^l--^j-log(^l-^-...-log(^l--^-j.
> (m + l)/& + (7tt + !)/(/; + 1) + . . .+(m + l)/?i,
by § 13, Cor. 5. Also, by § 13, Example 2, the limit of {m + 1)/A; + {m + 1)/(A; + 1)
+ . . . +(»i + l)/7t is infinite when n^cc . It follows, therefore, that LP = 0,
and therefore that L„jC„=0.
Next, let m< - 1, say m= - (1 + a), where a is a positive finite quantity.
We may now write
TO^n~i^ 1.2. . . .^ ~^ ' ^) ^^y-
Now
l„gP=-log(l-^-^)-l<.g(.-,j^)-. . .-1ob(i-.„-»-^),
> a/(l + a) + a/(2 + a) + . . . + al{n + o),
>a/(l+jj) + o/('2+iJ)+. . .+al(n+p),
where p is the least integer which exceeds a. But the limit of a/(l+2))
+ a/(2 + p)+. . . +al{n+p) is infinite. Hence LP =co.
When m= -1, ^6\=(-l)", and the question regarding the limiting
value does not arise.
§ 16.] T/ie fundamental tlieorem for tlie form 0" is that
L af>=l.
«e=+o
This follows at once from last paragraph ; for we have
JLaf = Le^^'^^^ = e^^"'^^ — e" — 1.
Example 1. L {x^)=' = l.
a:=+0
For L(a;»)* = Lx''^=L(.c^)» = (Lar^)» = l™ = l.
Example 2. L a;^" = l (« positive).
z=+0
For La;*" = Le*"'°8='=e"'''»B=' = e° = l, by § lo, Example 1,
^.Ij. — If n be negative, L a;*" = 0°° = 0.
88 THE FORM 0° CH. XXV
§ 17.] Ifu and v be functions of x^ both, of which vanish when
x = a, and are such that L v/u^ = I, where n is positive and neither 0
nar 00 , and I is not infinite, then L u^ = \, provided the limit be so
a;=o
approached that u is positive*.
For Lu'> = L (m^")"/"" = (Zw"")^'""-
Now, by § 16, Example 2, since n is positive, L ti^' = 1. Hence
«=+0
Lu'' = V=l.
If L v/u^ = CO, this transformation leads to the form 1°°;
x=a
and therefore becomes illusory.
The above theorem includes a very large number of parti-
cular cases. We see, for example, that, if Lvju be determinate and
not infinite, then Lu^ = 1. Again, since, as we shall prove in
chapter xxx., every algebraic function vanishes in a finite ratio
to a positive finite power of x — a, it follows that every such
function vanishes in a finite ratio to a positive finite power of
every other such function. Hence LijC = 1 whenever u and v
are algebraic functions of a?t.
Example. Evaluate I, {x- 1+V(a:^- l)}"^"""^' when x=l.
Here u=^{x-l)y{x-l) + ^{x^ + x + 1)}, v = ^(x-l), u^/sy^ = y (x - 1)
+ ^{x^ + x + l)Yri.
Hence Lu^/^Jv = ^S. Therefore Lm" = L (m"^'"-')''/"^^ = 1^'"^' = 1 .
§ 18.] In cases where the last theorem does not apply, the
evaluation of the limit can very often be effected by writing u"
in the form g"'"*", and then seeking by transformation to deduce
the limit of v logu from some combination of standard cases J.
Example. Evaluate x'Aok («'-i) ^hen a; = 0. '
It is obviously suggested to attempt to make this depend on
L {(e*-l)/x} = l. This may be effected as follows. We have
* See Franklin, American Journal of Mathematics, 1878.
t See Sprague, Proc. Edinb. Math. Soc, vol. in., p. 71 (1885).
t At one time an erroneous impression prevailed that the indeterminate
form 0" has always the value 1. See Crelle's Jour., Bd. xii.
17-21 THE FORMS 00 0,1
Now ^^S'' - ^°^^
log (6=^-1) log{(e==-l)/a;}+loga;'
1
~ log {(e* - l)/x}/log x + 1'
Since L log {(e*- l)/a;} =0, by § 13, Cor. 4, and L log a;= - qo , we see that
I,loga;/log(e"=-l) = l.
Hence ix^/ios («'~^) = e .
§ 19.] Since m" = 1/(1/^)", indeterminates of the form oo"
can always be made to depend on others of the form 0", and
treated by the methods already explained.
Example. Evaluate (1 + xyi^ when a; = oo .
Let l + x = lly, so that y = 0 when a; = Qo ; then we have
L (l + x)V«=i {llyvH^-v)} = llL{yV)Vi^-y).
X=co y=0
Novf Lyy = 1 and Lll{l-y) = l; hence L (l+x)V*=l.
a;=oo
§ 20.] The fundamental case for the form 1" is L (l + Ijccf
X=oo
= L {\+ xY'^-e, already discussed in § 13. A great variety of
a:=0
other cases can be reduced to this by means of the following
theorem.
If u and V he functions of x such that u=\ and v= cc when
x = a, then Lu^ = e^<"-^), provided Lv (u—l) be determinate.
We have in fact
W" = {(1 + ^^3Y)V(«-i)}MM-l),
Hence, by § 7,
Xw" = X {(1 + w - i)V(«-i)|i''(«-i)^
provided Lv {u - 1) be determinate.
Example 1. L xV(^-i) = L (1 + ^^)V(*-i)=c.
x=l a:=l
Example 2. Evaluate (1 + log x)V(a;-i) when a; = 1.
We have
l = L (1 +logx)V(a:-i)= L{(1 + log a.)i/ioKX}loga;/(a:-i)^
— gtIoga;/(x-l)_
Now L log xl{x -1)=L log xV(*-i) = log La;V(a:-i) = log e = 1. Hence I = e.
TRIGONOMETRICAL LIMITS.
§ 21.] We deal with this part of the subject only in so far
as it is necessary for the analytical treatment of the Circular
Functions in the following chapters. We assume for the present
that these functions have been defined geometrically in the usual
manner.
90 TRIGONOMETRICAL INEQUALITIES CH. XXV
We shall require the following inequality theorems : —
If xhe the number of radians {circular units) in any positive
angle less than a right angle, then
I. iaxix>w>s\n x\
II. x>mix>x-\aP\
III. l>cos^>l-i.^.
If PQ be the arc of a circle of radius r, whicli subtends the
central angle 2x, and if PT QT be the tangents at P and Q,
then we assume as an axiom that
PT+ TQ> a.Tc PQ> chord PQ.
Hence, as the reader will easily see from the geometric defini-
tion of the trigonometrical functions, we have
2r tan x > 2rx > 2r sin x ;
that is, tan x> x> sin x,
which is I,
To prove IL, we remark that sin x = 2 sin ^x cos ^x
^ 2 tan Ix cos^ |a7 = 2 tan ^x (1 - sin" ^x). Hence, since, by I.,
tan^;»>|.r and sin|a;<^a;, we have
sin a;>2 . ^a; {1 - {^xf},
>x-la^.
The first part of III. is obvious from the geometric
definition of cos x. To prove the latter part, we notice that
cos a; = 1 - 2 sin'^ ^x ; hence, by I.,
cosa;>l - 2(|a')"
>1-K-
§ 22.] The fundamental theorem regarding trigonometrical
limits is as follows: —
If X be the radian measure* of an angle, then L (sin xjx) = 1.
1=0
This follows at once from the first inequality of last para-
graph. For, if x<^ir, we have
tan x>x>8inx;
therefore sec x > ^r/sin x>l.
* In all that follows, and, in fact, in all analytical treatment of the trigono-
metrical functions, the argument is assumed to denote radian measure.
^21-23 Lsinx/x, Ltaxixjx 91
If we diminish x sufficiently, sec x can be made to differ from
1 by as little as we please. Hence, by making x sufficiently
small, we can make xj^m x lie between 1 and a quantity differing
from 1 as little as we please. Therefore
Lxjmix^l.
Hence also L sin xjx = 1.
Cor. 1. L tan xjx = 1.
For L tan xjx = L (sin ^/^)/cos x = L sin xjx x L 1/cos x = \ x 1 = 1.
Cor, 2. Zsin-/-
a;=oo X X
L tan - /- = 1 provided a is either a con-
stcmt, or a function of x which does not become infinite when a? = qo .
This is merely a transformation of the preceding theorems.
It should also be remarked that
i (sin ?/")'= i('tan-/°y=l.
a;=»V ^/ OCJ x=m\ XJ Xj
provided a and /? are constants, or else functions of x which
do not become infinite when x= co.
If, however, a were constant, and y3 a function of x which
becomes infinite when a? = co , then each of the two limits would
take the form 1*, and would require further examination.
§ 23.] Many of the cases excepted at the end of last para-
graph can be dealt with by means of the following results, which
we shall have occasion to use later on : —
If a. be constant, or a function of x ivhich is not infinite when
x= cc , then
i(sm2/-y = i;
a;=«\ X/ x/
L (cos -) = 1:
xftanV-y=l-
a:==oV Xl xJ
To prove the first of these, we observe that for all values of
a[x less than ^tt we have, by § 21, II.,
1>
(^^Ht)>{^<t)]'
92 L{Hml/iy, L^coslY CH. xxv
Now
L{1- a^A^r^L {(1 -aV4;r^)-*^^''V'^'^.
= e«=l, by §§7 and 13.
Hence Xfsin-/-) =1.
In exactly the same way we can prove that Z- ( cos - ) =1.
Finally, since
the third result follows as a combination of the first two.
Example. Evaluate (cos xfl^ when a;=0. By § 20, we have L (cosx)V»:
=:eZ,(co8x-i)/ar', Now (cos X - l)lx^ = - 2 sin^^xlx^^ - ^(sin i^xf^xf. Hence
L (cos.r-l)/a;2= _^_
We therefore have L (cos xfl^ = e-i.
SUM OF AN INFINITE NUMBER OF INFINITELY
SMALL TERMS.
§ 24.] If we consider the sum of n terms, say, Wi + Wa + • • •
+ Un , each of which depends on n in such a way that it becomes
infinitely small when n becomes infinitely great, it is obvious
that we cannot predict beforehand whether the sum will be finite
or infinite. Such a sum partakes of the nature of the form
0 X CO ; for we cannot tell a priori whether the smalluess of the
individual terms, or the iufiniteness of their number, will ulti-
mately predominate. We shall have more to do with such cases
in our next chapter ; but the following instance is so famous in
the history of the Infinitesimal Calculus before Newton and
Leibnitz that it deserves a place here.
I/r+1 be positive, then
L (V + 2''-h. . . +w'-)/7i'-+» = l/(r + 1).
In the case where r is an integer this theorem may be
deduced from the formula of chap, xx., § 9.
§§ 23, 24 Z (I'- + 2'- + . . . 4- rf)J'nr+'- 93
The proofs usually given for the other cases are not very-
rigorous ; but a satisfactory proof may be obtained by means of
the inequality
{r+\)x-{x-y)^x'^'~if''%{r^-l)f{x-y) (1),
which we have already used so often.
If we put first X =p, y -p - 1, and then x =p + 1, y -p, we
deduce
{p + l)'-+i - io'-+i ^{r + l)p'' $/■+' -{p- 1)'"+' (2)
where the upper or the lower signs of inequality are to be taken
according as the positive number r + 1 is > or <1.
If in (2) we put for p in succession 1, 2, 3, . . ., n and add
all the resulting inequalities we deduce
(w +!)'•+» -l>(r+l)(r + 2'-+. . . +wO<w''"''-
Hence
{(1 + 1/^)''+^ - llnr+'}/(r + 1) ^ (!'• + 2'- + . . . + nr)/nr+'
>l/{r + l).
That is to say, (l*" + 2'" + . . . + n^)/n^+^ always lies between l/(r+l)
and {(1 + l/7iY+' - l/w'-+^}/(r + 1). But L {1 + l/nY+' = 1 ;
n=oo
and L l/w*""*"^ = 0, since r + 1 is positive. Hence the second of
the two enclosing values ultimately coincides with the first, and
our theorem follows.
It may be observed that, if r + 1 were negative, the proof
would fail, simply because in this case L 1/rf'*'^ = co .
Cor. 1. If she any finite integer, and r +\ he positive,
L{V+2^+ . . .+(n- syyn'+' = !/(/• + 1).
n=ao
This is obvious, since L{V'+2''+ . . . + (n-sYl/n'''^^ differs
from L{V + 2^ + . . . + n^)/n'''^^ by a finite number of infinitely
small terms.
Cor. 2. j[fa he any constant, and r + 1 he positive,
L {{a + !)•• + (a + 2)'- + . . . + (a + w)'*}/w'-+^ = l/(r + 1).
This may be proved by a slight generalisation of the method
used in the proof of the original theorem.
94 dirichlet's limit ch. xxv
Cor. 3. If a and c he constants, and r + 1 4= 0,
L {{na + cy + (na + 2cY + . . . + (na + ncY}ln''+^
= {(rt + c)''^'-a'-+^}/c(r+l).
This also may be proved in the same way, the only fresh point
being the inclusion of cases where r + 1 is negative.
§ 25.] Closely connected with the results of the foregoing
paragraph is the following Limit Theorem, to which attention
has been drawn by the researches of Dirichlet: —
If a,h, p be all positive, the limit, when n=cc,ofthe sum of n
terms of the series
1 1 1 1 .
a^+p^ {a + Vy^"^ {a+2by+p^ ' ' ' ^ (a + nby+p'^ ' " * ^ ^'
is finite for all finite values of p, however small; and, if
2 l/(a + nbY'^f denote this limit, then
n-O
Lp^l/(a + nby+p^l/b (2).
p=0 n=0
By means of the inequality (1) of last paragraph, we readily
establish that
{a+ (]}- 1) b}''P- {a+pb}~p>pb {a+pb}~p-^>{a + pb}~P
-{a + (p+l)b}-P (3).
Putting, in (3), 0, 1, 2, . . ., n successively in place oi p,
adding the resulting inequalities, and dividing by bp, we deduce
li_l L__U5 1 ..Ml 1 1
bp\{a- b}p {a + 7ib}pj p=o {a + pby+p bp W \a + (n + l) b]p)
(4).
Since Ll/{a + nb}p = 0, and L l/{a + (n + l)b}p = 0, when
» = 00 , we deduce from (4),
pb (a - b)p ^ ^oia +pby+p ^pba? ■ ^^^*
From (5) the first part of the above theorem follows at
once ; and we see that i/pb{a-b)p and 1/pbp'^^ are finite upper
and lower limits for the sum in question.
§§ 24-26 GEOMETEICAL APPLICATIONS 95
We also have
1 s 1 1
;>P2 7-— nTiT->,
h{a- by ^p=o {a +pby+p hafi '
whence it follows, since L 1/b {a - b)p = L 1/baP = l/b, when p = 0,
that
p^o^p=o{a+pby+p b'
From the theorem thus proved it is not difficult to deduce
the following more general one, also given by Dirichlet : —
Ifkx.k^,. . -ykn, . ■ .be a series of positive quantities, no one
of which is less than any following one, and if they be such that
L Tjt - a, where T is the number of the k's that do not exceed t,
tJien ^Ijkn'^p is finite foi' all positive finite values of p, however
small; and L p^l/kn'^p = a*.
p=0 1
Cor. It follows from (5) that
p (^"^Jp^ n=4^p ^ WTiy+p + • • • + (a + ny+p\ ^p^ ^^^'
an inequality which we shall have occasion to use hereafter.
GEOMETRICAL APPLICATIONS OF THE THEORY OF LIMITS.
§ 26.] The reader will find that there is no better way of
strengthening his grasp of the Analytical Theory of Limits than
by applying it to the solution of geometrical problems. We may
point out that the problem of drawing a tangent at any point of
the graph of the function y =f{a)) can be solved by evaluating the
limit when A = 0 of {f(x + h)-f(a;)}/h; for, as will readily be
seen by drawing a figure, the expression just written is the
tangent of the inclination to the axis of x of the secant drawn
through the two points on the graph whose abscissae are a^ and
x + h; and the tangent at the former point is the limit of the
* See Dirichlet, Crelle's Jour., Bd. 19 (1839) and 53 (1857) ; also Heine,
ibid., Bd. 31.
96 GEOMETRICAL APPLICATIONS CH. XXV
secant when the latter point is made to approach infinitely close
to the former*.
Example. To find the inclination of the tangent to the graph of y=e'
at the point where this graph crosses the axis of y.
If 0 he the inclination of the tangent to the x-axis, we have
tane=L{e<^''-e^)lh,
= L{e^-l)jh,
=loge = l.
Hence 0 = \ir.
§ 27.] The limit investigated in § 24 enables us to solve a
problem in quadratures ; and thus to illustrate in an elementary
way the fundamental idea of the Calculus of Definite Integrals.
We may in fact deduce from it an expression for the area in-
cluded between the graph of the function y = aflV'~^, the axis of
X, and any two ordinates.
Let A and B be the feet of the two ordinates, a, b the corresponding
abscissae, and b -a=cf. Divide AB into n equal parts; draw the ordinates
through A, B, and the n-1 points of division ; and construct — 1st, the series
of rectangles whose bases are the n parts, and whose altitudes are the 1st,
2nd, . . ., nth ordinates respectively; 2nd, the series of rectangles whose
bases are as before, but whose altitudes are the 2nd, 3rd, . . ., (7i+l)th
ordinates. If I„ and J„ be the sums of the areas of the first and second series
of rectangles, and A the area enclosed between the curve, the axis of x and
the ordinates through A and B, then obviously I^<A<J„.
Now
J„ = c{a'-+{a + cjiiY +(a+ IcjnY + . . . + (a + n - lclnY]lnV''^ ;
J„=c{(a + c/n)'"4-(a + 2c/n)'"+ . . . +(a + 7ic/n)''}/ni'"-i.
Since e7„ - 1„ = c (6'' - aT)lnV~^ , which vanishes when n = ao , Lin = LJ„ , and
therefore A=LJn, when n — ao. Hence
c (na + lcY+(na + 2cY+ . . .+(na + ncY
-M'^i^^l-"^^^^-'-'--
Hence A = (fc'^-i - a»-+i)/(r + 1) i""-'.
This gives, when r—i^, and a~0, the Archimedian rule for the quadrature
of a parabolic segment.
* We would earnestly recommend the learner at this stage to begin (if
he has not already done so) the study of Frost's Curve Tracivg, a work which
should be in the hands of every one who aims at becoming a mathematician,
either practical or scientific.
t The reader should draw the figure for himself.
§§ 2G-28 THEORY OF IRRATIONALS 97
NOTION OF A LIMIT IN GENERAL. ABSTRACT
THEORY OF IRRATIONAL NUMBERS.
§ 28.] In the earlier part of this chapter limiting values have
been associated with the supply of values for a function in special
cases where its definition fails owing to the operations indicated
becoming algebraically illegitimate. This view naturally sug-
gested itself in the first instance, because we have been more
concerned with the laws of operation with algebraic quantity than
with the properties of quantity regarded as continuously variable.
It is possible to take a wider view of the notion of a limit ;
and in so doing we shall be led to several considerations which
are interesting in themselves, and which will throw light on the
following chapter.
Although in what precedes we defined a limit, it will be
observed that no general criterion was given for the existence of
a finite definite limit. All that was done was to give a demon-
stration of the existence of a limit in certain particular cases.
When the limit is a rational number, the demonstrations present
no logical difficulty ; but when this is not the case we are brought
face to face with a fundamental arithmetical difficulty, viz. the
question as to the definition of irrational number. For example,
in proving the existence of a finite definite limit for (1 + '[jxf
when X is increased indefinitely, what we really proved was not
that there exists a quantity e such that [e-(l + l/a?)*] can be
made smaller than any assignable quantity, but that two rational
numbers A and B can be found differing by as little as we please
such that (1 + IjxY will lie between them if only x be made
sufficiently large. From this we infer without farther proof that
a definite limit exists, whose value may be taken to be either
A or B. For practical purposes this is sufficient, because we can
make A and B agree to as many places of decimals as we choose :
but the theoretical difficulty remains that the limit e, of whose
definite existence we speak, is any one of an infinite number of
different rational numbers, the particular one to be differently
selected according to circumstances, there being in fact* no single
* See chap, xxvni., § 3.
C. 11, 7
98 THEORY OF IRRATIONALS CH. XXV
rational number which can claim to be the value of the limit.
The introduction of a definite quantity e as the value of the
limit under these circumstances is justified by the fact that we
thus cause no algebraic contradiction. Such quantities as ^"2,
^4, &c. have already been admitted as algebraic operands on
similar grounds.
§ 29.] The greater refinement and rigour of modern mathe-
matics, especially in its latest development — the Theory of
Functions— have led mathematicians to meet directly the logical
difficulties above referred to by giving a priori an abstract defi-
nition of irrational real quantity and building thereon a purely
arithmetic theory. There are three distinct methods, commonly
spoken of as the theories of Weierstrass, Dedekind and Cantor*.
A mixture of the two last, although perhaps not the most elegant
method of exposition, appears to us best suited to bring the issues
clearly before the mind of a beginner. We shall omit demon-
strations, except where they are necessary to show the sequence
of ideas, the fact being that the initial difficulties in the Theory
lie not in framing demonstrations, but in seeing where new
definitions and where demonstrations are really necessary. For
a similar reason we shall at once assume the properties of the
onefold of Rational Numbers as known ; and also the theory of
* The theory of Weierstrass, earliest in point of time, was given in his
lectures, but not published by himself. An account of it will be found in
Biermann, Theorie der Analytischen Functionen (Leipzig, 1887), pp. 19 — 33.
A brief but excellent account of Dedekind's theory is given by Weber,
Lehrbuch der Algebra (Braunschweig, 1895, 1898), pp. 4 — 16 : see also
Dedekind's two tracts, Stetigkeit und irrationale Zahlen (Braunschweig,
1872, 1892); and Was sind und icas sollen die Zahlen? (Braunschweig,
1888, 1893). For expositions of Cantor's theory see 3Iath. Ann., Bd. 5
(1872), p. 128, and lb. Bd. 21 (1883), p. 565; also Heine, Crelle's Jour.,
Bd. 74 (1872): and Stolz, AUgemeine Arithmetik, I. Th. (Leipzig, 1885),
pp. 97—124.
Meray, in his Nouveau Precis d'Analyse Infinitesimale (Paris, 1872),
published independently a theory very similar to Cantor's, which will be
found set forth in the first volume of his Lemons Nouvelles sur I'Analyse
Infinitesimale (Paris, 1894).
A good general sketch of the whole subject is given by Pringsheim in his
article on Irrationalzahlen, drc, Encyclopddie der Mathematischen Wissen-
schaften (Leipzig, 1898), Bd. i., p. 47.
§§ 28-31 THE RATIONAL ONEFOLD 99
terminating and repeating decimals, which depends merely on the
existence of rational limits.
§ 30. J Starting with 1 and confining our operations to the
four species +, -, x, h-, we are led to the onefold of Rational
Quantity
. . ., -mjn, . . . - 1, . . . 0, . . . + 1, . . . +mln, . . . {R)
in which every number is of the form + 7^^/w, where m and n are
finite integral numbers.
The onefold R possesses the following properties.
(i) It is an ordered onefold, in the sense that each number
is either greater or less than every other. The onefold may
therefore be arranged in a line so that each number occupies a
definite place, all those that are less being to the left, all greater
to the right.
(ii) R is an arithmetic onefold, in the sense that any con-
catenation of the operations +, — , x, -f- in which the operands are
rational numbers (excepting always division by 0) leads to a
number in R.
(iii) a and h being any two positive quantities in R, such
that Q<a<h, we can always find a positive integer n so that
na>b*; and consequently b/n<a.
(iv) Between any two unequal quantities in R, however
nearly equal, we can insert as many other quantities belonging
to R as we please. We express this property by saying that R is
a compact onefold. This follows at once from (iii), since the
rational numbers
a, a + {b-a)ln, a + 2{b-a)/n, . . ., a + {n-l) (b - a)/n, b
are obviously in order of magnitude, and the integer n may be
chosen as large as we please.
§ 31.] Dedekind's Theory of Sections. Any arrangement of
all the rational numbers into two classes A and B, such that
every number in A is less than every number in B, we may call
a section i of R. We denote such a section by the symbol (A, B).
It is obvious that to every rational number a corresponds a
* This is sometimes spoken of as the Axiom of Archimedes.
t Dedekind uses the word Schnitt.
7—2
100 THEOBY OF SECTIONS CH. XXV
section of B ; for we may take A to include all the rational
numbers which are not greater than a, and B to include the rest,
viz. all that are greater than a. Conversely, if in the class A
there be a number a which is not exceeded by any of the others
in A, then the section may be regarded as generated by a. The
same is true if in the class B there be a number a which is
not greater than any of the others in B ; for we might without
essential alteration transfer a to the class A, in which it would
then be the greatest number. The case where there is a greatest
number a in A and a least number ^ in B\& obviously impossible.
For a and fi must be different, since the two classes A and B are
exhaustive and mutually exclusive ; but, if a and /S were different,
we could, since R is compact, insert numbers between them which
must belong either to J. or to ^ ; so that a and /3 could not be
greatest and least in their respective classes as supposed.
But it may happen that there is no greatest rational number
in A, and no least rational number in B. There is then no
rational number which can be said to generate the section. Such
a section is called an empty or irrational section. It is not
difficult to prove that, if mjn be any positive rational number
which is not the quotient of two integral square numbers, and A
denote all the rational numbers whose squares are less than mjn,
and B all those whose squares are greater than min, then the
section {A, B) is empty.
§ 32.] An ordered onefold which has no empty sections is
said to be contimwus. It will be observed that the onefold of
rational numbers is discontinuous although it is compact.
Starting with the discontinuous onefold of rational numbers
B, we construct another onefold S by assigning to every empty
or irrational section a symbol which we shall call by anticipation
a number, adding the adjective irrational to show that it is not a
number in B. As the section and the number are coordinated,
we may use the symbol {A, B) to denote the number as well as
the section. We can also without contradiction re-name all the
rational numbers by attaching to each the corresponding sectional
symbol.
Naturally we define the number {A, B) as being greater than
§§ 31-33 SYSTEMATIC REPRESENTATION OF A SECTION 101
the number {A', B) when A contains all the (rational) numbers
in A' and more besides ; and consequently B' contains all the
numbers in B and more besides. The numbers {A, B) {A\ B')
are equal when A' contains all the numbers in A, neither more
nor less, and the like is consequently true of B' and B.
0 is the section in which A consists of all the negative and
B of all the positive rational numbers.
{A, B) is positive when some of the numbers in A are
positive ; negative when some of the numbers in B are negative.
Also, if we understand - A to mean all the numbers in A each
with its sign changed, then {- B, - A) =-{A, B).
The new manifold S is therefore obviously an ordered mani-
fold ; and it is clearly compact, since B is compact. It is also
continuous, i.e. every section in S is generated by a number in ^S* ;
for, if a, /8 be a classification of all the numbers (or sections) of S
such that every number in a is less than every number in fi, then
(a, P) determines a section in S of the most general kind. But,
if A contain all the rational sections in a and B all the rational
sections in yS, then {A, B) is a section in R, i.e. a number in 8 ;
and it is obvious that every number in S<{A, B) is a number in
a, and every number m. S>{A, B) au number in (i. Hence (a, ^)
corresponds to the number {A, B), which is a number in 8.
% 33.] Systematic representation of a number, rational or
irrational. Consider any number defined by means of a section
{A, B) of the rational onefold B. We are supposed to have the
means, direct or indirect, of settling whether any rational number
belongs to the class A or to the class B. Suppose {A, B) positive.
Consider the succession of positive integers 0, 1, 2, . . .; and
select the greatest of these which belongs to A, say a^. Then
/!>o = ao+l belongs to B. The two rational numbers ««, ^o de-
termine two sections in B between which there is a gap of
width 1. Within this gap the section {A, B) lies, i.e. ao<{A, B)
Next divide the unit gap into ten parts by means of the
rational numbers a^ + 1/10, «« + 2/10, . . . , ao + 9/10, and select
the greatest of these numbers, say a^ = a^ + jOj/lO, which belongs
to A ; then hi = ax + 1/10 belongs to B. We have now a gap in
102 SYSTEMATIC REPRESENTATION OF A SECTION CH. XXV
B of width 1/10, determined by tlie numbers cfj, hi within which
{A, B) lies.
We next divide the gap of 1/10 into ten parts by means of
the numbers al+l/10^ ai + 2/10', . . ., «! + 9/10^ ; and so on.
Proceeding in this way, we can determine two rational numbers
(terminating decimals in fact),
a» = ao+WlO + - • .+W10", &„ = »„ + 1/10'' (1)
between which {A, B) lies, the width of the gap between a^ and hn
being 1/10". It is obvious that a^, «!,...,»„ are a non-decreas-
ing succession of positive rational numbers ; and it can easily
be proved that ho, hi, . . ., hn are a non-increasing succession.
1°. At any stage of the process it may happen that a„ is the
greatest possible number in A, in other words that J3n+], and all
successive jo's are zero. The section {A, B) is then determined
by the number a„ ; and {A, B) is the rational number a„.
If the process does not stop in this way, two things may
happen.
2°. The digits pi, p^, . . ., Pn, > > > may form an endless
succession but repeat, say in the cycle j^^, Pr+i, • • ., Pw In this
case there exists a rational number a to which a„ = a,, +pil\Q + . . .
-i-j9m/10" approximates more and more closely as we increase n ;
and, since J„ = a„ + 1/10", hn also approaches the same limit. It
follows that the rational numbers of class A might be defined as
the numbers none of which exceeds every number of the succession
a^, ai, . . ., an, however large wbe taken. Hence, if we agree to
attach the number a to the class A, it will be the greatest number
of that class, and the section {A, B) is generated by a.
3°. The digits Pi, p^, ■ • -, Pn may form an endless non-
repeating succession. Since the gap hn-an= 1/10" can be made
as small as we please, it follows as before that the rational
numbers of class A may be defined as all the rational numbers
none of which exceeds every number in the endless succession
ao, Oi, •..,«»,... . This statement does not as in last case
enable us to identify {A, B) with any rational number; but, since
n may be as large as we please, we can by calculating a sufficient
number of the digits jOj, p^, . . . separate {A, B) from every other
§§ 33, 34 CONVERGENT SEQUENCES 103
number, rational or irrational, no matter how near that number
may be to (A, B).
Conversely, it is obvious from the above reasoning that every
terminating or repeating decimal determines a rational section in
B, and therefore a rational number ; and every non-terminating
non-repeating decimal an irrational section in U, i.e. an irrational
number.
It is an obvious consequence of the foregoing discussion that
between any two distinct numbers, rational or irrational, we can
find as many other numbers, rational or irrational, as we please,
§34.] Cantor's Theory. The rational numbers ««> «i, • • •,
«„,... in § 33 evidently possess the following property. Given
any positive rational number e, however small, we can always find
an integer v such that la,j-a„+r|<€ when n^^v, r being any
positive integer whatever.
We are thus naturally led to consider an infinite sequence of
rational numbers
w,, lu, ...,«„,... (2)
which has the property that for every positive rational value of e,
however small, there is an integer v such that \ Un ~ Un+r I < « when
n-^v, r being any positive integer whatever.
Such a sequence is called a convergent sequence ; and Ui, u^,
&c. may be called its convergents. It should be observed that we
no longer, as in § 33, confine the convergents to be all (or even
ultimately all) of the same sign ; nor do we suppose that they
form a non-decreasing or a non-increasing (monoclinic) sequence.
To every convergent sequence corresponds a definite section of
the onefold of rational numbers {R) : so that every such sequence
defines a real number, rational or ir?'atio7ial.
We may prove this important theorem as follows.
Let €i be any positive rational number whatever; then we can
find vi such that, when n'^v^, \un-Un+r\<^i' Iii particular, we
shall have, if w? > I'l, | m^, - u„, | < ei , whence
U^^-ei<Um<U^, + £i (2).
In other words, the two rational numbers «! = Uy^ - Cj, &i = w„, + Ci
determine two sections in 11 such that all the numbers of -the
104 CONVERGENT SEQUENCES CII. XXV
sequence 2 on and after m^, lie in the gap of width 2ej between
those two sections.
Next choose any rational number (2<^i' We can then es-
tablish a gap of width 2c2, whose bounding sections are given by
ttj = iiya — *2j ^2 = tifi + fg. The number v^ will in general be greater
than vi ; but it might be less. Also the gap a^bo might partly
overlap the gap Uibi. But, since all the convergents on and
after w^, lie within the gap aibi, we can throw aside the part of
a^bi, if any, that lies outside Uibi, and determine a number v2-«^vi
such that
a2<Um<b2
when w-^vj. Then, all the convergents on and after w^, lie
within the gap aa^a, whose width ;:^2€2<2ci. This process may
be repeated as often as we please; and the numbers Ci, C2> • • •
may be made to decrease according to any law we like to choose.
The numbers ai, a2, . . . form a non-decreasing and the numbers
bi, b^, . . . & non-increasing sequence : and each successive gap
lies within the preceding, although it may be conterminous with
the preceding at one of the two ends. Since f], Cj, . . . can be
made as small as we please, it is clear that by carrying the above
process sufficiently far we can assign any given rational number
to one or other of the two following classes : — (A ) numbers which
do not exceed every one of the numbers ttm, Um+i , . . . when m is
taken sufficiently large, (B) numbers which exceed any of the
numbers Um, Um+i, . . . when m is taken sufficiently large.
Hence every convergent sequence determines a section of B ;
and therefore defines a number, rational or irrational.
Conversely, as we have seen in § 33, every number, rational or
irrational, may be defined by means of a convergent sequence. If
the sequence is Ui, ti^, . . ., w„, . . . we shall often denote both
the sequence and the corresponding number by («„). Since it is
only the ultimate convergents that determine the section, it is
clear that we may omit any finite number of terms from a con-
vergent sequence without affecting the number which it defines.
In particular, the sequences Ml, u^, . . . tir, ...,«„,... and
Mr, ...,«„,.. . define the same number. It should be noticed
that in the case of rational numbers the convergents on and after
§§ 34-36 ARITHMETICITY OF IRRATIONAL ONEFOLD 105
a particular rank may be all equal : in fact we may define any
rational number a by the sequence a, a, . . ., a, . . ., and call
it (a).
Since each gap in the above process lies within all preceding
gaps, and the section in R which is finally determined within
them all, we have, if v he such that |M,i-w„+rl<« when n-^v,
U^-i1^.{Un)'^U^ + € (3),
an important inequality which enables us to obtain rational
approximations as close as we please to the number which is
defined by the sequence Ui, u^, . . ., Un, ....
§ 35.] Null-sequence. If by taking n sufficiently great we
can make | u^ \ less than any given positive quantity e, however
small, it follows from (3) that (?<„) must be between 0 and a
rational number which is as small as we please. We therefore
conclude that in this case tbe sequence u^, lu, . . ., Un, . . .
corresponds to 0 ; and we call it a null-sequence.
§ 36.] Definition of the four species fm- the generalised onefold
of real numbers S.
If (Un) (vn) be any two numbers, rational or irrational, defined
by convergent sequences, it is easy to prove that the sequences
{un + Vn), (Un-Vn), (tinVn), (un/vn), are Convergent sequences*,
provided in the case of (un/vn) that (Vn) is not a null-sequence.
We may therefore define these to mean (iCn) + (vn), (%) - (%),
(Un) X (Vn), (un) "^ (%) respectively. For it is easy to verify that,
if we give these meanings to the symbols +, -, x, -f- in connection
with the numbers (tin) and (■y„), then the Fundamental Laws of
Algebra set forth in chap. i. § 28 will all be satisfied.
For example!,
(un) - (vn) + (vn) = («„ - Vn) + («„), by definitions
^{{Un-Vn}+Vn), by dcf.
= (Un), by laws of operation for R.
* The reasoning is much the same as in § 6 above.
t The plain bracket ( ) is appropriated to the definition of the number by
a sequence ; the crooked bracket has reference to operations in R.
106 ARITHMETICITY OF IRRATIONAL ONEFOLD CH. XXV
Again,
(Un) X {(v„) + (Wn)} = (Un) X (v^ + «;„), by def.
= (un {Vn + iCn}), by def.
= {unVa+UnWn), by laws of Operation for (R),
= (UnVn) + (UnWn), by def.
= (m») (««) + (Un) (Wn), by def.
and so on.
In order that two numbers (m„) and («„) may be equal it is
formally necessary and sufficient that (m„) - (v„) = 0, in other
words, that (M„-'y„) = 0, that is, that Ui-Vi,U2-V2, . . ., Un-Vn,
. . . shall be a null-sequence. This from the point of view of
our exposition might also be deduced from the fact that (m„) and
(vn) must correspond to the same section in E. We can also
readily show that all null-sequences are equal, as they ought to
be, since they all correspond to 0.
We have now shown that the onefold of real quantity (S)
built upon E by the introduction of irrational numbers is an
arithmetic manifold. The proof that S has the property iii. of
§ 30 is so simple that it may be left to the reader. Henceforth,
then, we may operate with the numbers of S exactly as we do
with rational numbers.
§ 37.] It is worthy of remark that the properties of the
rational onefold R can, by means of appropriate abstract defini-
tions, be established on a purely arithmetical basis. It is not
even necessary to introduce the idea of measurement in terms of
a unit. The numbers may be regarded as ordinal ; and addition
and subtraction, greaterness and lessness, &c. interpreted merely
as progress backwards and forwards among objects in a row, which
are not necessarily placed at equal or at any determinate distances
apart*.
Following the older mathematicians since Descartes, we have
in the earlier part of this work assumed that, if we choose any
point on a straight line as origin, every other point on it has for
• See, for example, Harkness and Morley, Introduction to the Theory of
Analytic Functions. (Macmillau, 1898.)
§§ 36-39 GENERAL CONVERGENT SEQUENCE 107
its coordinate a definite real quantity : and conversely that every
real quantity, rational or irrational, can be represented in this way
by a definite point. The latter part of this statement, viz. that
to every irrational number in general* there corresponds a definite
point on a straight line, is regarded by the majority of recent
mathematicians who liave studied the theory of irrationals as an
axiom regarding the straight line, or as an axiomatic definition
of what we mean by "points on a straight line."
§ 38.] Generalisation of the notion of a Convergent Sequence.
It is now open to us to generalise our definition of a convergent
sequence by removing the restriction that « and Ui, u^, . . .,
Un, . . . shall be rational numbers. Bearing in mind that we
can now operate with all the quantities in S just as if they were
rational, we can, exactly as in § 34, establish the theorem that
everi/ convergent sequence oi real numbers Ui, u», . . ., Un, . . .
defines a real number (?«„).
Also we can show that, if e be any real positive quantity,
however small, we can always determine v so that
Wm-«<K)<Mm + C (4),
when w<|:v.
For we have merely, as in § 34, to determine v so that
l«m-Mm+r I <«'<«, when ?»<j:v.
Then we have
and therefore
when m<^v.
§ 39.] General Definition of a Limit and Criterion for its
Existence.
Returning now to the point from which this discussion
started, we define the limit of the infinite sequence of real
quantities
til, «2, . . ., w„, . . . (2),
as a quantity u such that, if c he any real quantity however small.
* We do not speak of special irrationalities, such as ^2, which arise in
elementary geometrical constructions.
108 LIMIT OF A SEQUENCE CH. XXV
then there exists always a positive integer v such that | w,i — w|<e
when n*^v. And we prove the following fundamental theorem.
The necessary and sufficient condition that the sequence, 2, have
a finite definite limit is that it be a convergent sequence ; and the
limit is the real number which is then defined by the sequence.
The condition is necessary ; for, if a limit u exist, then
I Un - Un+r I S j M„ - M + M - M„+r |,
:^\Un-u\ + \Un+r-u\.
Now, since u is the limit of the sequence, we can find v such
that I M„ - M I < I c when n^^v; and , d, fortiori, \tin+r-u\<^e
when w<^v. Hence we can always find v so that \un-Un+r\<f,
where c is any positive quantity as small as we choose. Hence 2
is convergent.
Also the condition is sufficient. In fact, we can show that
(un), the number defined by the sequence when it is convergent,
satisfies the definition of a limit. For, given c, we have seen that
we can find v so that
Um-€<(Un)<Um + e
when m<^v: whence it follows that | u^ - (w„) | <« when m'^v.
Moreover there cannot be more than one finite limit ; for, if
there were two such, say u and v, we should have
\U-V\ = \u-Un + Un-V\,
:!f>\Un-u\ + \nn-v\.
But, since both u and v are limits we could, by sufficiently
increasing n, make |ttft-w| and |Mn — •»! each less than ^c, and
therefore | w - « | < e, i.e. as small as we please. Hence u and v
cannot be unequal.
The reader will readily prove that, {/* Ml, Wa, . . .,«»,• • • ^^
a non-decreasing {non-increasing) infinite sequence, no number of
which is greater than (less than) the finite number I, then this
sequence has a finite limit not greater than {not less than) I.
§ 40.] Let us now consider any function of x, sa.yf{x), which
is well defined in the sense that, for all values of x that have to
be considered, with the possible exception of a finite number of
isolated critical values, the value of f(x) is determined when the
value of X is given. We define the limiting value, I, off{x) when
§§ 39-41 CONDITION FOR EXISTENCE OF A LIMIT 109
X is increased up to the value a, by the property that, when any
positive quantity e is given, there exists a finite quantity $<a such
that
\f{x)-l\<.
when t1^x<a.
This obviously includes our former definition of a limiting
value ; and we may denote I hy L f{x).
a;=o-0
Let ai, a2, . . ., an, . ■ . be any ascending convergent
sequence which defines the number a ; and let us suppose, as
we obviously may, that there is no critical value of x in the
interval ai^x<a. Then, if we consider the sequence Ui =f{a^,
ih-fia^), • • ', w„ =/(«„), . . ., the results of last paragraph
lead us at once to the following theorem.
The necessary and sufficient condition that L f{x) be finite
x=a-0
and definite is that it be possible to find a finite quantity i<a
such that, wlien $^x<x'<a,
\/(x)-/(x)\<.,
where c is any finite positive quantity however small.
The reader will easily formulate the corresponding proposition
regarding L f{x).
a;=a+0
§ 41.] There is one more point to which it may be well to
direct attention before we leave the theory of limits.
L f{x) is not necessarily equal to the value of f{x) wJien
x=a±0
x = a. For example, L{af- l)/(x - 1) = 2 ; but (aP - l)/(x - 1)
a:=l±0
has no value when x-1.
A more striking case arises when f(x) is well defined when
x = a, but is discontinuous in the neighbourhood of x = a.
Thus, if
/{x) = L {sin xjl - sin 2^/2 + . . . + (- 1)""^ sin nxjn],
n=oo
then it is shown in chap, xxix., § 40, that L f{x) - + 7r/2,
a;=7r-o
L f{x) = - 'jr/2 ; whereas /(tt) = 0.
110
EXERCISES VII
CH. XXV
Exercises VII.
Limits.
Find the limiting values of the following functions for the given values of
the variables : —
(1.) (3xi + 2x^ + 3xi)/(a;i+a;7 + a;^), x = Q, and x=<x>.
(2.) (a;4-x3-9x2+16a;-4)/(,r''-2a;2-4x + 8), x = 2.
(3.) log(x3-2x'''-2x-3)-log(x3-4a;2 + 4x-3), x = 3.
(4.) {a;-(n + l)a;"+'+n.T™+2}/(l-x)2, a;=l (n a positive integer). (Euler,
Diff. Gale.)
(5.) {V(x-l)-(x-l)}/{4/(x-l)-V(x-l)}, x=l.
(6.) (x'"+"-a'"x")/(xP+9-ai'x«), x = a.
(7.) {(a + x)"»-(a-x)™}/{(a + x)»-(a-x)"}, x = 0.
(8.) {(x'»-l)P-(x™-l)9}/{(x-l)P-(x-l)«}, x = l.
(x'»-l)^-(x'»-l)(x"-l) + (x«-l)''
^ ■* (a;"*-l)2+(a;"'-l)(x™-l) + (x"-l)2*
(10.) {a-V(a«-a;2)}/x2, x=0. (Euler, Z)/J. Cafc.)
(11.) {i:J{a-\-x)-^{a-x)}l{;>l{a + x)-^{a-x)}, x = 0.
(12.) {(a2 + ax + x2)i-(a2-ox + x2)^}/{(a + x)i-(a-x)i}, x=:0. (Euler,
Z)//f. Cafc.)
(13.) {(2a-''x-x^)i-a(a-x)^}/{a-(ax3)i}, x = a. (Gregory, JBxamjj/fs in
D7j. Gale.)
(14.) {a + V(2a2-2ax)-V(2ax-x2)}/{a-x4-v'(a2-x2)}, x = a. (Euler,
Biff. Gale.)
X - ^{x^ - j/2), when x = oo , i/ = oo , but j/'/x finite = 22).
Sx" (y - 2)/n (y - z), x = y=z.
Sx'"(?/"-z")/SxP(j/«-2«), x=y = z = a.
nx»-V(x" - a") - l/(x - a), x = <j.
(15.
(16.
(17.
(18.
(19.
(21.
(23.
(25.
(27.
(29.
(31.
(32.
(33.
(34.
(36.
(36.
(87.
(1 + 1/x'y, X=QO.
(1 + 1/x)^ x = 0.
xV(^-i)', a; = l.
a^"lx, X = 00 ,
(log x/x)V*, X = CO .
(20.) xV*. x=ao.
(22.) x2^/(l+x2)^, x = ao..
(24.) (l + l/x)< x = (».
(26.) .tV(x«-i), a;=l.
(28.) (logx)V^, x = oo.
(30.) log^x/log^x, x = a).
a*/(x), x=cc, where /(x) is a rational function of x, and a a
constant.
(ax" + 6x"-J+ . . .)'/*, x = oo, (Cauchy.)
3.1/(1+2 log x)^ a;=0.
{(x2 + x + l)/(x2-x + l)}«, x=x..
{i(a'^+&'=)}'/^, x=0.
{l + 2/V(x2 + l)}\^(»«*+i), x = oo. (Longchamps.)
/an + a-,x+ , . , +a_x''\Ao-("A,a;
{^^h^^TTTTb^) ' ' = ^- (M-^th. Trip., 1886.)
§41
EXERCISES VII
111
(38.
(39.
(40.
(41.
(42.
(44
(45
(47.
(49.
(50.
(52.
(54.
(56.
(57.
(58.
(60.
(61.
(62.
(64.
{!/(«*- 1)}V*, x = aa.
{log(l + a;)}'»«(i+A a; = 0.
log(l + aa;)/log(l + 6a;), x = 0.
(e«_e-x)/log(l + x), x = 0. (Euler, D?/. Cai!c.)
(^7r-a:)tanaj, a; = ^. (43.) tan-ix/a;, a; = 0.
(l-sina; + cos3;)/(sina; + cosx-l), x = \ir. (Euler, Diff. Calc.)
sina;/(l-a^/7r2), x-tt. (46.) x {cos (ajx) -1}, x=<x>.
(amx-8ma)l{x-a), x = a. (48.) sec a; -tan a;, x-\ir.
(sin* x - tan* x)j{l + cos x){l- cos x)^, a; = 0.
sinhx/x, a;=0. (51.) (cosh x - l)/x2, x=0.
tanh-'x/x, x=0. (53.) 8inx/log(l + x), x = 0.
sin X log X, x=0. (55.) cos x log tan x, x = ^w.
log tan 77ix/log tan Tix, x = 0.
(log sin mx - log x)/(log sin nx - log x), x = 0.
sinx^'"*, x=0. (59.) sinxt*"^, x = 0.
(sinhx)'*""', x = 0.
{(x/a)sin (a/x)}=^"'(?ra<2), x = oo.
(cosmx)"/^, x = 0. (63.) (cos mx) <=<>'««''«, x = 0.
(2 - xja) **" '^'^/2<», x-a.
(65.) logJlogeX)/cos
2x'
(66.) Show that sin x cot (a/x) log (1 + tan (a/x)) has no determinate limit
when X = 00 .
(67.) If l^x stand for log„(log„x), l^H for log„(log„(log„x)), &c., show
that L [1 - {;aPx/Z„P(x + l)}'"]xi„xZ„2x . . . /„J'x = m(\e)P (Schlomilch,
Z=oo
Algebraische Analysis, chap, n.)
(68.) Show that I, 2 (a + s)^/"/n = l.
71=00 g=l
(69.) Show that L S { (a + 8)/7i}" lies between e« and e«+i.
n=oo 8=1
g—n
(70.) Show that L S {(a + sc/n)/(a + c)}"isfiniteif a + cbenumerically
n=oo g=l
s=n
greater than a, and that L S {(a + sc/n) /a}" is finite if a + c be numerically
less than a. ™=°° «=i
(71.) Trace the graph of ?/ = (a^- l)/x, when a>l, and when a<l.
(72.) Trace the graph of y==x^l' for positive values of x; and find the
direction in which the graph approaches the origin.
* For the definition and elementary properties of the hyperbolic functions
cosh X, sinh x, tanh x, &c., see chap. xxix. All that is really wanted here is
cosh X = i (e=>= + e-'') , sinh x = ^ (e^^ - e-»=).
112 EXERCISES VII CH, XXV
(73.) Trace the graph of y = {l + llx)'; and find the angle at which it
crosses the axis of y.
(74.) Find the orders of the zero and infinity vaUies of y when determined
as a function of x by the following equations* : —
(a) X (rc2 -ayf~y'^ = 0. (Frost's Curve Tracing, § 155, Ex. 3.)
(/3) x2j/« + ah/ - x^y^ + axhj - aV = 0. (76. , Ex. 7.)
(7) (x-l)?/ + (a;2-l)2/2_(a;_2)2j/ + a;(.T-2)=0.
(75.) If u and v be functions of the integral variable n determined by the
equations m„=m„_^ + v„_i, v„ = m„_i , show that L w„/r„=(l±v'5)/2. How
ought the ambiguous sign to be settled when «„ and t/j are both positive ?
(76.) Show that
(77.) Show that L f ■^" f V> " ' ' ""'^"'I'^^l.
^ ' n=oo I 1 . 2 , . . n J
(78.) I/log(l-x) loga;=0, when a;=0.
For a general method for dealing with such problems, see chap. xxx.
CHAPTER XXVI.
Convergence of Infinite Series and of Infinite
Products,
§ 1.] The notion of the repetition of an algebraical operation
upon a series of operands formed according to a given law
presents two fundamental difficulties when the frequency of the
repetition may exceed any number, however great, or, as it is
shortly expressed, become infinite. Since the mind cannot over-
look the totality of an infinite series of operations, some defi-
nition must be given of what is to be understood as the result of
such a series of operations ; and there also arises the further
question whether the series of operations, even when its meaning
is defined, can, consistently with its definition, be subjected to
the laws of algebra, which are in the first instance laid down for
chains of operations wherein the number of links is finite. That
the two difficulties thus raised are not imaginary the student
will presently see, by studying actual instances in the theory of
sums and products involving an infinite number of summands
and multiplicands.
§ 2.] One very simple case of an infinite series, namely, a
geometric series, has already been discussed in chap, xx., § 15.
The fact that the geometric series can be summed considerably
simplifies the first of the two difficulties just mentioned*; never-
theless the leading features of the problem of infinite series are
all present in the geometric series ; and it will be found that
most questions regarding the convergence of infinite series are
ultimately referred to this standard case.
* The second was not considered.
c. II. 8
114 CONVERGENCY, DIVERGENCY, OSCILLATION CH. XXVI
The consideration of the infinite geometric series suggests
the following definitions.
Consider a succession of finite real summands Ui, U2, th, • • •>
Un, . . ., unlimited in number, formed according to a given law,
so that the nth term Un is a finite one-valued function of n ; and
consider the successive sums
When n is increased more and more, one of three things must
happen : —
1st. Sn may approach a fixed finite quantity S in suck a way
that by increasing n sufficiently we can make Sn differ from S by as
little as we please; that is, in the notation of last chapter, L Sn = S.
In this case the series
«i + W2 + W3 + . . . + M„ + . . .
is said to be convergent, and to converge to the value S, which is
spoken of as the sum to infinity.
Example. I + 0 + T+ • • • +n;i+ • • • HereS= L -S'„=2.
2nd. Sn may increase with n in such a way that by increasing
n sufficiently we can make th£ numerical value of S^ exceed any
quantity, however large ; that is, L Sn = ±^ . In this ca>se the
series is said to be divergent.
Example. 1 + 2 + 3+ . . • Here L S„=oo.
3rd. Sn may neither become infinite nor approach a definite
limit, but oscillate between a number of finite values the selection
among which is determined by the integral character of n, that is,
by such considerations as whether n is odd or even; ofthefwm 3m,
Sm + 1, 3m + 2, o&c. In this case the series is said to oscillate.
N.B. If all the terms of the series have the same sign, then Sn
contintially increases {or at least neve)' decreases) in numerical value
as n increases : and the series cannot oscillate.
Example. 3-1-2 + 3-1-2 + 3-1-2+ . . . Here L S„ = 0, 3, or2,
according as n is of the form 'dm, 3;n + 1, or 3;n + 2. »=*
§§ 2, 3 CEITERION FOR CONVERGENCY 115
In cases 2 and 3 the series
U1 + U.2 + U3+ . . . + tin + . . .
is also said to be non-convergent*. In many important senses
non-convergent series cannot be said to have a sum ; and it is
obvious that infinite series of this description cannot, except in
special cases, and under special precautions, be employed in
mathematical reasoning.
Series are said to be more or less rapidly convergent according
as the number of terms which it is necessary to take in order to
get a given degree of approximation to the sum is smaller or
larger. Thus a geometric series is more rapidly convergent the
smaller its common ratio. Rapid convergency is obviously a
valuable quality in a series from the arithmetical point of view.
It should be carefully noticed that the definition of the con-
vergency of the series
U1 + U2 + U3+ . . . + Un + . . .
involves the supposition that the terms are taken successively in
a given order. In other words, the sum to infinity of a con-
vergent series may be, so far as the definition is concerned,
dependent upon the order in which the terms are written. As a
matter of fact there is a class of series which may converge to one
value, or to any other, or even become divergent, according to the
order in which the terms are written.
§ 3.] Two essential conditions are involved in the definition
of a convergent series — 1st, that Sn shall not become infinite
for any value of n, however great ; 2nd, that, as n increases,
there shall be continual approach to a definite limit S. If we
introduce the symbol m^n to denote Un+i + Un+2+ • ■ - +w«+m,
that is, the sum of m terms following the wth, following Cauchy
we may state the following criterion : —
The necessary and sufficient condition for the convergence of a
series of real terms is that, by taking n sufficiently great, it he
possible to make the absolute value of mRn «« small as we please, tig
matter what the value of m may he.
* Some writers use divergent as equivalent to non-convergent. On the
whole, especially in elementary exposition, this practice is inconvenient.
8 9
116 CRITERION FOR CONVERGENCY CIT. XXVI
This condition may be amplified into the following form.
Given in advance any positive quantity f, however small, it must
be possible to assign an integer v such that for n = v and all
greater values |m-^^n|<c : or it may be contracted into the form
J^ml^n = 0 when n= zc , for all values of m.
The condition is necessary; for, by the definition of con-
vergency, we have L Sn = S, where ;S^ is a finite definite quantity;
n=oo
therefore also, whatever m, L Sn+m = S. Hence
n=oo
-^ {^n+m ~ Sn) = S— S=Q '.
n=w
that is, L mP'n - 0.
Also the condition is sufficient : for, if we assign any positive
quantity c, it is possible to find a finite integer v such that, when
w <|; V, I mRn 1 < f J that is I Sn+m - Sn\<(. In particular, therefore,
|/S^v+m->S',, |<€. Since S^,, being the sum of a finite number of
finite terms, is finite, and m may have any value we please, it
follows that for no value of n exceeding v can Sn become infinite.
Hence L Sn cannot be infinite.
Also the limit of S^ cannot have one finite value when n has
any particular integral character, and another value when n has
a different integral character ; for any such result would involve
that for certain values of m L Sn and L Sn+m should have
n=oo n=«)
diff'erent values ; but this cannot be the case, since for all values
of m L {Sn+m — Sn)= L m^„ = 0*.
It should be noticed that, when all the terms of a series have
the same sign, there is no possibility of oscillation ; and the
condition that Sn be finite for all values of n however great
is sufficient. In case the subtlety of Cauchy's single criterion
should puzzle the beginner, he should notice that the proof which
shows that i/mjB„ = 0 can usually be readily modified so as to
show that LSn is not infinite. In fact some of our earlier
* A more rigorous demonstration of the above criterion is obtained
by applying the result of § 39, chap. xxv. to the sequence S^, S^, . . .,
8^, . . . We have given the above demonstration for the sake of readers
who have not mastered the Theory given in chap. xxv. , §§ 28—40.
§ 3 RESIDUE AND PARTIAL RESIDUE 117
demonstrations are purposely made redundant, by proving both
Lmlin = 0, and LSn not infinite.
Cor. 1. In any convergent series L w,i = 0.
n=co
For Un = ^n — ^n-i = iRn-i, and, by the criterion for con-
vergency, we must have L ii2„_i = 0. This condition, although
n=oo
necessary, is not of itself sufficient, as will presently appear in
many examples.
Cor. 2. If Bn= L mRn, cb^d S and Sn have the meanings
»n=oo
above assigned to them, then Sn = S-Iln.
For Sn+m = ^n + mRn, therefore L Sn+m = Sn+ L mRn\ and
L Sn+m = ^> hence the theorem.
Bn is usually called the residue of the series, and mUn a
partial residue.
Obviously, the smaller RnjSn is for a given value of n, the
more convergent is the series ; for R^ is the difference between
Sn and the limit of Sn when n is infinitely great.
Rn is, of course, the sum of the infinite series
^re+l + ^n+2 + Wn+3 +••'',
and it is an obvious remark that the residue of a convergent series
is itself a convergent series.
Cor. 3. The convergency or divergency of a series is not
affected by neglecting a finite number of its terms.
For the sum of a finite number of terms is finite and definite ;
and the neglect of that sum alters L Sn merely by a finite
n=oo
determinate quantity ; so that, if the series was originally con-
vergent, it will remain so ; if originally oscillating or divergent,
it will remain so.
Example 1. Consider the series 1/1 + 1/2 + 1/3+ . . . +1//1+ . . .
Here ^R„=l/(n + l) + !/(« + 2)+ . . . +l/(n + 7n),
>l/()f + m) + l/(n + ??i)+ . . . +1/(w + 7h),
>m/(H, + m),
>l/(?i/m + l).
Now, however great n may be, we can always choose in so much greater that
»t/m shall be less than any quantity, however small. Hence we cannot cause
„jR„ to vanish for all values of m by sufficiently increasing n. We therefore
conclude that the series is not convergent ; hence since all the terms are
118 EXAMPLES CH. XXVI
positive it must diverge, notwithstanding the fact that the terms ultimately
become infinitely small. We shall give below a direct proof that Z-S„=qo .
Example 2.
1, 22 1, 32 1, (n + l)2
jlog j-3 + 2log— ^+ . . . +-log^(^^.
Since (ri + l)-/;i(n + 2) = (l + l/n)/{l + l/(n + l)}, we have
JL_ l + l/(n + l) 1 l + l/(n + 2)
»" "~n + l ^^l + l/(n + 2J'*'7i + 2 ^^l + l/(u + 3)
1 . 1 + 1/(h + wi)
+ . . . +— — log '^ '-
n + m ** l + l/(n + m + l)'
n + 1 V°''l + l/(n + 2)^'°'^l + l/(n + 3)^ • ^ '°^ l + l/(M + m+l)r '
1 . l + l/(ra + l)
^n + l^^l + l/(n + m + l) - ^ ^•
Now, whatever m may be, by making n large enough we can make l/(n+ 1),
and, a fortiori, l/(n + m + l), as small as we please, therefore L ^R^—0 for
all values of m. "="
If in (1) we put 0 in place of n, and n in place of vi, and observe that
iS_=_Iio, we see that
r, , 1 + 1/1
. • ^""^°°l + l/(n + l)-
so that fif„ can never exceed log 2 whatever n may be.
Both conditions of convergency are therefore satisfied.
Putting 7)1 = 00 in (1), we find for the residue of the series
J?„<[log{l + l/(n + l)}]/(n + l);
a result which would enable us to estimate the rapidity of the convergency,
and to settle how many terms of the series we ought to take to get an
approximation to its limit accurate to a given place of decimals.
§ 4.] The following theorems follow at once from the
criterion for convergency given in last paragraph. Some of
them will be found very useful in discussing questions regarding
convergence. We shall use 2«,i as an abbreviation for Mj + u^
+ . . . +Mn+ • . ., that is, " the series whose nih. term is ?f„."
I. If Un and v„ be positive, Un<Vn for all values of n, and
2»„ convergent, then 2w„ is convergent.
If Un and Vn be positive, w„>'y„ for all values of n, and 2v„
divergent, then 2m„ i? divergent.
For, under the first set of conditions, the values of ^„ and
,„i?„ belonging to 2m„ are less than the values of the correspond-
ing functions S'n and ^R'n belonging to 2«j„. Hence we have
0<Sn<S'n, 0<mIin<mR'n- But, by hypothcsis, S\ is finite for
§§3,4 ELEMENTARY COMPARISON THEOREMS 119
all values of w, and L ^B'n - 0 ; hence 8n is finite for all values
of n, and L rn.Rn = 0 ; that is, 2% is convergent.
Under the second set of conditions, Sn>S'n. Hence,
since L S'n = qo , we must also have L JSn- <x) ; that is, 2?«„ is
divergent.
11. ^f, for all values of n, Vn> 0, and Un/vn is finite, then
%Un is convergent if Sv„ is convergent, and divergent if 5v„ is
divergent.
By chap, xxiv., § 5, if ^ be the least, and B the greatest of
the fractions, ?«„+i/v„+i, t«;i+2/??„+2, . . ., M„+mK+m, then
A Un+l + UnJr2 + • • . + Un+m n
1'n+l + Vn+i + . . . + Vn+m
Now, since ujvn is finite for all values of ;^, A and B are
finite. Hence we must have in all cases mBn = Cm-R'n, where C
is a finite quantity whatever values we assign to m and n.
Hence 8n (that is, nRo) will be finite or infinite according as
S'n is finite or infinite ; and if L ^R'n = 0, we must also
n=oo
have L mBn-^-
n=oo
HI. If iin and Vn he positive, and if, for all values of n,
Un+\/un < Vn+i/v„ , and '^Vn is Convergent, then %Un is convergent; and
if Un+i/un>Vn+i/vn, and 5v„ is divergent, then ^u^ is divergent.
We have, if Un+i/Un<Vn+i/Vn,
a f-, Wo th Uo 1
1 . r . . . r
Ul U2 Ui )
Vi, V2 Vi
<->S„. ^
Now, by hypothesis, Z/S",i is finite : hence LR^ must be finite.
Also, since all the terms of 2m„ are positive, the series cannot
oscillate, therefore %Un must be convergent.
In like manner, we can show that, if Un+ilun>Vn+ilvn, and
%Vn be divergent, then Swn is divergent.
N,B. — In Theorems I., II., III. we have, for simplicity,
stated that the conditions must hold for all values of n ; but
120 ABSOLUTE CONVERGENCE CH. XXVI
we see from § 3, Cor. 3, that it is sufficient if they hold for all
values of n exceeding a certain finite value r ; for all the tenns up
to the rth in both series may be nej,flected.
Also, when all the terms of a series have the same sign, we
suppose, for simplicity of statement, that they are all positive.
This, clearly, in no way affects the demonstration.
It is convenient to speak of m»+i/m„ as the Ratio of Con-
vergence of 2«<„. Thus we might express Theorem III. as
follows : — Any series is convergent (divergent) if its ratio of
convergence is always less (greater) than the ratio of convergence
of a convergent (divergent) series.
IV. If a series which contains negative terms be convergent
when all the negative terms have their signs changed, it will be
convergent as it stood o^'iginally.
For the effect of restoring the negative signs will be to
diminish the numerical value both of >S'„ and of mRn-
Definition. — A series which is convergent when all its terms are
taken positively is said to be absolutely convergent.
It will be seen immediately that there are series whose
convergency depends on the presence of negative signs, and
which become divergent when all the terms are taken positively.
Such series are said to be semi-convergent. In §§ 5 and 6, unless
the contrary is indicated, we suppose any series of real terms to
consist of positive terms only, and convergence to mean absolute
convergence.
SPECIAL TESTS OF CONVERGENCY FOR SERIES WHOSE TERMS
ARE ULTIMATELY ALL POSITIVE.
§ 5.] If we take for standard series a geometric progression,
say 2r", which will be convergent or divergent according as
r< or > 1, and apply § 4, Th. I., we see that 2m„ will be con-
vergent if, on and after a certain finite value of ti, u,^<7^,
where r<l ; divergent if, on and after a certain finite value of
n, Un>r^, where r>l. Hence
I. 2m„ is convergent or divergent according as Un"" is
ultimately less or greater than unity.
§§ 4, 5 GEOMETRIC STANDARD 121
This test settles nothing in the case where u^'^ is ultimately
unity, or where L Un''^ fluctuates between limits which include
n=oo
unity.
Example. 21/(1 + !/«)" is a convergent series ; for
L u„i/» = l/L(l + l//i)" = l/e,
by chap, xxv., § 13, where e > 2, and therefore 1/e < 1.
If, with the series 2r" for standard of comparison, we apply
§ 4, Th. III., we see that %Un is convergent or divergent according
as Un+i/un is, on and after a certain finite value of n, always < 1
or always > 1. Hence
11. ^Un is convergent or divergent according as its ratio of
convergency is ultimately < or >\.
Nothing is settled in the case where the ratio of convergency
is ultimately equal to 1, or where L tin+i/un fluctuates between
limits which include unity.
The examination of the ratio Un+i/un is the most useful of
all the tests of convergence*. It is sufficient for all the series
that occur in elementary mathematics, except in certain extreme
cases where these series are rarely used. In fact, this test, along
with the Condensation Test of § 6, will suffice for the reader
who is not concerned with more than the simpler applications of
infinite series.
Notwithstanding their outward difference, Tests I. and II. are
fundamentally the same when L Un+i/un is not indeterminate.
n=«)
This will be readily seen by recalling the theorem of Cauchy, given
in chap, xxv., § 14, which shows that L Un+i/un- L Un''^. It is
useful to have the two forms of test, because in certain cases I. is
more easily applied than II.
Example 1. To test the convergence of 2n''a;", where r and x are
constants. We have in this case
= (1 + 1/h)'"x.
Hence Lu„_^j/w„ = x. The series is therefore convergent if a: < 1, and divergent
if x>\.
* We here use (as is often convenient) " convergence" to mean " the quality
of the series as regards convergency or divergency."
122 EXAMPLES CH. XXVI
If x = 1, we cannot settle the question by means of the present test.
Example 2. If tp{n) be any algebraical function of n, 'S<f>{n)x^ is con-
vergent if a:<l, divergent if a;>l.
This hardly needs proof if L 0 (n) be finite. It L <p (n) be infinite, we
n—<D n=oo
know (see chap, xxx.) that we can always find a positive value of r, such
that L <p{7i)jn^ is finite, =A say. We therefore have
n— 00
^) I L^^l L^"^^)'"
=x{AIA}xl,
= x.
This very general theorem includes, among other important cases, the
integro-geometric series
</){l)x + (f>(2)x'+ . . . +0(n)a;"+ . . .
where ip (») is an integral function of n ; and the series
X x^ x"
1+2 + - ••+¥+••• W-
which, as we shall see in chap, xxvin., represents (when it is convergent)
-log(l-x). It follows, by § 4, Th. IV,, that, since the series (1) is con-
vergent when x<l, the series
is also convergent when .r < 1.
When (2) is convergent, it represents log (1 -f x).
Example 3. Sx^/n! (the Exponential Series) is convergent for all values
«»-«/«»= {•^"+Vf« + 1)!}/MH!},
=x/(n-f 1),
Hence, however great x may be, since it is independent of n, we may always
choose r so great that, for all values of n>r, .t/(«-i-1)<1. Since the limit
of the ratio of convergence is zero in this case, we should expect the con-
vergency for moderate values of x to be very rapid ; and this is so, as we
shall show by examining the residue in a later chapter. We have supposed
x to be positive ; if x be negative the series is convergent a fortiori ; the
convergence is in fact absolute, § 4, Th, IV.
Example 4. S(-)"m(m-l) . . . (m-n + l)^"/;/! (x positive), where m
has any real value*, is convergent if x< 1, divergent if x> I.
* If wt were a positive integer, the series would terminate, and the
qnestion of convergency would not arise.
5, 6 cauchy's condensation test 123
m-n
71+1 '
•n T I ^ m-n
For LUn^Ju,^= -xL
^ mln - 1
=x.
Hence the theorem.
The series just examined is the expansion of (1-x)'" when a;<l. It
follows, by § 4, Th. IV., that the series Sm(m-l) . . . (m-n + l)x'7«!.
whose terms are ultimately alternately positive and negative, is convergent
if a;<:l; this series is, as we shall see hereafter, the expansion of (1 + ar)"'
when x<l.
§ 6.] Cauchy's Condensation Test. — The general principle of
this method, upon which many of the more delicate tests of
convergence are founded, will be easily understood from the
following considerations : —
Let 2m„ be a series of positive terms which constantly
decrease in value from the first onwards. Without altering the
order of these, we may associate them in groups according to
some law. li Vi, v^, . . . Vm, . . . be the 1st, 2nd, . . . mih, ... of
these groups, the series ^v^ will contain all the terms of Sm^ ;
and it is obvious from the definition of convergency that 2«*„
is convergent or divergent according as Sv^ is convergent or
divergent ; we have in fact L ^Un- L 'Xvm,- It is clear that the
n=oo m=«>
convergency or divergency of 2v,„ will be more apparent than
that of 2m„, because in Sv^ we proceed by longer steps towards
the limit, the sum of n terms of 2v,„ being nearer the common
limit than the sum of % terms of 2m». Finally, if 2v'„ be a new
series such that 'y'n^'Wn. then obviously 2m„ is ,. , if %v'n
^ J ™ divergent
. convergent
divergent
We shall first apply this process of reasoning to the following
case : —
Example. The series 1/1 + 1/2+ . . . +l/7i+ ... is divergent.
Arrange the given series in groups, the initial terms in which are of the
following orders, 1, 2, 22,... 2"', 2"'+i, . . . The numbers of terms in the
successive groups will be 2 - 1, 2^ - 2, 2» - 2*, . . . 2'»+i - 2"*, 2'»+2 - 2'»+i, . . .
respectively. Since the terms constantly decrease in value, if 2"*+^ be the
greatest power of 2 which does not exceed w, then
124 CAUCHY'S condensation test CH. XXVI
1/1 iv /i 1 1 l\ /_L _L 1 \
"^l"*'V2''"3J ■^' V22"^5"^6'^7/'*' * ' * ■^V2"»'^2»» + l'*'* * '■^2'»+'-iy'
,11 1
Hence, by making n sufficiently great, we can make S„ as large as we please.
The series 1/1 + 1/2 + 1/3 + . . . is therefore divergent. This might also be
deduced from the inequality (6) of chap, xxv., § 25.
Cauchy's Condensation Test, of which the example just
discussed is a particular case, is as follows : —
Xf f{n) he positive for all values of n, and constantly/ decrease
as n increases, then "^fin) is convergent or divergent according
as '^a^f{a^) is convergent or divergent, where a is any positive
integer <j: 2.
The series 2/(w) may be arranged as follows : —
+ {/(a^)+/(«^ + l)+. . .+/(»=> -1)}
+ {/K) +/(«'" +1)+. . . +/(a"'+^-l)}
Hence, neglecting the finite number of terms in the square
brackets, we see that '^{n) is convergent or divergent accord-
ing as
2 {f{ar) +f{a^ + 1) + . . . +/(a™+^ - 1)} (1)
is convergent or divergent. Now, since f{a'")>f(a^ + 1)>. . .
>f{a"'+' - l)>/(a'"+^), we have
(a'»+^-a"')/(a'") >/(«'") +/(»'" + 1) + . . . +/(a'"+^- 1)
that is,
(a - 1) «•»/(«'") >/(«"•) +/(«"' + 1) + . . . +/(a"'+' - 1)
> {(a - l)/a} a'»+y(a'»+^).
Hence, by § 4, Th. I., the series (1) is convergent if 2 (a- 1)
a'"/(a'") is convergent, divergent if 2 {(a- l)/a}a'"+7'(a'"+*) is
§6 CRITERIA OF DE MORGAN AND BERTRAND 125
divergent. Now, by § 4, Th. II., 2 (a- l)«™/(c*'") is convergent
if :Sa"'/(a"0 is convergent, and % {{a - !)/«} a'"+y(a"'+0 is
divergent if 2«'""^V'('^*"'^') is divergent ; and for our present
purpose %a'"J\d"') and 2«'"+V"(a'"+^) are practically the same
series, say 'Za'^fia^). Hence Caucliy's Theorem is established.
N.B. — It is obviously sufficient that the /miction f{n) be
positive and constantly decrease for all values of n greater than
a certain finite value r.
Cor. 1. The theorem will still Iwld if a have any positive
value not less than 2*.
Let a lie between the positive integers b and b + \, (6 <|; 2).
If 2ay(a") be convergent, then L ay(a") = 0, that is, L ccf{x) = Q.
n=oo a;=oo
Hence, on and after some finite value of x, the function xf{x) will
begin to decrease constantly t as i» increases. We m\ist therefore
have (6 + 1)V'{(6 + !)"}< «"/(«")> on and after some finite value
of w. If, therefore, %a^f{a^) is convergent, a fortiori, will 2(6 + 1)"
f{{b + 1)"} be convergent, and therefore, by Cauchy's Theorem,
2/(w) will be convergent.
If %a"f{a"') be divergent, xf(x) 1° may, or 2° may not decrease
as X increases.
In case 1°, 6"/ (6") > a"'f(a"'). Hence the divergence of 5a"/(a")
involves the divergence of '^b'f{b^'') ; and the divergence of %f{n)
follows by the main theorem.
In case 2°, the divergence of ^f(n) is at once obvious ; for,
if L xf{x)^0, then ultimately xf{x)>A, where A>0. Hence
x=oo
f{x)>A/x. Now %A/n is divergent, since 2l/w is divergent;
therefore 2/(w) is divergent.
In what follows we shall use ex, ^x, ... to denote a*,
a"*, . . ., a being any positive quantity <j; 2 ; and \x, ^?x, . . .
Ix, Px,... to denote logaX, hgai^ogax), . . . loge^, loge(logeir), . . .,
where e is Napier's Base.
• Also if l<a<:2, see Kohn, Grunert's Archiv, Bd. 67 (1882) and Hill,
Mess. Math., N. S., 307 (1896).
t This assumes that xf{x) has not an infinite number of turning values;
so that we can take x so great that we are past the last turning value, which
must be a maximum.
126 CRITERIA OF DE MORGAN AND BERTRAND CH. XXVI
Cor. 2. %f{n) is convergent or divergent according as
^fne^n . . . €^n/{e^n) is convergent or divergent.
This follows, for integral values of the base a, by repeated
application of Cauchy's Condensation Test ; and, for non-integral
values of a, by repeated applications of Cor. 1. Thus ^/(n) is
convergent or divergent according as ^mf{m) is convergent or
divergent. Again, '^mf{m) is convergent or divergent according
as 2€Wc(ew)/{€(£»)}, that is %m€-n/(€^n), is convergent or divergent;
and so on.
Cor. 3. 2/(w) is convergent or divergent according as the first
of the functions
T, = \f{x)lx,
T,^H^f{x)}lXx,
T^ = \{x\xf{x)]lX\v,
Tr==\{x\xX^X . . .\'--^xf{x)}lyX,
which does not vanish when x=co, has a negative or a positive limit.
By Cor. 2, ^f(n) is convergent or divergent according as
^ene^n . . . €^nf(e^n) is convergent or divergent.
Now the latter series is (by § 5, Th. I.) convergent or
divergent according as
L {m^n . . . e'vf(e'-n)Y"'<OT>l ;
n=oo
that is, according as
Lhgaiene'n. . . e'-nf(e-n)Y"'<>0;
n=oo
that is, L loga{ew€^» . . . €''nf{e^n)}ln<>0.
n=oo
If we put X = i^n, so that \x = ("'hi, \^x = f^~hi, . . .
X^~^x = en, X'".2; = w, and x-cc when n-<x>, the condition for
convergency or divergency becomes
L X{xXxX''x . . . X^-'xf{x)}/X^x<>0 (1).
2=00
If, on the strength of Cor. 1, we take e for the exponential
base, the condition may be written
L l{xlxPx . . . l''-'xf(x)}/l'-x<>0 (2),
where all the logarithms involved are Napierian logarithms.
§ 6 DE morgan's logarithmic scale 127
We could establish the criterion (2) without the intervention
of Cor. 1 by first establishing (1) for integral values of a,
and then using the theorem of chap, xxv., § 12, Example 4,
that L >TxlVx = l//a.
a;=oo
Cor. 4. Each of the series
2l/'«^+» (1),
21/w{/w}^+<' (2),
%l/nln{Pny+'' (3),
^l/nlnPn . . . T-^w {/'•wp+» (^' + 1),
is convergent if a>0, and divergent i/a = or<0.
As the function nlnPn . . . l^'n frequently occurs in what
follows, we shall denote it by Pr (n) ; so that Po (n) = n, Pi (n) =
nln, &c.
1st Proof. — Apply the criterion that 2/(w) is convergent or
divergent according as Ll{Pr{x)f{x)}IV''^^x<>Q. In the pre-
sent case, fix) = IjPr (x) {l^'x)"-. Hence
/ {Pr {x)f{x)]IV+'x = I{l/(1'-Xy}/1'-+'X,
— — a.
It follows that (r+l) is convergent if a>0, and divergent
if a<0. If a = 0, the question is not decided. In this case,
we must use the test function one order higher, namely,
I {Pr^i {x)f{x)}ll'+''x. Since f(x) = 1/Pr (x), we have
/ {Pr+i {x)f{x)W-''x = / {1-+'X]IV^'X,
= 1>0.
Hence, when a = 0, (r+l) is divergent.
'2nd Proof. — By the direct application of Cauchy's Condensa-
tion Test, the convergence of (1) is the same as the convergence
of '2,0^ 1(0^)^+", that is, 2 (l/a*)™. Now the last series is a geo-
metrical progression whose common ratio is l/«" ; it is therefore
convergent if a>0, and divergent if a= or <0. Hence (1) is
convergent if a>0, and divergent if a= or <0.
Again, the convergence of (2) is by Cauchy's rule the same
as the convergence of 2aVa'M/«'T+", that is, 2l/(/ay+«w^+» ;
128 I>E morgan's logarithmic scale CH. XXVI
and the convergence of this last the same as that of %l/n''\
Hence our theorem is proved for (2).
Let us now assume that the theorem holds up to the senes
(r) We can then show that it holds for (r+ 1). In fact, the
convergence of (r+ 1) is the same as that of 2aV«;/«"/V • ;;
Z'-«"UVr",thatis,2l/0i/a);(«?a) . . . l^-^nlaW'' {nla)^ .
First suppose a>0, and a >^. Then la>l, nla>n. Hence
ll{nla)l{nla) . . . I''-' (nla) {l^-Hnla)}''"
<\lnln . . . V-'n{V-'nY^''.
But, since a>0, 2l/P.-i(«) [I'-'nY is convergent, a fortiori,
21/Pr {n) {VnY is convergent.
Next suppose al^O, and 2<a<e. Then nla<n; and, pro-
ceeding as before, we prove 21/P,(«) {r»}* more divergent than
the divergent series %\IPr-M{l^-^n\\
Logarithmic Scale of Convergency.-lh^ series just discussed
are of great importance, inasmuch as they form a scale with
which we can compare series whose ratio of convergence is
ultimately unity. The scale is a descending one ; for the least
convergent of the convergent series of the rth order is more
convergent than the most convergent of the convergent series of
the (r+l)th order. This will be seen by comparing the «th
terms, w. and u\, of the rth and (r + l)th series. We have
ii\lii^ = {l^-^nYl{VnY^'^\ where a is very small but >0, and
a' is very large. _
If we put x = V-''n, we may write ^2/^ tt „/«„ = ^x/^ i^ ^ 7
IxY^"'. Hence, however small a, so long as it is greater than 0,
and however large a', Luju^ = qo .
If we suppose the character of the logarithmic scale estab-
lished by means of the second demonstration given above, we
may, by comparing 2?<„ with the various series in the scale, and
using § 4, Th. I., obtain a fresh demonstration of the criterion
of Cor. 3. We leave the details as an exercise for the student.
This is perhaps the best demonstration, because, apart from the
criterion itself, nothing is presupposed regarding /(.r), except
that it is positive when x is greater than a certain finite value.
§ 6 DE MORGAN AND BERTRAND'S SECOND CRITERION 129
By following the same course, and using § 4, Th. III., we
can establish a new criterion for series whose ratio of con-
vergence is ultimately unity, as follows, where Px=f{^+ l)//(^)-
Cor. 5. If f{x) be always positive when x exceeds a certain
finite value, '%f(^i) is convergent or divergent according as the first
of tJie following functions —
'''o = Pa - 1 ;
T-t_ = Po{x+\)px-Po{x)\
T^ = Pi{x+l)p^-Pi(x);
Tr = Pr-i(x+l)px~Pr-i{x);
which does not vanish when x= cc has a negative or a positive limit.
Comparing 2/(w) with '^llPr{n){VnY, we see that 2/(w)
will be convergent if, for all values of x greater than a certain
finite value,
p<,<Pr (X) {VxflPr {X + 1) {V {X + l)}» (l),
where a>0.
Now (1) is equivalent to
Pr {X +l)p^- Pr {X) < Pr {x) [{/'>//'• (^ + l)}» - l].
Also LPr {x) [{/'•^/r (^ + 1 )}'' - 1]
=-£i'._.(.)ir(..i)-r.).^^. j^:-;^:|::;>f:;,
= — lxlxa = — a,
by chap, xxv., §§ 12 and 13.
Hence a sufficient condition for the convergency of 2/(w) is
L [Pr (x +l)px- Pr (^)} < - a (a positive),
<0.
In like manner, the condition for divergency is shown to be
L {Pr (x + l)pa. -Pr{x)}>-a (a negative),
a=oo
>0.
Example 1. Discuss the convergence of 2e~i~'/2-' • •-V»/n'*.
Here , T, = l{f{n)}ln,
_ 1 + 1/2+ . . . + lln + rln
n
Now, by chap, xxv., § 13, Example 1,
l + (r + l)ZM>l + l/2+. . . + l//i + r!n>rZ/i + Z(n + l).
c. II. 9
130 EXAMPLES CH. XXVI
Hence Lru = 0. We must therefore examine Tj , Now
Ti=l{7if{n)}lln,
= -{1 + 1/2+. . .+lln + (r-l)ln}lln,
= -{1 + 1/2+. . . + l/«}/Jn-(r-l).
By chap, xxv., § 13, Example 2, 1,(1 + 1/2+. . , + l/w)/Zn = l. Hence
LTj= -l-r + l= -r. The given series is therefore convergent or divergent
according as r> or <0.
If r=0, LTo=0, and LT^^O. But we have
2\=l{Hlnf{n)}jl-^n,
= l-{l + l/2+. . .+lln-ln}ll'n.
Now, when n is very large, the value of 1 + 1/2 + . . . + 1/h - In approaches
Euler's Constant. Hence I,2'2=l>0. In this case, therefore, the series
under discussion is divergent.
Example 2. To discuss the convergence of the hypergeometric series,
a^ a(a + l).p (/3 + 1) „
^+7-5 7(7+l)-'5(5 + l) * •
The general term of this series is
_ a(a+l) . . . (a + n-l).j3(^ + l) . . . (^ + n-l)
•'^^"7(7 + 1) . . . (7 + n-l). 5(5 + 1) . . . (5 + 71-1) '
The form of / (n) renders the apislication of the first form of criterion
somewhat troublesome. We shall therefore use the second. We have
(a + u)(/3+?t)
''»~(7 + «)(5 + ")
(a + n)(/3 + ?i)
.r-l,
" (y + 7i)(d + ny
Ltq = x-1.
Hence the series is convergent if a;<l, divergent if a;>l.
If a; = l, Ltq = 0, and we have
{7i + l)(a + n)(^ + n)
^1 {y + n)(5+Ji)
- {a + P-y-S + l)n^ + An+B ^
^n^ + Cn + D '
LTi = a + (3-7-5 + l.
If^ therefore, x = l, the hypergeometric series is convergent or divergent
according asa + /3-7-5 + l< or >0.
If o + /3-7-5 + l = 0, Lri = 0. But we have
To={n + l)l(n + 1) ) , {,. ( - nln,
= [n{l(n + l) -bi} + (a + ^ + 1) \l(n + l)-hi} + {Al{n + 1) + Bln}[n
+ CI (« + l)/7i2]/[l + Ejn + Fjifl].
Hence, since Ln{l{n + l)-ln}==l, L {l(n + l)-ln}=0, Ll{n + l)ln>=0,
Llnln'=0 («>0), (fee, we have
Lr2=l>0.
In this case, therefore, the series is divergent.
§ 6 HYPERGEOMETRIC AND BINOMIAL SERIES 131
Example 3, Consider the series
m . m{m-l) , ,^„)»()»-l) . . . (m-n + l) ,
^"r+ 1.2 +---+(-A) 1 . 2 . . . 7t ^•••
This may be written
^ --m (-"')(-?» + !) 4. _ ^ (-n,)(-m + l) . . . (-m + n-1) _^ ^ ^ ^
1 1.2 • • • • 1 . 2 . . . n
It is therefore a hypergeometric series, in which a=-7n, fi=y, 5 = 1,
x = l. It follows from last article that the series in question is convergent or
divergent according as -m<>0, that is, according as m is positive or
negative.
This series is the expansion of (1 - x)^, when x = l.
Example 4. Consider the series
^ + 1'*— 172^+' • •+ 1.2 . . .n "^- • • ^^'-
In this series the terms are ultimately alternatively positive and negative
in sign. Hence the rules we have been using are not directly applicable.
Ist. Let III be positive ; and let m - r be the first negative quantity among
m, m-1, m-2, . . . &c., then, neglecting all the terms of the series before
the (j' + l)tb, we have to consider
m(m-l) . . . (m-r + 1) f m-r {m-r)(m-r-l) \
1.2 . . . r |^"*",. + l"^ (r + l)(r + 2) + • • • [ ^^^
If we change the signs of the alternate terms of the series within brackets,
it becomes
, , r-m , (r-m)(r-m + l) ,
^■^Tn"*" (r + l)(r + 2) +••• (^^
Now (3) is a hypergeometric series, in which a = r-m, P = y, 5 = r+l,
x = l. Hence a+(3-7-5 + l = r-7H-(r+l) + l= -?H<0. Therefore (3) is
convergent. Hence (2), and therefore (1), is absolutely convergent.
2nd. Let m be negative, = - /j. say. The series (1) then becomes
M /iOi+l)_ , ,x(n+l) . . . (m + »-1)
1^ 1.2 •••-^V ; 1.2... ^^ " "^ ■ ' ■ ^^•
Since /* is positive, the hypergeometric series
M m(m + 1) m(m+1) ■ ■ ■ (M + n-1) ,5>
^1^ 1.2 ^^ ^ 1.2 ... 7j ^ ' ' ' ^ ''
is divergent.
Hence (4) cannot be absolutely convergent in the present case.
Since /3„= - (/* + ?«)/(« + 1), the terms will constantly increase in numerical
value if M > 1. Hence the series cannot be even semi-convergent unless ix<\.
If fjL be less than 1, />„<!, and the series will be semi-convergent provided
l-u„=0.
Now logM„ = 2:log^ = 2:log jl-f-'^l.
Since I,log{l-f (/^-l)/(/n-l)}/{(|x-l)/(w-f-l)} = l (see chap, xxv., § 13),
the series 21og {l-t-(^- l)/(7t-f-l)} and S (/x-l)/(n-f 1) both diverge to an
infinity of the same sign. But the latter series diverges to - oo or -foo,
according as /i< or >1. Hence iM„=0 or oo , according as yu< or >1.
9—2
132 HISTORICAL NOTE CH. XXVI
Hence the series (1) is divergent if /*>!, semi-convergent it fjL<l,
It obviously oscillates if /x = l. Hence, to sum up, the series (1)
is absolutely convergent, if 0 ^ m < + oo ;
semi-convergent, if -l<m<0;
oscillating, if ~l = m;
divergent, if - oo <j;i< - 1*.
SERIES WHOSE TERMS HAVE PERIODICALLY RECURRING NEGATIVE
SIGNS, OR CONTAIN A PERIODIC FACTOR SUCH AS SI^ llO.
§ 7.] Series wliich contain an infinite number of negative
terms may or may not be absolutely convergent. The former
class falls under the cases already discussed. We propose now
to give a few theorems regarding the latter class of series, whose
convergency depends on the distribution of negative signs
throughout the series.
The only cases of much practical importance are those — 1st,
where the infinity of negative signs has a periodic arrangement ;
* Historical Note. — If we except a number of scattered theorems, given
chiefly by Waring in his Meditationes Anahjticce, and Gauss in his great
memoir on the Hypergeometric Series, it may be said that Cauchy was the
founder of the modern theory of convergent series ; and most of the general
principles of the subject were given in his Resumes Analytiques and in
Analyse Algihrique. In his Exercices de Mathematiques, t. ii. (1827), he gave
the following integral criterion from which most of the higher criteria have
sprung : — If, for large values of n,f(n) be positive and decrease as n increases,
fm+n
then S/(n) is convergent if L j dxf{x)=0 (m arbitrary), otherwise divergent.
The second step of the r-criteria was first given by Eaabe, Crelle's Jour.,
Bd. XIII. (1835). De Morgan, in his Differential Calculus, p. 323 et seq. (1839),
first gave the Logarithmic Scale of Functional Dimension, established the
Logarithmic Scale of Convergency of Cor. 4, and stated criteria equivalent
to, but not identical in form with, those of Cor. 3 and Cor. 5. Continental
writers, nevertheless, almost invariably attribute the whole theory to Bertrand.
Bertrand, Liouv. Jour. (1842), quotes De Morgan, stating that he had obtained
independently part of De Morgan's results. His Memoir is very important,
because it contains a discussion of various forms of the criteria and a demonstra-
lion of their equivalence; we have therefore attached his name, along with De
Morgan's, to the two logarithmic criteria. Bonnet, Liouv. Jour. (1843), gave
elementary demonstrations of Bertrand's formulee ; and Malmsten, Grunert's
Archiv (1816) , gave an elegant elementary demonstration, depending essentially
§§ G, 7 SEMI-CONVERGENT SERIES 133
2nd, where the occurrence of negative signs is caused by the
presence in the nth. term of a factor, such as sin 7iO, which is a
periodic function of n.
In the former case (which might be regarded as a particular
instance of the latter) we can always associate into a single term
every succession of positive terms and every succession of negative
terms. Since the recurrence of the positive and negative terms
is periodic, we thus reduce all such series to the simpler case,
where the terms are alternately positive and negative.
We may carry the process of grouping a step farther, and
associate each negative with a preceding or following positive
term, and the result will in general be a series whose terms are
ultimately either all positive or all negative.
The process last indicated often enables us to settle the con-
vergence of the series, but it must be remembered that the series
derived by groiiping is really a different series from the original
one, because the sum of n terms of the original series does not
always correspond to the sum of m terms of the derived series.
The difference between the two sums will, however, never exceed
on the inequality of chap, xxv., § 13, Cor. 6, that Sl/P^(m + n) {^(m + ??)}"
(where Fm is positive) is convergent or divergent, according as a < or <t 0 ; and
thence deduces Cor. 3. Paucker, Crelle's Jour., Bd. xlii. (1851), deduces both
Cor. 3 and Cor. 5 from Cauchy's Condensation Test, much as we have done,
except that the actual form in which we have stated the rule of Cor. 5 is
taken from Catalan, Traite El. d. Series (1860). Du Bois-Reymond, Crelle's
Jour., Bd. Lxxvi. (1873), gives an elegant general theory embracing all the
above criteria, and also those of Kummer, Crelle's Jour., xiii. (1835). Abel
had shown that, however slightly divergent 2!(„ may be, it is always possible
to find 7i, 72) • • •> 7n> • • • ^^^^ ^^^^ J^7n = 0 and yet S7„?(,j shall be
divergent. Du Bois-Reymond shows that, however slowly 2»„ converge, we
can always find 7i , 72 , . . .,7n> • • . such that L7,j=qo and 27„m„ neverthe-
less shall be convergent. He shows that functions can be conceived whose
ultimate increase to infinity is slower than that of any step in the logarithmic
scale ; and concludes definitely that there is a domain of convergency on
whose borders the logarithmic criteria entirely fail — a point left doubtful by
his predecessors. Finally, Kohn, GrunerVs Archiv (1882), continuing Du Bois-
Eeymond's researches, gave a new criterion of a mixed character; and
Pringsheira (Math. Ann. 1890, 1891) has discussed the whole theory from a
general point of view. The whole matter, although not of great importance
as regards the ordinary applications of mathematics, illustrates an exceedingly
interesting phase in the development of mathematical thought.
134 EXAMPLE OF SEMI-CONVERGENT SERIES CH. XXVI
the sum of a finite number of terms of tlie original series ; and
this difference must vanish for w = oo ,.if the terms of the original
series ultimately become infinitely small.
Example. Consider the series
1 2 •6'^ i 5 6"^' ■ ■"''3/1-2 3h-1 a» "^ ' * ' ^ ''
Compare this with the series
i-G + l) + ^G4)+---+3-;^-(B;r:i + i) + - ••(-)'
that is, the series whose {2n- l)th term is l/(3«-2), and whose (27t)th term
is -(l/(3«-l) + l/3«).
If S^ S„' denote the sums of n terms of (1) and (2) respectively, then
Ssn-^=S,„'_„ S3n-i = ^2n'-i-l/(3't-l), 5^3.=^V- Since Ll/(3n - 1) = 0, we
have in all cases I,S„=L/S„'. Hence (1) is convergent or divergent according
as (2) is convergent or divergent. That (1) is really divergent may be shown
by comparing it with the series
2 {1/(3h - 2) - l/(3u - 1) - l/3«} (3).
If .9,," denote the sum of n terms of this last series, we can show as before
that LSn" — LSn- But the Jith term of (3) can be written in the form
( - 9 + 12/h - 2//t"'^)/(3 - 2/m) (3 - Ijn) 3n ; and therefore bears to the ?ith term of
Sl/n a ratio which is never infinite. But Sl/u is divergent.
By § 4, II., (3) is therefore also divergent. Hence (1) is divergent.
It should be noticed that in the case of an oscillating series,
where Liin + 0, the grouping of terms may convert a non-convergent
into a convergent series; so that we cannot in this case infer tJie
^onvergency of tlie oi'iginal from the convergency of the derived
series*.
Example.
(.,>y.(.,j)%..,,(.,>)'.(,,^)V...
is obviously a non-convergent oscillating series. But
](-i)'-(-3)M(-f-(-0}--{(-^.y-
^ + 2nTT
whose 7ith term is (Sn- + %n + l)l(\ifi ■\-2nY, i.e. (8 + 8/k + 1/«2)/16(1-(- l/27i)V,
is convergent, being comparable in the scale of convergency with Sl/zt-.
* This remark is all the more important because the converse process of
pplitting up the ?ith term of a series into a group of terras with alternating
signs, and using the rules of § 8, often gives a simple means of deciding as to
its convergency. The series 1/1 . 2 + 1/3 . 4 + 1/5 . 6 + 1/7 . 8 + . , . may be tested
in this way.
§§7-9 iii — u.^ + U3 — t('i+. .. 135
§ 8.] The following rule is frequently of use in the discus-
sion of semi-converging series : —
If Ui>U2>U3> • • • >w„> . . . and fill be positive, then
ih-ih + u-i-. . . (-)''-X + (-)Xm + - . • (1)
converges or oscillates according as L Un = or 4= 0.
Using the notation of § 3, we have
= ± {(««+! -«ft+2) + (2«n+3- "71+4) + . • •}•
Hence we have
numerical values being alone in question. If, therefore, Lun ~ 0,
we have LunJr\ = Lun+2 = 0 ; f^nd it follows that L m,Rn = 0 for all
n=«>
values of m. Also
so that 8n is finite for all values of n. The series (1) is there-
fore convergent if Lun = 0.
If Lun = a + 0, then L ^Rn = « or =0 according as m is odd
n=<»
or even. Hence the series is not convergent. We have, in fact,
L(SM+i-S2n) = Lu2n+i = (^, which shows that the sum of the
series oscillates between S and S + a, where S=LS2n-
Cor. The series
(til - U2) + {«3 - M4) + • • • + (^^271-1 - «2ft) + . . .
where «i, ih, • • • «''^ «« be/ore, is convergent.
Example 1. The series S ( - 1)"-^ is convergent, nolwitlistanding the
fact, ah-eady proved, that Sl/)i is divergent.
Example 2. S( - 1)"'-i(h + 1)/(i is an oscillating series; but SC-l)""!
{{n + l)ln- (n + 2)l{n + l)} is convergent.
§ 9.] The most important case of periodic series is 2a,i cos
{nO + <f)), where a,i is a function of 7i, and ^ is independent of n,
commonly spoken of as a Trigonometrical or Fourier s Series. The
question of the convergence of this kind of series is one of great
importance owing to their constant application in mathematical
physics.
l3d Abel's inequality ch. xxvi
We observe in the first place that
I. If 2«„ he an absolutely converging series then "^ancos (nO + <^)
is convergent.
This follows from § 4, I.
II. 1/6 = 0 or 2k7r {k being an integer), 2a„ cos (nd + 4>) is
convergent or divergent according as ^a,i is convergent or divergent.
This is obvious, since the series reduces to Sa^ cos <^.
III. 1/6^0 or 2kTr, then Ian cos (nO + <f>) is convergent if, for
all values of n greater than a certain finite value, «„ has the same
sign and never increases as n increases, and if L an- 0.
n=oo
This is a particular case of the following general theorem,
which is founded on an inequality given by Abel : —
IV. If 'Xunbe convergent or oscillatm'y , anda^, ^2, . . ., a„, . . •
be a series of positive quantities, v^hich never increase as n increases,
and if i/a„ = 0, the7i lanUn is convergent.
Abel's Inequality is as follows : — If, for all values of n,
A>U^ + Ui + . . . + Un>B,
where Ui, u^, . . ., Un are any real quantities whatever, and
if Oi, fflj, . . ., ttrt be a series of positive quantities which never
increase as n increases, then
ttiA > aiUi + a-iU^ + . . . + «„% > cti^-
This may be proved as follows: — Let Sn=iii + ih + ' • ■ + «»,
Sn = aiUi + a-iU^ + . . . + antin- Then Ui = Si, W3 = /Sj-/Si, &c. ;
and
Sn =aiSi + a^ iS.j,-Si) + . . .+ a„ (S^ - Sn-i),
= >Si («! - ttj) + /S^2 (^2 - tts) + • . .+Sn-i{an-i-an) + Snan.
Hence, since Si, S^, . . ., /S'„ are each <A and >B, and (ai-a^),
(aj - Oa), . . . , {a„-i — ttn), an are all positive or zero,
{(ai-«2) + («2-a3) + . . . + {an-i - an) + an} A
>Sn'>{{ai-a2)-\-{a2-a3)+. . . + («„_i - a„) + «„} 5 ;
that is,
ajA>Sn'>aiB (1).
Theorem IV. follows at once, for, since 2m„ is not divergent,
§§ 9, 10 TRIGONOMETRICAL SERIES 137
Sn is not infinite for any value of n. Hence, by (1), S^ is not
infinite. Also, by Abel's Inequality,
where (7 and D are the greatest and least of the values of
r^Rn (= M„+i + M„+2 + . . . + Un+m = ^n+m " ^n) for all different
positive values of m. Now, since 2m,» is convergent or oscillatory,
^n+m - Sn is either zero or finite, and L a„+i = 0, by hypo-
n=oo
thesis. Therefore, it follows from (2), that L m,Rn = 0 for all
values of m. Hence ^a^Un is convergent.
We shall prove in a later chapter that, when
Un = cos {n6 + (fi),
Sn = sin ^n6 cos {^{n + l)9 + </>}/sin |^.
If, therefore, we exclude the cases where 6 = 0 or 2kir, we see
that Sn cannot be infinite. Theorem HI. is thus seen to be a
particular case of Theorem IV.
Cor. If a^he as above, 2(— l)"~'a„cos(w^ + ^), 2a„siw(w^+<^),
and 2 ( - l)"~^a„s«« (n^ + <^) are all convergent.
CONVERGENCE OF A SERIES OF COMPLEX TERMS.
§ 10.] If the wth term of a series be of the form Xn + yJ,
where i is the imaginary unit, and x^ and yn are functions of n,
we may write the sum of n terms in the form &n + TJ, where
Tn=yi+y2 + ' . '+yn-
By the sum of the infinite series % (xn + yj) is meant the limit
when n= CO of /S„ + TJ ; that is, (LSn) + (LTn) i-
The necessary and sufficient condition for the convergency of
^(xn + yni) is therefore that 2ir„ and lyn he both convergent.
For, if the series 2^„ and 2y„ converge to the values 8 and
T respectively, S {xn + yJ) will converge to the value S+ Ti;
and, if either of the series 2ir„, %„ diverge or oscillate, then
(LSn) + {LTn) i will not have a finite definite value.
138 CONVERGENCE OF COMPLEX SERIES CH. XXVI
§ 11.] Let Zn denote a^n + ^J; and let \zn\ be the modulus
of Zn* ; so that | ;2n T = I ^» P + I ^n P- We have the following
theorems t, which are sufficient for most elementary purposes : —
I. The complex series %Zn is convergent if tlie real series '^\zn\
is convergent.
For, since 2 1 2;„ ] is convergent, and | Xn \ and | ?/„ | are each less
than \zn\, it follows from § 4, L, that '^\xn\ and %\yn\ are both
convergent ; that is, 2a?„ and %y,^ are both convergent. Hence,
by § 10, Izn is convergent.
It should be noticed that the condition thus established,
although sufficient, is not necessary. For example, the series
(1 - i)/l - (1 - i)/2 + (1 - i)/3 - ... is convergent since 1/1 - 1/2
+ 1/3 — . . . and - 1/1 + 1/2 — 1/3 + . . . are both convergent;
but the series of moduli, namely, *y2/l + J^/'Z + j2/3 + . . . ,
is divergent.
When ^Zn is such that '^\zn\ is convergent, 2;r„ is said to be
ahsolutehj convergent. Since the modulus of a real quantity Un is
simply Un with its sign made positive, if need be, we see that
the present definition of absolute convergency includes that
formerly given, and that the theorem just proved includes
§ 4, IV., as a particular case.
Cor. 1. If mRn denote Zn+i + Zn+2 . . . + Zn+m, then the necessary
and sufficient condition that the complex series 2^„ converge is that
it be possible, by taking n sufficiently great, to make \mRn\ «s small
as we please, whatever the value of m.
Cor. 2. If Xn be real or complex, and z^ a complex number
whose modulus is not infinite for any value of n, however great, then
2(^>i*n) '^m be absolutely convergent if 2X„ is absolutely convergent.
For I XnZn I = I ^n 1 1 ^» I ; and, since 2A„ is absolutely con-
vergent, 2 I X„ I is convergent. Hence, since \zn\ is always
finite, 2 1 X,i 1 1 z„, \ is convergent by § 4, 11. ; that is, 2 | A.„;2r„ | is
convergent. Hence % {K.Zn) is absolutely convergent.
Example 1. The series 2j'7n! is absolutely convergent for all finite
values of z.
Example 2. The series S«"//t is absolutely convergent provided | * ] < 1.
* See chap, xii,, § 13.
+ Cauchy, R6sum6s Anahjtiques, § xiv.
^ 11, 12 LAW OF ASSOCIATION FOR SERIES 139
Example 3. The series S (cos 6 + i sin ^)"/)i is convergent if ^ 4= 0 or 2krr.
For the series 2 cos ndjii and S sin v0ln are convergent by § 9, III.
Example 4. The series (cos ^ + i sin 0)"ln" is absolutely convergent. For
the series of moduli is Sl/ft^, which is convergent.
II. Let O be the fixed limit or the greatest of the limits^ to
ivhich l^^np^"' tends wJien n is increased indefinitely, or a fixed limit
to which I Zn+i/Zn I tends when n is increased indefinitely ; then tJie
series IZn will he convergent ifQ,<l and divergent ifQ,>\.
For, if fi<l, then, by §5, I. and II., the series '^\zn\ is
convergent; and therefore, by § 11, I., 2;5„ is convergent.
If 0>1, then either some or all of the terms of the series
2 I s^n I ultimately increase without limit. In any case, it will be
possible to find values of n for which | Zn \ exceeds any value
however great ; and, since | 2r„ | = (j a;„ |^ + | y„ ffi"^, the same must
be true of one at least of | x^ \ and \yn\- Hence one at least of
the series 2ir„, %„ must diverge; and consequently %{xn + yni),
i.e. ^z.,i, must diverge.
APPLICATION OF THE FUNDAMENTAL LAWS OF ALGEBRA
TO INFINITE SERIES.
§ 12.] Law of Association. — We have already had occasion to
observe that the law of association cannot be applied without
limitation to an infinite series; see the remarks at the end of § 7.
It can, however, be applied without limitation provided the series
is convergent. For let SJ denote the sum of m terms of the new
series obtained by associating the terms of the original series into
groups in any Avay whatever. Then, if /S'^ denote the sum of n
terms of the original series, we can always assume m so great that
S,n' includes at least all the terms in Sn- Hence S^' - Sn=^pJin,
where p is a, certain positive integer. Now, since the original
* It will be noticed that this includes the case where L |2,J'/" has
n=ao
different values according to the integral character of n: but the corre-
sponding case where L \ s,i+i/*„ | oscillates is not included. We have
n=x
retained Cauchy's original enunciation ; but it is easy to see that some
additions might be made to the theorem in the latter case.
140 LAW OF COMMUTATION FOR SERIES CH. XXVI
series is convergent, by taking n sufficiently large we can make pRn
as small as we please. It follows therefore that L SJ= L Sn-
Hence the association of terms produces no effect on the sum of the
infinite convergent series.
§ 13.] Law of Commutation. — The law of commutation is even
more restricted in its application than the law of association.
We may however prove that the law of commutation can he
applied to absolutely convergent series.
"We shall consider here merely the case where each term of
the series is displaced a finite number of steps*. Let 2m„ be
the original series, 2^^' the new series obtained by commuta-
tion of the terms of 2m„. Since each term is only displaced by
a finite number of steps, we can, whatever n may be, by taking
m sufficiently great always secure that Sm contains all the
terms of Sn at least. Under these circumstances SJ — Sn con-
tains fewer terms than pRn, where p is finite, since m is finite.
Now, since 2w„ is absolutely convergent, even if we take the
most unfavourable case and suppose all the terms of the same
sign, we shall have L pEn - 0 ; and, a fortiori, L Sm - L Sn = 0.
Hence L SJ = L Sn', which establishes our theorem.
The above reasoning would not apply to a semi-convergent series
because the vanishing of L pRn does not depend solely on the
individual magnitude of the terms, but partially on the alterna-
tion of positive and negative signs.
Cauchy, in his Resumes Analytiques, § Vii. (1833), seems to
have been the first to call explicit attention to the fact that the
convergence of a semi-convergent series is essentially dependent on
the order of its terms. Dirichlet and Ohm gave examples of the
effect of the order of the terms upon the sum.
Finally Riemann, in his famous memoir on Fourier's Seriest,
showed that the series 2 (-1)"~%„, where Lun = 0, and ^u^+xOxA
lu^n are both divergent, can, by proper commutation of its terms,
• See below, § 33, Cor. 2.
t Written in 1854 and published in 1867. See his Gesammclte Math.
Werke, p. 21L
§§ 12, 13 LAW OF COMMUTATION FOR SERIES 141
be made to converge to any sum we please ; and Dirichlet has
shown that commutation may render a semi-convergent series
divergent.
When the sum of an infinite series is independent of the order
of its terms it is said to converge unconditionally. It is ohvious
from what has been said that unconditional convergence and
absolute convergence are practically synonymous.
1 !_
Example 1.
The
series
1
1
1
1
is convergent by § 8 ;
but the series
V(4m+1) v/(4m + 3) ^(2m+2),
+ ... (2),
v?hich is evidently derivable from (1) by commutation (and an association
which is permissible since the terms ultimately vanish), is divergent. For,
if «^ = 1/J(4m + 1) + 1/^(4771 + 3) -l/v'(2m + 2), and v^ = lljm, then
LuJv^=L {1/V(4 + 1/m) + ly (4 + 3/«0 - 1/^(2 + 2/m)} = 1/2 + 1/2 - 1/^2 =
1 - i^2. Hence w^/^'ot i^ always finite ; and 2v^ is divergent, by § 6, Cor. 4.
Hence 22t„ is divergent. (Dirichlet.)
Example 2. The series
11 11 1_ 1 1
1 2'*'3 4'*'5 •'•"^(2«-l) (2)1)'^''' ^''
\1^3/ 2^V5^7y 4^ •^V4»i + 1^4m + 3y
(2),
are both convergent; but they converge to different sums. For, by taking
successively three and four terms of each series, we see that the sum of (1) lies
between -583 and -833; whereas the sum of (2) lies between -926 and 1-176.
Addition of two infinite series. If 2m„ and Sv^ i>^ both con-
vergent, and converge to the values S and T respectively, then
'%{Un + v^ is convergent and converges to the value S+T.
We may, to secure complete generality, suppose u^ and v„ to
be complex quantities. Let 8n, Tn, Un represent the sums of
n terms of 2m„, 2v„, 2 («„ + Vn) respectively ; then we have, how-
ever great n may be, V'n = Sn+ Tn. Hence, when n=cc,
LU'n = LSn + LTn, which proves the proposition.
142 LAW OF DISTRIBUTION FOR SERIES CH. XXVI
§ 14.] Law of Distribution. — The application of the law of
distribation will be indicated by the following theorems : —
If a he any finite quantity^ and 2m„ converge to the value S,
then '^aun converges to aS.
The proof of this is so simple that it may be left to the
reader.
If %Un and %Vn converge to the values S and T respectively, and
at least one of tlie two series he absolutely convergent, then the series
lhVi+{lhV2 + U-iVi) + . . .+{tliVn + U./On-i + . • . + Un'«l) + • • • (l)
converges to the value ST*.
Let Sn, Tn, Un denote the sums of n terms of 2?^„, %Vn,
2 {UiVn + ii'^n~\ + . • . + UnVi) respectively ; and let us suppose that
%Un is absolutely convergent. We have
where L^ = ii-iV^ + Ws'^n-i + . . . + UnV-i
+ UiCj, + . . . + U,,V.i
+ w^vn
= U.Vn + Wj {Vn + Vn_i) + . . . + U^ (??„ + . . . + Vo) (2).
If therefore n be even, = 2w say,
Ln - [WoVsm + Ih (Vin, + V2m-i) + . . . + U,„, {v.„, + . . . + V,a+o)]
+ [Um+1 {V-im + . • ■ • + V,„+i) + . . . + lh,n (V-zm + ■ • • "+ «-)] (3).
If n be odd, =2ni+ 1 say,
+ [Um+l{Vzm+l + - - '+Vm+2) + . . • + «2;ft+l (V27n+1 + • • • + ''^2)] (4).
Now, since 2v„ is convergent, it is possible, by making m
sufficiently great, to make each of the quantities \v2m\, \v^-i+V2m\,
. . ., \Vm+2 + ' . .+V2„,\, \Vo,a+i\, \v.,n + V.2m+i\, ■ . ., | ^m+a + • • •
* The original demonstration of this theorem given by Cauchy in his
Analyse Algebrique required that both the series S?/„, Sr„ be absolutely con-
vergent. Abel's demonstration is subject to the same restriction. The more
general form was given by Mertens, C reliefs Jour., lxxix. (1875). Abel had,
however, proved a more general theorem (see § 20, Cor.), which partly in-
cludes the result in question.
§§ 14, 15 THEOREM OF CAUCHY AND MERTENS 143
+ v.2m+i I as small as we please. Also, since \Ti\, \T2\,\Ts\, . . .,
|y;i, ... are all finite, and \Tr-Ts\< \T,.\ + |2;|, therefore
. 1 'y^+i + . . . +V.2,n\, ' - . , \'C2 + ' . . + •»2m I,
are all finite. Hence, if c^ be a quantity which can be made as
small as we please by sufficiently increasing m, and p a certain
finite quantity, we have, from (3) and (4), by chap, xii., § 11,
\Ln\<^ra{\u2\ + \u3\ + ' - • + | «m| )
+ /3 ( I M^+i I + I ^^,„+2 1 + . . . + I Wft I ).
If, therefore, we make n infinite, and observe that, since
2w„i is absolutely convergent, Im,] + l^sl + • • . + Iwjil is finite, and
L{\um+i\ + |w,„+2| + . . . + \un\) = 0, we have (seeing that L(m = 0)
L\Ln\ = 0. Hence LSnTn = LUn, that is, LUn = ST.
Cauchy has shown that, if both the series involved be semi-
convergent, tlie multiplication rule does not necessarily apply.
Suppose, for example, u„=i;„=r ( - l)'*-^". Then both 2m„ and 2v„ are
semi-convergent series. The general term of (1) is
^"="U
1 1 1 , .-.
Now, since r{»i-r + l) = J(n + 1)2 -{i(n + l)-r}2, therefore, for all values
of r, r(r^-r^-l)<|(J^ + l)-, except in the case where r=^ (w + 1), and then
there is equality. It follows that \io^\>nl\ (n-(-l)>2/(l + l/)i). The terms of
2w„ are therefore ultimately numerically greater than a quantity which is
infinitely nearly equal to 2. Hence 2i^„ cannot be a convergent series.
UNIFORMITY AND NON-UNIFORMITY IN THE CONVERGENCE
OF SERIES WHOSE TERMS ARE FUNCTIONS OF A VARIABLE.
§ 15.] Let X for the present denote a real variable. If the
?2th term of an infinite series be f{n, x), where /(w, x) is a single
valued function of n and of x, and also for all integral values of n
a continuous function of x within a certain interval, then the
infinite series 2/(%, x) will, if convergent, be a single valued
finite function of x, say <i>{x). At first sight, it might be
supposed that 4*{x) must necessarily be continuous, seeing that
each term of f{n, x) is so. Cauchy took this view ; but, as
144 UNIFORM AND NON-UNIFORM CONVERGENCE CH. XXVI
Abel* first pointed out, (f> {x) is not necessarily continuous.
No doubt 2/(w, X + h) and 2/(w, x), being each convergent, have
each definite finite values, and therefore 2 {/{n, x + h) —/{n, x)}
is convergent, and has a definite finite value ; but this value is
not necessarily/ zero when h = 0 for all values of x. Suppose, for
example, following Du Bois-Reymond, thsd, f(n, x) = x/{nx + 1)
(nx -x+1). Since /{n, x) = nx/{nx +l)-{n-l) x/{n -lx + 1},
we have, in this case, Sn = nxl{nx+1). Hence, provided x + 0,
LSn = l. If, however, x=0 then Sn = 0, however great n may
be. The function tf> (x) is, therefore, in this case, discontinuous
when x=0.
The discontinuity of the above series is accompanied by
another peculiarity which is often, although not always, asso-
ciated with discontinuity. The Residue of the series, when
x=¥0, is given by
En=l-Sn=l/(nX+ 1).
Now, when x has any given positive value, we can by making n
large enough make l/{nx+l) smaller than any given positive
quantity e. But, on the other hand, the smaller x is, the larger
must we take n in order that l/(nx + 1) may fall under € ; and,
in general, when x is variable, there is no finite lower limit for n,
independent of x, say v, such that if w>i' then Rn<^. Owing
to this peculiarity of the residue, the series is said to be non-
uniformly convergent in any interval which includes 0 ; and,
since, when x approaches 0, the number of terms required to
secure a given degree of approximation to the limit becomes
infinite, the series is said to Converge Infinitely Slowly near x = 0.
These considerations lead us to establish the following
important definition, where we no longer restrict ourselves
to functions of a real variable. If, for all values of z within
a given region R in Argand^s Diagram, we can for every
positive value of €, however small, assign a positive integer v
INDEPENDENT OF z, such that, when n>v, |i?„|<€, then the series
* Eecherches sur la Serie 1 + y^+^ J^^Z x^+ • • • Crelle's Jour.
Bd. 1. (182G).
I 15 UNIFORM CONVERGiENCE 145
2/(w, x) is said to be Unifokmly Convergent within the region
in question.
Stokes*, who in his classical paper on the Critical Values of
the Sums of Periodic Series was the first to make clear the
fundamental principle underlying the matter now under dis-
cussion, has pointed out that the question of uniformity or
non-Tiniformity of convergence always arises when we consider
the limiting value of a function of more than one variable.
Consider, for example, the function f{w, y) ; and let us suppose
that, for all values of y in a given region R, f{x, y) approaches
a finite definite limit when x approaches the value a ; and let us
call this limit/(«, y). Then if we assign in advance any positive
quantity e, however small, we can always find a positive quantity
A, such that, when |.'r-al<X, \f{x,y)-f{a,y)\<€. If it be
possible to determine A. so that the inequality
\/{'^>y)-f{a,i/)\<^
shall hold for all values of y contained in li, then the approach
or convergence to the limit is said to be uniform within R. If,
on the other hand, A depends on y, the convergence to the limit
is said to be non-uniform.
Example 1. Consider the series l + z + z'^+ . . .+z"+. . .; and let
l2|<p<l. We have \I\n\ = \z^'^^l{l-z)\<p^'^^l(l- p). Hence, in order to
secure that iJ„ < e, we have merely to choose n > - 1 + log (e - e/3)/log p.
Since - 1 + log (e - ep)/logp is independent of z, we see that within any circle
whose centre is the origin in Argand's Diagram, and whose radius is less
than unity by however little, the series Sz" is uniformly convergent.
On the other hand, as p approaches unity log (e- €p)llog p becomes larger
and larger. Hence the convergence of Sz" becomes infinitely slow when | z
approaches unity. We infer that the convergence of 2S^" is not uniform
within and upon the circle of radius unity. And, in fact, when the upper
limit of 1 2 I is 1, it is obviously impossible when e is given to assign a finite
value of n such that 1 2"+'/(l -z)\<e shall be true for all values of z.
* Trans. Camb. Phil. Soc, Vol. viii. (1847). Continental writers have
generally overlooked Stokes' work; and quote Seidel, Abld. d. Bayerischen
Akad. d. Wiss. Bd. v. (1850). For exceptions, see Reiff, Geschichte der
uiiendlichen ReUien, p. 207 (1889) ; and Pringsheim, Eiic. d. Math. Wiss.
Bd. II. p. 95 (1899). In his first edition the writer, although well acquainted
with Stokes' great paper, by an unfortunate lapse of memory, fell into the
same mistake. The question of uniformity of convergence is now a
fundamental point in the Theory ©f Functions.
c. 11. 10
146 CoNTtNUlTY AND tfNlPORM CONVEHGENcE CH. XXVI
Example 2. Osgood* has shown that, if
'Pn (^) = V(2e) n sinVa; . c-»'^»i"*'f*,
the infinite series which has ^„ {x) + 0„ (2! x)j2\ + . . . + ^„ (n! x)jn\ for the
sum of n terms converges non-uniformly ia every interval.
From the definition of Uniform Convergence we can at once
draw the following conclusions.
Cor. 1, If the terms of 2 \f{n, z) \ are ultimately less than
the terms of a converging series of positive terms wlwse values are
independent of z, then %f{n, z) converges uniformly.
For, if %Un be the series of positive terms in question, and Rn
the residue of "^{n, z), then
\B,\1^\f{n+\,z)\ + \f{n + 2,z)\+ . . .,
< M„+i + Un+2 + . . .
Since Swn is convergent, we can find an integer v so that, when
w> V, Un+i + Mn+2 + • • '<^\ and V will be independent of z, since
Un+i, w„+2, ... are independent of z. Hence we can find v
independent of ^: so that |i2„|<e, when n>v, c having the usual
meaning.
Cor. 2. If % \f{n, z) \ is uniformly convergent, then ^{n, z)
is unifoi'mly convergent.
§ 16.] We now proceed to establish a fundamental theorem
regarding the Continuity of a Uniformly Converging Series.
Let f{n, z) he a finite single valued function of the complex
variable z and the integral variable n, which is continuous as
regards z for all values of n, Jwwever large, and for all values of
z within a region B in Argand's Diagram. Farther, let ^{n, z)
converge uniformly within R, say to <t> (z). Then ^ (z) is a con-
tinuous function of z at all points within the region R.
Let the sum to n terms and the residue after n terms of
2/(7i, z) be Sn and Rn ; and let Sn and Rn be the like for
%f{n, z'), where z and z' are any two points within the region R.
Then we have
^(z)=Sn + Rn, ^{z) = S: + R: (1).
* Bull. Am. Math. Soc, Ser. 2, iii. (1S96). This paper is well worthy of
study on account of the interestiug geometrical methods which the author
uaes.
§§ 15, 16 CONTINUITY AND UNIFORM CONVERGENCE 147
Since '^/{n, z) is uniformly convergent within R, given any
positive quantity c, however small, we can find a finite integer v,
independent of z, such that for all values of z within R, Rn<€ and
Rn<i, when n>v. Let us suppose n in the equations (1) chosen
to satisfy this condition. Since the choice of z is unrestricted we
can by making \z-z'\ sufficiently small cause the absolute value
of each of the differences /(I, z)-f{\, z), . . .,f(n, z)-f{n, z)
to become as small as we please, and, therefore, since n is finite
we can choose \z-z'\ so small that \Sn-Sn\, which is not greater
n
than % \f{n, z) -f{n, z') |, shall be less than c.
Now
\<i>{z)-^{z')\ = \8r,-S,: + R,,-R:\
1^\Sn-Sn\ + \Rn\ + \ Rn \
<3e,
which proves our theorem ; for c, and therefore 3c, can be made as
small as we please.
It follows from what has been proved that discontinuity of
%f{n, z) is necessarily accompanied by non-uniformity of con-
vergence ; but it does not follow that non-uniformity of con-
vergence is necessarily accompanied by discontinuity. In fact,
Du Bois-Reymond has shown by means of the example
% {xjii {nw + 1) {nw -x+l)- w'^/(naf +1) (naf -x+1)}
that infinitely slow convergence may not involve discontinuity.
The sum of this series is always zero even when ^ = 0 ; and yet,
near x^O, the convergence is infinitely slow.
It should also be noticed that the fact that a series converges
at a point of infinitely slow convergence, does not involve that
the sum is continuous at that point. Thus the series
%xl{nx + 1) {nx -x-\-\)
converges at x = Q; but, owing to the infinite slowness of con-
vergency at x-0, the sum is discontinuous there, being in fact
0 at i» = 0, and 1 for points infinitely near to a? = 0. In such
cases it is necessary to state the region of uniform convergence
with some care. The fact is that the series in question is
convergent in the real interval pl^xl^b, where b is any finite
10—2
148 DU BOIS-REYMOND's theorem CH. XXVI
positive quantity and p is a positive quantity as small as we
please but not evanescent. This is usually expressed by saying
that the series is uniformly convergent in the interval 0<x^b.
Such an interval may be said to be 'open' at the lower and
* closed ' at the upper end*.
Examplef. If /*„ be iDdei)endent of z, and u\^{z) be a single valued
function of n and z, finite for all values of n, Lowever great, and finite and
continuous as regards z within a region R, then, if 2,/j.^ be absolutely con-
vergent, l,flJ^w,^ (z) is a continuous function of z within jR.
It will be sufficient to prove that the series 2/x„it'„ (2) is uniformly
convergent within E.
Since w„(2) is finite for all points within li, we can assign a finite
positive quantity, g, independent of z, such that, for all points within R,
Consider i?„, the residue of S/U,ji<j„ (2) after n terms. We have
■Kn = M„+i Jf„+i (2) + /J^n+^J^n+i (z) + • • •
Hence
1 ^L I > I M»+l I 1 Wn+l (2)1 + 1 Mn+2 I I «''«+2 {^)\+ ' • • .
Since S/i„ is absolutely convergent, S | ;«„ | is convergent, and we can assign
an integer v such that, when n>v, \ fi^^-y \ + \ /j.^^^„ | + . . . <e/f7, where e is a
positive quantity as small as we please.
Both /x„ and g being independent of 2, it is clear that v is inde-
l^endent of z. Hence we have, Avhen n>v, |JJ,J<e, v being independent
of 2. The series is therefore uniformly convergent : and it follows from the
main theorem of this paragraph that its sum is a continuous function of z.
SPECIAL DISCUSSION OF THE POWER SERIES Sfln^".
§ 17.] As the series 2«,iC;" is of great importance in Algebraic
Analysis and in the Theory of Functions, we shall give a special
discussion of its properties as regards both convergence and
continuity. We may speak of it for shortness as the Power
Series ; and we shall consider both «„ and z to be complex
numbers, say «„ = ?'n (cos a,^ + i sin a,i), z = p (cos 0 + i sin 6), where
Tn and o„ are functions of the integral variable n, but p and 0 are
independent of n.
* Harkness and Morley use these convenient words in their Introduction
to the Tliconj of Analytic Futictiotus, Macmillan (1898).
t Du Bois-Reymond, Math. Ann. iv. (1871).
§§ 16, 17 CIRCLE AND RADIUS OF CONVERGENCE 149
The leading property of the Power Series is that it has what
is called a Circle* of Convergence, whose centre is the origin in
Argand's Diagram, and whose radius {Radius of Convergence) may
be zero, finite, or infinite. For all values of z within (but not
upon) this circle the series is absolutely and uniformly con-
vergent ; and (if the radius be finite) for all values of z without
divergent. On the circumference of the circle of convergence
the series may converge either absolutely or conditionally,
oscillate, or diverge ; but on any other circle it must either
converge absolutely or else diverge.
The proof of these statements rests on the following theorem.
If the series "^anZ^ be at least semi-convergent when z = Zq,
then it is absolutely and uniformly convergent at all points within
a circle whose radius <\zq\.
Since %anZj^ is convergent, none of its terms can be infinite
in absolute value, hence it is possible to find a finite positive
quantity g such that | anZ^^\<g, for all values of n however large.
Hence \a,z''\ = \a,,z,''{zlz,Y\,
'^Aa,,z-\\{zlz,Y\,
Now, since z is within the circle \zo\, \zIzq\<1. Hence the
series g%{zlz^'^ is absolutely convergent. Therefore (§ 4, I.)
2 I anZ^ I is absolutely convergent.
The convergence is uniform. For, since |2;|<|co|, we can
find z' such that |;c;]<|2;'|<|2'o|. Now, by the theorem just
established, 2 1 Unz"^ \ will be convergent, and its terms are inde-
pendent of z. But, since | ;3 ] < 1 2;' | , | ofj^^;"' | < | «„«'"■ | . Hence, by
§ 15, Cor. 1, %anZ^ is uniformly convergent.
Circle of Convergence. Three cases are in general possible.
1st. It may not be possible to find any value Zq of z for which
the series Sa^s" converges. "We shall describe this case by saying
that the circle of convergence and the radius of convergence are
infinitely small. An example is the series 2»!a;".
2nd. The series may converge for any finite value of z
* When in what follows we speak of a circle {R), we mean a circle of
radius B whose centre is the origin in Argand's Diagram.
150 RADIUS OF CONVERGENCE, CAUCHY'S RULES CH. XXVI
however large. We shall then say that the circle and the radius
of convergence are infinite. An example of this very important
class of series is ^x^jnl.
3rd. There may be finite values of z for which 2a„2;" con-
verges, and other finite values for which it does not converge.
In this case there must be a definite upper limit to the value
of \zq\ such that the series converges for all points within the
circle l^^ol and diverges for all points without. For the series
converges when |2;|<|^;olj f^iid it must diverge when |2;|>|2;o|;
for, if it converged even conditionally for |«;'|>|2ro|, then it
would converge when |2;|<|2;'|. We could, therefore, replace
tlie circle \zq\ by the greater circle \z'\, and proceed in this way
until we either arrive at a maximum circle of convergence,
beyond which there is only divergence, or else fall back upon
case 2, where the series converges within any circle however great.
We shall commonly denote the radius of the circle of con-
vergence, or as it is often called the Radius of Convergence, by R.
It must be carefully noticed that both as regards uniformity and
absoluteness of convergency the Circle of Convergence is (so far
as the above demonstration goes) an open region, that is to say,
tlie points on its circumference are not to be held as being within
it. Thus, for example, nothing is proved as regards the con-
vergence of the power series at points on the circumference of
the Circle of Convergence ; and what we have proved as regards
uniformity of convergence is that ^a^z^ is uniformly convergent
within any circle whose radius is less than R by however little.
§ 18.] Cauchys Rules for determining the Radius of Con-
vergence of a Power Series.
I. Let to be tlie fixed limit or tlie greatest of the limits to
ivhich I ttn P'" tends when n is increased indefinitely, then l/a>
is the radius of convergence of "ZanZ^.
For, as we have seen in § 11, II., 2a„2;" is convergent or
divergent according as X|a„«''p^<or>l; that is, according as
w I « I < or> 1 ; that is, according as \z\< or > 1/<d.
II. Let (o be a fixed limit to which \ a„+,/a„ ] tends when n is
increased indefinitely; then l/w is the radius of convergence of
^anz"".
§§17-19 CONVERGENCE ON CIRCLE OF CONVERGENCE 151
The proof is as before. The second of these rules is often
easier of application than the first ; but it is subject to failure in
the case where L \ an+ildn \ is not definite.
Example 1. 1 + 2/1 + 2^/2+...
Here, by the first rule, w= I, (l/n)V"= L m'»=l (chap, xxv., §16).
n=«o »n=0
Hence R = l.
By the second rule, w= L {l/(n + l)}/{l/n} = L n/(n + l) = l. Hence
R = l, as before.
Example 2. z + 2z^ + z^ + 2z*+ . , .
Kereifn = 2??i, L |a„V»]= L lVn = l,
n=» n=ot)
if n = 2m + l, L |a„V»|= L 2V«=:1.
jl=oo n=aJ
Hence w=l, and JJ = 1. The second rule would fail.
§ 19.] Convergence of a Power Series on its Circle of Con-
vergence,
The general question as to whether a power series converges,
oscillates or diverges at points on its circle of convergence is
complicated. For series whose coefficients are real the following
rule covers many of the commoner cases.
I. Let a„ be real, such that ultimately «„ has the same sign
and never increases; also that Lan = 0, and Lan+i/an=l, when
n- <^. Then the radius of convergence of^a^z^ is unity ; and
1st. If 2a„ is convergent, %anZ"' converges absolutely at every
point on its circle of convergence.
2nd. If %an is divergent, 'ZanZ^ is semi-convergent at every
point on its circle of convergence, except z= 1, where it is
divergent.
If we notice that on the circle of convergence Sa„s" reduces
to 2a„ (cos nO + i sin nO) = 2a„ cos nO + i^an sin nO, we deduce the
above conclusions at once from § 9.
Cor. Obviously the above conclusions hold equally for
2 ( - l)"'anZ^, except that the point z = — l takes the place of
the point z=l.
The following Rule, given by Weierstrass in his well-known
memoir Ueber die Theorie der Analytischen Facultdten* , applies
♦ Crellt'a Jour., Bd. 51 (185G).
152 Abel's continuity theorems ch. xxvi
to the more general case where the coefficients of the power series
may be complex. By § 6, Cor. 5, it is easy to show that it
includes as a particular case the greater part of the rule already
given.
II. If on and after a certain value of n we can expand
dn+i/ftn in the form
an+1 , a + hi a^
an n n^
where g and h are real, then the behaviour of S^n^" on its
circle of convergence, the radius of which is obviously unity, is
as follows : —
1st, Jf g<i;iO the series diverges.
2nd. If g<-l the series converges absolutely.
3rd. If — 1;{>^<0 the series is semi-convergent, except at the
point z=l, where it oscillates if g — —l and h = 0, and diverges
if g>-l.
For the somewhat lengthy demonstration we refer to the
original memoir.
§ 20.] Abel's Theorems* regarding the continuity of a power
series.
Since (§18) Sa^^" converges uniformly at every point within
its circle of convergence, we infer at once that
I. The sum of the power series "^a^z^ is a continuous function
of z, say (f) (z), at all points within its circle of convergence.
This theorem tells us nothing as to what happens when we
pass from within to points on the circumference of the circle of
convergence, or when we pass from point to point on the circum-
ference. Much, although not all, of the remaining information
required is given by the following theorem.
II. If the power series ^a^s" be convergent at a point Zq on
its circle of convergence, and z be any point within the circle, then
provided the order of the terms in '^a^^z^^ be not deranged in cases
where it is only semi-convergent.
• CrelU's Jour., Bd. i. (1826).
§§ 19, 20 Abel's continuity theorems 153
In the first place, we can show that in proving this theorem
we need only consider the case where z and ^o lie on the same
radius of the circle of convergence. For, if z and ^o be not on
the same radius, describe a circle through z, and let it meet the
radius Oz^ in z^. Then it is obvious that, by making |2;-2?o|
sufficiently small, we can make | s - ;2;i ] and j 2^1 — «o | each smaller
than any assigned positive quantity however small.
Since z and Zx are both within the circle of convergence, we
can, by making 1 ;2 - ;2;i | sufficiently small, make | ^ (2;) - <^ (^^i) |
less than any assigned positive quantity c, however small. But
1 <^ (2;) - </> (^0) I = I <^ (^) - «^ (;2a) + «^ (^1) - </> (^0) I,
>l<^(«)-<^(;^OI+l'^(^i)-<^(^o)!,
If, therefore, we could prove that by making | «i - 2^0 1 sufficiently
small we could make ] ^ {z^ - <f) (zo) \ as small as we please, it
would follow that by making \z- Zo\ sufficiently small we could
make \(f>(z) — <f> (zq) \ as small as we please.
Let us suppose then that z and Zo have the same amplitude 6,
then we may put z = p (cos 6 + i sin 0), Zq = po (cos B + i sin 0), and
we take a„ = r„ (cos a„ + i sin a„). Hence
ttnZ^ - rn (cos a„ + i sin a„) p" (cos 110 + i sin nO),
= (-) '''nPo' {cos {nO + a„) + i sin {nd + a„)} ,
\Po/
where x = p/po, and becomes unity when 2; = 2^0 5 and ?7„ and F„
are real and do not alter when z is varied along the radius of the
circle of convergence.
It is now obvious that all that is required is to prove that if
the series of real terms ^x^Un remains convergent when a;=l,
00 00
then L %af^ Un = '^Un, or, what is practically the same thing,
a:=-=l-0 1 1
to prove that, if 5 ?7„ bo a convergent series, then
L i(l-^")C^„ = 0.
J-=l-0 1
Let 8n = {l-a;)Ux + {l-af)lh + . . .+{\-x'')Un,
154 ABEI4'S CONTINUITY THEOREMS CH. XXVI
Since a;^l, \ — x^, 1 — ^"~\ . . ., 1 — x satisfy the conditions
imposed on a^, a.2, . . ., «« in Abel's Inequality (§ 9). Also,
since ^dn is convergent, Un, Un-i, . . .,U-^ satisfy the con-
ditions imposed on Wi, ^^, . . ., u^. Hence, A and B being two
finite quantities, we have
(l-^»)^>>S;>(l-a?")5.
This inequality will hold however large we may choose n ; and
we may cause x to approach the value 1 according to any law we
please. Let us put x-\- Ijii^. Then we have, for all values
of n, however great,
{1 - (1 - 1/viT} yi >^„> {1 - (1 - lAOn B.
But L {I- Ijny = L {{I- l/w^)-» V'^" = ^"^ = 1.
Therefore, since A and B are finite, L /Sf^ = 0 ; that is,
L i (1 - ^») ZJ^ = 0.
X=l-0 1
It will be observed that, in the above proof, each term of
%x^^ Un is coordinated with the term of the same order in 2 Z7„.
Hence the order of the terms in S Un must not be deranged, if it
converges conditionally.
It follows from tlie above, by considering paths of variation
within the circle of convergence and along its circumference, that,
if a power series converge at all points of the circumference of its
circle of convergence, then as regards continuity of the sum the
circle of convergence may be regarded as a closed region. This
does not exclude the possibility of points of infinitely slow con-
vergence on the circumference of the circle of convergence,
because such points are not necessarily points of discontinuity.
On the other hand, if at any point P on the circumference
of the circle of convergence the series either ceases to converge
or is discontinuous, then the series cannot at such points be
continuous for paths of variation which come from within. If
however the series converge on both sides of P at points on the
circumference infinitely near to P, it must converge to the same
values.
It would thus appear to be impossible that a power series
§20 CONVERGENCE OF X (UnV^ + Un-iVz + . . . + U^Vn) 155
should converge infinitely near any point P of the circumference
of its circle of convergence to one finite value and to a different
finite value at P itself. It follows that, if a power series is
convergent, generally speaking, along the circumference of its
circle of convergence, it cannot become discontinuous at any
point on the circumference unless it cease to converge at that
point.
By considering the series Swn^", S-Wnz", and the series
which is their product when both of them are absolutely con-
vergent, and applying the second of the two theorems in the
present paragraph, we easily arrive at the following result, also
due to Abel.
Cor. If each of the series 2m„ and 2v„ converge, say to_ limits
u and V respectively, then, if the series % (unVi + Un-iV^ + . . . + Ui^n)
be convergent, it will converge to uv ; and this will hold even if
all the three series he only semi- convergent.
Example 1. The series l+z+ . . . +z"+ . . . has for circle of con-
vergence the circle of radius unity. Within this circle the series converges
to 11(1 -z). On the circumference the series becomes 2 (cos nd + i sin nd),
which oscillates for all values of 6, except 6 = 0 for which it diverges. At
points within and infinitely near to the circle of convergence the series
converges to i + tcot^.
Example 2, The radius of convergence for 2/1+ . . . +«"/«+ ... is
unity. Within the unit circle, as we shall prove later on, the series con-
verges to - Log (1 - z). On the circumference of the unit circle the series
reduces to S(cosn0-Fisinn^)/n. This series (see §9,111.) is convergent
when ^ + 0; but only semi-convergent, since Sl/n is divergent. When 0 = 0,
the series diverges. The sum is therefore continuous everywhere at and on
the circle of convergence, except when ^=0. At points within the circle
infinitely near to 2 = 1 the series converges to a definite limit, which is very
great; but at 0=1 the series diverges to -f oo .
Example 3. S^^/n* converges absolutely at every point on the circum-
ference of its circle of convergence (i? = l): and consequently represents a
function of z which is continuous everywhere within that circle and upon
its circumference.
Example 4. S?!2" is divergent at every point on its circle of convergence
(J? = l); and its sum is a continuous function at all points within its circle
of convergence, but not at points upon the circumference.
156 INDETERMINATE COEFFICIENTS CH. XXVI
Example 5. Pringsheim * has established the existence of a large class
of series which are semi-convergent at every point on the circumference of
their circle of convergence : a particular case is the series S ( - l)'^»z^jn log n,
where \„=1 when 22"' }> «< 22'»+i, x^=0 when 22'»+i > n < 2-''»+2.
§ 21.] Principle of Indeterminate Coefficients.
If Uo^O, there is a circle of non-evanescent radius within
which the convergent power series ^a^z^ cannot vanish.
Since the evanescence of the series implies a^ = - UiZ- a,,z^. - . . .,
it will be sufficient to show that there exists a finite positive
quantity A such that, if p = \z\<'K, then
\— aiZ - a<iZ'^ - . . .|<|aoI.
Now, since the series %anZ^ is absolutely convergent at any
point Za within its circle of convergence, there exists a finite
positive quantity ^ such that for all values of n, \ a^zi^ \ ^ a^p^' <g.
Hence |a„|<^/po".
Now
\- a-^z - a<2,z^ - . . . |:j>|ai2;| + |a2;:;^| + . . .
>l«i|p + I«2|p'' + . . .
<9{{plp^) + {plpof + . . .}
<9p/{Po-p)-
Hence, if we choose A. so that g^/(po - X) = | a^ J, that is A. = j a^ | p^/
(^ + 1 «(, I ), we shall have
I - aiZ - a-iz- — . . . I < I tto I
for all values of z within the circle X.
Cor. 1. If am^O, there is a circle of non-evanescent radins
within which the convergent power series am^'" + «m+i^"'"'"' + - • •
vanishes only when z = 0.
For a,nZ"' + ar„+^z^+^ + . . .
= z'^{am + ara+iz + . . .).
Now, since am + 0, by the theorem just proved there is a circle
of non-evanescent radius within which a,„ + am+iZ + . . . cannot
vanish : and z^ cannot vanish unless z-0.
* Math. Ann., Bd. xxv. (1885).
^20-22 INFINITE PRODUCTS 157
Cor. 2. If ao + a^z + a.^z"^ + . . . vanish at least once at some
point distinct from z = 0 within evei'y circle, however small, then
must ao-0, «! - 0, a., — 0, . . ., that is, the series vanishes
identically/.
Cor. 3. If for one value of z at least, differing from 0, the
series ^a^z^ and ^bnZ"' converge to the same sum within every
circle, however small, then must a^ = bo, tti-bi, . . ., that is, the
series must be identical.
INFINITE PRODUCTS.
§ 22.] The product of an infinite number of factors formed
in given order according to a definite law is called an Infinite
Product. Since, as we shall presently see, it is only when the
factors ultimately become unity that the most important case
arises, we shall write the nth. factor in tlie form 1 + m„.
By the value of the infinite product is meant the limit of
(1 + Ml) (1 + th) . . . (1 + lln),
n
(which may be denoted by 11 (1 + n,^\ or simply by Pn), when n
is increased without limit.
It is obvious that if Lun were numerically greater than unity,
then LPn would be either zero or infinite. As neither of these
cases is of any importance, we shall, in what follows, suppose
I w„ I to be always less than unity. Any Jlnite number of factors
at the commencement of the product for which this is not true,
may be left out of account in discussing the convergency. . We
also suppose any factor that becomes zero to be set aside ; the
question as to convergency then relates merely to the product of
all the remaining factors.
Four essentially distinct cases arise —
1st. LPn may be 0.
2nd. LPn may be a finite definite quantity, which we may
denote by 11 (1 + m„), or simply by P.
3rd. LPn may be infinite.
4th. LPn may have no definite value ; but assume one or
other of a series of values according to the integral character of n.
158 ZERO, CONVERGENT, DIVERGENT, CH. XXVl
In cases 1 and 2 the infinite product might be said to be
convergent ; it is, however, usual to confine the term convergent
to the 2nd case, and to this convenient usage we shall adhere ;
in case 3 divergent ; in case 4 oscillatory.
§ 23.] If, instead of considering P„, we consider its logarithm,
we reduce the whole theory of infinite products (so far as real
positive factors are concerned*) to the theory of infinite series ;
for we have
log P„ = log (1 + Wi) + log (1+^2) + . . . + log(l + «^)
n
= % log (1 + Un) ;
and we see at once that
n
1st. If S log (1 + u^ is divergent, and L% log (1 + m„) = - 00 ,
then n (1 + w„) = 0 ; and conversely.
2nd. If 2 log (1 + w») be convergent, then n (1 + m„) converges
to a limit which is finite both ways ; and conversely.
n
3rd. If 2 log (1 + %) is divergent, and L% log (1 + Uj) = + 00 ,
then n (1 + «„) is divergent; and conversely.
4th. If 2 log (1 + M„) oscillates, then n (1 + m,^) oscillates ;
and conversely.
§ 24.] If we confine ourselves to the case where w„ has
ultimately always the same sign, it is easy to deduce a simple
criterion for the convergency of n (1 + w„).
If Lun < 0, then S log (1 + m„) = - 00 , and n (1 + m„) = 0.
If Lun >0,% log (1 + Un) = + 00 , and n (1 + m„) is divergent.
It is therefore a necessary condition for the convergency of
n (1 + Ur) that LUn — 0.
Since Lun = 0, 2/ (1 + m,J'/"»i = e ; hence L log (1 + M„)/Mn = 1-
It therefore follows from § 4 that 2 log (1 + u^ is convergent or
divergent according as Sm^ is convergent or divergent. More-
over, if Un be ultimately negative, the last and infinite parts of
5w» and 51og(l + u^ will be negative ; and if m„ be ultimately
* The logarithm of a complex number has not yet been defined, much
less discussed. Given, however, the theory of the logarithm of a complex
variable there is nothing illogical in making it the basis of the theory of
infinite products, as the former does not presuppose the latter.
§§ 22-24 AND OSCILLATING INFINITE PRODUCTS 159
positive, the last and infinite parts of 2m„ and 2 log (1 + w„) will
be positive. Hence the following conclusions —
If the terms of 2m„ become ultimately infinitely small, and
hate ultimately the same sign, then
1st. n (1 + w„) is convergent, if 2m„ he convergent ; and con-
versely.
2nd. n (1 + w„) = 0, */* %Un diverge to - ao ; and conversely.
3rd. n(l + w„) diverges to + co, if ^Un diverge to + cc ; and
conversely.
Since in the case contemplated, where u^ is ultimately of
invariable sign, the convergency of n (1 + m„) does not depend on
any arrangement of signs but merely on the ultimate magnitude
of the factors, the infinite product, if convergent, is said to be
absolutely convergent. It is obvious that any infinite product in
which the sign of Un is not ultimately invariable, but which is
convergent when the signs of w„ are made all alike, will he,
a fortiori, convergent in its original form, and is therefore said
to be absolutely convergent ; and we have in general, for infinite
products of real factors, the theorem that 11 (1 + w„) is absolutely
convergent when %Un is absolutely convergent ; and conversely.
Cor. If either of tJie two infinite products n (1 + m„), n (1 - u,)
be absolutely convergent, the other is absolutely convergent.
For, if %Un is absolutely convergent, so is S ( - «„) ; and
conversely.
Example 1. (1 + l/P) (1 + 1/22) . . . (i + i/n-') ... is absolutely conver-
gent since 21/h^ is absolutely convergent.
Example 2. (1 - 1/2) (1 - 1/3) ... (1 - l/?t) . . . has zero for its value
since S ( - l/;i) diverges to - oo .
Example 3. (1 + 1/^2) (1 + 1/^3) . . . (1 + 1/^/0 • • . diverges to +qo
since S (If/Jn) diverges to + oo .
Example 4. (1 + 1/^1) (1 - l/v/2) (1 + 1/^3) (1 - l/x/4) . . . Since the
sign of «„ is not ultimately invariable, and since the series 2 ( - l)"~'/\/'^ i^
not absolutely convergent, the rules of the present paragraph do not apply.
We must therefore examine the series S log (1 + ( - 1)"~VV«)- The terms of
this series become ultimately infinitely small ; therefore we may (see § 12)
associate every odd term with the following even term. We thus replace the
series by the equivalent series
Slog {1 + 1/V(2tt - 1) - 1/V(2n) - 1/V(4n« - 2h)}.
160 INDEPENDENT CRITEEIA CH. XXVl
It is easy to show that
1/V(2» - 1) - 1/V(2h.) - y^iin' - 2;0 <0,
for all values of n>l. Hence the terms of the series in question ultimately
become negative. Moreover, 1/V(2m - 1) - 1/\/(2«) - 1/V(47i-' - 2n) is ulti-
mately comparable with -\j'2n. Hence S log (l + ( - 1)"~VV") diverges to
- 00 . The value of (1 + lyi) (1 - 1/^2) (1 + l/V^) (1 - W^) • • • is there-
fore 0. This is an example of a semi-convergent product.
Example 5. e^+^e-'^~ie^^ie~'^~^ . . . The series 2log(l + M„) in this
case becomes
(l + l)-(l + i) + (l + i)-(l + |)+. . .
which oscillates. The infinite product therefore oscillates also.
Example 6. 11 (1 - a;"'"^//*) is absolutely convergent if a; <1, and has 0 for
its value when x — 1.
§ 25.] We have deduced the theory of the convergence of
infinite products of real factors from the theory of infinite series
by means of logarithms ; and this is probably the best course for
the learner to follow, because the points in the new theory are
suggested by the points in the old. All that is necessary is to
be on the outlook for discrepancies that arise here and there,
mainly owing to the imperfectness of the analogy between the
properties of 0 (that is, +a- a) and 1 (that is, x a h- a).
It is quite easy, however, by means of a few simple inequality
theorems*, to deduce all the above results directly from the
definition of the value of n (1 + m„).
If Pn have the meaning of § 22, then we see, by exactly the
same reasoning as we used in dealing with infinite series, that
the necessary and sufficient conditions for the convergency of
n (1 + M„) are that P^ be not infinite for any value of n, however
large, and that L (Pn+m — J^n) = 0 ; and that the latter condition
includes the former.
If we exclude the exceptional case where L P„ = 0, then,
n=»oo
since Pn is always finite, the condition L (Pn+m - Pn) ^ 0 is
equivalent to L (P„+m/P„ - 1) = 0, that is, LPn+m/Pn = 'i-
* See Weierstrass, Abhandlungen aus d. Functionenlehie, p. 203 ; or
Crelle's Jour., Bd. 51.
§§ 24-26 COMPLEX PRODUCTS 161
If, therefore, we denote (1 + Un+\) (1 + M„+a) . . . (1 + ^,1+^)
by mQn , we may state the criterion in the following form, where
Un may be complex : —
Tlie necessary and sufficient condition that 11 ( 1 + m„) converge
to a finite limit differing from zero is that L \ mQn - 1 1 = 0, for
71=00
all values of m.
For, since L \ ^Qn - 1 1 = 0, given any quantity e however
small, we can determine a finite integer v such that, if %<j;v,
1^^,1-1 |<c. Therefore, since w.Qn = Pn+mlPn, we have in
particular
l-€<P;,+JP^<l + €.
Since V is finite, Pv is finite both ways by hypothesis. Therefore
(l-e)P,<P,+^<(l + c)P,.
Since m may be as large as we please, the last inequality shows
that P„ is finite for all values of n however large.
Again, since P„ is not infinite, however large n, the con-
dition L 1 raQn " 1 1 = 0, wMch is cquivalcut to L mQn = 1, leads
n=Qo n=Qo
to L Pn+m= L P„. The possibility of oscillation is thus ex-
n=oo n=co
eluded. The sufficiency of the criterion is therefore established.
Its necessity is obvious.
We shall not stop to re-prove the results of § 24 by direct
deduction from this criterion, but proceed at once to complete
the theory by deducing conditions for the absolute convergence
of an infinite product of complex factors.
§ 26.] n (1 + «i„) is convergent ?/ n (1 + | w„ | ) is convergent.
Let Pn~\un\, so that p„ is positive for all values of n, then,
since n (1 + p„) is convergent,
L {(1 + pn+i) (1 + Pn+2) . . . (1 + Pn+m) " 1} = 0 (1).
Now
mQn - 1 = (1 + «f™+i) (1 + W«+2) . . . (1 + W„+m) " 1,
= 2tUnJf.x + AUn+iUn+2 + . . . + Un+iUn+2 • • • ^»+ni«
Hence, by chap, xii., §§9, 11, we have
O^lmQn - 1 I ^2p,i+i + ^Pn+lPn+2 + • •. . + Pn+lPH+2 • • • Pn+m,
^-(1 + Pn+l) (1 + Pn+2) . . . (1 + Pn+m) " 1-
C. II. 11
162 ASSOCIATION AND COMMUTATION CH. XXVI
Hence, by (1),ZU$„-1 1 = 0.
Remark. — The converse of this theorem is not true ; as may
be seen at once by considering the product (l + l)(l-^)(l + 5)
(1 - -J) . . . , which converges to the finite limit 1 ; although
(1 + 1)(1 + ^)(1 + ^) (1 + ^) ... is not convergent.
When n(l + M„) is such' that n(l + |w„|) is convergent,
11(1 + Wre) is said to he absolutely convergent. i/'n(l + M„) he
convergent, hut n (1 + 1 ?^,i |) non-convergent, n (1 + w„) is said to he
semi-convergent. The present use of these terms includes as a
particular case the use formerly made in § 24.
§27.] If '^\un\ be convergent, then n(l + z^„) is absolutely
convergent; and conversely.
For, if %\Un\ be convergent, it is absolutely convergent, seeing
that \un\ is by its nature positive. Hence, by § 24, n (1 + [ «„ |)
is convergent. Therefore, by g 26, n (1 + w„) is absolutely con-
vergent.
Again, if n(l+w„) be absolutely convergent, n(l + |M„|)
is convergent; that is, since |m„| is positive, n(l + |M„|) is
absolutely convergent. Therefore, by § 24, 2 1 tin \ is absolutely
convergent.
Cor. If %Un he absolutely convergent, n (1 + Unx) is absolutely
convergent, where x is either independent of n or is such a function
of n that X I a; 1 4= CO when n= co.
Example 1. 11 (1 - a;"/n) is absolutely convergent for all complex values
such that I a; I < 1, but is not absolutely convergent when | x | = 1.
Example 2. ^.{l-xjn^), where x is independent of n, is absolutely
convergent.
§ 28.] After what has been done for infinite series, it is not
necessary to discuss in full detail the application of the laws of
algebra to infinite products. We have the following results —
I. Ths law of association may be safely applied to the factors
^ n (1 + w„) provided Lun = 0 ; hut not otherwise.
H. The necessary and sufficient condition that n (1 + w„) shall
converge to the same limit {finite both ways), whatever the ordsr of
itsfactws, is that the series %Un be absolutely convergent.
When Un is real, this result follows at once by considering the
series 2 log (1 + m„) ; and the same method of proof applies when
§§ 26-28 UNIFORM CONVERGENCE 163
Un is complex, the theory of the logarithm of a complex variable
being presupposed*.
An infinite product which converges to the same limit what-
ever the order of its factors is said to be unconditionally convergent.
The theorem just stated shows that unconditional convergence and
absolute convergence may be taken as equivalent terms. A con-
ditionally convergent product has a property analogous to that of
a conditionally convergent series ; viz. that by properly arranging
the order of its terms it may be made to converge to any value
we please, or to diverge.
III. If both n (1 + w„) and Tl {I + v^^ be absolutely convergent^
then n {(1 + Uj) (1 + v^] is absolutely convergent, and has for its
limit {n (1 + Un)] X {n (1 + Vr^} ; also n {(1 + w„)/(l + v^] is abso-
lutely convergent, and has for its limit {n(l + w„)}/{n (1 +«?„)},
provided none of the factors q/II (1 +Vn) vanish.
If Qn denote (1 + w„+i) (1 + w^+s) • . ., the total residue of
the infinite product n (1 + w„) after n factors, then, if the product
converges to a finite limit which is not zero, given any positive
quantity e, however small, we can always assign an integer v such
that \Qn-l\<€, when w<|;i'.
If Un be a function of any variable z, then, when c is given,
V will in general depend on z.
If however, for all values of z within a given region R in
Argand's diagram an integer v independent of z can be assigned
such that
\Qn-l\<^,
when n<^v, then the infinite product is said to be uniformly
CONVERGENT witMn U.
IV. If fin, z) be a finite single valued function of the integral
variable n and of z, continuous as regards z within a region R,
and if Il{l +f(n, z)} converges uniformly for all values of z
within R to a finite limit <j>{z), then ff> (z) is a continuous function
of z within R.
Let z and z' be any two points within R, then, since
* See Harkness and Morley, Treatise on the Theory of Functions (1893),
§ 79 ; or Stolz, Allgemeine Arithmetik, Thl. ii. (1886), p. 238.
11—2
164
CONTINUITY OF INFINITE PRODUCT CH. XXVI
(f> (z) and <^ (z') are each finite both ways, it is sufficient to prove
that L I <l> {z')/cf> (z) I = 1.
z—z'
Let
<l>{z) = PnQn, i>(z') = P'nQ'n,
where P„, Qn, &c. have the usual meanings.
Since the product is uniformly convergent, it is possible to
determine a finite integer v (independent of z or z) such that,
when fi'^v, we have
\Q„-1\<€, and 1Q'„-1|<€,
where c is any assigned positive quantity however small. Hence,
in particular, we must have
\Qn\ = i + ee, \Q'n\ = i + x^;
where 6 and x are real quantities each lying between - 1 and + 1.
Now
<}>(z')
<i>{z)
=
P.
Also, since L \P'v/Pv\ = l, v being a finite integer, and, z
z=i!
being at our disposal, we can without disturbing v choose \z- z'\
so small that \ P'vjPx
Hence
^{z)
-1
1 + i/'e, where - 1 <i/^< + 1
(1 + i/^e) (1 + xe)
l + ^e
i + e€
< e
3 + 6
1-e"
Since € (3 + €)/(! — c) can, by sufficiently diminishing e, be
made as small as we please, it follows that L \ 4> {z')/<t> (z)\ = l.
Cor. 1. 1/ fin and Wn (z) satisfy the conditions of the example
in § 16, then n {1 + finWn {z)} is a continuous function of z within
the region E.
For, if we use dashes to denote absolute values, we have
I Q« - 1 |<(1 + f^'n+lW'n+i) (1 + /n+2«'^'n+2) ... - 1.
§ 28 CONTINUITY OF INFINITE PRODUCT 165
Since Wn{z) is finite for all values of n and z, we can find a finite
upper limit, g, for w'n+i, w'n+2, . . . Therefore
I Q„ - 1 1 < (1 + gix^+^) (1 + ^M'n+2) . . . - 1.
Since 2ju.'n is absolutely convergent, ^g/n is absolutely con-
vergent. Hence n (1 + g/x^) is absolutely convergent ; and we
can determine a finite integer v (evidently independent of z,
since g and fi'„ do not depend on z), such that, when n^^v,
(1 + gfx.'n+i)(l +gfJ-'n+2) . . . -1<€. Hence we can determine v,
independent of z, so that | ^„ - 1 1 < e, where e is a positive
quantity as small as we please. It follows that n {1 + finWn(z)}
is uniformly convergent, and therefore a continuous function of
z within B.
Cor. 2. If 'XanZ" be convergent when \z\ = Il, then n (1 + a^s")
converges to <^ (z), where </> {z) is a finite continuous function of z
for all values of z such that \z\<B.
Cor, 3. If f{n, y) he finite and single-valued as regards n,
and finite, single-valued, and continuous as regards y within the
region T, and if "^fin, y) z^ be absolutely convergent when \z\=B;
then, so long as \z\<E, n (1 +f(n, y) z^) converges to \f/ (y), where
\^{y) is a finite continuous function of y within T.
Cor. 4. If %an be absolutely convergent, then n (1 + anz)
converges to ij/ (z), where if/ (z) is a finite and continuous function
of z for all finite values of z.
We can also establish for infinite products the following
theorem, which is analogous to the principle of indeterminate
coefficients.
V. If, for a continuum of values of z including 0, 11 (1 + anZ^)
and n (1 + bnZ^) be both absolutely convergent, and n (1 + anZ^) =
n (1 + bnZ% then ax = bi, az^b^, . . ., an = bn, . . .
For we have
2 log (1 + «„2;") = 2 log (1 + bnZ"),
both the series being convergent.
Hence for any value of z, however small, we have, after
dividing by z,
^a^z""-^ log (1 + ttr^zy-'" = ^bnz''-' log (1 + bnz'y'''n''\
166 ROOTS OF AN INFINITE PRODUCT CH. XXVI
Since L log (1 + a^^;")^''"''''" = 1, we have, for very small
values oi z, ' ' '
a^A^ + a^A^z + aiA^z^-^ . . . = h^Br + KB<iZ + h3B3Z^ + . . . (1),
where Ai, A2, . . ., B^, B^ difier very little from unity, and all
have unity for their limit when z-Q.
Hence, since Sa„;2;""^ and :S6„«""^ are, by virtue of our
hypotheses, absolutely convergent, we have
L {a^A^z + a^A^z^ + . . . ) =^ 0
L {hB^z + h^B^z" + . . . ) = 0.
Hence, if in (1) we put z = Q, we must have
ai L Ai = bi L Bi.
But LAi = LBi = l; therefore ai = bi. Removing now the
common factor 1 + aiZ from both products, and proceeding as
before, we can show that (Zg = &2 ; and so on.
§ 29.] The following theorem gives an extension of the
Factorisation Theorem of chap, v., § 15, to Infinite Products.
If iff {z) = 11(1 + anz) be convergent for all values of z, in the
sense that L \ mQn - 1 1 = 0, when n= ^ ,no matter what value m
may have, then \j/ (z) will vanish if z have one of the values — l/ai,
- 1/«2, . . . , - l/ctr, • • • ) (^nd, if >/' (z) = 0, then z must have one
of the valties — l/ai, - l/a^, . . ., - l/ar, . . .
In the first place, we remark that, by our conditions, the
vanishing of L^Qn when w = qo is precluded. The exceptional
case, mentioned in §23, where Slog(l +a„;2;) diverges to -co,
and n (1 + ttnz) converges to 0 for all values of z, is thus excluded.
Now, whatever n may be, we have
^{z)=FnQn (1).
Suppose that we cause z to approach the value — l/ar. We
can always in the equation (1) take n greater than r ; so that
1 + arZ will occur among the factors of the integral function P„.
Hence, when z = - l/ar, we have P„ = 0, and therefore, since
Qn+QO, ilf{-l/ar) = 0.
Again, suppose that ^{z)^0. Then, by (1), PnQ» = 0.
But, since u may be as large as we please, and LQ^-l when
§§ 28, 29 FACTORS OF EQUAL PRODUCTS 167
n='X), we can take n so large that Qa + 0. Hence, if only n
be large enough, the integral function P„ will vanish. Hence z
must have a value which will make some one of the factors of
Pn vanish ; that is to say, z must have some one of the values
It should be noticed that nothing in the above reasoning
prevents any finite number of the quantities ai, a^, . . ., ar, . . .
from being equal to one another ; and the equal members of the
series may, or may not, be contiguous. If there be /*„ contiguous
factors identical with 1 + ttnZ, the product ip (z) will take the form
n (1 + anzY" ; and it can always be brought into this form if it be
absolutely convergent, for in that case the commutation of its
factors does not affect its value.
Cor. 1. If z lie within a continuum (z) w/iich includes all the
values
-1/^1, -1/^2, . . ., -!/«»,. . . (A),
and -i/bi, -l/b^, . . ., -1/bn, . . . (B),
if 11(1 + anzY" and n (1 + bnzy» be absolutely convergent for all
values of z in {z), iffiz) and g{z) be definite functions of z which
become neither zero nor infinite for any of the values (A) or (B),
and if, for all values of z in {z),
f{z) n (1 + anzY'^ = g{z)U{l^. b^z)"' (1),
then must each factor in the one product occur in the other raised
to the same poiver ; and, for all the values of z in (z),
f(^) = g{^) (2).
For, since, by (1), each of the products must vanish for each
of the values (A) or (B), it follows that each of the quantities
(A) must be equal to one of the quantities (B) ; and vice versa.
The two series (A) and (B) are therefore identical.
Since the two infinite products are absolutely convergent, Ave
may now arrange them in such an order that ai = bi, az^ba, . . .,
&c., so that we now have
f{z) (1 + aizyi (1 + a2zy* . . .=g{z){l + a^z^ (1 + a<2zy^ . . . (3).
Suppose that )u.i + vi, but that /aj, say, is the greater; then
we have, from (3),
/(«)(! +ai;2;)'^>-''i(l+a,<£)'*2. . . = g {z) {\ -v a^zy^ . . . (4).
168 FACTORS OF EQUAL PRODUCTS CH. XXVI
Now this is impossible, because the left-hand side tends to 0
as limit when z = — l/ai, whereas the right-hand side does not
vanish when z = — l/ai. We must therefore have aii = Vi; and,
in like manner, fta = ''2 ; and so on.
We may therefore clear the first n factors out of each of the
products in (1), and thus deduce the equation
f{z)Qn = g{z)qn (5),
where Qn and Q'n have the usual meaning. The equation (5) will
hold, however large n may be. Hence, since LQn = LQ'n = 1, we
must have
/(^)=^(4
Cor. 2. From this it follows that a given function of z which
vanishes for any of the values (A) and for no others within the
continuum (z), can he expressed within {z) as a convergent infinite
product of the form f {z) n (1 + anZ)^"" {where f{z) is finite and not
zero for all finite values of z within (z)), if at all, in one way only.
If the infinite product be only semi-convergent, the above
demonstration fails.
It may be remarked that it is not in general possible to
express a function, having given zero points, in the form described
in the corollary. On this subject the student should consult
Weierstrass, Ahhandlungen aus der Functionenlehre, p. 14 et seq.
ESTIMATION OF THE RESIDUE OF A CONVERGING SERIES OR
INFINITE PRODUCT.
§ 30.] For many theoretical, and for some practical purposes,
it is often required to assign an upper limit to the residue of an
infinite series. This is easily done in what are by far the two
most important cases, namely: — (1) Where the ratio of converg-
ence {pn = Un+i/un) ultimately becomes less than unity, and the
terms are all ultimately of the same sign ; (2) Where the terms
ultimately continually diminish in numerical value, and alternate
in sign.
Ca^e (1). It is essential to distinguish two varieties of series
§§ 29, 30 RESIDUE OF A SERIES 169
under this head, namely : — {a) That in which p„ descends to its
limit p ; {b) That in which p„ ascends to its limit p.
In case {a), let n be taken so large that, on and after n, Pn is
always numerically less than 1, and never increases in numerical
value. Then
Rn = Un+\ + Wri+2 + Un+'i + .
= Un4.^ i 1 + + • +
•■}-
^re+1 Wre+2 W^+1
= Un+\ {1 + Pra+1 + Pn+\ Pn+2 + Pn+1 Pn+2 Pn+3 + • • •}•
Therefore, if dashes be used to denote the numerical values,
or moduli, of the respective quantities, we have
R'n'^u'n-\-\ {1 + p'n+\ + P n-\-\ + • • •},
>m'„+i/(1-p'„+i),
>>m'„+,/(1 - U'n+2fu'n+,) (1).
And also, for a lower limit,
n'n<u'n+.Kl-p) (2).
In case (b), let n be so large that, after n, p» is numerically
less than 1, and never decre'ases in numerical value. Then
Rn - Un+i {1 + Pn+l + Pn+2 Pn+l + ...}•
B'n:^u'n+i {1 + p + p' + . . .},
:^u'n+J{l-p) (3);
and we have also
R'n^u'n+i/(l — p'n+i), ,
<t:M'„+i/(l - u'n+^/u'n+i) (4).
Case (2). When the terms of the series ultimately decrease
and alternate in sign, the estimation of the residue is still
simpler. Let n be so large that, on and after n, the terms never
increase in numerical value, and always alternate in sign. Then
we have
>u'n+i (5);
<^Un+i~u'n+2 (6).
170 RESIDUE OF AN INFINITE PRODUCT CH. XXVI
§ 31.] Residue of an Infinite Product. Let us consider the
infinite products, 11 (1 + w„) and 11 (1 — m„), in which w„ becomes
ultimately positive and less than unity. If the series %Un converge
in such a way that the limit of the convergency-ratio p„ is a
positive quantity p less than 1, then it is easy to obtain an
estimate of the residue. Let Q^, Q'n denote the products of all
the factors after the nth. in n (1 + m„) and 11 (1 -?0 respectively,
so that Qn>l, and Q'n<l. We suppose n so great that, on
and after n, Un is positive, Pn less than 1, and either («) Pn never
increases, or else {b) p„ never decreases. In case {a), 2m„ falls
under case (1) (a) of last paragraph ; in case {b), %Un falls under
case (1) {b) of last paragraph. We shall, as usual, denote the
residue of %Un by Rn ; and we shall suppose that n is so large
that \Rn\<l. ■
Now (by chap, xxiv., § 7, Example 2),
.'■ ' ) Q« = (1 + Mn+l) (1 + %+2) . • • ,
> 1 + Un+x + Un+2 + . . . ,
>l + Rn ' (1).
Q\ = (1 - Un+l) (1 - Un+^ . . . , .
>l-Rn (2).
Also,
IjQn = {1 - Mn+]/(l + <+l)} {1 - M„+2/(l + Un^^)] • • .,
> 1 - M„+i/(l + W«+i) - Un^ilil + Mft+s) - • • • ,
> 1 — Un+i — Un+2 ""•••>, . ,
>1-Rn.
Whence Qn-K Rn/il - Rn) (3).
In like manner,
l/Q'n = {1 + Un+lRl - Un+i)} {l + 2««+2/(l " Un+2)} • • -,
> 1 + M„+i/(l - Un+i) + Un+oRl - Ua+2) + . • • ,
> 1 + Un+i + Un+2 + . . . ,
>1+Rn.
Whence 1 - Q'n>Rnl{l + i^n) (4).
§§31,32 DOUBLE SERIES DEFINED l7l
From (1), (2), (3), and (4) we have
En<Qn-l<Itn/{l-Bn) (5);
Rnl{l+Rn)<l-Q'n<Rn (6).
Since upper and lower limits for B^ can be calculated by-
means of the inequalities of last paragraph, (5) and (6) enable us
to estimate the residues of the infinite products 11 (1 + Un) and
n(i-%).
Example. Find an upper limit to the residue of 11 (1 - a;"/)t), x<l.
Here u„=x^ln, /)„=a;/(l + l/n), p=x. The series has an ascending con-
vergency-ratio ; and we have -Rn<Mn+i/(l~p)<^"'*'V(" + l) (1- ^)' There-
fore, 1 - Q'„<a;"+i/(n + l) (1 -a;). Hence, if P'„ be the nth approximation to
n(l-x"/n), P'„ differs from the value of the whole product by less than
100a;»+V(n + l)(l-^) °lo of P'n itself.
CONVERGENCY OF DOUBLE SERIES.
§ 32.] It will be necessary in some of the following chapters
to refer to certain properties of series which have a doubly in-
finite number of terms. We proceed therefore to give a brief
sketch of the elementary properties of this class of series. The
theory originated with Cauchy, and the greater part of what
follows is taken with slight modifications from note vin. of the
Analyse Algehrique, and § 8 of the Resumes Anahjtiqiies.
Let us consider the doubly infinite series of terms repre-
sented in (1). We may take as the general, or specimen term,
Um,n, where the first index indicates the row, and the second the
column, to which the term belongs. The assemblage of such
terms we may denote by 2m;„, „; and we shall speak of this
assemblage as a Double Series*.
A great variety of definitions might obviously be given of
the sum to a finite number of terms of such a series ; and,
corresponding to every such definition, there would arise a
definite question regarding the sum to infinity, that is, regarding
the convergency of the series.
There are, however, only four ways of taking the sum of the
double series which are of any importance for our purposes.
Sometimes the term " Series of Double Entry" is used.
172
DIFFERENT DEFINITIONS OF THE CH. XXVI
First Way. — We may define the finite sum to be the sum of
all the mn terms within the rectangular array OKMN. This
we denote by 8m,, n- Then we may take the limit of this by
first making m and finally n infinite, or by first making n in-
finite and finally m infinite. If the result of both these limit
operations is the same definite quantity 8, then we say that
"^Um, n converges to 8 in the first way.
0
A B C D
K
A'
B'
C
D'
"i.i
«1.2
W].2
"1.4
"i,«
"l.n+1
"2,1
«2.2
«2,3
"2,4
"2.™
"2.W+1
%1
«E,2
"3.3
"3,4
"3.»
"3,»l+l
"4.1
"4.2
"4,3
"4.4
"4,™
"4.»+1
•
•
•
K'
"«,1
"«,2
"n,3
"n.4
"«.n
"n.Ji+1
•
•
L
'',n,l
"«.,2
"m.3
",n.4
"»>,»
"m.n+1
N
"w+l. 1
"m4-], 2
"»n+l,3
"to+1.4
"m+1. n
M
"m+l.n+l
(1).
It may, however, happen — 1st, that both these operations
lead to an infinite value ; 2nd, that neither leads to a definite
value ; 3rd, that one leads to a definite finite value, and the
§ 32 SUM OF A DOUBLE SERIES 173
other not ; 4th, that one leads to one definite finite value, and
the other to another definite finite value*. In all these cases
we say that the series is non-convergent for the first way of
summing.
Second Way. — Sum to n terms each of the series formed by
taking the terms in the first m horizontal rows of (1) ; and call
the sums T-i^n, Ti,n, • • ., ^m,»- Define
^ m, «— -* I, n+ -* 2, n + • • • + -'m, n \^)
as the finite sum.
Then, supposing each of the horizontal series to converge
to Ti, 7^2, . . ., jTot respectively, and %Tm to be a convergent
series, define
>Sf' = Ta+jr2 + . . . + 7;, + . . . ad 00 (3)
as the sum to infinity in the second way.
Third Way. — Sum to m terms each of the series in the first
n columns; and let these sums be Ui^m, £^2, m, • • •, Un,m'
Define
S"m, n= U'l,m+ Uo,m + - • •+ Un,m (4)
as the finite sum.
Then, supposing these vertical series to converge to Ui, U^,
. . ., Un respectively, and %Un to be a convergent series,
define
^"=J7i+ t72 + . . .+ Z7„ + . . . ad 00 (5)
as the sum to infinity in the third way.
So long as m and n are finite, it is obvious that we have
C*' _ C" o
*J m, n " m,n — *J»i, n j
so that, for finite summation, the second and third ways of
summing are each equivalent to the first.
The case is not quite so simple when we sum to infinity. It
is clear, however, that
8'= L { L Sm, n\ (6) ;
and S"=L{LS^,n} (7);
* Examples of some of these cases are given in § 35 below.
174 DOUBLE SERIES OF POSITIVE TERMS CH. XXVI
SO that S' and S" will be equal to each other and to S when the
two ways of taking the limit of Sm,n both lead to the same
definite finite result*.
Fourth Way. — Sum the terms which lie in the successive
diagonal lines of the array, namely, AA', BB', CC, . . ., KK' \
and let these sums be i)2, A, • ■ ., ^n+i respectively ; that is,
A = «^i,l, A = «^l,2 + M2, X, . . • , -On+i = «<i, n + ^2, n-i + . • • + «»,!•
Define
>S""„=A + A + . . . + A (8)
as the finite sum ; and, supposing SZ)„ to be convergent, define
>S""=A + A + . . .+-^« + - . . adoo (9)
as the sum to infinity in the fourth way.
The finite sum according to this last definition includes all
the terms in the triangle OKK' ; it can therefore never (except
for w = w = 1) coincide with the finite sum according to the
former definitions. Whether the sum to infinity {8'") according
to the fourth definition will coincide with S, S', or S", depends
on the nature of the series. It may, in fact, happen that the
limits S, S', S" exist and are all equal, and that the limit S'" is
infinite t.
§ 33.] Double series in which tJie terms are all ultimately of
the same sign. By far the most important kind of double series
is that in which, for all values of m and n greater than certain
fixed limits, Um,n has always the same sign, say always the
positive sign. Since, by adding or subtracting a finite quantity
to the sum (however defined), we can always make any finite
number of terms have the same sign as the ultimate terms of
the series, we may, so far as questions regarding convergency
are concerned, suppose all the terms of %Um, » to have the same
(say positive) sign from the beginning. Suppose now (1) to
represent the array of terms under this last supposition ; and let
us farther suppose that %Um, n is convergent in the first way.
Then, since L (S^+p, n+q - Sm,n) = S-S = 0, when w = qo ,
n= CO whatever p and q may be, it follows that the sum of all
* For an illustration of the case when this is not so, see below, § 35.
t See below, § 35.
§§ 32, 33 DOUBLE SERIES OF POSITIVE TERMS 175
the terms in the gnomon between NMK and two parallels to
NM and MK below and to the right of these lines respectively,
must become as small as we please Avhen we remove NM suffi-
ciently far down and MK sufficiently far to the right.
From this it follows, a fortiori, seeing that all the terms of
the array are positive, that, if only m a,nd n be sufficiently great,
the sum of any group of terms taken in any way from the residual
terms lying outside OKMN will be as small as we please.
Hence, in particular,
1st. The total or partial residue of each of the horizontal
series vanishes when n= <x>.
2nd. The same is true for each of the vertical series.
3rd. The same is true for the series 5Z>„.
The last inference holds, since S"'n obviously lies between
^q, n-q and. On-i, n-1'
Hence
Theorem I. ^f all the terms of %Um, n be positive, and if the
series he convergent in the first sense, then each of the horizontal
series, each of ths vertical series, and the diagonal series will be
convergent, and the double series will be convergent in the re-
maining three ways, always to the same limit.
If we commutate the terms of a double series so that the
term Um, n becomes the term Um', %', where m =f{m, n), n' = g {m, n),
f{m, n) and g {m, n) being functions of m and n, each of which has
a distinct value for every distinct pair of values of m and n (say
non-repeating functions), and each of which is finite for all finite
values ofm and n (Restriction A*), then we shall obviously leave
the convergency of the series unaffected. Hence
Cor. 1. If ^Um, n be a series of positive terms convergent in
the first way, then any commutation of its terms {under Re-
striction A) will leave its convergency unaffected; that is to say, it
will converge in all the four ways to the same limit S as before.
* No such restriction is usually mentioned by writers on this subject ;
but some such restriction is obviously implied whep it is said that the terms
of an absolutely convergent series are commutative ; otherwise the character-
istic property of a convergent series, namely, that it has a vanishing residue,
would not be conserved.
176 DOUBLE SERIES OF POSITIVE TERMS CH. XXVI
Cor. 2. If the terms {all positive) of a convergent single series
^Un be arranged into a double series ^u^.', n'> where m' and n a/re
functions of n subject to Restriction A, then %Um',n' will converge
in all four ways to the same limit as ^Un.
It should be noticed that this last corollary gives a further
extension of the laws of commutation and association to a series
of positive terms ; and therefore, as we shall see presently, to
any absolutely convergent series.
Let us next assume that the series %Um, » is convergent in the
second way. Then, since ^T^ is convergent, we can, by suffi-
ciently increasing m, make the residue of this series, that is, the
sum of as many as we choose of the terms below the infinite
horizontal line NM, less than ^e, where e is as small as we
please. Also, since each of the horizontal ^ries is, by our
hypothesis, convergent, we can, by sufficiently increasing n, make
the residue of each of them, less than €/2w ; and therefore the
sum of their residues, that is, as many as we please of the terms
above iVJ/ produced and right of MK, less than ^e. Hence, by
sufficiently increasing both m and n, we can make the sum of
the terms outside OKMN, less than c, that is, as small as we
please. From this it follows that ^m^, m is convergent in the
first way, and, therefore, by Theorem I., in all the four ways.
In exactly the same way, we can show that, if %Um, % is con-
vergent in the third way, it is convergent in all four ways.
Finally, let us assume that 2^^, n is convergent in the fourth
way. It follows that the residue of the diagonal series %Dp can,
by making p large enough, be made as small as we please.
Now, if only m and n be each large enough, the residue of S^., %,
that is, the sum of as many as we please of the terms outside
OKMN, will contain only terms outside OKK', all of which are
terms in the residue of 8'" p. Hence, since all the terms in the
array (1) are positive, we can make the sum of as many as we
please of the terms outside OKMN as small as we please, by
§§ 33, 34 cauchy's test for absolute convergency 177
sufficiently increasing both m and n. Therefore ^tim,n is con-
vergent in the first way, and consequently in all four ways.
Combining these results with Theorem L, we now arrive at
the following : —
Theorem 11. If a double series of positive terms converge in
any one of the four ways to the limit S, it also converges in all the
other three ways to the same limit S ; and the subsidiary single
series, horizontal, vertical, and diagonal, are all convergent.
Cor. Any single series %Un consisting of terms selected from
^'U'm.n {under Restriction A) will be a convergent series, if Sw„i,n
be convergent.
Restriction A will here take the form that n' must be a
function of m and n whose values do not repeat, and which is
finite for finite values of m and n.
Example. The double series Sx'"^/'* is convergent for all values of x
and y, such that 0<a;<+l, 0<j/<+1.
For the [m + l)th horizontal series is x^S?/", which converges to a;'"'/(l - y)
since 0<j/< +1. Also Sa;'"/(1 - y) converges to 1/(1 -x)(l-i/) since 0<x< +1.
§ 34.] Absolutely Convergent Double Series. — When a double
series is such that it remains convergent when all its terms are
taken positively, it is said to be Absolutely Convergent.
Any convergent series whose terms are all ultimately of the
same sign is of course an absolutely convergent series according
to this definition.
It is also obvious that all the propositions which we have
proved regarding the convergency of double series consisting
solely of positive terms are, a fortiori, true of absolutely con-
vergent double series, for restoring the negative signs will, if it
affect the residues at all, merely render them less than before.
In particular, from Theorem II. we deduce the following,
which we may call Cauchy's test for the absolute convergency of a
double series.
Theorem III. If u'm,n be the numerical or positive value of
Um,n, and if all the horizontal series of %u'm,n be convergent, and
the sum of their sums to infinity also convergent, then
1st. The Horizontal Series of '%Um,n are all absolutely con-
c. II. 12
178 EXAMPLES OF CAUCHY's TEST CIl. XXVI
vergent, and the sum of their sums to infinity converges to a
definite finite limit S.
2nd. ^Um,n converges to S in the first way.
3rd. All the Vertical Series a/re absolutely convergent, and
the sum of their sums to infinity converges to S.
4th. The Diagonal Series is absolutely convergent, and con-
verges to S.
5th. Any series formed by taking terms from %Um,n {under
Bestriction A) is absolutely convergent.
The like conclusions also follow, if all the vertical series, or if
the diagonal series of'Zu'm,n be convergent.
Cor. If %Un and S'Wn be each absolutely convergent, and con-
verge to u and v respectively, then % (u^Vi + Un-iV^ + . . . + UiV^) is
absolutely convergent, and converges to uv.
For the series in question is the diagonal series of the double
series %UmVn, which, as may be easily shown, satisfies Cauchy's
conditions.
This is, in a more special form, the theorem already proved
in § 14
Example 1. Find the condition that the double series S ( - )™„(7^a;'*-^y"*
(n<tni, qGq=1) be absolutely convergent; and find its sum.
The series may be arranged thus : —
1 + x+ x''+ . . . +a;"+. . .
~y - 2yx - Byx^ - . . . - (n + 1) j/x" - . . .
+ 2/2+ By"-x+ 6j/2a;2 + .,. + i(/i + l)(n + 2)r/2a;»+, . .
( _ )mym + ( _ )rr^^+^C,y^x + ( - I'^^+s^al/'^x^ +...+(- )'"„,4^(7„2/™a;» + . . .
If x' and y' be the moduli, or positive values, of x and y, then the series
^^'m,n corresponding to the above will be
1 + x'+ x'2+. . . +a;'»+. . .
+y'+2y'x' + 3y'x'''+. . . + (n + 1) j/'x'™ + . . .
In order that the horizontal series in this last may be convergent, it is
necessary and sufficient that x' < 1.
Also T'^^y=y'^j{l — x'y^'^^; hence the necessary and sufficient condition
that Sr'^ be convergent is that y'<.l-x', which implies, of course, that
The given series will therefore satisfy Cauchy's conditions of absolute
convergency if | a; | < 1, | a; | + 1 j/ 1 < 1, and consequently also | y [ < 1.
These being fulfilled, we have r^+i= ( - )"^'"/(l - x)'»+i ;
34, 35 EXCEPTIONAL CASES 179
1-x+y '
and the sum of the series, in whatever order we take its terms, is 1/(1 -x + ij).
Example 2. If Ur=x'' + X''^^ + x^^+ . . ., where x<l, show that
Mo , Wl "2 , _0,, _?!! _ ^' _ ^'_
2b'*'2i 22 ° 2** 21 2^ *'*
Let S denote the series on the left. Then S may be written as a double
series thus,
l(x2» + x2Va;2V. . .+x'''+. . .)
+ ^(0 +a;2Va;2'+. . .+x^- +. . .)
+ ^(0 +0 + X2-+. . . + a;2''+. . .)
Now each of the vertical series is absolutely convergent, and we have
J7„=x2" (1 - l/2"+i)/(l - i) = a;2'' (2 - 1/2»). SC7„ is of the same order of con-
vergence as Sx'^", hence it is absolutely convergent. Also all the terms of the
double series are positive. The double series therefore satisfies Cauchy's
conditions ; and its sum is the same as that of Sf7„, or of Sr„. Now
ST„=Wo/20 + Mi/2i + «2/22+. . .;
and Sl7„=2x2"(2-l/2«),
= 22x2" -2x272",
= 2wo-a;2"/2»-x2V2i-. . .
Hence the theorem.
§ 35.] Examples of tlie exceptional cases that arise when
a double series is not absolutely convergent. It may help to
accentuate the points of the foregoing theory if we give an
example or two of the anomalies that arise when the conditions
of absolute convergency are not fulfilled.
Example 1. It is easy to construct double series whose horizontal and
vertical series are absolutely convergent, and which nevertheless have not a
definite sum of the first kind ; but, on the other hand, have one definite sum
of the second kind and another of the third kind.
If the finite sum of the first kind, /S^.^, of a double series be A +f{m, n),
where A is independent of m and n, then it is easy to see that
'i^m,n=f{m, n)-f{m-l, n)-f{m,n-l)+f{m-l,n-l).
Hence we have only to give/(ni, n) such a form that
L { Lf(m,n)}dr L { Lf(m,n)},
12—2
180 EXCEPTIONAL CASES CH. XXVI
and we shall have a series whose sums of the second and third kind are not
alike, and which consequently has no definite sum of the first kind.
Suppose, for example, that/(m, n) = (?n + l)/(m + n + 2), then
u^„=(m + l)/(m + n + 2)-m/(m + n + l)-(m + l)/(TO + n + l) + m/(m + n),
= (m-n.)/(m + n) (m + n + 1) {m + n + 2).
It is at once obvious that the sums of the second, third, and fourth kind
for this series are all different. For in the first place we observe that
^m,n— ~'"'rum- Hence there is a "skew" arrangement of the terms in the
array (1), such that the terms equidistant from the dexter diagonal of the
array and on the same perpendicular to this diagonal are equal and of opposite
sign, those on the diagonal itself being zero. Each term of the diagonal series
2D„ is therefore zero ; and the sum of the fourth kind is 0.
Also, owing to the arrangement of signs, we have Tm,„= - U^„; and,
since each of the horizontal and each of the vertical series in this case is
convergent, Tjn= -U^, and therefore S'= -S".
Now
Tm,n= 2 [(m + l){l/(ni + n + 2)-l/(jn + n + l)}-m{l/(m+n+l)-l/(wi + n)}],
n=l
= (m + l){l/(m + 7i + 2)-l/(m + 2)}-m{l/(m + 7i + l)-l/(m + l)}.
Hence
Tm=- ('» + !)/("* + 2) + ml{m + 1) = - ll(m + 1) (m + 2).
The series ZT^ is therefore absolutely convergent ; and its sum to infinity
is obviously - 1 + 1/2 = - 1/2. Hence the double series has for its sum
- 1/2, + 1/2, or 0, according as we sum it in the second, third, or fourth way.
At first sight, the reader might suppose (seeing that the horizontal series
are all absolutely convergent, and that the sum of their actual sums is also
absolutely convergent) that this case is a violation of Cauchy's criterion.
But it is not so. For, if we take all the terms in the mth horizontal series
positively, and notice that the terms begin to be negative after m = n, then
we see that T'^ the sum of the positive values of the terms in the mth series
is given by
m <"
n=l n=in+l
= (m + l){l/(2m + 2)-l/(m + 2)}-m{l/(2?tt + l)-l/(m + l)}
-(m + l){0-l/(2m + 2)} + m{0-l/{2m + l)},
= l-2m/(2?;i + l)-(m + l)/(m + 2) + OT/(m + l),
= (m2 + m + l)/(ni + 1) (m + 2) (2m + 1).
Now the convergence of ST'^ is of the same order as that of Sl/m, that is
to say, ST'^ is divergent. Hence Cauchy's conditions are not fully satisfied ;
and the anomaly pointed out above ceases to be surprising. The present case
is an excellent example of the care required in dealing with double series
which are wont to be used somewhat recklessly by beginners in mathematics*.
* Before Cauchy the reckless use of double series and consequent
perplexity was not confined to beginners. See a curious paper by Babbage,
Phil. Trans. R.S.L. (1819).
§§ 35, 36 COMPLEX DOUBLE SERIES 181
Example 2. The double series S ( - )'"+™l/mn, whose horizontal and
vertical series are each semi- convergent, converges to the sum (log 2)* in the
second, third, or fourth way (see chap, xxviii., § 9, and Exercises xiii. 14).
But alteration in the order of the terms in the array would alter the sum
(see chap, xxviii. , § 4, Example 3).
Example 3. If the two series Sa„ and 2&„ converge to a and h respectively,
and at least one of them be absolutely convergent, then it follows from § 14
that the double series 2a„&„ converges to the same sum, namely ah, in all
the four ways, although it is not absolutely convergent, and its sum is not
independent of the order of its terms.
The same also follows by § 20, Cor., provided Sa„, 26„, 2 {0'rfii + o,n-i\
+ . . . + aj &„) be all convergent, even if no one of the three be absolutely
convergent*.
If, however, both Sa„ and 2&„ be semi-convergent, then the diagonal series
may be divergent, although the series converges to the same limit in the
second and third way. This happens with the series S( - )"»+"l/(m7i)* where
a is a quantity lying between 0 and \. This series obviously converges to the
finite limit (1 - 1/2* -f 1/3"* — ...)"■' iii the second and third ways. For the
diagonal series we have
D„= S l/r''(n-r)«.
r=l
Now, since 0<o<l, we have, by chap, xxiv., § 9, r"-|-(w-r)«<2i-<*{r
+ (n -»•)}"< 21-" R«.
Therefore
y. ^ 1 ^ ^'"""^ 1 r°-h(w-r)°
n 2i-*ft* r* {n - »•)* 2i-«n« r" (n - r)» '
2 » 1 2 n
2 — ;
Hence, if a = J, LD„<t2''; and, if a<J, I/D„ = qo, when n = cc. Therefore
2D„ diverges if 0 < a J> ^.
IMAGINARY DOUBLE SERIES.
36.] After what has been laid down in § 10, it will be
obvious that, in the first instance, the convergency of a double
series of imaginary terms involves simply the convergency of
two double series, each consisting of real terms only.
It is at once obvious that each of the two double series,
2a^„, %Pm,n, will be absolutely convergent if the double series
* See Stolz, Allgemeine Arithmetik, Th. i., p. 248.
182 'S.am,nOC'""y'^ CH. XXVI
^Ji'^^m.n + (i\n) IS Convergent. Hence, if u'm.n denote the
modulus of Um,n = ttm.n + i^m.n, We 866 that S^m.n will C0nV6rg6
to the same limit in all four ways if ^u'm,n he convergent.
In this case we say that the imaginary series is absolutely
convergent.
Since all the terms u'm,n are positive, we deduce from
Theorem II. the following : —
Theorem IV. If all the horizontal series in the double series
formed by the moduli of the terms of 2m;„,„ be convergent, and the
sum of their sums to infinity be also convergent, then the series
5wto,» is absolutely convergent, and all its subsidiary series are also
absolutely convergent.
Here subsidiary series may mean any series formed by
selecting terms from 2«m.» under Restriction A. Theorem IV.,
of course, includes Theorem III. as a particular case.
§ 37.] The following simple general theorem regarding the
convergency of the double series Sa^.m^^i/" will be of use in a
later chapter.
If the moduli of the coefficients of the series 2am,„a?™3/" have a
finite upper limit \, then 2a„,^„a;'"2/'' is absolutely convergent for
all values of x and y such that \x\<l, \y\<\.
For, if dashes be used to indicate moduli, we have, by
hypothesis, a'm,™^^- Hence the series ^a!m,nx''^y"^ is, a fortiori,
convergent if the series %\x'^y"' is convergent ; that is, if
^^'wiyn. jg convergent. Now, as we have already seen (§ 33),
this last series is convergent provided x <1 and7/'<l. Hence
the theorem in question.
Exercises VIII.
Examine the convergency of the series whose «th terms are the
following : —
(1.) (l + n)/(l + n-^). (2.) nP/(nP + a).
(3.) e-»'»^. (4.) l/(n2±l).
(5.) iy(n2-n){Vn-\/(n-l)}. (6.) a»/(a» + x»).
(7.) (nl)2x»/(2ji)l. (8.) n*ln\.
(9.) {(y + x»)/(2-a;»)}V», (10.) nlog{{2« + l)/(2n-l)} -1.
(11.) 1.3.5. . . (2h-1)/2.4.6. . . 2n.
(12.) {l/l»+l/2''+. . .+l/n«}/«».
^ 36, 37 EXERCISES VIII 183
(13.) ll{an + b). (14.) nl{an^+b).
(15.) m(m-l) . . .(m-n + l)ln\ (16.) {(7i + l)/(n + 2)}"/n.
/ir,\ en ii, J. "* m(wi + l) m(m + l)(wi + 2) . .
(17.) Show that - H -r-' H — -, :rrA ^r- + ... IS convergent or
^ ' n n(n + l) n{n + l) {n + 2)
divergent according as 7i - m > or }> 1.
(18.) Show that aV»» + aV™+i/{»n-i) + aV^+iAm+iJ+i/Cw+a) + ... is conver-
gent or divergent according as a< or <tl/e. (Bourguet, Notiv. Ann., ser.
II., t. 18.)
(19.) Exataine the convergency of Sl/zif^+i)/".
(20.) Show that "Ln^jin + !)'*+<» is convergent or divergent according as
a > or > 1 . (Bertrand. )
(21.) Show that Sl/n log ?i {log log n}* is convergent or divergent accord-
ing as a> or <1.
(22.) Show that Sl/(/H-l + cos mrY is convergent. (Catalan, Traite El.
d. Series, p. 28.)
Examine the convergency of the following infinite products : —
(23.) n{l+/(?i)r"}, where/(?j) is an integral function of n.
(24.) n{(a;2» + a;)/(a;2»+l)}. (25.) H {n'+^ I (n - 1^ {n + z)} .
(26. ) If 2/ (n) be convergent, show that, when ?i = oo ,
L{n{x+f{n))yi''=x.
1
(27.) If p denote one of the series of primes 2, 3, 5, 7, 11, . . . , then
2/(2)) is convergent if S/(p)/log jj is convergent. (Bonnet, Liouville's Jour.,
VIII. (1843), and Tchebichef, ih., xvii. (1852).)
(2d.) If a;<l, show that the remainder after n terms of the series
Vx + 2'rx^ + ^-rx^+ . . .
is <(n + l)''a;"+V{l-(l + l/n)'"a;}.
(29.) If Wfl, Mj, . . ., w„ be all positive, and 2w„a;" be convergent for all
values of x^ < «'■*, then
S^» |w„ - (n + 1) au^+, + (!i±_lH^±^ a-^u,+, - &c. |
will be convergent between the same limits of x.
(30.) Point out the fallacy of the following reasoning : —
Let 2 = 1 + 1 + ^4- ... ad 00,
then log,2 = l-^ + ^-i-H. . .
= (l + l+i+. . .)-2(i + i + K. . .)
= 2-22/2=0.
(31.) If p and p' be the ratios of convergence of 21/P^_i (n) {V-^ny+'^ and
21/P^ (n) {lrnY+'^' (see § 6), then L (p'„ - pj P^_^ (n) = a, when n = oo . What
conclusion follows regarding the convergence of the two series ?
(32.) If 2m„ is divergent, then 2u„/5f„_i" is divergent it a>l (where
S„=2tj + W2+ . . . +«„), and 2u„/S,j"'+i is convergent if a>0. Hence show
184 EXERCISES VIII CH. XXVI
that there can be no function 0 (n) such that every series Sm„ is convergent
or divergent, according as L 0(n)u„= or #=0. (Abel, CEuvres, ii., p. 197.)
n=oo
(33.) If Sw„ be any convergent series whose terms are ultimately positive,
we can always find another convergent series, Su„, whose terms are ultimately
positive, and such that LvJu„ = co.
If St{„ be any divergent series whose terms are ultimately positive, we
can always find another divergent series whose terms are ultimately positive,
and such that Lujt\ = oo .
(These theorems are due to Du Bois-Eeymond and Abel respectively; for
concise demonstrations, see Thomae, Elementare Theorie der Analytischen
Functionen. Halle, 1880.)
(34.) If w„+i/M„=(n« + ^n''-i+. . .)/("" + ^'"""^ + • • •). t^^n 2m„ will
be convergent or divergent according as ^ -^'> or J>1. (Gauss, Werke,
Bd. III., p. 139.)
(85.) If M„^.i/u„=a-/3/7i + 7/n2 + S/n^+ . . ., then 2m„ is convergent or
divergent according as a<: or >1, If a = l, 2m„ is convergent only if /3>1.
(Schlomilch, Zeitschr. f. Math. , x., p. 74.)
(36.) S1/m„ is convergent if m„+2 - 2u„^i + «„ is constant or ultimately
increases with n. (Laurent, Nouv. Ann., ser. ii., t. 8.)
(37.) If the terms of 2u„ are ultimately positive, then —
(I.) If \{/{n) can be found such that \l/(n) is positive, L^(n)M„=0, and
L {xp {n) M„/«„+i -\f/(n + l)}>0, 2m„ is convergent.
(II.) If xp{n) be such that ^^^(n) w„=0, L {^{n)uju„+i-xp(n + l)}=0,
and L\p {n) m„/{ V' («) uju^+i - vt- (ra + 1) } + 0, 2m„ is divergent.
(III.) If w„/w„+i can be expanded in descending powers of n, 2m„ is
convergent or divergent according as L {«uju„+i- (n + l)}> or t>0.
(IV.) If m„/m„+i can be expanded in descending powers of n, 2w„ is
convergent or divergent according as Lnu,^= or 4=0. (Rummer's Criteria,
Crelle's Jour., xiii. (1835) and xvi.)
(38.) If the terms of 2u„ be ultimately positive, and if, on and after a
certain value of n, o„Mn/"n+i~''^n+i>A'> where a„ is a function of n which
is always positive for values of n in question, and ;tt is a positive constant,
then 2m„ is convergent.
From this rule can be deduced the rules of Cauchy, De Morgan, and
Bertrand. (Jensen, Comptes Rendus, c. vi., p. 729. 18S8.)
Discuss the convergence of the following double series : —
(39.) 2 (-)»-! r^/n. (40.) 2 ( - l)»-i r"'/"! .
(41.) 2 { (n - l)'»/n'»+i - 7i'»/(n + l)"»+i }.
(42.) 2a;'»t/"/(m + n). (43.) 21/(ni + n)2.
(44.) 21/(m + n). (45.) 21/(771" - ?{'■').
(46.) Under what restrictions can 1/(1 + a; + j/) be expanded in a double
series of the form 1 + 2^,„, „a;'«i/" ?
(47.) If ^u^,n converge to S in the first way, and if its diagonal series be
convergent, show that the diagonal series converges to S also.
§ 87 EXERCISES VIII 185
Deduce Abel's Theorem regarding the product of two semi-convergent
series. (See Stolz, Math. Ann., xxiv.)
(48.) If uju^_^ can be expanded in a series of the form 1 + ajn + ajir + . . . ,
show that
1°. If ai = 0, ajj=0, . . ., a(n_i = 0, a^^O, then u^=u + vjn, where w is a
definite constant +0 and +x, and Lv^^ is finite when Ji = co .
2°. If Oj + O, and the real part of a^ be positive, then Lm„ = qo when
n=QO.
3°. If aj + O, and the real part of a^ = 0, then I,w„ is not infinite, but is
not definite.
4^ If Oj + O, and the real part of a^ be negative, then I,m„=0.
Apply these results to the discussion of the convergency of 2;j/,^x", and,
in particular, to the Hypergeometric Series, and to the following series : —
^t^+H^ni^ + yir, 2x«/«^+>'», 2^GJ(m + n)P, 2 ( - )»^CJ(m + «F.
(See Weierstrass, Ueber die Theorie der Analytischen Facultat. — Crelle's
Jour., LI.)
(49. ) Discuss the convergence of S ^C„ (a - n/3)"-^ (x + n/3)».
(50.) If M„ and i;„ be positive for all values of n, never increase when n
increases, and be such that I/M„=0, Lt;„=0, when n = Qo, find the necessary
and sufiicient condition that 2 (m„Vi + Un-it^^ + • • • + "i^n) = ^Wn ^ 2i>„. (See
Pringsheim, Math. Ann., Bd. xxi.)
(51.) If 0 < ilf „ < iH„+i and LM^=0 when n=oo , show that every diver-
gent series of real positive terms can be expressed in the form 2 (^„+i - M^) ;
and every convergent series of real positive terms in the form 2 (il/„+i - M„)/
Also that the successions of series
2(M„+i-lfJ/Pr(M„), r=0, 1, 2, . . .
^(M,+,-MJIPAM^+,){lrM^+,r, r=0, 1, 2, . . .,
where 0<p<l, and P^ (x) has the meaning of § 6 above, form two scales, the
first of slower and slower divergency ; the second of slower and slower
convergency. (Pringsheim, Math. Ann., Bdd. xxxv., xxxix.)
CHAPTER XXVII.
Binomial and Multinomial Series for any Index.
BINOMIAL SERIES.
§ 1.] We have already shown that, when w is a positive
integer,
{l + xf=\+,„C^x + ^C^x' + . . . + »C„a;'' + . . . + ^(7„a;™ (1),
where raCn = m(m-l) . . . (w - w + l)/w! (2).
When m is not a positive integer, mOn, although it has still a
definite analytical meaning, can no longer be taken to denote
the number of ;i-combinations of m things ; hence our former
demonstration is no longer applicable. Moreover, the right-hand
side of (1) then becomes an infinite series, and has, according
to the principles of last chapter, no definite meaning unless the
series be convergent. In cases where the series is divergent
there cannot be any question, in the ordinary sense at least,
regarding the equivalence of the two sides of (1).
As has already been shown (pp. 122, 131), the series
1 + rriCiO! + ^dar" + . . .+^C„iP™ + . . . (3)
is convergent when a has any real value between — 1 and + 1 ;
also when w = + l, provided m>-l; and when w = -l, pro-
vided 'm>0. We propose now to inquire, whether in these cases
the series (3) still represents (1 + a;)™ in any legitimate sense.
In what follows, we suppose the numerical value of m to be
a commensurable number* ; also, for the present, we consider
* If m be incommensurable we must suppose it replaced by a commensur-
able approximation of sufficient accuracy.
^ 1, 2 FIRST PROOF 187
only real values of x, and understand {l+xY^ to be real and
positive.
§ 2.] If we assume that (1 + a?)"* can be expanded in a con-
vergent series of ascending powers of x, then it is easily shown
that the coefficient of x^ must he m{m-\) . . . (m-n+ l)/n\.
For, let
(l+x)'^ = ao + aiX + a2a^ + . . .+anX^ + . . . (1)
where fto + «i^ + «2^ + . . . + cinX^ + . . . (2)
is convergent so long as \x\<R (it will ultimately appear that
R = 1). Then, if h be so small that \x + h\<B, we have
{1 + x^- hy- ^ ao+ ai{x + h) + a2{x + hY+ . . . + a„(^ + ^)'*+. . . (3),
the series in (3) being convergent by hypothesis.
Hence by the principles of last chapter, we have
{l+x + h)'^-{l+x)"' _ {x + h)-x {x + hf-x^
{\+x-^h)-{\+x) ~ ^ {x + h)-x ^ {x + h)-x
the series in (4) being still convergent. Hence, if we take
the limit when h-O, and observe that
(l+i» + A)-(l+^) ^ ' ' {x + h)-x '
by chap, xxv., § 12, we have
m{l+x)'^-'^ = ai + 2a.x + . . .+nanC(f-^ + . . . (5),
where the series on the right must still be convergent, since
L {n + 1) an+i/nun = Za„+i/a„ when w = c» *. Hence, multiplying
by 1 +x, we deduce
m (1 + x)'^ = ai + («! + 2a.2) x + . . . + {na^ + (w + 1) an+i} ^" + . . . ,
that is,
muo + maix + . . . + manX^ + . . . = «! + («! + '2a^ x+. . .
+ {nan + (n + 1) ttn+i} x"" + . • . (6).
* We here make the farther assumption that the Hmit of the sum of the
infinite number of terms on the right of (4) is the sum of the limits of these
terms.
188 euler's proof ch. xxvii
By chap, xxvi., § 21, the coefficients of the powers of x on
both sides of (6) must be equal. Hence
a^-ma^, 2a2 = {m-l)ai, . . ., (n + l)an+i = {m-n)an, . . . (7).
From (7) we deduce at once
ai = mao, a2 = m(m-l)ao/2\, . . .
an = m(m-l) . . . {m-n + l)ao/n\, . . .
To determine ao we may put a; = 0. We then get from (1),
a„ = !»» = 1 (if we suppose, as usual, the real positive value of
any root involved to be alone in question). We therefore have
(1 + «)'"=1 + :S„C„^" (8).
The theorem is therefore established ; and we see that the
hypothesis under which we started is not contradicted provided
|a7|<l, this being a sufficient condition for the convergency of
§ 3.] Although the assumption that (1 + a;)'^ can be expanded
in a series of ascending powers of x leads to no contradiction in
the process of determining the coefficients, so long as | a; | < 1 ;
this fact can scarcely be regarded as sufficient evidence for the
validity of a theorem so fundamentally important. We proceed,
therefore, to establish the following theorem, in which we start
from the series in the first instance.
Whenever the series 1 +%mCnX^ is convergent, its sum is the
real positive value o/ (1 + x)"^.
The fundamental idea of the following demonstration is due
to Euldr* ; but it involves important additions, due mainly to
Cauchy, which were necessary to make it accurate according to
the modern view of the nature of infinite series.
Let us denote the series
1 + J]^x + JJ^ar + . . . + JJuX"" + . . . (1)
by the symbol f{m).
So long as -l<a?<+l, f{m) is an absolutely convergent
series, and (by chap, xxvi., § 20) is a continuous function both
of m and of x.
• }f(ov. Comm. Petrop., t. xix. (1775).
§§ 2, 3 BINOMIAL ADDITION THEOREM 189
Hence, mi and m-i, being any real values of m, we have
= 1 + 2 (mi^ri + 7n,f^lmif^»-l + m2f^2mi^n-2 + • • • "^ rn^^n) X (2),
where the last written series is convergent (by chap, xxvl, § 14),
since the two series, 1 + ^rnfinX^ and 1 + Sm,(7„^", are absolutely
convergent.
Now, by chap, xxiii., § 8, Cor. 5,
hence f{mi)/{m2) = 1 + ^m,+m,C„w'\
=f(mi + m^) (3).
In like manner, we can show that
f{mi + m2)/{ms) =/(w^l + ^2 + ma).
Hence f{mi)f{m2)f{m^ =/(wi + mj + m^) ;
and, in general, v being any positive integer,
f{mi)f{mi) . . . f{m„) =f{mi + m^ + . . .+m„) (4).
This result may be called the Addition Theorem for the
Binomial Series.
If in (4) we put mi^m^^. . . = m„=l, then we deduce
{f{l)Y=fiv) (5),
where v is any positive integer.
If in (4) we put mx = m^ = . . . = m„ =p/q, where p and g
are any positive integers, and also put v = q, we deduce
{/(p/q)}'=/ip) (6).
Hence, by (5), {/(p/q)}' = {/(l)}^ (7).
Again, if in (3) we put mi = m, m^ =-m, we deduce
/(m)/{- m) =f{m - m) =/(0) (8).
Hence f{-m)=f(0)//{m) (9).
These properties of the series (1) hold so long as -l<x<+ 1,
and they are sufficient to determine its sum for all real com-
mensurable values of m.
190 SUMMATION OF l.JJnX'' CH. XXVII
For, since aCi=l, ,C; = 0, . . ., ^ = 0, . . . o^^^^O, 0^2 = 0,
. . . , oC^n = 0, . . .we have
f{l) = l + x, f{0) = \.
Suppose, now, 7?i to be a positive integer. Then, by (5),
(1 + xY =f{m) = 1 + r^Cx + mC^x' + . . . + ^C^^™ (10),
where the series terminates, since mCm+i = 0, m^m+2 = 0, . . .,
when w is a positive integer. This is another demonstration of
that part of the theorem with which we are already familiar.
Next, let m be any positive commensurable quantity, say
p/q, where p and q are positive integers. Then, by (7),
{Ap/q)}' = ii + ^y (11).
Hence/ (p/q) is one of the g'th roots of the positive* quantity
(1 + xy. But /(p/q) is necessarily real ; hence, if (1 + x)^''^
denote, as usual, the real positive qth. root of (1 + xy, we must
have
f(p/q)^±{l+a;y"' (12).
The only remaining question is the sign of the right-hand side
of (12).
^'mce /(p/q) is a continuous function both oi p/q a.nd of x, its
equivalent ± (1 + x)^'^ must be a continuous function both of
p/q and of x. Now (1 + x)^'^ does not vanish (or become in-
finite) for any values of p/q or of x admissible under our present
hypothesis ; and being the equivalent of a continuous function it
cannot change sign without passing through 0. Hence only one
of the two possible signs is admissible ; and we can settle which
by considering any particular case. Now, when ^2; == 0, /(p/q) = + 1.
Hence the positive sign must be taken ; and we establish finally
that
/(p/q)^ + (l + x)P'%
that is,
(1 + X)"^ = 1 + mCiX + mCx" + . . . + ^CnX^ + . . . (13),
when m is any positive commensurable quantity.
* Positive, since -l<x<l, by hypothesis.
§ 3 CASES WHERE X=±l 191
Finally, let m be any negative commensurable quantity, say
m = - m', where m' is a real positive commensurable quantity.
By (9) we have
/(-m')=/(0)//(m') = l//W-
Hence, by (13),
/(- m) = 1/(1 + x)^',
= (1 + ;2;)-™',
that is,
where m is any commensurable negative quantity.
The results of (10), (13), and (14) establish the Binomial
Theorem for all values of x such that - l<a;<+ 1. It remains
to consider the extreme cases.
When a; = +l, the series (1) reduces to
1 + mf^i "*" m^2 + . . . + m^^n + • . .
This series is semi-convergent if —l<m<0, absolutely con-
vergent if m>0. Hence, by Abel's Second Theorem, chap, xxvr.,
§20,
(1 + 1-0)'"= L {l+mCrW + ^C,aP+. . ,+^CnX- + . . .},
a;=l-0
that is,
2'»=l+„,(7i + ™(7, + . . . + M. + . . . (15),
provided m>-l, with the condition that, when -l<w<0, the
order of the terms in the series of (15) must not be altered.
If 0<i»<l, we have, by the general case already established,
(l-^)™=l-™Ci^ + ^C2^-^-. . .(-)\Cna;'' + . . .
Hence, since the series
l~mW + mf^2~' • • (~)'^>lf'n + • • •
is convergent if wz > 0, we have, by Abel's Theorem,
(l-r^)'»= L {l-rr^Cw + mC.x'-. . .(-)»^C„^» + . . .),
x=l-0
that is,
0=^—mOi + mC2-k . • ( — )\Cn + . . . (16),
provided m be positive.
The results of (15) and (16) complete the demonstration of
192 PARTICULAR CASES CH. XXVII
the Binomial Theorem in all cases where its validity is in
question.
Cor. If x^y, it follows from the above result that we can
always expand {x + yY' in an absolutely convergent series. We
have in fact, if | ^ | > 1 3/ 1 , that is, | y\x \ < 1,
(a; + y)™ = ic"* (1 + ylxY,
= x''{l+r.GAylx)+raG.,{ylxY + . . . + „(7„ (y/^)" + . . .},
= x'^ + ^G,x^-'y + m.C,x^-Y^- ' .+^^^^-"2/" + . . . (17);
and if I a; I < 1 2/ 1 , that is, | xjy | < 1,
{x + y)'^ = y'^{l-¥xly)'^,
= y'^{l+MCiix/y) + ^C,(xlyy + . . .+raGn{xlyY + . . .},
= 3^™ + raG^y'^-^X + rnG^y'^-'^x'' + . . . + JJ^y'^-'^ x"" + . . . (18).
If 7W be a positive integer, both the formulae (17) and (18) will
be admissible because both series terminate. But, if m be not a
positive integer, only one of the two series will be convergent.
§ 4.] The general formulae of last paragraph contain a vast
number of particular cases. To help the student to detect these
particular cases under the various disguises which they assume,
we proceed to draw his attention to several of the more com-
monly occurring. The difficulties of identification are in reality
in most cases much smaller than they at first sight appear. We
assume in all cases that the values of the variables are such that
the series are convergent. ■ ■
Example 1.
(l + a-)-' = l-x + a;2-. . , + (-)»a;" + . . .;
(l-a;)-i = l + a; + x2 + . . . + a;" + . . .
For
(l + x)-i=l+S_iC„a;";
and
_iC„=-l(-l-l)(-l-2) . . . (-l-n + l)/nl.
= (-)"!. 2. 3 . . . n\n\.
= (-)»!.
(l-x)-i = l + S_i(7„(-x)»;
and
-iC» {- ^)" = (-)"(-)"«'''=(- P^"
Example
2.
(l+a;)-2 = l-2a; + 3x2-. . . +( - )»(n+l)a:'' + . . .;
(l-a;)-2=l + 2x + 3x'' + . . .+(n + l)x" + . . . *
For
^C„=-2(-2-l) . . . (-2-n + l)/nI,
= (-)»{« + !).
§§ 3, 4 ULTIMATE SIGN OF THE TERMS 193
Example 3.
(l + a;)-»=l-3a; + 6.T2- . . . +(- )»i (w+ 1) (n + 2) a,"+ . . .;
(l-x)-8=l + 3x + 6a;2+ . . . +^(n + l)(7i + 2)a;»+ . . .
Example 4.
Example 5.
Example 6.
/I x^» 1 '""■ X m{jH-2) fx\^
■ m(m-2)(m-4) . . . (7?i-2n + 2) /xN"
"*■ 7j1 V2J "^ ■ ■ ■ "'
-1 , "^y I "^("^-2) o m(m-2)(m-4). . . (ot-271 + 2)
~ ^ 2 2.4 +•••+ 2.4.6. ..2« '^ ' ' '
(l + x)-n./.= l + S(-)n"^(" + ^n"; t^ • • • l" + ^"-^)x".
^.4.0 • • • a'H
Example 7.
(l + xr/^^l + Z^^^-^^^V;^- • •^^-"^ + ^^xn;
^ ' q .2q .2q . . . nq
(1 - x)-P/«= 1 + S P(P + g)(^ + ^'?)-- -(P + ^g-g) ^».
^ ' q .2q .Sq . . , nq
Example 8.
It will be observed that the coefficient of x" in this last expansion, when
m la integral, is (see chap, xxiv., § 10) the number (^HJ of n-combinations
of m things when repetition is allowed. It is therefore usual to denote this
coefficient by the symbol ^iT„, m being now unrestricted in value. We
shall return to this function later on.
Example 9.
i{(l + xr+(l-x)'"}=l + ^C2x''+^(74X*+ . . . +„C'2„x2»+ . . . ;
^{(l+x)"'-(l-xr}=^CiX + „C3x3+ . . . +^C^^_,x^n-l+ . . ,
Ultimate Sign of the Terms. — Infinite Binomial Series belong
to one or other of two classes as regards the ultimate sign of
the terms — 1st, those in which the signs of the terms are
ultimately alternately positive and negative ; 2nd, those in
which all the terms are ultimately of the same sign.
c. n. 13
194 INTEGRO-BINOMIAL SERIES CH. XXVII
If X and m denote positive quantities (m of course not a positive integer),
1st. The expansions of (l + a;)"* and (1 + a;)-'" both belong to the first
class. In (l + x)'" the first negative term will be that containing a;"+^, where
n is the least integer which exceeds m. In (1 + x)~'^ the first negative term
is of course the second.
2nd. The expansions of (l-a;)"», (l-x)-'", both belong to the second
class. In (1 - a;)"* the terms will have the same sign on and after the term
in a;" n being the least integer which exceeds m, and this sign wUl be + or
- according as n is even or odd. In (1 - a;)""* all the terms are positive
after the first.
§ 5.] A great variety of series suitable for various purposes
can be readily deduced from the Binomial Series ; and, conversely,
many series can be summed by identifying them with particular
cases of the Binomial Series itself, or with some series deducible
from it.
The following cases deserve special attention, because they
include so many of the series usually treated in elementary text-
books as particular cases, and because the methods by which the
summation is effected are typical.
Consider the series S^r(^)mC»i«", where <i>r{n) is any integral
function of n of the rth degree. Such a series stands in the
same relation to the simple Binomial Series as does the Integro-
Geometric to the simple Geometric Series. We may therefore
speak of it as an Integra- Binomial Series.
We may always, by the process of chap, v., § 22, establish
an identity of the following kind,
^r(w) = J.o+^iW+yl2w(w-l)+. . .+Arn{n-\) . . . (w-r+1) (1),
where Ao, Ai, A^, . . ., Ar are constants, that is, are independent
of n.
We can therefore write the general term of the Integro-
Binomial Series in the following form : —
^r(»)m^»«" = Ao m^n^" + Ain^CnX'^ + . . .
■\-Arn{n-\) . . . (w-r+l)^(7„a;",
= ^Om^n'^ "T niJ±i3/ fn—ifyfi—ilV
+ m(?w-l) J.2^m-2^n-2^"~^+ . . . +in(m-l) . . .
{m-r+l)Araf^-rCn-ra;'*-'' (2).
§§ 4, 5 'Z<f>r {n)mCnX^I{n ^-a){n^h) . . . {n + h) 1 95
Hence, if the summation proceed from 0 to go , we evidently
have
Mr{n)raCnX''=A3'mGnX^+rnA^X%ra-lCn-^X'^~'^+. . .
0 0 1
+ m{m, — l) . . . {m-r+\)ArX^%m-rGn-rX'^~'^ (3),
r
+ m{m-\) . . . {m-r+\)ArX'''{l+xY-'',
since all the Binomial Series are evidently complete*. Hence
2(^r {n)r„.GnCc^^{AQ + mA^xl{l + x) + m{m-\) A^x^il +xy + . . .
0
+ m{m-l) . . .{m-r+1) Arx''/{1 + xY] (1 + x)"^ (4) ;
and the summation to infinity of the Integro-Binomial Series is
effected!.
The formula will still apply when w is a positive integer,
although in that case the series on the left of (4) has not an
infinite number of terms. The only peculiarity is that a number
of the terms within the crooked bracket on the right-hand side
of (4) may become zero.
• °°
Cor. We can in general sum the series ^i>r(n)mCna!"l(n + a) (n + b)
. . . {n + k), where a, h, . . ., k are unequal positive integers,
in ascending order of magnitude.
For, by introducing the factors w+1, w + 2, . . ., n + a-\,
w + a+1, w + a + 2, . . .,w + 6-l, &c., we can reduce the general
term to the form
«A {n)rr.^uGn^ica;''^''l{m + 1) (m + 2) . . . (m + k) a^ (5) ;
where i/' {n) is an integral function of n, namely, 4*r (n) multiplied
by all the factors introduced which are not absorbed by m+kGn+k-
* If the lower limit of summation be not 0, then the Binomial Series on
the right-hand side of (3) will not all be complete, and the sum will not be
quite so simple as in (4).
t It may be remarked that the series is evidently convergent when x<l.
The examination of the convergence when x=l will form a good exercise on
chap. XXVI.
13—2
196 EXAMPLES CH. XXVII
Hence
00
^i>r {n)mGnCffl{n + o) (fl + b) . . . (fl + k)
= {i./. (w) ,„+fc(7„+,^"+'=}/(m + l){m + 2). . .{m + k)o^ (6).
The summation of the series inside the crooked bracket may
be effected ; for it is an Integro-Binomial Series. Hence the
summation originally proposed is always possible.
We have not indicated the lower limit of the summation,
and it is immaterial what it is. Even if the lower limit of
summation be 0, the Binomial Series into which the right-
hand side of (6) is decomposed will not all be complete (see
Example 6, below).
It should also be noticed that this method will not apply if
m be such that any of the factors m + \, m + 2, . . ., m + k
vanish. In such cases the right-hand side of (6) would become
indeterminate, and the evaluation of its limit would be trouble-
some.
The above method can be varied in several ways, which
need not be specified in detail. It is sufficient to add that by
virtue of Abel's Second Theorem (chap, xxvi., § 20) all the
above summations hold when a? = ± 1, provided the series in-
volved remain convergent.
Example 1. To expand (a; + ?/)"* in a highly convergent series when x
and y are nearly equal. From the obvious identities
{{x+y)l2x}^={2xl{x + y)}-"^={l + (x-y)l(x + y))-^,
{{x + y)l2y}«'={2yl{x + y)}-«*={l-{x-y)Hx + y)}-^,
(x+j/)"* {l/(2xr ± l/(2j/n = {1 + (x - i/)/(x + j/)}-'»± {1 - (x - 2/)/{a; + y)}-».
•we deduce at once
(i + !,)"=2"i". |l + S(-)VH.(^)"[ ,
where ^fl'„=m(m + l) . . . (m + n-l)/n!,
_2V^+ix^y«* I m(m + l) fx-yy m(w + l)(m + 2) (ot + 3) fx-y\*
t^+ 21 \x + y) + 41 [IT^)
xm + yn
^ 2^+1 a-mym J jn fx-y\ TO(w + l)(m + 2) fx-yy 1
All these series are highly convergent, since (x-y)l(x+y) is small.
}■
EXAMPLES 197
Example 2. To sum the series
2
9-*-
2 /2y 2^ /2\3 2.5.8 /2\*
If we denote this series by Wj+M2 + %+ • • •• ^^ s^e that
+ (ra-2)3} ^
32n»
(-l + T^-1) /2\»
_2.5. . .{2 + (ra-2)3} 2^
"" ~ nl 32" '
«1
(!)"•
_ ^ )ni(i-i)a-2)- • -g-^+i) m"^
!ti.GIlC6
= 1/4/3.
Therefore, «t^ + M2 + M3+ . . . =1-1/4/3.
Example 3. To sum the series
m
m (m - 1) m (wi - 1) (m - 2)
+ J + j-y2 + • • M
whenever it is convergent.
Here we have
_m{m-l){m-2) . . . (m-n)
"n+i - -I ,
_m(m-l) (m-1-1) . . . (m-l-n + l)
~ ni •
Hence
Mi+M2 + «3+ • • . ='n{l+m-iCi + «-iC2+ . . .}
= ni{l + l}'"-i=m2"*-\
provided m - 1 > - 1, that is m > 0.
It should be observed that we have at once from §2(5) the equation
m(l + x)^-^ = l^Ci + 2^C^x+ . . . +n^C„a;™-i+ . . . (1),
from which the above result follows by putting x=l.
By repeating the process of § 2, we should deduce the equation
m(m-l) . . . (m-ft + l)(l + a;)'"-*=1.2 . . . fc^Cfc+2.3 . . . (fc + 1)
mCk+i^+ • • • +{n-k + l){n-k + 2) . . . 7i„(7„a;«-*+ . . . (2),
whence it follows that
m(m-l). . .(m-fc + l)2'»-* = 1.2. . . k^Gj,
+ 2.3. . .(k + l)^G,+,+ . . . (3),
provided m>k-l. These results might also be easily estabhshed by the
method first used.
Example 4. To sum the series
1.2. . .&^2.3. . .(/c + l)^3.4. . . {k + 2y • • •
198
Here we have
EXAMPLES
,X7„a;»
CH. XXVII
Hence
(l + .r)"
(ra+l)(n + 2) . . , (n + k)
(m+l) (ni + 2) . . .{vi + k)x'''
1
(»i + l)(m + 2) . . . (m + fe)a;fc (m + l)(m + 2) . . . (m + A)a;*^^'^'^*^^^
Therefore
_ (1 + a;r+fc-l- ^+,(7j X - ^+fcC2a;2 - ... - ^+^0^., a*-i
(4).
(m + l) (m + 2) . . . (7;i + /c)a^
If 7)t > - ft - 1, this gives as a particular case
S^C„/(n + l)(n + 2). . . {n + k) =
{2m+*_l-*°S '^+fc(7j/(7n + l)(m + 2) . . . (m + k) (5).
«=i
The formulffi (1), (2), (3), (4), and (5) contain of course a considerable
variety of particular cases.
CO
Example 5. Evaluate Sn^^C7„a;".
0
Let n^=AQ + A-^n + A^n (n - 1) + A^n (n - l){n - 2), then we have the follow-
ing calculation to determine Ag, A^, A^, A^ (see chap, v., § 22).
1 +0 +0|+0 ^=0,
^1 = 1,
A^ = ^, Jg=l.
Hence
Sn=i^C„x«=0 . I^C„x'' + l?7ia;l^_iC„_ia;"-i + 3m (m - 1) x'l^^_^C^_^x'^-^
0 0 1 2
+ m (m- 1) (m- 2) x3S^-3C„_3a;»-3,
3
= mx (1 + x)*"-! + 3m (m - 1) x2 (1 + x)"»-2 + m (m - 1) {m - 2) x^ (1 + x)'"-3,
= {w?x^ + m (3m - 1) a;2 + mx} (1 + x)™"'.
CO
Example 6. Evaluate S^C„x''/(n + 2) (n + 4).
^C^x" ^ (» + !)(» + 3) „^+40n+4a^""^*
(h + 2) (n + 4) ■ x^ (m + l) (m + 2) (m + 3) (m + 4) *
(7t + l)(n + 3) = 7i2 + 4« + 3,
= ^o + ^i(" + 4) + ^2(« + 4)(M + 3).
1 +4 +3
0 -4 +0
1
0 +1 +1
2
1 +1|+1
0 +2
11+3
-4
-3
1 +0|+3
0 -3
11 -3
EXERCISES IX 19^
We therefore have
» C a;" 1 ■»
?{n+V(« + 4) " xHm + l){m + 2)(m + 3)(m + i) ^^? rr^^n+i^''^* - 3 (m + 4) x
0 0
+ (m + 4) (m + 3) a;2 {(l + a;)»'+2-l -^+2010;}],
a* (m + 1) (m + 2) (m + 3) (m + 4) "• ^ ^ ' ^ ' ^ ' ^ ^ '
+ {i(m + 3)(m + 4);r2_3|].
Exercises IX.
Expand eacli of the following in ascending powers of a; to 5 terms ; and in
each case write down and simplify the coefficient of a;''.
(1.) {l + xfl\ (2.) (l-a;)-V2. (3.) (l_a;)-3/4.
(4.) {2-1x^1^. (5.) (a + 3a;)i/3. (G.) ^l{d^-x^).
(7.) ^(1-nx). (8.) l/(l-3a;2)V3. (9.) (^-l/x)-".
(10.) Write down the first four terms in the expansion of { (a + x)/(a - x) } V-^
in ascending powers of x.
Determine the numerically greatest term in
(11.) (3 + a;)2/3, x<3. (12.) (2-3/2)"/2. (13.) (1 - 5/7)-i3/s.
(14.) Find the greatest term in (l + a;)~", when x=^, n=4.
(15.) If 71 be a positive integer, find the greatest term in (n - l/?i)2n+i.
(16.) The sum of the middle terms of (l + a;)"» for all even values of vi
(including 0) is (1 - 4a;)-i/2.
(18.) Show that, if m exceed a certain value, then
»^. 1 I (»t + l)"t I (m + l)m{m-l){m-2) .
-i+ 2! "^ 41 ■*■•••
(19.) Sum the series
a-(a + &)m+(a + 26)— 5^j — i_(a + 36)-i ^ '+. . .,
for such values of m as render the series convergent.
(20.) N/27=2 + 24 + y+...
V**'i 24 3*^ a'*31 2*4r 2^51 *''
250 EXEKCISES iX CH. XXVll
(22.) Sum to infinity
1 1.4 1.4.7
6"*" 6.12"*' 6. 12. is"*"' ■ *
(23.) Sum the series
m(m IN i"^('"-^)("^-^) I m(m-l) . . . (m-r+1)
for such values of m as render the series convergent.
(24.) If n be even, show that
n(n + 2) . . . (2ra-2)/1.3 . . . (?i-l) = 2»-i.
(25.) In the expansion of (1 - x)~'"* no coefficient can be equal to the next
following unless all the coefficients are equal.
(26.) Prove by induction that
m{m + l) m{m + l) . . . {m + r-1) _{m + r)l
l+m+ 2j +...+ ^:j ^-i^ifd'
where r is a positive integer. Hence show that, if x< 1,
^ ''> -^ (m-l)Irl •
(27.) The sum of the first r coefficients in l/,;/(l - x) : the coefficient of
the rth term = 1 + n (r - 1) : 1.
(28.) If ir(a) = l4-^ + ^(^)..H^i^±4/^±^)x3+. . ..theseries
being absolutely convergent, then
F{a)F{b) = F{a + b).
What is the condition for the convergency of the series ?
(29.) Show that
j-nGij + nC,j-. . .=[l-{(n + l)a; + l}(l-x)»«]/(n + l)(n + 2).
Sum the following series, so far as they are convergent : —
(30.) 2(71- 1)2 jn (Hi -1) . . . (77i-n + l)x»/7il, from w = l to?i=oo.
(31.) S(-)»-i(ji + l)(n + 2)1.3.5 . . . (2n-5)a;"/n!, from ji^O to ■rt=oo ,
(32.) Sm(m+1) . . . (m + n-l)a;»/(n + 3)?i!, from ri=0 to n=oo .
(33.) S(?i-l)n.4.7 . . . (3n-2)/(?i + 2)(n + 3)n!, from n=l to 71=00.
(34.) Why does the method of summation given in § 5 not apply to
la;»/(n + l)?
SERIES DEDUCED BY EXPANSION OF RATIONAL FUNCTIONS OF X.
§ 6.] Since every rational function of w can be expressed in
the form I+F^, where / is an integral function of x, and i^ a
proper rational fraction, and since F can, by chap, vni., § 7, be
|§ 6, 7 EXPANSION OF (2 - pa;)/(l - px + qay^) 201
expressed in the form %A{x- a)-'^, where A is constant, it follows
that for certain values of a; a rational function of a; can be ex-
panded in a series of ascending powers of x, and for certain
other values of a; in a series of descending powers of a;*. "We
shall have occasion to dwell more on the general consequences of
this result in a later chapter, where we deal with the theory of
Recurring Series. There are, however, certain particular cases
which may with advantage be studied here.
§ 7.] Series for expressing a" + ^ and (a"+^ - ^"+^)/(a - /8) in
terms of a^ and a + fi, n being a positive integer.
If we denote the elementary symmetric functions a + y8 and
ajS by p and q respectively, it follows from chap, xviii., § 2, that
we can express the symmetric functions a'' + /3™, (a"+^ - y3"+^)/
(a - j3) as follows : —
ar'-\-^ = aoP^+a^p^-^q+ . . . + a^;?*'"'^?*" + • • • (1),
(„n+l _ ;8"+l)/(a - /3) = 6„j9" + 6i^«-V + . . . + brP^'-^-'q'- + . . . (2),
where both series terminate.
By the methods of chap, viii., § 8, or by direct verification
we can establish the identity
2-px _ 2-(a + (3)w _ 1 ^ 1 ,g.
1 -px + qa^ (1 — ax) (1 - (3x) 1 - aa; 1- fix ^ ^'
Now if X be (as it obviously always may be) taken so small
th.&tpx-qx'^Kl, we have by the Binomial Theorem
1 -pxTqa^ " ^^ ~-^^^ {l-{px- qx")}-' = (2 -px) {l + {px- qx")
+ (px - qx^y + . . . + (px - qxy- + . . . } (4).
Now (by chap, xxvi., § 34) if x be taken between - a and + a,
a being such that the numerical value of ±po.±qa?<\, that
arrangement of signs being taken which makes ±pa. + qa? greatest,
then each of the terms on the right-hand side may be expanded
in powers of x and the whole rearranged as a convergent series
proceeding by ascending powers of x.
* Strictly speaking, this is as yet established only for cases where a
is real. The cases where o is imaginary will, however, be covered by the
extension of the Binomial Theorem given in chap. xxix.
202 a" + ^" IN TERMS OF ayS, a + /3 CH. XXVII
We thus find that
+ (-)Vra.i3''-''-/+. . .)^n (5),
= 2 {1 + 2 &c.} -px {1+2 &c.} (6).
The coefficient of x^ on the right-hand side of (6) is
Now
2n-rCr-n-r-iCr = n{n — r-l)(n-r-2). . . (n-2r+l)/rl
Hence
^—p^ n ^ r « w «-9 n(n-3) „. „
Again
■--^ + :p^-{l + a^ + aV+. . . + a'^^'* + . . . ] + {1 + /3x
+ i8V+. . . +fi"x''+. . . },
= 2 + 2(a™ + y8'^)^™ (8).
All the series involved in (8) will be absolutely convergent,
provided x be taken so small that | ax \ and | ^x \ are each < 1.
Now, by (3), the series in (7) and (8) must be identical. Hence,
comparing the coefficients of x"^, we must have (by chap, xxvi.,
§21)
+ (_ ^y.n{n-r-l){n-r-2). . . (w - 2r + l)^^.^^^^ ^
(9).
As we have indicated (by using =), the equation (9) is an
algebraical identity, on the understanding that p stands for a + ^
§7 SERIES FOR a'^ + yS^ («"+>- ;8"+0/(a-/3) 203
and q for a/3. The last term will or will not contain p according
as n is odd or even.
In like manner, from the identity
sj^ ^—\JL
\—px + qa? 1 - (a + /8) i» + a.^a? \\-ax 1 - ^x) a — P
we deduce
(„«+i _ y8"+i)/(a -I3)=p^- ^^p^-"^q + ^^ '^l^""' ^^ p^-'q-" - . . .
I i\rO^~0(w-r-l) . . .(«-2r+l) „ o„ r /i/v\
+ (-1)'^^ ^^ '-^^ ^ ->»-V+- • • (i^X
subject to the same remarks as (9).
If we write the series (9) in the reverse order, and observe
that, when n is even, = 2w say, only even powers of p occur, and
that the term which contains p" is
i_\ra-» 2?;^(m + g-l)(m + g-2). . . (2^ + 1)
^ ^ _ (m-s)! ^ ^ '
that is,
. .„j_g2w(w + s-l)(w + s-2). . . (w+ l)w(m- 1) . . . (y»-s+ 1)
p^q"^-',
that is,
\ ) ^ (2s)! -P ^ »
then we have
a2'» + ^« = (_)'«2f/" -^jt?V'+^^^^^^l^^V<Z"'~'-- • •
Similarly, we have
^2,»+l + ^2m+l = (_)'« (2m + 1) [pq^^ - (^' + l)^^3^m-l
I 3!
5! ^^
5 ~TO-2
^ ^ (2s-l)! ^ ^ • • •/
(9").
204 SERIES FOR \x + s/{x^ + 2/")}" + {•'» " "Ji^"" + 2/")]'' CH. XXVII
^(-)-"^"^-^^|,;_,)f-^^\--v-'^...} (10').
a-^ -'^ M^ 2! -^^ 4!
p-m^m-s^ ... I
(10").
Since a and P are the roots of the quadratic function
!? -pz + q, we may replace a and /3 in the above identities by
i {p+ J{p^ - 4g')}, and ^ {p - sj{p^ - ^)} respectively. If
this be done, and we at the same time put p = x and —^q=y^,
we deduce the following : —
{x + V(^ + f)Y + {x- Jix" + f)Y
= 2« [x^ + ^, x^-Y + ^^^ ^'^-Y +
^K^-y-l)(^-^-2)...(?^-2r+l) ^„_,,^^,
r!2'^
+ ^— — ~'^" ' . ~r ' •^" "• • '' a;«-='Y' + . . . I ,
7Z-(7^''-2^)(;^^-4^). ■ .(^^--2^2-) 1
(2s)! ^^ ""•••/'
if w be even ;
-of n-i ^ *K«' - 1") ^ n 3 n {11^ - V) (n" - 3==)
^^ '^•••'^ (2^rri)!
^.+i^™-2*-i^^ . . j,ifwbeodd.
(9'").
§§ 7, 8 SERIES FOR {x + V(«^ + 2/^)1" -{a;- ^{x- + y'')Y
{x + J {a? + /)}™ -{x- ^(x^ + /)}»
205
a;"- V +
a;n-2r-l^2r _^,
{n-r-l)(n-r-2) . . . (n-2r)
}■
r!2-''
= 2 V(^ + 1/) 1^ xf-^ 4- ?ii^!f^ ^s^.-4 +
3!
_^7»(w^-2'^). . . (71^-28-2') ^_, ,,_„, 1
if n be even ;
= 2 V(^ + 2/ ) [y + 2! ^^ ^ 4!
^y +• • • +- (2i)!
^y%-1»-\
}■
if n be odd.
(10'").
These series are important in connection with tlie theory of
the circular and hyperbolic functions.
§ 8.] A slight extension of the method of last paragraph
enables us to find expressions for the sum and for the number of
r-ary products of n letters (repetition of each letter being allowed).
The inverse method of partial fractions gives us the identity
1/(1 - tti^;) (1 - ajir) . . . (1 -a„a7)s2^g(l-a^a;)-^ (l),
where ^g = a/-Y(a, - Oj) (a, - Oj) . . . (ttg-a^).
Also, since (1-0,3;;)"^= 1 +Sa/af, we have (by chap, xxvi.,
§ 14), provided x be taken small enough to secure the absolute
convergency of all the series involved,
1/(1 - a^x) (1 - a^X) ... (1 - a^x)
= (1 + ^a^^x"-) (1 + Sa/^'-) . . . (1 + Sa/^r'-) (2),
= 1 + ^JCrX^ (3),
where ^t is obviously the sum of all the r-ary products of
ttj, 05, . . . a„. Since the coefficients of x^ on the right-hand
sides of (1) and (3) must be equal, we have
„Z, = 2a/+'-V(a, - ai) (a, - a^) ... (a, - a„) (4).
206 SUM AND NUMBER OF r-ARY PRODUCTS CH. XXVII
If, for example, there be three letters, oj, a,, aj, we have
„ r+2 „ r+2 r+2
Z- _ "-1 , "2 . "3
r =
(tti - 02) (tti - as) (og - tti) (og - 03) (as - Oj) (a^ - a^)
_ <'^^ («2 - "3) + <"^^ (^3 - «i) •<■ <^^ («i - 02) /.x
~ (02 - "3) (tts - «l) («! - "2)
If we put ai = a2= . . . =a„ = l, then each of the terms in
nKr reduces to 1, and nJ^r becomes n^r- Hence, from (3),
(l-a;)-''=l + :^^Hrar (6).
Equating coefficients of oT on both sides of (6), we have
nffr =n{n + l) . . . {n+r~ l)/rl,
a result already found by another method in chap, xxiii., § 10.
§ 9.] Some interesting results can be obtained by expanding
l/(y + a;)(i/ + a; + l) . . . {^ + a; + n)m descending, and in ascend-
ing powers of ?/.
If we write
r=n
l/{y + x)(y + a;+l) . . . {y + x + n)= 1 Ar{y -^ x + r)-\
r=0
then we find, by the method of chap, viii., § 6, that
l=Ar{-r)(-r+l). . . (-1)1.2. . . (n-r).
Hence J.^ = ( - )*■« Cr/nl
Therefore
nl/{y + x)(y + x+l). . .(y + x + n) = -S,(-y^Cr(ij + x + r)-' (1).
Hence, if Pi, P2, P&, . , . denote respectively the sum of
X, X + ly . . ., x + n, and of their products taken 2, 3, . . . at a
time (without repetition), we have
= S(-)-.a{l + S(-)'(5^)} (2),
§§ 8, 9 EXPANSIONS OF 1/(2/ + x)(7j + OS +1) . . . (y+ as + n) 207
where we suppose t/ to have a value so large that all the series
involved are convergent.
Since there is no power of l/i/ less than the nth. on the left
of (2), the coefficient of any such power on the right must
vanish. Therefore
{a; + ny-nC,{a; + n-iy + nC^(iv + n-2y-. . .(-)"^ = 0 (3),
where s is any positive integer <n.
Equating coefficients of l/y", ll'}f"^\ and 1/3/"+^ we find
{x + nf-nG,{as + n-lf + nC!o{x + n-'2.f-, . .
{-fx'^ = n\ (4);
= {n+iy.{x + \n) (5);
{x + nf+^-nCx{x + n-lf'^^ + nC',{x + n-2f^^-. . .
{-fx^-^^=n\{P^^-P^\
= ^{n+ 2)! [x" + nx + ^^n (3w + 1)} (6) ;
and so on.
Again from (1) we have
x{x+l) . . . (x + n)'- ^ -^ ^ -^ *
= ^\-y-^\l + -^~' (7),
where Qi, Q2, Qs, . . . are respectively the sum of 1/x, l/{x + 1),
, . . , l/{x + n), and the sums of their products taken 2, 3, . . .
at a time. From (7), hy expanding and equating coefficients of
y, we get
n\ fl 1 11
x{x+l) . . . (x + n)\x X + 1 ' ' ' (x + n))
"iT* (^+1)2 (a; +2)^^ •••^M^ + w)' ^ ^*
If we put x=l, we get the following curious relation between
the sum of the reciprocals of 1, 2, . . ., w + 1, and the reciprocals
of their squares : —
208 EXAMPLES CH. XXVII
1 fl 1 _J_\ ^ 1 _ -^
w+1 ll ^2'^' ' ' n + 1} V "" "^
2' 3''
§ 10.] We have now exemplified most of the elementary
processes used in the transformation of Binomial Series. The
following additional examples may be useful in helping the
student to thread the intricacies of this favourite field of exercise
for the tyro in Mathematics.
Example 1. Find the coefficient of x" in the expansion of (1 - x)^l(l + a;)''/^
in ascending powers of x.
If (H-x)-3/2 = l + Sa„a;», then (l-x)2/(l + a;p = (l-2x + a;2)(H-Sa„x»).
Hence the coefficient required is o„ - 2a„_i + a„_2 . If we substitute the
actual values of a„, a„_i, a„_2, we find that
Example 2. If f{x)=zaQ + aiX + a2X^+ . • ., then the coefficient of x^ in
the expansion of / (x)/(l - a;)"» in ascending powers of x is a,, mHr+ «i m-^r-i
+ a2mHr-2 + ' • • + «?• This follows at once from the equation
/ {x)l(l - x)'"= (ao + Sarxn (1 + S^H.x'-).
In particular, if we put /(x) = (1 - x)~" and m — 1, we deduce that
and, if we put /(x) = (1 - x)~", we deduce that
m-hi^r — m^r + »n-"r-l »-" l + m " r-2 n-" 2 + • • ■ + n-Hp j
results which have already appeared, in the particular case where m and n are
integral (see chap, xxiii., § 10).
Example 3. Show that
m<?«/2 + ^+iC„/22 + ^+2^„/23+. . . adoo=l + „,Ci + ^C2+. . .+„C7„ (1).
The left-hand side of (1) is obviously the coefficient of x" in
^ = (1 + x)'»/2 + (1 + xr+V22 + ( 1 + x)'»+2/23 + . . . ad 00 .
Now Z=^(l + x)"'[l + {(l + x)/2} + {(l + x)/2}2+. . . ad 00],
= (l + x)'»/2{l-(l + x)/2}, if we suppose x<l.
= (l + x)"»/(l-x),
= l + S(l+m^l + mC2+- • •+m(?»)a;™,
by last example. Hence the theorem follows.
Example 4. Sum the series
«-3 , («-4)(n-5) (n-5)(n-6)(7i-7).
*-^ 2r+ 31 41 ■*■••••
n being a positive integer.
§§ 9, 10 EXAMPLES 209
The equations (9'") of § 7 being algebraical identities, we may substitute
therein any values of x and y we choose, so long as no ambiguity arises in
the determination of the functions involved. We may, for example, put
x= -1 and y = 2i. We thus find
Hence, if w and ufi denote, as usual, the two imaginary cube roots of + 1,
we have
/S ={ 1 + (-)"-! (w" + w^*^) } /re.
If we evaluate w^ + w^™ for the four cases where n has the forms 6m, 6ni±l,
6m±2, 6ni + 3 (remembering that ui^^=\, w-i = w2^ u-^ — ui), we find that
S has the values - Ijn, 0, 2/7i, and 3/« respectively.
Example 5. Sum the series
«(n-l) w(re-l)(re-2)(n-3) n{n-V)(n-2) (w-3) (n-4) (n-5)
''■2(2;+l)"^ 2.4(2r + l){2r + 3) "^ 2 . 4.6(2r + l) (2r + 3) (2?- + 5)
+ . . .
n being a positive integer.
If we denote the series by 1 + Mj + "a + W3 + . . . , then
_ n(ji-l) . . . (n-2s + l)
* 2.4 .. . 2s(2r+l)(2r + 3) . . . (2r + 28-l)'
_?tl(27-)!(r + l)(r + 2) . . . (r + s)
~ (re-2s)!(2r+2s)!sl '
restricting r for the present to be a positive integer. We may therefore write
n!j2r)!
"a — In + ^rW "+2'" 2r+28 • r+s^f
Now r+s^s is *^6 coefificient of x^*" in the expansion of ai^'^+^s (i + l/x^)*^*; that
is, in the expansion of a;^+2'{^(l + l/x'-^)}2''+2*. Hence 2«g is one part of the
coefiicient of x'^'^ in the expansion of
Hence 2S is the whole coefficient of rc^'" in the expansion of
Now, by § 7,
{l + V(l + a;^)}"+''' + {l-N/(l + ^2)}™+'"'
_ or^'zr ii 4. s (>^ + 2r)(re + 2r-«-l)(M + 2r-s-2) . . . (n^ 2;--2.- + l) x^\
\ {s)\ 2'4'
the coefficient of x"^^ in which is
(» + 2r) (ji+r -!)(?? + r- 2) . . . (» + l)
^T2^^^^ '
14
210 EXERCISES X CH. XXVIl
c _ on+2r-i nl(2r)!(n + 2r)(« + r-l)l
(n + 2r)!r!nl22'- '
_ (n+r-l)(n + r-2) . . . (r + 1)
~ (n + 2r-l){n + 2r-2) . . . (2r + l)'
The summation is thus effected for all integral values of r. So far, how-
ever, as r is concerned, the formula arrived at might be reduced to an
identity between two integral functions of r of finite degree. Since we have
shown that this identity holds for an infinite number of particular values of
r, it must (chap, v., § 16) hold for all values of r. The summation is there-
fore general so far as r is concerned.
Exercises X.
Find the coefficient of x*" in the expansion of the following in ascending
powers of x,
(1.) xl{x-a)(x-b){x-c). (2.) x^+^l(x-a){x-b){x-c).
(3.) x'"^+^l{x -a){x- b) (x-c), where m is a positive integer <r - 3.
(4.) (3 - x)l{2 -x)(l- x)\ (5.) ■ 2x2/ (x - 1)2 (x» + 1).
(6.) {l-px)"'{V-qx)-\
(7.) If (1 - 3a;)'*/(l - 2x)^ be expanded in ascending powers of x, the co-
efficient of x""!^"^ is ( - 1)" (r - 2n) 2^-^, n and r being positive integers.
(8.) Find the numerically greatest term in the expansion of (a - x)/(6 + x)^
in ascending powers of x.
(9.) Show that
(x + /3)(x-l-2;3) . . . (x + w/3)
(x-j3)(x-2^) . . , (x-n/3)
-14-T^ ^~-r»(» + >•)(»^-l^)("''-2') " • (n'-r-l') r/3 ,
and hence show that
*•;"( )„_.r»(n + r)(n«-l')(n'-2-^) . . . (,,2-7312)^^ ^^^ ^ ^^^
r=i {^^Y
(10.) If n be a positive integer, show that
■'■ ~ ni^l + m^2 ~ • • • \~)rrfin—\~rm-l^n'
(11.) If n be an even positive integer,
m^n ~ m^n-1 • m^l + »n^n-2 • nfii - . . . + ,„C/„ — \- r'^mPnJ% '
(12.) If m and n be positive integers, show that
m^O • m/2^n + to^2 • (»>i-2)/2^n-l + m^i ' (m-4)/2^n-2 + • • • + m^2n • (m-2n)/2^0
m2(wt''-2'') . . . {m^-2n-2"-)
(2n)l '
m^l • (m-l)l2^n + m^3 • (m-3)li^n-l + m^6 • (»n-«)/a^n-2 + • • • + »n^2n+l • (m-2n-l)/2^0
_ m(wt^-P)(m'-3'') . . . {m^-'JiT^^)
(2n + l)!
(See Schlomilcb, Hci«d6. d. Alg. Anal., § 38.)
§ 10 EXERCISES X 211
(13.) Show, by equating coefficients in the expansion of (1 - x~^)^{l - «)""*,
where m is a positive integer, that
l_™.+!5!Kri)+. . .^(-ir""(~'-i')...(..'-53I^^„,
(2!)^ ^ [miy
(14.) If n be a positive multiple of 6, then
(15.) li {l-\-x)-'^=l + a-^x + a^x'^-\-. . ., sum the series l-a^ + a^-a^-\- . . .
to n terms.
(16.) If (l + x)^'' = l + a^x + a^x^+ . . ., then l-a^^+a^^- . . . =
(-l)'»2ra(2ra-l) . . . (n+l)/Ml.
r\ 2^r + l)\ {-iy2^r(2r)\_{-l)r
^ ' rill (r-l)13l"^' • '"^ 01(2r + l)! ~2r + l'
(18.) ''s"l/4'- (r!)2 {2n - 2r)! = (4n)I/4"{ (2n)l\^.
(19.) Sum to n terms S (2n - 2)I/22'*-in {(jj - 1)1 }^.
(20.) Sum the series
,. ,.1 ,, „, 1.4 . „, 1.4.7, 1.4...(3n-5)
« + (n-l)3 + (n-2)— + (n-3)3-^g+. . • + 3 .^ . . . (3„-3) ♦
(21.) Find for what values of n the following series are convergent; and
show that when they are convergent their sums are as given below.
1 TC 1 w(n-l) 1 (m-l)!
m~llm + l"^ 21 m+2 ' ' ' ~ {n + l)(n + 2) . . . (n + m)'
1 n 1 w(n-l) 1 (m-l)!
m'^'llm + l"^ 21 m + 2''"* ■ •~(n + l)(7i + 2) . . . (n + m)^'"+"^'"-i^"
-^4^C^_22»+2 + . . . + (-)'«-i2'>-Hn + (_)mi},
i» in both cases being a positive integer.
/22 ) »^"(r + s)!(m+n-r-g-l)l _ (m + n)
«=o ^•Is! (m-r- 1)1 (n-s)! ~ ml?*!
(23 ) '"IT *s" (^ + ^)'("^+"-^-g)' _ (m + w + l)l
r=o «=o ris! (m-r)! (n-s)! mini
(24.) The number of the r-ary products of three letters, none of which is
to be raised to a power greater than the nth, where n < r < 2n, is
r(3?i-?-) + l-fK(n-l).
(25.) Prove, for a, b, c, that "LarKa-h) (a-c) = 0, if r-0, or r=l ; =1,
if r=2 ; and generalise the theorem.
(26.) Show that
a(b-c) (be - aa') {a^ - a'^) & (c - a) (ca - 66') (&»> - fc'"')
a -a' b-b'
c{a-b) {ab - cc') (c*» - c"»)
c-c'
= {b-c){c- a) {a - b) {be - aa') (ca - bb') (ab - cd) S^_^\ahc,
where aa' = bb'=ee'y and S^_^ is the sum of the (m-3)-ary products of
a, 6, c, a'y b', c'. (Math. Trip., 1886.)
14—2
212 EXERCISES X CH. XXVII
(27.) If S^ be the sum of the r-ary products of the roots of the equation
a;'»+aix"'-i + a2a;"-- + . . . + a„=0, then
0 = ^1 + 01,
O^Sa + S'iOi + fla,
(Wronski.)
(28.) If Sy be the sum of the r-ary products of n letters, P^ the sum of the
products r at a time, S^ the sum of their rth powers, then
Zr=nSr-(n-l)PiSr-i+. . . + {-lY{n-r)P^, if r<n-l.
= nS^-{n-l)P^Sr-.i+. . .+(-l)"-iP„_iSr-„+i, if r>n-l.
(Math. Trip., 1882.)
(29.) If i; = (1 - ax)~'^ (1 - ^x)~^ ■ • -, the number of ways of distributing n
things, X of which are of one sort, fj. of another sort, . . ., into p boxes
placed in a row is the coefficient of x^^a'^^'^ ... in the expansion of {v - 1)p
in ascending powers of x, namely,
ih-pCjii^+pC^Ui- . . .,
where Ug={p + \-s)\{p + iJ.-s)l . . . I(p-sy.\l{p -s)l fil . . .
(Math. Trip., 1888.)
(30.) With the same data as in last question, show that the whole number
of ways of distributing the things when the order in which they are arranged
inside each box is attended to is
nI(?i-l)I/(n-ij)!(2J-l)!X!Ai!»'! . . .
(Math. Trip., 1888.)
Show that
(31.) 1 + 1/2 + . . . + llx=^C^-h^C, + ^,C,-. . .
{m + l)m {m + 2){vi+l)m(m-l) _{-Vr
(32.) 1 ---2 + gj 2 2m+l-
/^Q^ 1 w^n. I m^(m^-P)„, "t" K - 1") K - 2") ^^ , -^-1^"*
(66.}i.-:^^+ ji ^- gj ^ +. . . V ; .
(34.) If 7n and n are both positive integers, and jn>n, then
2^ {m-n){m-n-l) _^ (m-n)(jn-n-l)(m-n-2){m-n-B) ^_^_^
n\ ■*" ll(n-l-l)! ■*" 21(n + 2)!
1.3.5 . . . (2nt-l)
" " ■ ~ (m + w)!
(35.) If r be a positive integer,
( ^2 _ 12 (^2 _ 12) (^2 _ 22) {f - 12) (f _ 22) (r^ - 3^) „ , )
rjl-f—3j-a; + ^ 5] ^^ 7! ^^'' + - ' •[
= (a; + 2)'-i-^_a(7i(a; + 2)'-3 + r-8t''2(x + 2)'-'>-r_4C3(a; + 2)'-7 + . . .
§ 11 CONVERGENCY OF MULTINOMIAL SERIES 213
MULTINOMIAL THEOREM FOR ANY INDEX.
§ 11.] Consider the integral function aiX + «2^ + . . . + arX^,
whose absolute term vanishes, the rest of the coefficients being
real quantities positive or negative. Confining ourselves in the
meantime to real values of x, we see, since the function vanishes
when a; = 0, that it will in all cases be possible to assign a posi-
tive quantity p such that for all values of x between - p and + p
we shall have
\aiX + a2a^+. . . ■<rarX^\<l (1).
In fact, it will be sufficient if p be such that
ap + ap^^ . . . + «/)'■< 1
where a is the numerical value of the numerically greatest
among ai, a.2, . . ., ar. That is, it will be sufficient if
«p(i-pO/(i-p)<i;
a fortiori (supposing p<l) it will be sufficient if
apl{l-p)<\\
that is, if p<l/(a + l)* (2).
p is, in fact, the numerically least among the roots of the
two equations
ttrX^ + . . . + aiX±l = 0,
as may be seen by considering the graph of arX'^ + . . . +aiX.
Therefore, whether m be integral or not, provided
-p<x< + p we can always expand (1 +aiX + a.2X^+ . . . +arX^)^
in the form
1 + 2m(7g (ttiX + a2X^ + . . . + arx'y (3) ;
and the series (3) will be absolutely convergent whether m be
positive or negative. Hence, since aiX+a.2af+. . . +arafh a
terminating series and therefore has a finite value for all values
of X positive or negative, it follows from the principle established
in chap, xxvi., § 34, that we may arrange (3) according to powers
* This ia merely a lower limit for p ; in any individual case it would in
general be much greater.
214 MULTINOMIAL COEFFICIENTS CH. XXVII
of X, and the result will be a power series which will converge to
the sum (l + aiir + a2^+ . • • + ar^O™ so long as -p<^< + p.
Since s is a positive integer, we can expand m.Gs{aiX + a^x^ ■\-
. . . ■varX^'Y by the formula of chap, xxin., § 12. The coefficient
of x^ in this expansion will be
XJJ.sXa^'a.^-" . . . ar^-'ja^W . . . a^!,
that is,
^a^'-a^ . . . a^m{m-\) . . . {m-s+l)/ai\a^l . . . a^! (4),
where the summation extends over all positive integral values of
«!, oa, . . ., Ur, including 0, which are such that
"1 + 02+ . . . +a;. = S|
ttj + 2a2 + . . . + ra^ = wj ^ ''
In order, therefore, to find the coefficient of of* in (3) we have
merely to extend the summation in (4) so as to include all
values of 5 ; in other words, to drop the first of the two restric-
tions in (5).
Hence, wliether m he integral or not, provided x he small
enough, we have
(1 + aix + a^x^ + . . . +arafy"'
^ J ^ 2 m(«,-l) . .(m-2...M) , __
the summation to he extended over all positive integral values of
"■1, (h, • • •, "rj including 0, su^h that
Oj + 2a2 + . . . + raj. = n.
The details of the evaluation of the coefficient in any parti-
cular case are much the same as in cbap. xxin., § 12, Example 2,
and need not be farther illustrated. It need scarcely be added
that when n is very large the calculation is tedious. In some
cases it can be avoided by transforming l + aiX + a^g^ + . . . + arOf
before applying the Binomial Expansion, but in most cases the
application of the above formula is in the end both quickest and
most conducive to accuracy.
§§ 11-13 CONDITIONS FOR GOOD APPROXIMATION 215
Example. To find the coefficient of .r" in (1 + x + x^ + . . . + a;*")"*.
We have
(l + x + x^+. . .+a;'')'"={(l-a;'*+i)/(l-a;)}"*,
= (l-a;''+i)™(l-a;)-"»,
Hence, if n<r + 1, the coefficient of a;" is simply
^fl'„=m(ffH-l) . . . (?« + n-l)/nI;
but, if n<tr + l, the coefficient of x" is
m"-n ~ m^l • m"n-r-l •" m^2 • m"«-2r-2 ~ • • •
NUMERICAL APPROXIMATION BY MEANS OF THE BINOMIAL
THEOREM.
§ 12.] The Binomial Expansion may be used for the purpose
of approximating to the numerical value of (1 + w)'^. According
as we retain the first two, the first three, . . ., the first n + 1
terms of the series 1 + nCiO! + nO^a^ + . . ., we may be said to
take a first, a second, ... an nth approximation to (1 + a?)"'.
The principal points to be attended to are—
1st, To include in our approximation the terms of greatest
numerical value ; in other words, to take n so great that the
numerically greatest term, at least, is included.
2nd, To take n so great that the residue of the series is
certainly less than half a unit in the decimal place next after
that to which absolute accuracy is required.
3rd, To calculate each of the terms retained to such a degree
of accuracy that the accumulated error from the neglected digits
in all the terms retained is less than a unit in the place next after
that to which absolute accuracy is required.
The last condition is easily secured by a little attention in
each particular case. We proceed to discuss the other two.
§ 13.] The order of the numerically greatest term.
In the case of the Binomial Series (1 -^-xY, if ^ denote the
numerical value of x, so that 0<^<1, we have for the numerical
value of the convergency -ratio u^^^lu^
216 NUMERICALLY GREATEST TERM CH. XXVII
m — n J. n — m
(r„ =
^'OT^--—i, (1),
n+l n+1
according a,8 m-n is positive or negative.
Hence it is obvious, in the first place, that, if - 1 ^ m<+ 1,
that is, if m he & positive or negative proper fraction, the condi-
tion o-„<l is satisfied from the very beginning, and the first
term will be the greatest.
If m>+l, the condition o-„<l is obviously satisfied for any
value of n which exceeds m ; in fact, the condition will be
satisfied as soon as
(m-n)i<n+ 1,
that is, n>(mi- 1)/(1 + i) (2),
the right-hand side of which is obviously less than m. This
condition is satisfied from the beginning if i<2l(m-l).
If m be <-l=-/A, say, where )u.>l, the condition o-„<l
will be satisfied as soon as
(fj. + n)i<n + 1,
that is, n > (fjii -l)/(l-i) (3).
This condition is satisfied from the beginning if ^<2/(/ti + 1).
§ 14.] Upper limit of the residue. We have seen that,
ultimately, the terms of a Binomial Series either (1) alternate in
sign or (2) are of constant sign.
To the first of these classes belong the expansions of (1 +a;)'"
and (1 + x)"^, where x and m are positive.
If n be greater than the order of the numerically greatest
term, and in the case of (1 +xy"' (see § 4) also >m, then the
residue may be written in the form
-Sn = ± {Un+i - ltn+2 + W»+3- • • •) (l),
where «„+!, %+2, ^n+a, ... are the numerical values of the
various terms, and we have Un+\>ti,i+2>Un+%> . . •
Hence, in the present case, the error committed by taking an
n\\\ approximation is numerically less than Un+\. In other words.
§§ 13, 14 UPPER LIMIT FOR RESIDUE 217
if we stop at the term of the nth order, the following term is an
upper limit for the error of the approximation.
Cor. A lower limit for the error is obviously m^+i - Un+2'
The expansions of (1 - a;)™ and (1 - a;)~™ belong to the
second class of series, in which the terms are all ultimately of
the same sign. It will be convenient to consider these two
expansions separately.
In the case of (l-ar)™, if we take n>m, then we shall
certainly include the numerically greatest term ; and o-„, the
numerical value of the convergency-ratio, will be {n - m) xl{n + 1),
that is, {\. — {m+l)l{n + \)}x. This continually increases as n
increases, and has for its limit x, when n=<xi. Hence
Therefore, m^+i, w„+2, ■ • • having the same meaning as before,
-^ra = ± («*»+! + Un+2 + «^n+3 + • • •),
— ± W»+i (1 + <^n+i + O^n+iO'n+2 + 0'n+iO'»+20"n+3 + • • •)•
Therefore
<Un+il{l-x) (2).
Hence the error in this case is numerically less than w„+i/(l - x),
and it is in excess or in defect according as the least integer
which exceeds m is even or odd (see § 4).
Cor. A lower limit for the errm^ is obviously %+]/(! — a-n+i),
that is, mC„+i^"+V{l -{n+l-m) x/{n + 2)}.
In the expansion of (l-x)'"^, all the terms are positive;
and, in order to include the greatest term, we have merely
to take n>{mx-l)/{l-x).
We have, in this case,
(T^ = (w + m) x/{n + 1) = {1 - (1 - m)/(n + 1)} x,
= {l + (m~l)/{n+l)}x.
Hence, if »i<l
0'n+l<0-n+2<' • .<X<1,
218
EXAMPLE
CH. XXVII
and an upper limit of Bn will he w»+i/(l - x) as in last case, a
lower limit being m„+i/(1 - o-„+i), that is, m^„+ia;'*+V{l - (w + 1 +
m) xl{n + 2)}.
If w>l,
l>0'»+l>0-«+2>. . '>X,
and an upper limit of Rn will be Un+i/(l - a-n+i), that is,
»n^«+ii»"'^V{l -{n + l+m) x/{n + 2)}, a lower limit being Un+j
(l-x).
The error for (1 - ar)-"* is, of course, always in defect.
Example 1. To calculate the cube root of 29 to 6 places of decimals.
The nearest cube to 29 is 27. "We therefore write
4/29 = (33 + 2)1/3 = 3 (1 + 2/33) V»,
= Mq + Mi-M2 + M3-M4 . . . .
The first term is here the greatest ; and the terms alternate in sign after m^.
Also Uf, written in the most convenient form for calculating successive terms, is
«r=3 (A) (tI.) (,*A) (t.V\) (A\) . . . (^) .
Therefore
+
-
«o=
Mi=«o2/81 =
M2= Ml 4/162 =
«3= 1*2 10/243 =
W4=M3 16/324 =
3 000,000,00
74,074,07
75,27
•001,828,99
3,72
3 074,149,34
•001,832,71
•001,832,71
«5=M4 22/405
3 072,316,63
20
Hence the error in defect, due to neglect of the residue, amounts to less
than 2 in the seventh place. The error for neglect of digits does not exceed
1 in the seventh place. Therefore, the best 6-place approximation to
^29 is 3^072,317. In Barlow's Tables we find 3-072,316,8 given as the
value to 7 places.
Example 2. To calculate (1 - x)'"/(l + x + x*)*" to a second approximation,
X being small.
(l-x)'"(l + a; + x2)-'"
(, 7n(7»-l) „1 I, , , „, m(?tt + l) 2]
= < l-mx + -^ — ' x^V X i l-Tn(x + x^) + —^ — '-x^V ,
§ 14 EXERCISES XI 219
where we have already neglected all powers of x above the second in each of
the two series ;
(, m(m-l) „) ( ^ m(m-l) „)
= \l- mx + — 5_ ' a;2 I Jl-mx+ — ~ — '- x^ \ ,
, Ivi(m-l) „ m (m - 1)1 „
= l + {-m-m)x+ \—^ — '- + m^+—^ — '-V x",
where higher powers of x than x^ have again been neglected in distributing
the product ;
= 1 - 2mx + m (2m - 1) x*.
Exercises XI.
(1.) The general term in the expansion of {l + x + y + xy)l{l + x + y) ia
( _ l)m-H» (^ 4. „ _ 2)1 a;"'?/»/(m - 1)! (n - 1)1.
Determine limits for x within which the following multinomials can be
expanded in convergent series of ascending powers of x ; and find the
coefficients of
(2.) X* in (1 - 2x + a;2 _ Sx^)-V*. (3.) x« in (1 - 3x - Ix^ + x^)-^l^.
(4.) a;8 and x^ in (a; + 3a;» + Sa;" + 7a;^ + . . . )-\
(5.) x7in(l-3a; + a;8-x')-3/2. (6.) x'" in (2 + 3a; + x2)-2.
(7.) Show that in (Qa^ + 6ax + ix^)-^ the coefficient of x^^ is 23'- (3a)-*'-2 ;
and that the coefficient of every third term vanishes.
(8.) The coefficient of x"* in (1 + a; + x"^)^ (m a positive integer) is
^ m(m-l) m(m-l)(?/t-2)(m-3)
■*■ (11)2 "^ (2ip +... .
(9.) The coefficient of 3^^+^ in (1 + x)/(l + x + x^ is - (r + 1),
(10.) Evaluate «2/( 100/99), and 1^(1002/998), each to 10 places of deci-
mals ; and demonstrate in each case the accuracy of your approximation.
Find a first approximation to each of the following, x being small: —
/„. {x+^jx^ + l)}'^- {x- V(x2+1)}^»
^ ■'' {x+^{x^+i)y^'n+i-{x-^(x-'+i)y^+i'
(12.) (l + aj)(l + rx)(l + r2a;). . ./(l-a;)(l-x)'-(l-x)'^. . . .
(13.) V(2-x/(2-v/(2- . . . -^(1 + x). . .))); where ^ is repeated
n times.
(14.) If X be small compared with N^, then ^{N^ + x) = N+xl4:N +
Nxl2 (2N^ + x), the error being of the order x'^jN'^. For example, show that
^(101) = 10^0^, to 8 places of decimals.
(15.) If p differ from N^ by less than 1 per cent, of either, then i^p differs
from ^N+IpIN^ by less than ^'/90000. (Math. Trip. , 1882.)
220 EXERCISES XI CH. XXVII
(16.) Itp=N* + x'where x is small, then approximately
show that when J/=10, x = \, this approximation is accurate to 16 places of
decimals. (Math. Trip., 1886.)
(17.) Show that L {1/V«- + 1/V(«^ + 1) + • • . +\l^{n^ + 2n)} = 2.
n=oo
(Catalan, Nouv. Ann., sec. i., t. 17.)
(18.) Find an upper limit for the residue in the expansion of (l + a;)"*
when m is a positive integer.
CHAPTER XXVIII.
Exponential and Logarithmic Series.
EXPONENTIAL SERIES.
§ 1.] We have already attached a definite meaning to the
symbol a* when a is a positive real quantity, and x any positive
or negative commensurable quantity. We propose now to discuss
the possibility of expanding (f in a series of ascending powers
of X.
If we assume that a convergent expansion of a'' in ascending
powers of X exists, then we can easily determine it^ coefficients.
For, let
a'' = AQ-^AiX + A^ + . . .+Anx'^+. . . (1),
then, proceeding exactly as in chap, xxvii., § 2, we have
L{a''+^-d")lh = Ai + 2A^ + . . . + nAnX''-'^ + . . .;
and the series on the right will be convergent so long as x lies
within limits for which (1) is convergent. Now (by chap, xxv., § 13)
L ia"^^ - d')lh - a'^XL (e^" - 1)/XA,
where X = log^a, and e is Napier's Base, namely, the finite quantity
L{l + llnf. Hence
Xa'' = lAi + 2A^x + . . .+nAnX''-'^ + . . . (2).
Therefore, by (1),
\{Ao + AiX-¥. . . + An-idf"-^ + . . .)
= lAi + 2A2X + . . .+nAnaf~'^ . . . (3).
Since both the series in (3) are convergent, we must have
lAi = XAo, 2Aa = kAi, . . ., nAn = XAn-i.
222 DETERMINATION OF THE COEFFICIENTS CH. XXVIH
Using these equations, we find, successively,
A^ = AMl\, A, = Ao\y2\, . . ., An = AoX''/n\ (4).
Also, since, by the meaning attached to a% a° = + 1, putting
a; = 0 on both sides of (1), we have
+ 1=^ (5).
Hence, finally,
a'' = l+XxfU + {Xx)y2l + . . .+{Xxfln\ + . . . (6).
We see, a posteriori, that the expansion found is really con-
vergent for all values of x (chap, xxvi., § 5), and also that the
series in (2) is convergent for all values of x. Our hypotheses
are therefore justified.
This demonstration is subject to the same objection as the
corresponding one for the Binomial Series : it is, however, interest-
ing, because it shows what the expansion of d" must be, provided
it exist at all. We shall next give two other demonstrations,
each of which supplies the deficiency of that just given, and each
of which has an interest of its own.
§ 2.] Deduction of the Exponential from the Binomial Expansion.
By the binomial theorem*, we have, provided z be numeric-
ally greater than 1,
i}^W-
1 zx(zx-l) 1
z 2! z^
+
zx{zx-\) . . . (zx-n+1) 1
. x^l- 1/zx) x"" (1 - Hzx) ... (1 - w - \\zx)
= 1 + a; + — ^—^ — - + . . . + — ^ — '- — :— ^^ ■ — -
2! n\
where
+ i2» (1),
„ _x''^^{\-\lzx)...{\-n\zx) x"^^"" {l-ljzx) . ..{\-n^\lzx)
(^i+1)! "" (« + 2)!
+• • . (2).
* In what follows we have restricted the value of the index zx. Since
z is to be ultimately made infinite, there is no obj jction to our supposing it
always so chosen that zx is a positive integer. We then depend merely
on the binomial expansion for positive integral indices. This will not affect
the value of L(l + l/z)**, for it has been shown (chap, xxv., § 13) that this
has the same value when z becomes + or - oo , and whether z increases by
integral or by fractional increments.
§§ 1, 2 DEDUCTION FROM THE BINOMIAL THEOREM 223
Suppose now ar to be a given quantity ; and give to n any fixed
integral value whatever. Then, no matter what positive or
negative commensurable value x may have, we can always choose
z as large as we please, and at the same time such that zx is a
positive integer, p say, where p>n. The series (2) will then
terminate; and we shall have IjzxK^jzxK. . .<nlzx . . .
<{p- l)/zx<l. With this understanding, it follows that
^"^{71 + 1)1^ {n+ 2)1'^' ' •%!'
af+^ ( X id' J I
(w + 1)! i w + 2 (w + 2f J '
< a;"+ V(w + 1 ) ! { 1 - xl{n + 2)} (3) ;
and we have
V 2;/ 2! * ' * n\
+ Rn (4),
where ^„ satisfies the condition (3).
Now let z, and therefore also p, increase without limit (;*
remaining fixed as before). Then, since
X(l-1/^) . . . (1-72-1/^) = !,
we have
Bn being still subject to (3).
We may now, if we choose, consider the effect of increasing
n. When this is done, a;"+Y(w + 1)!{1 -a;/(w + 2)} (see chap.
XXV., § 15) continually diminishes, having zero for its limit when
« = Qo ; we may therefore write
c^ x^
l+a;+-+. . . + —: + . . . ad CO (6).
2! n\ ^ '
Thus the value of Z(l + I/2;)** is obtained in the form of an
infinite series, which converges for all values of x. For most
purposes the form (5) is, however, more convenient, since it gives
an upper limit for the residue of the series.
z~\ z)
224 CALCULATION OF e CH. XXVIII
§ 3.] The conditions of the demonstration of last paragraph
will not be violated if we put x=\. Hence, using e, as in chap.
XXV., to denote i/ (1 + llzf, we have
where Rn<{n + 2)/(n + 1) (% + 1)! (8).
This formula enables us to calculate e with comparative rapidity
to a large number of decimal places. We have merely to divide
1 by 2, then the quotient by 3 ; and so on. Proceeding as far
as 11 = 12, we have
1 + 1 =2-000000000
1/2! =
•500000000
1/3! =
166666667
1/4! =
41666667
1/5! =
8333333
1/6! =
1388889
1/7! =
198413
1/8! =
24802
1/9! =
2756
1/10! =
276
1/11! =
25
1/12! =
2
2718281830
Here the error in the last figure owing to figures neglected in the
arithmetical calculation could not exceed the carriage from 10x5,
that is, 5. Also the residue i2i2<i^(l/13!)<if '0000000002
< '0000000003, so that the neglect of Ei^ would certainly not
affect the eighth place. Hence we have as the nearest 7-place
approximation for e
e=: 2-7182818.
It is usual to give a demonstration that the numerical constant e
is incommensurable. The ordinary demonstration is as follows : —
Let us suppose that e is commensurable, say =pjq, where jp and q are
finite positive integers. Then we have by (7)
p/e = 2 + l/2!+. . .+l/(?! + 2?„
where ■B«<(3 + 2)/(3 + iP3l.
§§ 3, 4 INCOMMENSURABILITY OF e 225
Hence, multiplying by g!, we get
2>.(g-l)I=I + glEg,
where p (gf - 1)! and I are obviously integral numbers. Hence gliJg must be
integral.
Now 3!i^g<{'Z + 2)/('? + l)^
<(3 + 2)/{3('Z + 2) + l},
that is, glEj is a positive proper fraction.
The assumption that e is commensurable therefore leads to an arithmetical
absurdity, and is inadmissible.
Another demonstration which gives more insight into the
nature of this and some other similar cases of incommensurability
in the value of an infinite series is as follows : —
If ?'i, ?'2, . . .,?*«»• • . be an infinite series of integers <7ivere in magnitude
and in order, then it can be shown (see chap, ix., § 2) that any commen-
surable number pjq (where p and q are prime to each other, and p<q) can
be expanded, and that in one way only, in the form
P=Pl+P2.+.Is_+_, + PiL +. .. (9),
q Ti r^r^ r^r^r^ r^r^ • • • »«
where 2'i<»'i, i'2<''2> • • •» i'n<^n» • • •; ^^^ *^^* ^^^ series will always
terminate when either q or all its factors occur among the factors of the
integers r-i,r2, • • •, r„, . . . Hence no infinite series of the form (9) can
represent any vulgar fraction whose denominator consists of factors which
occur among r j , r^ , . . .,»•«,• • •
In particular, t/ J*i,?'2, • • •,'!'%>• • • contain all the natural primes,
and, a fortiori, if they be the succession of natural numbers {excepting 1),
namely, 2, 3, 4, 5, . . ., n + 1, . . ., then the series in (9) cannot represent
any commensurable number at all*.
The incommensurability of e is a mere particular case of the last con-
clusion ; for we have in the series representing e - 2
ri = 2, r2=3, . . ., r„=n + l, , . .;
i'i = l. i'2=l i'n = l. ....
Hence e - 2 is incommensurable, and therefore e also.
§ 4.] Returning to equation (5) of § 2, since L(l + l/z)" has
a finite value e, we have L(l + l/zf^ = {L{1 + l/zf}'' - e^, there-
fore
* It should be noticed that an infinite series of the form (9) may
represent a fraction whose denominator contains a factor not occurring
among TitT^, . . •> J'ni • ■ •» for example,
112 3 4
2 = 3+375 + 375?7 + 3.5.7.9+---^^^-
This point seems to have been overlooked by some mathematical writers.
c. II. 15
226 cauchy's summation ch, xxviii
^=1 + ^ + 1; + . .•+f;+^« (10),
where Rn is subject to the inequality (3).
Finally, since a"' = e^% where \ — \<dgea, we have
a^-l + ^^ + ^-^+. . . + ^+i2„ (11),
where Rn<{^f+^l{n + 1)!{1 - A;r/(w + 2)} (12).
Since LRn'^O when n^cc, the series (10) and (11) may of
course each be continued to infinity.
This completes our second demonstration of the exponential
theorem.
§ 5. ] Summation of the Exponential Series for real values of x.
A third demonstration was given by Cauchy in his Analyse
Algehrique. It follows closely the lines of the second demonstra-
tion of the binomial theorem ; and no doubt it was suggested
by the elegant process, due to Euler, on which that demonstra-
tion is founded. This third demonstration is of great import-
ance, because we shall (in chap, xxix.) use the process involved in
it to settle the more general question regarding the summation
of the Exponential Series when a? is a complex number.
Denote the infinite series
' x" af
l+X+~. + . . . + -:+. . .
2! n\
by the S3anbol f{x). Then, since the series is convergent for all
values of x, f{x) is a single valued, finite, continuous function
of X (chap. XXVI., § 19).
Also, since f{x) and f(i/) are both absolutely convergent
series, we have, by the rule for the multiplication of series
(chap. XXVI. , § 14),
/(^)/(2/) = l + (^+2/)+(^+^^ + |-j) + . . .
\n\ {n-\)\l\ {n-2)\2\ n\J
§§ 4, 5 EXPONENTIAL ADDITION THEOREM 227
Now
x^ x^~\l x^'^y^ y^
= {x^ + nC,x^'-'y + nC^x'^-hf + . . .+2/")^!,
=^{x + yfln\,
by the binomial theorem for positive integral exponents.
Hence f{x)f{y)---l + ^{x+ yfjii ! ,
=/(^ + 2/) (1).
Hence f{x)f{y)f{z) =f{x + y)f{z),
=f{x + y + z);
and, in general, x, y, z, . . . being any real quantities positive or
negative,
f{^)f{y)m- ■ .=f{x + y + z + . . .) (2).
This last result is called the Addition Theorem for the
Exponential Series.
From (2), putting x=y = z, . . ., =1, and supposing the
number of letters to be w, we deduce
{f{l)r=f(n) (3).
Also, taking the number of the letters to be q, and each to
be p/q, we deduce
{/(p/q)}'=f{p) (4),
where p and q are any positive integers. From (4), by means of
(3), we deduce
{/(plq)}'-{/{l)}' (5).
Finally, from (1), putting y = — x, we deduce
/{x)f{-x)=f{0) (6).
The equations (5) and (6) enable us to sum the series /(x)
for all commensurable values of x.
From (5) we see that /(p/q) is a g'th root of {/(l)}^. Now,
since p/q is positive, the value of /(p/q) is obviously real and
positive. Also /(I), that is, 1 + 1/1! + 1/2! + . . , , is a finite
positive quantity, which we may call e. Therefore {/(l)p, or e^,
is real and positive. Hence /(p/q) must be the real positive
g-th root of e^, that is, e^''^. Hence
15—2
228 cauchy's summation ch. xxviii
1+^+^ + ...=^" (7).
p and q being any positive integers.
Finally, since /(0)= 1, we see from (6) that
f{-p/q) = i/f{p/q),
= e~P'^.
Hence
l^Lz£lSlJjz£lsy^...=e-m (8),
J. 1 Jil
where p/q is any positive commensurable number.
By combining (7) and (8) we complete the demonstration of
the theorem, that
1! 2! n\
for all commensurable values of a;, e being given by
,11 1
1! 2! nl
The student will not fail to observe that e is introduced and
defined in the course of the demonstration.
The extension of the theorem to the case where the base is
any positive quantity a is at once effected by the transformation
a" = e^, as in last demonstration.
§ 6.] From the Exponential Series we may derive a large
number of others ; and, conversely, by means of it a variety of
series can be summed.
Bernoulli's Numbers. — One of the most important among the
series which can be deduced from the exponential theorem is
the expansion of xl{\-e~'^), the coefficients in the even terms
of which are closely connected with the famous numbers of
Bernoulli.
We shall first give Cauchy's demonstration, which shows, a
priori, that xl{\-e~'^) can he expanded in an ascending series of
powers of X, provided x lie within certain limits.
§§ 5, 6 EXPANSIBILITY OF x/(l - e"^) 229
We have
' ' (1),
1-e-'' {l-e~^)lx l-y
where 2/=l-(l-0/-» (2).
Now, from (1), we have
^/(l-0 = l+2/+y' + - • .ad 00 (3);
and this series will be absolutely convergent provided - 1 <?/< + 1.
Also, from (2), using the exponential theorem, we have
3/ = ^/2!-^/3! + ar'/4!-. . . ad 00 (4);
and this series is absolutely convergent for all values of x, and
therefore remains convergent when all the signs are taken alike.
If, therefore, we can find a value of p such that
p/2! + pV3! + /3V4! + . . . ad ox 1 (A),
then, for all values of x between — p and + p, Cauchy's condi-
tions of absolute convergency (chap, xxvi., § 34) will be fulfilled
for the double series which results, when we substitute in (3) the
value of y given by (4). This double series may therefore be
arranged according to powers of x, and the result will be a
convergent expansion for a;/(l — e~*).
It is easy to show that a value of p can be found to satisfy
the condition (A) ; for we have
p/2!+pV3! + . . .=^{e<>-\)lp-l.
We have, therefore, merely to choose p so that
e^-K^p (5).
If the graphs of e^ — 1 and of 'ix be drawn, it will be seen
that both pass through the origin, the former being inclined to
the a?-axis at an angle whose tangent is 1, the latter at an angle
whose tangent is 2, that is to say, at a greater angle. There-
fore, since e^—1 increases as x increases, and that ultimately
much faster than 'ix, the graph of e*— 1 will cross the graph of
2x just once. Therefore the inequality (5) will be satisfied pro-
vided p be less than the unique positive root of the equation
e'—l^'lx. Since e^ - 1 < 2 x 1, nnd r — 1 > 2 x 2, this root lies
230 COEFFICIENTS IN EXPANSION OF xj{l - e'") CH. XXVIII
between 1 and 2.* It will, therefore, certainly be possible to
expand ir/(l-e~^) in a convergent series of powers of x if
If we make the substitution for y, and calculate the co-
efficients of the first few terms, we find that
1-e-^ 2^^ 62! 304! "^426! •'• ^^^•
Knowing, a priori, that the expansion exists, we can easily
find a recurrence formula for calculating the successive co-
efficients. Let
a;/(l-e-'')=Ao + AiX + A^aP + A3a:^ + . . . (7).
Then, putting - ^ in place of x, we must have, since
- x/(l -0") = e-^^/(l - e-%
e-''xl{l-e-'°) = Aa-A^x + Aoa^-A3C(^ + . . . (8).
Since both the series are convergent, we have, by sub-
tracting,
x = 2AiX + ^Aia? + . . . (9).
Hence Ai = ^; and all the other coefficients of odd order
must vanish.
Therefore, from (7), we have
x = {Ao + ix + A2aP + AiX^ + . . .)(l-e-^),
= (Ao + ix + A2iv'^ + AiX* + . . . + A2nx'^'' + . . .)
""U! 2! "^3! ••• (2nj\^{2n + l)r- ' '}
The product of these two convergent series will be another
convergent series, all of whose coefficients, except the coefficient
of X, must vanish. Hence, equating coefficients of odd powers of
X, we deduce Ao= 1, and
■^•2n A<2n-2 A^ 1 1
1-' 3! • * ■ {2n ~l)\~ 2(2^ "^ {2n + 1)! ^ "'
* More nearly, the root is 1-250 . . . ; but the actual value, as will be
seen presently, is not of much importance.
§ 6 Bernoulli's numbers 231
that IS, IT + -gr + • • • + (2^^31)1 = 2T2;r;:i)! ^^^^'
In like manner, if we equate the coefficients of even powers
of iT, we deduce
2! 4! (2^2)! ~ 2 (2w + 2)! ^ ^'
If, as is usual, we put ^2)i = ( - r~^^n/(2w)!, our expansion
becomes
"" ; = l + ^^ + i^-^^^ + #^-. .. (12);
1-e-^ 2 2! 4! 6!
and the equations (10) and (11) may be written
a»+l ^271 -Dn ~ 271+1 ^271-2 -On-1 + • . • ( ~ ) 271+1 ^2 -0 1 = ( — ) (W — ^)
and
(10')
271+2 ^271 -On ~"2n+2 ^271-2^77-1 + • • •("/ 271+2^2^1 — \ ) W (11)
respectively.
If we put n-l, 11 = 2, n = 3, . . . , successively, either in
(10') or in (11'), we can calculate, one after the other, the
numbers Bi, B2, . . ., Bn, • • -, which are called Bernoulli's
numbers*. Since we know, a priori, that the expansion exists,
the two equations (10') and (11') must of necessity be con-
sistent. Neither of them furnishes the most convenient method
for calculating the numbers rapidly to a large number of decimal
places ; but it is easy to deduce from them exact values for a
few of the earlier in the series, namely,
•^^'"^6' ^'^30' ^'^42' ^'"30'
7?_5 o_691 J. J D_3617
^'■'66' ^«~2730' ^'~6' ^'~ 510'
43867 „ 1222277 n
^^^"79F' ^''~ 2310 ' ^'''
* There is considerable divergence among mathematical writers as to the
notation for Bernoulli's numbers. What we have denoted by B^ is often
denoted by Bj^, or by -Ban-i- ^^^ further properties of these numbers, and
for tables of their values, see Euler, Inst. Diff. Calc. Cap. 5, § 122 ; Ohm,
Crelle's Jour., Bd. xx. p. 11 ; J. C. Adams, Brit. Assoc. Rep., 1877, p. 8,
also Cambridge Observations, 1890, App. i. ; Staudt, Crelle's Jour., Bd. xxi. ;
Boole's Finite Dijj'erences (ed. by Moulton) ; and, for a useful bibliography
of the relative literature, Ely, Am. Jour, Math. (1882),
232 EXPANSlONSOF^(e*^+e"^)/(e*'-e"'")ANDa;/(l+e-^) CH.xxviii
We shall return to the properties of these numbers in
chap. XXX.
Remark regarding the limits within which the expansion of a;/(l-c~*) is
valid. — If we denote the series
,1 B, ^ Bo ,
by <f) (x), we may state the problem we have just solved as follows : — To find
a convergent series <f> (x) such that (1 - e~^) <p (x) — x, that is, such that (x - a;^/2!
+ a;3/3!- . . . )<p(x) = x.
Now, since x~ x^l2l + x^l^\-is absolutely convergent for all values of x,
and the coefficients of (p (x) satisfy (10') and (11'), <p {x) will satisfy the con-
dition (a;- a;2/2! +a;^/3! - . . .) ^(x) = a; so long as ^(a;) is convergent. Hence,
so long as ^{x) is convergent, it will be the expansion of a;/(l-e~^). As a
matter of fact, it follows from an expression for Bernoulli's numbers given in
chap. XXX. that <p(x) is convergent so long as -27r<x< +2ir. The actual
limits of the validity of the expansion are therefore much wider than those
originally assigned in the a priori proof of its existence.
Cor. 1. Since x{e" + e-^)/(e* - g"^) = ^/(l - e"^) - ^/(l - e^),
we deduce from (12)
Cor. 2. Since «/(l + e"^) = 2^/(1 - e"'^) - ^/(l - g-*),
j-^,=.|(2^-l)^ + §(2^-l)^='-§^(2^-l)^^ + . . . (14).
§ 7.] Bernoulli's Theorem. — We have already seen that the
sum of the rth powers of the first n integers {nSr) is an integral
function of ii of the r + 1th degree (see chap, xx., § 9).
We shall now show that the coefficients of this function can
be expressed by means of Bernoulli's numbers.
From the identity
(g"* - 1)1 {e^ -l) = l + e^ + e^ + . . .+ e(™-i)%
that is,
(e"^ - 1)/(1 - e-"=) = e*= + e^ + e*^ + . . . + g"*,
we deduce at once
(nx 7iV n^'x" , "1 f^
.^^+!1^ + . . .4-"A^ + . .. (1),
§§ 6-8 Bernoulli's expression for Xn^ 233
wherein all the series are absolutely convergent, so long as n
is finite, provided a: do not exceed the limits within which
1 + ^a? + ^i^Y2! -^2^7^' + • • • is convergent. The coefficient
of af'^'^ on the right of (1) must therefore be equal to the co-
efficient of af'^^ in the convergent series which is the product of
the factors on the left. Hence
„Sr_ 11""+' jf^ B^Tf-^ B,n^-' ^^n"--'
r\ (r+1)! 2.H 2!(r-l)! 4!(r-3)! Ql{r-5)l'
Therefore
^ n''+^ 1 „ ^ D r 1 r(r-l)(r-2) „ ,. ,
"'^r = ,r^ + 2 '' "" 2\ ^'"^ " 4! ' ^''"
r (r-l)(r-2)(r-3)(r-4)
(2),
the last term being ( - )H''-2) i?^,.w, or ^ ( - )H''- V ^j (r_i)»^ accord-
ing as r is even or odd.
This formula was first given by James Bernoulli {Ars Conjectandi, p. 97,
published posthumously at Basel in 1713). He gave no general demonstra-
tion ; but was quite aware of the importance of his theorem, for he boasts
that by means of it he calculated intra semi-quadrantem hoiw ! the sum of
the 10th powers of the first thousand integers, and found it to be
91,409,924,241,424,243,424,241,924,242,500.
It will be a good exercise for the reader to check Bernoulli's result.
SUMMATION OF SERIES BY MEANS OF THE EXPONENTIAL
THEOREM.
§ 8.] Among the series which can be summed by means of
the Exponential Series, two, related to it in the same way as the
series of chap, xxvil, § 5, are related to the Binomial Series,
deserve special mention.
CO
We can always sum the series %<f)r (n) x^jnl, where <}>r (n) is an
integral function of n of the rth degree. {Integro-Exponential
Series.)
234 ^(f)r(n)/n\, ^<f)r(n)/n\(n+a)(n+h) ...(n + Jc) CH. xxviii
For, as in chap, xxvii., § 5, we can always establish an identity
of the form
clir{n)-Ao + Ain + A2n(n-l) + . . . + Arn{n-1) . . . (n-r + l).
Then we have, taking, for simplicity of illustration, the lower
limit of summation to be 0,
^tlM^^Ao%'^^+A,a;^y^^+A,a^%j^^~-^, + . . .
0 w! 0 w! 1 (»-l)! '2 (w-2)!
{n-r)V
= (Ao + Aiiv + A.x^ + . . . + Araf)e^.
QO
Cor. We can in general sum the series %<f)r(n)af^/n\(n + a)
(n + b) . . . (n + k), where a,b, . . ., k are unequal positive integers.
The process is the same as that used in the corollary of
chap, xxvn., § 5, only the details are a little simpler. (See
Example 5, below.)
Example 1. To deduce the formulae (3), (4), (5) of chap, xxvii., § 9, by
means of the exponential theorem.
(x + n)'-nCi{x + n-lY+. . . {-)r^^C^{x + n-r)'+. . . (-)»a^
is evidently the coefficient of 2' in
s!|g(a;+n)?_^(7^e(a;+n-l)2 4.. . . { - )\C ^ e(='+'>'-^)^ + . . . (-)"«=:'}
==s!e^^(e^-l)»,
The lowest power of z in the product last written is 2", and the coeflScienta
of 2", 2"+i, 2"+2 are si, s! (x + ^n), 4s!{.x;2 + n.'c + TV"(^" + l)} respectively.
Hence
(x + 7i)»-„Ci(x + n-l)»+. . . (-)r^fi^{x + n-ry + . . . {-y>x'
— 0, if s<n;
= ji!, if s = 7i;
= (n + l)!(x + ^n), if 8 = 71 + 1;
= ^(n + 2)l{x^ + nx + -^^n{3n + l)}, if s=:n + 2.
Example 2. If n and r be positive integers, show that
„(1, n n{n-l) . . .{n-s + 1) . n{n-l) . . .1 J
{r\ l!(r + l)! sl(r + s)! nl (»• + «)! |
r! l!(r+l)! s!(r + s)!
§ 8 EXAMPLES
The right-hand side is the coefficient of z'^+^ in
^ ' 1! n (r + s)I
= (z + .t)» «^+*,
= eMz» + ,^(7j2"-ix+. . .+„C7„a;»}x |l + ^ + ^ + . . . + ^ + .
Now the coefficient of z™+'' in this product is
235
■{^
ra; + .
.+
71 (W - 1)
l!(r + l)!
Hence the theorem.
If we put r =0, and x=l, we have
, n+1 (ra + l)(n + 2) ,
^ + Il!F+ (2!)^ ^ + ..-adoo
( , n n(7i-l)
' "*"(!! )2'*'" (2!)2 ^'
.!(r + 7i)! T
Example 3. Sum the series
n{n-l)
(«!)^
-I-
13 13 + 23
13 + 23+. .
-X'^+.
ad 00,
We have (by chap, xx., § 7)
13 + 23 + , . . + ri3=(ra4 + 27l3 + n2)/4,
=J{^o + ^ira + ^2«(«-l) + ^3"("- 1) {n-2)+A^n{n-l) (n-2) (w-3)},
where ^q, A-^^, . . ., A^ may be calculated as follows: —
A,= 0,
+ 1
+ 2
+ 3
Hence
1+ 2+ 1+ 0 + |0
0+ 1+ 3+ 4
1+ 3+ 4+|4
0+ 2+ 10
1+ 5 + |14
0+ 3
1 + 1
_13+23 + ... + n3 „ ^ x«-i 7 ,^
n!
^= 4,
^3= 8, ^4=1.
rn-3 1
+ 2.!;3S,-^-^ + ;a;*S
(n-l)!"^2 ^(n-2)l
= (x + |a;2 + 2a;3 + Jx^)e^.
If we put a; = 1, we have
S(13 + 23 + . . . + n3)/ji! = 27e/4.
n=«o
Example 4. Show that S n^lnl = 5e.
n=l
Since n^^n + 3n{n-l)+n {n- 1) {n- 2),
S?i3/ral = 21/(n - 1)! + 3Sl/(rt - 2)! + Sl/(?t - 3)!,
'(ra-3)!^4-^ ^(n-4)!'
236 EXERCISES XII CH. XXVIII
Example 5. Evaluate S (n - 1) x"/(n + 2) nl.
1
(w-l)a;" _ly, (n2-l[x^
(n+2)n!~x2 (n + 2)! *
Now 7i2-l = 3-3(n + 2) + (n + 2)(n + l).
Therefore
1 (7r:r2) nl - ^2 I'^f („ + 2)1 ^"^ t (n + 1)! a n! r
= {3 (e^ - 1 - X - ia;2) - 3x (e=' - 1 - x) + x^ {c'' - 1)} jx-,
= { (a;2 - 3a; + 3) e^ + (^a;2 _ 3) } ^^2.
Exercises XII.
(1.) Evaluate 1/e to six places of decimals.
(2.) Calculate x to a second approximation from the equation
501oge(l + x) = 49x.
(3.) If e*= 1 + xe^^, and x* be negligible, show that
;iz=l/2! + x/4!-x3/4!5!.
(4.) Show that, if n be any positive integer,
(1-1/k)-"> 1 + 1/11 + 1/2! + . . . + l/n!>(l + l/w)».
(5.) Sum from 0 to oo S (1 - 3n + n^) x»/n!.
Sum to infinity
(6.) 12/21 + 22/3! + 3'^/4! + . . . .
(7.) 13/2! + 23/31 + 3'V41 + . . . .
(8.) 1-23/11 + 33/21-43/3! + . . . .
(9.) l< + 2*/2! + 3*/31 + . . . .
Show that
(10.) 1/(271)1 - 1/1! (2k - 1)1 + 1/21 (2h - 2)1 - ... - 1/11 (2ra - 1)1 + l/(2n)I = 0.
(11.) If >i>3, n3 + „C2(/i-2)3 + „C4(n-4)3 + . . .^ji^ (n + 3) 2"-*.
(12.) n"-„(7i(7i-2)» + „C2(n-4)™-. . .=2"nl.
(13.) By expanding e^/l^"''), or otherwise, show that, if
Jr="i°°(n + 7--l)!/?i!(n-l)!, then J^j-(2r + l)^^ + r(r-l)^^_i = 0.
"=i (Math. Trip., 1882.)
(14.) Prove that
(x-x3/31 + x5/51- . . .)(l-x2/21 + a;'»/4!-. • . ) = S(-)'-22'-x2'+i/(2r+l)!.
(15.) Solve the equation x2 - x - Ifn—O ; and show that the ?ith power of
its greater root has e for its limit when n — ao.
(16.) For all positive integral values of n
"-cw("-a-r---(«^)-"'"-'^-
(17.) If
x»=^„ + y|(x-l) + ^(x-l)(x-2) + . . . + ^(x-l)(x-2) . . . (x-n),
show that A,={s + 1)» - ,C^ s" + .Cj (s - 1)» -...(- )%C, 1».
Also that
EXERCISES XII 237
(18.) Show that S(n3 + 2n2 + n-l)/ji! = 9e + l.
(19.) Sum S(7i + a)(7i + 6)(7i + c)x"/7i! from 7J = 0 to n=QO .
(20.) Show that e cannot be a root of a quadratic equation having finite
rational coefficients.
(21.) Sum the series 2x"/(» + 3) n\ from n = 0 to ji=qo .
(22.) Sum to infinity the series 13/3. 1I + 3S/4. 2! + 5'V5. 3! + .. . .
If i?i, JSg, . . ., jB„ denote Bernoulli's numbers, show that
(23.) 2)i+1^2n-l^n~2n+l<^2n-3^n-l + ' ' • ( ~ )"~ 2ii+l^l^l = ( " l)"" •
^94^ r n 2n+1^2n-2 -^w-1 , / _ w-i 2n+1^2 -^1 _ / _ w-l ^
\^*-) 2?i+l'^2n-"n 22 ' • \ I ^in ~\ I ^in,'
(25.) hnCiB^-i^C^B^ + l^C^B^-. . . = (n-l)/2 (71 + 1), the last term on
the left being (-)**""''' -B^2> or 4( - )^'""'^' "^(»-i)/2 - according as n is even or
odd.
(26.) By comparing Bernoulli's expression for 1'* + 2'" + . . . + ti*" with the
expressions deducible from Lagrange's Interpolation Formula, show that
1 '
'-r,-).-.„«c.&±..=o.
1 >-
r ^"^ 2p+i*^'«(t+i)-°-
(Kronecker, Crelle's Jour., Bd. lxxxiv. ; 1887.)
(27.) a; (6== - e-»=)/(e»= + e-»=) = |i (22 - 1) 22x2 + 1? (2* - 1) 2*x J + |j^ (26 - 1) 26x« + . . .
LOGARITHMIC SERIES.
§ 9.] Expansion of log (1 + x). — It is obvious that no function
of X which becomes infinite in value when x^O can be expanded
in a convergent series of ascending powers of x. For, if we
suppose
f{x)=^AQ + A-^x + A2X^ + . . .,
then on putting a; = 0 we have <x> = A^; and the attempt to
determine even the first coefficient fails.
There can therefore be no expansion of log^ of the kind
mentioned.
238 EXPANSION OF LOG (l+.'r) CH. XXVIII
We can, however, expand log{\ + x) in a series of ascending
powers of x, provided x he numerically less than unity.
The base in the first instance is understood to be e as usual.
By § 4, we have
{l + xy=l^z {log (1 + x)] + z" {log (1 + x)fl2\ + . . . (1) ;
and this series is convergent for all values of z.
Again, by the binomial theorem, we have, provided the
numerical value of x be less than 1,
{l^xY = l+zx + z{z-l)x'l2\+z{z-l){z-2)a?l?>\ + . . .,
= 1 + zx-z{l- z/1) xy2 +z(l- z/l)(l- z/2) x'/S + . . . (2).
If we arrange this as a double series, we have
(l + xy = l+zx- {zxy2 - «V/2} + {zaf/3 - (1 + i)2V/3 + ^ z'a^/S} +
{-y-^ {zx'^/n - n-iPiz'x''/n+ n-iP^z^^/n- . . .
{-f-\-,Pn-,z'x^ln]
(3),
where n-\Pr stands for the sum of all the r-products of 1/1,
1/2, . . . , l/(w - 1), without repetition.
In order that Cauchy's criterion for the absolute convergency
of the double series (3) may be satisfied, it will be sufficient if
the series
zx^'jn + „_iPi z^x'^jn + . . . + n-iPn-i z'^x^'ln (4)
and
l+zx + zil + zll)a^/2 + z{l + z/l){l + zl2) 0^/3 + . . . (5)
be both convergent when z and x are positive.
Now the sum of (4) is always z{z + l) . . . (z + n-l) x^'/nl ;
and this has 0 for its limit when n= cc, provided x<l. Also,
the series (5) is absolutely convergent when x<l.
Hence, by chap, xxvi., § 34, we may rearrange the series (3)
according to powers of z, and it will still converge to (1 + x)".
Confining our attention to the first power of z, for the
present, we thus find
{l+xY=l + {x/l-x'/2+ar'/S-. . .}z+. . . (5).
Now, since there can only be one convergent expansion of
§§9,10 EXPANSION OF {log (1+ a;)] '^ 289
(1 + xf in powers of z, the series in (1) and (5) must be
identical. Therefore
log(l + a;)=^/l-a^/2 + ar'/3-. . .{-f-^ x'^jn^- . . . (6).
The series on the right of (6) is usually called the logarithmic
series. It is absolutely convergent so long as — 1 <;»<!, and it
is precisely under this restriction that the above demonstration
is valid.
If we put x = l on the right of (6), we get the series
1/1 - 1/2 + 1/3- . . . (-l)"~Yw + . . ., which is semi-con\er-
gent. Hence, by Abel's Theorem (chap, xxvi., § 20), equation
(6) will still hold in this case ; and we have
log 2 = 1/1-1/2 + 1/3-. . . + ( - 1)"-V7i + . . . (7),
provided the order of the terms as written be adhered to.
If we put a! = -l in (6), the series becomes divergent. It
diverges, however, to — co ; so that, since log 0 = - qd , the
theorem still holds in a certain sense.
Cor. If we arrange the coefficients of the remaining powers
of z in (5), and compare with (1), we find
{log(l+^)P=2!{iPi^/2-2Pi^/3+3Pi;2?V4-. . .},
{log (1 + X)Y = n\ {n-iPn-i as'' In - nPn-i ^"+70* + 1)
+ «+iP«-i^"+V(w + 2)-. . .} (8).
These formulae and the above demonstration are given by
Cauchy in his Analyse Algebrique.
§ 10.] A variety of expansions can be deduced from the
logarithmic theorem. The following are some of those that
are most commonly met with : —
We have
log {l+x) = x/1 - a?l2 + ^73 - . . . ( - f-^x^ln + . . . ;
also
log(l-;r) = -;r/l-^/2-/r'/3-. . .-^7«-. . . .
Hence, by subtraction, since log {\ + x)- log (1 - x) = \og
{(1 + x)l{l - x)], we deduce
log{(l + ^)/(l-.r)} = 2{^/l + ^73+. . . +^^"-7(2»-l) + . . .} (9).
240 VARIOUS LOGARITHMIC EXPANSIONS CH. XXVIII
Putting in (9) y = {l + x)l{l-x), and therefore x-{y — \)l
{y + 1), we get
^^ \l\y + l) 3\?/ + l/ 2n~l\y+l) ^- • •/
(10),
an expansion for log y (but not, be it observed, in powers of y)
which will be convergent if y be positive — the only case at
present in question.
Again, since 1 + x = x{\ + llx), and log(l + a^) = loga; + log
(1 + 1/a;), putting in (10) y=l + l/x, so that (y-l)/(y+l) =
l/{2x + 1), we have
log(l+a;) = log« + 2{l/l(2^+l) + l/3(2.r+l)^ + . . .} (11).
Finally, since x+l = ar(l — \jaP)j{x — 1 ),
log {x+\)~'2\ogx- log {x — 1)
-2 {1/1(2^^-1) + 1/3(2^^-1)^ + . . .} (12).
If, in any of the above formulce, we wish to use a base a
different from e, we have simply to multiply by the " modulus "
l/logett (see chap, xxi., § 9). Thus, for example, from (10) we
derive
ON THE CALCULATION OF LOGARITHMS.
§ 11.] The early calculators of logarithms largely used
methods depending on the repeated extraction of the square
root. This process was combined with the Method of Differences,
which seems to have arisen out of the practical necessities of the
Logarithmic Calculator*.
* See Glaisher, Art. "Logarithms," Encyclopcedia Britannica, 9th ed.,
from which much of what follows is taken.
10, 11
CALCULATION OF L0G^2
241
Thus, Briggs used the approximate formula
logio 2 = (2'"'" - 1) 2^10 log, 10,
depending on the accurate formula
L {af-l)Jz = hgeX,
2=0
which we have already established in the chapter on Limits,
and which might readily be deduced from the exponential
theorem. The calculation of logio 2 in this way, therefore, in-
volved the raising of 2 to the tenth power and the subsequent
extraction of the square root 47 times !
Calculations of this kind were infinitely laborious, and nothing
but the enthusiasm of pioneers could have sustained the calcu-
lators. If it were necessary nowadays to calculate a logarithmic
table afresh, or to calculate the logarithm of a single number to
a large number of places, some method involving the use of
logarithmic series would probably be adopted.
The series in § 10 enable us to calculate fairly rapidly the
Napierian Logarithms of the small primes, 2, 3, 5, 7.
Thus, putting 3/ = 2 in (10) we have
log 2 = 2 {1/1 . 3 + 1/3 . 3=^+ 1/5 . S'' + . . . }.
The calculation to nine places may be arranged thus : —
1/3
•333,333,333
1/1 .3
•333,333,333
1/3='
37,037,037
1/3 .3='
12,345,679
l/3«
4,115,226
1/5 .3'
823,045
1/3^
457,247
111 .3^
65,321
l/3»
50,805
1/9 .3"
5,645
1/3"
5,645
1/11 . 3"
513
1/3^^
627
1/13 . 31^
48
1/3^^
70
1/15 . 3'^
5
1/3'^
8
1/17 . 3^^
0
•346,573,589
2
±4
•693,147,178
±8
By the principle of chap, xxvi., § 30, the residue of the series
is less than
{l/l9.3-}/(l-?),
c. II. 10
242 NAPIERIAN LOGARITHMS OF 1, 2, . . ., 10 CH. XXVIII
that is, less than '000,000,000,06 ; and the utmost error from
the carriage to the last line is ±4. The utmost error in our
calculation is ± 8. Hence, subject to an error of 1 at the utmost
in the last place, we have
log 2 = -693,147,18.
Having thus calculated log 2, we can obtain log 3 more
rapidly by putting w = 2 in (11), Thus
log3 = log2 + 2{l/l. 5 + 1/3.5^ + 1/5.5^ + . . . }.
Knowing log 2 and log 3, we can deduce log4 = 2log2, and
log 6 = log 3 + log 2. Then, putting x = 4:m (12), we have
log5 = 2log4-log3-2{l/31 + l/3.3P + . . . }.
Also, putting a) = 6 m (12), we have
log7 = 2log6-log5-2{l/71 + l/3.7P + . . . }.
It will be a good exercise in computation for the student to
calculate by means of these formulae the Napierian Logarithms
of the first 10 integers. The following table of the results to
ten places will serve for verification : —
No.
Logarithm,
1
0 000,000,000,0
2
0-693,147,180,6*
3
1-098,612,288,7
4
1-386,294,361,1
5
1-609,437,912,4
6
1-791,759,469,2
7
1-945,910,149,1
8
2079,441,541,7
9
2197,224,577,3
10
2-302,585,093,0
From the value of log^lO we deduce the value of its re-
ciprocal, namely, M= -434,294,481,903,251 ; and, by multiplying
by this number, we can convert the Napierian Logarithm of
* 6 means that the 10th digit has been increased by a unit, because the
llta exceeds 4.
§§11,12 FACTOR METHOD OF CALCULATING LOGARITHMS 243
any number into the ordinary or Briggian Logarithm, whose base
is 10.
Much more powerful methods than the above can be found
for calculating log 2, log 3, log 5, log 7, and M.
By one of these (see Exercises xiil, 2, below) Professor
J, C. Adams has calculated these numbers to 260 places of
decimals.
§ 12.] Tlie Factor Method of calculating Logarithms* is one
of the most powerful, and at the same time one of the most
instructive, from an arithmetical point of view, of all the methods
that have been proposed for readily finding the logarithm of a
given number to a large number of decimals.
This method depends on the fact that every number may, to
any desired degree of accuracy, be expressed in the form
io>o/(i-Wio)(i-Wio^)(i-Wio') . . . (1),
where Pq, Pi, p^y • • • each denote one of the 10 digits, 0, 1,
2, . . ., 9, jt?o being of course not 0.
Take, for example, 314159 as the given number. First
divide by 10' . 3, and we have
314159 = 10^ 3. 1-047,196,666,666 ....
Next multiply 1-047,196,666,666 by 1-4/10^ that is, cut
off two digits from the end of the number, then multiply by 4
and subtract the result from the number itself. The effect of
this will be to destroy the first significant figure after the
decimal point. We have in fact
1-047,196,666,666 x (1-4/10')= 1-005,308,800,000.
Next multiply 1-005,308,800,000 by 1-5/10^ and so on
till the twelve figures after the point are all reduced to zero. The
actual calculation can be performed very quickly, as follows : —
* For a full history of this method see Glaisher's article above quoted ;
or the Introduction to Gray's Tables for the Formation of Logarithms and
Anti-Logarithms to Twenty-four Places (1876).
16—2
244 FACTOR METHOD OF CALCULATING LOGARITHMS CH. XXVIII
1-0 4 7, 19 6,6 6 6, 6|6 6
41,8 8 7,866,6 66
5,308,800,1000
5, 0 2 6, 5 4 4, 0 0 0
2 8 2,2 5|6,0 00
20 0, 05 6, 451
8 2, 1|9 9, 549
8 0, 0 0 6, 5 7 6
2,|19 2,9 73
2, 0 0 0, 0 0 4
|1 9 2, 9 6 9
100,000
4/10=
5/10''
2/10*
8/10''
2/l0«
1/10^
92,969 9/10^ 2/10", 9/10", 6/10", 9/10^1
The remaining factors being obvious without farther calcula-
tion. Hence we have
314159 x(l-4/lO')(l- 5/10^) . . . (1-9/10^^)
= lo^3(l + ^/lO"), ^:|>9.
Therefore
314159 = 10^ 3 (l+^/10^^)/(l- 4/10^) (I-S/IO'') . . . (1-9/10^^)
(2).
Since log(l ^- xlW^)<a;IW\ it follows from (2) that, as far
as the twelfth place of decimals,
log 314159 = 5 log 10 + log 3 - log (1 - 4/10") - log (1 - 61 W)
- log (1 - 2/10*) - log (1 - 8/10') - log (1 - 2/10")
- log (1 - 1/100 - log (1 - 9/100 - log (1 - 2/100
- log (1 - 9/10") - log (1 - 6/10") - log (1 - 9/10^0-
All, therefore, that is required to enable us to calculate
log 314159 to twelve places is an auxiliary table containing the
logarithms of the first 10 integers, and the logarithms of l-pjKf
for all integral values oi p from 1 to 9, and for all integral values
of r from 1 to 12. To make quite sure of the last figure this
auxiliary table should go to at least thirteen places.
§ 13.] It should be noticed that a method like the above is
suitable when only solitary logarithms are required. If a com-
plete table were required, the Method of Differences would be
employed to find the great majority of the numbers to be entered.
§§ 12-14 FIRST DIFFERENCE OF LOG X 245
A full discussion of this method would be out of place here* ;
but we may, before leaving this part of the subject, give an
analytical view of the method of interpolation by First Differ-
ences, already discussed graphically in chap. xxi.
We have
logio (a; + h) - logio w = logjo (1 + h/a;)
= M{h/x - -^ (kiwf + I (h/a-y - . . . } (1).
Hence, if h < x, we have approximately
logio (^ + A) - logio ^ ^ Mk/a; (2),
the error being less than ^M(h/wy.
The equation (2) shows that, if \M{klxf do not affect the
nih. place of decimals, then, so long as h'if^k, the differences of
the values of the function are proportional to the differences of
the values of tlie argument, provided we do not tabulate beyond
the wth place of decimals.
Take, for example, the table sampled in chap, xxr., where the numbers
are entered to five and the logarithms to seven places. Suppose x = 30000 ;
and let us inquire within what limits it would certainly be safe to apply the
rule of proportional parts. We must have
I X -4343 (/i/30000)2< 5/108,
if the interpolated logarithm is to be correct to the last figure, that is,
/i< 3^23-04,
<14.
It would therefore certainly be safe to apply the rule and interpolate to
seven places the logarithms of all numbers lying between 30000 and 30014.
This agrees with the fact that in the table the tabular difference has the
constant value 144 within, and indeed beyond, the limits mentioned.
SUMMATION OF SERIES BY MEANS OF THE LOGARITHMIC
SERIES.
§ 14.] A great variety of series may, of course, be summed
by means of the Logarithmic Series. Of the simple power series
that can be so summed many are included directly or indirectly
under the following theorem, which stands in the same relation
* For sources of information, see Glaisher, I.e.
246 X(f) (n) x^l{n + a) (w + i) . . . {n + h) CH. xxviii
to the logarithmic theorem as do the theorems of chap, xxvii., § 5,
and chap, xxviii., § 8, to the binomial and exponential theorems : —
The series whose general term is (f> (n) x^l{n + a){n + h) . . .
(n + k), where <{> (n) is an integral function of n, and a, b, . . . ,
k are positive or negative'^ unequal integers, can always he
summed to infinity provided the series is convergent.
It can easily be shown that the series is convergent provided
X be numerically less than unity, and divergent if x be
numerically greater than unity.
If the degree of <^ {n) be greater than the degree of {n + a)
(n + b) . . . (n + k), the general term can be split into
i/' {n) x^ + x (n) x''l(n +a)(n + b) . . . {n + ^) (1),
where ^{n) and x{n) are integral functions of n, the degree of
the latter being less than the degree of {n + a){n + b) . . . {n + k).
Now %\l/{n)x'^ is an integro-geometric series, and can be
summed by the method of chap, xx., § 13.
By the method of Partial Fractions (chap, viii.) we can
express x {n)l{n + a){n + b) . . . (n + k) in the form
AI{qi + a) + B/(n + b) + . . . + IC/{n + k),
where A, B, . . ., K are independent of n. Hence the second
part of (1) can be split up into
Ax^l{n + a) + Bx^'lin + h) + . . . + Kaf'l{n + k) (2) ;
and we have merely to sum the series
^Sa;7(w + a), B^x^'lin + b), . . ., K%x'%n + k) (3).
Now, supposing, for simplicity of illustration, that the sum-
mation extends from w = 1 to w = oo , we have
A ix^'/in + a) = Ax-^^x^+^lin + a),
= -Ax-''{xJl + xy2+ +x^/a+hg{l-x)} (4).
Each of the other series (3) may be summed in like manner.
Hence the summation can be completely eft'ected.
* When any of the integers a, b, . . ., k are negative, the method
requires the evaluation of limits in certain cases.
§ 14 !</) (n) x''l(n + a)(n + h) .. .(n + Jc) 247
If a; = 1, the series under consideration will not be convergent
unless the degree of <^ (n) be less than the degree of {n + a)
(n + b) . . . (n + k). It will be absolutely convergent if the
degree of ^ {n) be less than that oi {n + a){n + h) . . . (n + k) by-
two units. If the degree of ^ {n) be less than that of {n + a)
(n + b) . . . (n + k) by only one unit, then the series is semi-
convergent if the terms ultimately alternate in sign, and divergent
if they have ultimately all the same sign.
In all cases, however, where the series is convergent we can,
by Abel's Theorem, find the sum for x=l by first summing for
x< 1, and then taking the limit of this sum when x=\.
In the special case where </> {n) is lower in degree by two
units than {n + a)(n + b) . . . (n+ k), and a,b, . . ., k are all
positive, an elegant general form can be given for 2</» {n)l{n + a)
(n + b) . . . {n + k).
From the identity
<i> (n)/{n + a){n + b) . . . (n + k)
= AI{n + a) + Bl{n + b) + . . . + K,(n + k),
we have
ff>{n) = A{7i + b)(n + c) . . . {n + k) + B{n + a)(n + c) . . . (n + k)
+ . . . + K{n + a)(n + b) . . . (n +j) (5),
and, bearing in mind the degree of <^ (n), we have
A + B + . . .+K=0 (6).
Also, putting in succession n = -a, n=-b, . . , , n~-k, we
have
A^^{- a)l{b -a){c-a) . . . {k-a)\
B = 4>{-b)l{a-b){c-b) . . . ik-b)[ (7)^
K=cl>(-k)l{a-k){b-k) . . . (j-k)
Reverting to the general result, we see from (4) that
'h^{n)x'^l{n + a){n + b) . . . {n + k)
= -%Ax-''{xl\ + x'l2 + . . .+afla)-\og{\-x).^Ax-'' (8),
where the S on the right hand indicates summation with respect
to a,b,. . . , k.
248 EXAMPLES CH. XXVIII
JN^ow, since A+B + . . .+K=0, '^Ax'"' is an algebraical
function of x which vanishes when x = l. Also \-x is an
algebraical function of x having the same property. Therefore,
by chap, xxv., § 17, we have
L \og{l-x).'^Ax-''= L log {(1-^^)2^'=-"},
x=\ x=l
= log 1,
= 0.
Hence, taking the limit on both sides of (8), we have, by Abel's
Theorem,
^c{>{n)/{n + a){n + b) . . . {n + k) = -%A (l/l + 1/2 + . . . + 1/a),
_ ^^(-a){l/l + l/2 + . . . + l/a)
{b-a){c-a) . . . (c-k) ^^^'
the S on the right denoting summation with respect to
a, b, c, . . ., k
Example 1. Evaluate 2,n^x^l{n - 1) (n + 2).
2
We have ri?x'^j{n - 1) (n + 2) = (n - 1) a;" + \x'^l{n - 1) + |a;'7(7i + 2),
Now 2(n-l)a;"=la;2 + 2x3 + 3a;* + . . .,
(1 - xf-Z {n - 1) a;» = Ix^ + 2x^ + Zx*+.
-2.1x3-2.2a;4-.
+ lx*+.
= x\
Hence
2(ji-l)x'»=x2/(l-a;)2.
2
Also
i la;»/(n - 1) = 4a;2a;"-V{n - 1),
= -lx log (1 - a;) ;
1 2a;»/(n + 2) = |a;-2 Sa;»+2/(n + 2),
2 2
= -fa:-2{a;/H-a^/2 + .-c3/3 + log(l-a;)}.
Hence the whole sum is
x-'lil - xf - Sx-i - 1 - «x - i (x + 8x-2) log (1 - x).
Example 2. Evaluate 2 l/(n-l)(n + 2).
2
By the same process as before, we find
Sx»/(n - 1) (n + 2) = Jx-i + ^ + Jx + ^ (x-2 - a;) lo- (1 - a;}.
§ 14 EXAMPLES 249
Now, since L {l-xY~'-^-l (chap, xxv., § 17), L {x-"^- x)\og{l- x) = 0.
Theref ore Sl/(n - 1) (n + 2) = ^ + i + i = -H-
2
This result might be obtained in quite another way.
It happens that Sl/(7i- 1) (n + 2) can be summed to n terms. In fact,
we have
l/(«-l)(7i + 2) = Hl/(«-l)-l/(« + 2)}.
Hence, since the series is now finite and commutation of terms therefore
permissible,
3|l/(„-l)(« + 2)=.J4 + l3+...+-i_ + ^3 + ^2 + ^^
1 _Jl 1 1 1_
~4 **' n-4 n-'d n-2 n-1
_i_ J^ i_
n n + 1 n + 'A'
_1 1 1_ 1__1 1_
~i'^2"^3 n 11 + 1 n + 2'
Hence, taking the limit for «=qo , we have
|_1/1 1 1\_11
2~3Vl'*'2'^3y"18'
Example 3. To sum the series
(Lionnet, Nouv. Ann., ser. ii.,t. 18.)
Let the (n + l)th term be m„, then, since m„=0, association is permitted
(see chapter xxvi., § 7), and we may write
■ +
4w + l 4n + 3 2)i + 2'
4n + l 4?i + 2 4n + 3 4?t + 4 47z + 2 4?i + 4'
^ /^L 1_ _1 ]_\ 1 /^ 1_\
~\,4n + l 471 + 2"^ 4n + 3 4ji + 4/ "^ 2 ^271 + 1 2n + 2/'
= u„ + w„, say.
Now, as may be easily verified, v^ and w„ are rational functions of n, in
which the denominator is higher in degree than the numerator by two units
at least. Hence (chap. xxvi. , § 6) Zv^ and Xtu^ are absolutely convergent
series. Therefore (chap, xxvi., § 13)
Sw„=S(v„ + «;„),
0 0
=Sv„+2w„.
0 0
250 INEQUALITY THEOREMS CH. XXVIII
Hence, again dissociating v„ and w^ (as is evidently permissible) we have
« ,1111111
l{
,1111111
2 3 4^5 6 7 8
=log,2 + ilog,2, by §9 above,
This example is an interesting specimen of the somewhat delicate opera-
tion of evaluating a semi-convergent series. The process may be described
as consisting in the conversion of the semi-convergent into one or more
absolutely convergent series, whose terms can be commutated with safety.
It should be observed that the terms in the given series are merely those of
the series 1 - 1/2 -i- 1/3 - 1/4 + 1/5 - . . . written in a different order. We
have thus a striking instance of the truth of Abel's remark that the sum of
a semi-convergent series may be altered by commutating its terms.
APPLICATIONS TO INEQUALITY AND LIMIT THEOREMS.
§ 15.] The Exponential and Logarithmic Series may be
applied with effect in establishing theorems regarding inequality.
Thus, for example, the reader will find it a good exercise to
deduce from the logarithmic expansion the theorem, already
proved in chapter xxv., that, if ic be positive, then
a;-l>\oga;>l-llw (1).
It will also be found that the use of the three funda-
mental series — Binomial, Exponential, and Logarithmic — greatly
facilitates the evaluation of limits. Both these remarks will be
best brought home to the reader by means of examples.
Example 1. Show that
nil 1 1 , n+1
loe - — > — I 1 h . . . -f - > log .
^m-1 m^m + 1 m + 2^ n ^ m
If we put 1 - l/x = 1/to, that is, x=mj{m- 1), in the second part of (1) above,
and then replace m by m-hl, to + 2, . , ., n successively, we get
log m - log (m - 1) > l/»i,
log {m + 1)- log m > l/(m + 1),
log n - log (« - 1) > 1/w.
Hence, by addition,
logn-log(m-l)>l/y)i-l-l/(Ht + l)+ . . . +l/« (2).
§ 15 LIMIT THEOREMS, EXERCISES XIII 251
Next, if we put x - l = l/?n in the first part of (1), and proceed as before,
we get
log (m + 1) - log m < 1/m,
log {m + 2) - log (m + 1) < ll{m + 1),
log (n + 1) - log n < 1/n.
Hence
log(»i + l)-log?n<l/m + l/(Ht + l)+ . . .+1/71 (3).
From (2) and (3),
log{K/(m-l)}>l/m + l/(m + l)+ . . .+!/«> log {(n + l)///i}.
Example 2. If ^ and q be constant integers, show that
L {l/m+l/(m + l)+ . . . +ll{pm + q)}=\ogp.
ni=oo ,
(Catalan, Traite Elementaire des Series, p. 58.)
Put n=pm + q in last example, and we find that
log{(p»i + 3)/(TO-l)}>l/wi + l/(m + l) + . . . + ll{pm + q)>log{{pm + q + l)lm}.
Now L log {{pm + q) I (m-l)} = logp,
m=oo
and L log{(2J)n + g + l)/»t} = logiJ.
Hence the theorem.
Example 3. Evaluate L (e*-l)2/{a;-log(l + a;)} when x = 0.
Since {e''-l)^={x + ^x^+ . . .)^=x^{l + ^x+ . . .)2;
x-log(l + x):^ix^-^x^+ . . .=ia;2(l-|a;+ . . .).
Therefore
(e''-lfl{x-log(l + x)}=2(l + isX+ . . .)2/(l-|x+ . . .).
Since the series with the brackets are both convergent, it follows at once
that L(e»-l)2/{x-log(l + a;)} = 2.
Exercises XIII.
(1.) If P = l/31 + l/3.31» + l/5.3Pf . . .,
Q=l/49 + l/3. 493 + 1/5. 49''+. . .,
iJ = 1/161 + 1/3. 16P + 1/5. 161» + . . .,
then log2 = 2(7P+5g + 3i?),
log 3 = 2 (11P + 8(3 + 512),
log5 = 2(16P + 12(3 + 7i?).
(See Glaisher, Art. "Logarithms," Ency. Brit., 9th ed.)
(2.) If a= -log (1-1/10), &=- log (1-4/100), c = log (1 + 1/80), d =
-log (1-2/100), e = log (1 + 8/1000), then log 2 = 7a - 26 + 3c, log 3 = 11a- 3&
+ 5c, log 5 = 16a -46 + 7c, log7 = i(39a- 106 + 17c-d) = 19a- 46 + 8c + e.
(Prof. J. C. Adams, Proc. R.S.L. ; 1878.)
(3.) Calculate the logarithms of 2, 3, 5, 7 to ten places, by means of the
formulaj of Example 1, or of Example 2.
(4.) Find the smallest integral value of x for which (1-01)*> lOar.
252 EXERCISES XIII CH. XXVIII
Sum the series : —
(5.) 21/1 (a;3 - 3x)i + 2^/3 (x^ - 3x)^ + . . .
(7.) a;V1.2-a;2/2.3 + a;3/3.4- . . . (-)»-ix"//i(ji + l) . . .
(8.) a;2/3 + a;^/]5+. . . + a;2"/(4ra2 - 1) + . . .
(9.) a;/P + a;2/(12 + 22) + a;3/(12 + 22 + 32)+. . .+x^l{l^ + 2'^ + . . . + n'^) + . . .-,
also l/P + l/(P + 22) + l/(12 + 22 + 32) + . . . + 1/(12 + 22 + , . . + ,f-)+ . . .
(10.) 4/1.2.3 + 6/2.3.4 + 8/3.4.5+ .. .
(11.) If x>100, then, to seven places of decimals at least, log(a; + 8) =
2 log (x + 7) - log (a; + 5) - log (x + 3) + 2 log a; - log (x-3)- log (a; - 5) + 2 log
(x-7)-log(a;-8).
(12.) Expand log (1 + a; + a;2) in ascending powers of x.
(13.) From log (a;3 + l) = log (x + l) + log(a;2-cB + l), show that, if m be a
positive integer, then
6»t - 2 (6»i - 3) (Got - 4) {6m - 4) (6nt - 5) (6m - 6) _
1- 21 + 3! 4! +...-0.
(Math. Trip., 1882.)
(14.) {loge(l + a;)}2 = 2a;2/2-2(l/l + l/2)a;3/3 + . . . (-)»2{l/l + l/2 + . . .
l/(n- l)}a;"/7i . . . Does this formula hold when x = 1 ?
where Q^n-i = 1/1 - 1/2 + 1/3 - . . . + l/(2?j - 1).
(16.) If a; <1, show that
X + ^a;2 + ^x* + ^W^. . . = log{l/(l - a;) } - 1P3 - ^P^ + IP^ - ^Py - ^^9 + tV^ 10 • • • :
where P„=x" + x2» + x'"' + a[:8™ + a;i^"+ . . ., and the general term is (-)"P„/7i,
unless n is a power of 2, in which case there is no term.
(Trin. Coll., Camb., 1878.)
(17.) li e'^xe'^'^xe''''^ . . . = Ao + A^x + . . ., then A^r = ^2r+2 = '^-^-^ ■ • •
(2r-l)/2.4.6. . .2r.
(18.) lix + asa^ + a^x^+. . . + y + a.^y^ + a^i/ + . . . = {{x + y)l{l-xy)y +
a3{{x + y)l(l- ocy)}'^ + a^{{x + y)l{l - xy)y+ . . ., for all values of x and y
which render the various series convergent, find a^, a^, . . .
Show that
(19.) log(4/e) = l/l. 2-1/2. 3 + 1/3. 4-1/4. 5+. . .
(20.) log 2 = 4(1/1. 2. 3 + 1/5. 6. 7 + 1/9. 10. 11 + 1/13. 14. 15 + ...) (Euler.)
(21.) (1-1/2 -1/4) + (1/3 -1/6 -1/8) + (1/5 -1/10 -1/12) + . . . = ilog2.
(See Lionnet, Nouv. Ami., ser. 11., t. 18.)
(22.) ffi/1! -7J(ro/2! + n(n-l) 0-3/3! - . . . to n + 1 terms = l/(n + 1)2, where
(7^=1/1 + 1/2 + 1/3+. . . +l/r. (Math. Trip., 1888.)
(23.) e~(l + l/m)'" lies between e/(2m + l) and e/(2m + 2), whatever vi
may be. (Nouv. Ann., ser. 11., t. 11.)
(24.) L{x/(x-l)-l/logx}=i, when x = l. (Eulei, Inst. Calc. Dif.)
(26.) I,{e»=-l-log(l + .T)}/x2=l, whenx=0. (Euler, Z.c.)
(26.) L(a;*-x)/(l-x + logx)= -2, whenx = l. (Euler, Z.c.)
§ 15 EXERCISES XIII 253
(27.) I, (1 + l/?i)Vn (1 + 2/71) V». . . (l + n/n)V»=4/e, when 71 = 00.
(28.) L{(27i- I)l/n2™-i}i/n = 4/e2, when « = oo .
(29.) ^>l + x, for all real values of x.
(30.) a;-l>loga;>l-l/a;, for all positive values of a; ; to be deduced
from the logarithmic expansion.
(31.) «"> (1 + n)"/7il, n being any integer,
(32.) If n be an integer > e, then ?i"+i > {n + 1)".
(33.) If A, B, a, h be all positive, then {a-h)l{A-B) + {Aa -Bh)
log (BIA)I{A - BY is negative. (Tait.)
(34.) llx>y>a, then {{x+a)l{x - a)}''<{{y + a)l{y -a)}y.
(35.) L{l/(ji + l) + l/(n + 2) + . . . + l/27i} = log2,whenn = Qo, (Catalan.)
(36.) log{(n + i)/(m-i)}>l/m+l/(m + l) + . . . + l/7i>log{(n + l)/m}.
(Bourguet, Nouv. Ann., ser. ii,, t. 18.)
(37.) log3=5/l. 2. 3 + 14/4. 5.6 + . . . + (97i-4)/(3n-2) (3?i-l)37i + . . .
(38.) If S( -)"-V (")/(« + «)(?i + ^) • . • (n+fc), where a,b,...,k are
1
all positive integers and (p(n) is an integral function of n, be absolutely
convergent, its sum is
S= S 0(-a){l/a-l/(a-l). . . (-)«-il/l}/(6-a) (c -a) . . .(k-a);
a,b, . ..,k
and, if it be semi-convergent, its sum is
S + log2 S {-)<'<p{-a)l{b-a){c-a) . . .(k-a).
a,b,...,k
(39.) Show that the residue in the expansion of log {1/(1 -a;)} lies
between
a;i+i{l+(n + l)a;/(n + 2)}/(7! + l)
and x^+^{l + {n + l)xl(l-x)(n + 2)}l{n + l).
(40.) In a table of Briggian Logarithms the numbers are entered to
5 significant figures, and the mantissae of the logarithms to 7 figures.
Calculate the tabular difference of the logarithms when the number is near
30000 ; and find through what extent of the table it will remain constant.
(41.) Show that (1 + l/a;)*^+i continually decreases as x increases.
(42.) Show that S I/71 (4n2 - 1)" = f - 2 log 2.
CHAPTEE XXIX.
Summation of the Fundamental Power Series for
Complex Values of the Variable.
GENERALISATION OF THE ELEMENTARY TRANSCENDENTAL
FUNCTIONS.
§ 1.] One of the objects of the present chapter is to generalise
certain expansion theorems established in the two chapters which
precede. In doing this, we are led to extend the definitions of
certain functions such as «% loga^, cos a:, &c., already introduced,
but hitherto defined only for real values of the variable x ; and
to introduce certain new functions analogous to the circular
functions.
Seeing that the circular functions play an important part in
what follows, it will be convenient here to recapitulate their
leading properties. This is the more necessary, because it is
not uncommon in English elementary courses so to define and
discuss these functions that their general functional character is
lost or greatly obscured.
§ 2.] Definition andProperties of the Direct Circular Functions.
Taking, as in chap, xii.. Fig. 1, a system of rectangular axes, we
can represent any real algebraical quantity 6, by causing a radius
vector OP of length r to rotate from OX through an angle con-
taining 6 radians, counter-clockwise if 6 be a positive, clockwise
if it be a negative quantity. If {cc, y) be the algebraical vahies of
the coordinates of P, any point on the radius vector of 6, then
xjr, yjr, yjx, xjy, rjx, rjy are obviously all functions of 0, and
of B alone. The functions thus geometrically defined are called
§§ 1, 2 EVENNESS, ODDNESS, PERIODICITY 255
COS 0, sin 6, tan 6, cot 6, sec &, cosec 6 respectively, and are spoken
of collectively as the circular functions.
All the circular functions of one and the same argument, 6,
are algebraically expressible in terms of one another, for their
definition leads immediately to the equations
tan 6 - sin ^/cos 6, cot 6 = cos ^/sin d ; \
sec 6 = 1/cos 0, cosec 0 - 1/sin 6 ; Y (1) ;
cos^ 6 + sin^ ^ = 1, sec- 6 — tan^ ^ = 1 ; i
from which it is easy to deduce an expression for any one of the
six, cos 0, sin 6, tan 6, cot 6, sec 6, cosec 6, in terms of any other.
When F{6) is such a function of 6 that F{-0) = F(6), it is
said to be an even function of 0 ; and, when it is such that
F{-0) = -F(0), it is said to be an odd function of 0. For
example, 1 + 6^ is an even, and ^-^^* is an odd function of 6.
It is easily seen from the definition of the circular functions
that cos 0 and sec 6 are even, and sin 0, tan 9, cot 0, and cosec 0
odd functions of 6.
When F(0) is such that for all values of e,F{e + nX) = F(6),
where A. is constant, and n any integer positive or negative, then
F{0) is said to be a periodic function of 0 having the period A.
It is obvious that the graph of such a function would consist
of a number of parallel strips identical with one another, like the
sections of a wall paper ; so that, if we knew a portion of the
graph corresponding to all values of 6 between a and a + X, we
could get all the rest by simply placing side by side with this an
infinite number of repetitions of the same.
Since the addition of ±27r to 6 corresponds to the addition
or subtraction of a whole revolution to or from the rotation of
the radius vector, it is obvious that all the circular functions are
periodic and have the period 27r. This is the smallest period,
that is, the period par excellence, in the case of cos 6, sin 9, sec 9,
cosec (y. It is easily seen, by studying the defining diagram, that
tan 9 and cot 9 have the smaller period tt. Thus we liave
256
ZERO AND TURNING VALUES
CH. XXIX
COS {6 + 2w7r) = COS 6, sin {6 + 2mr) = sin 6, \
sec {6 + 2w7r) = sec ^, cosec {6 + 2w7r) = cosec 9, Y (2).
tan (^ + nir) = tan ^, cot {6 + n-n) = cot 6. )
Besides these relations for whole periods, we have also the
following for half and quarter periods : —
cos {tt+O) =- cos 6, sin (■rr±6) = + sin ^ ;
cos(^ir±6) = + sm6, sin(^7r+ ^) = + cos^; I /^\
tan {^Tr±0) = + cot 6, cot (^■n-±0)= + tan 6 ;
&c.,
all easily deducible from the definition.
We have the following table of zero, infinite, and turning
values : —
e
0
i^
IT
Itt
27r
(fee. '
cos 6
+ 1
0
-1
0
+ 1
sin^
0
+ 1
0
-1
0
tan^
0
GO
0
00
0
&c.
cot^
GO
0
GO
0
CO
sec^
+ 1
CO
-1
GO
+ 1
cosec 6
00
+ 1
GO
-1
GO
/
which might of course be continued forwards and backwards
by adding and subtracting whole periods
Hence cos 0 has an infinite number of zero values correspond-
ing to 0 = ^{2n + 1) TT, where n is any positive or negative integer;
no infinite values ; an infinite number of maxima and of minima
values corresponding to ^ == 2mr and 6 = {2n + 1) tt respectively ;
and is susceptible of all real algebraical values lying between
- 1 and + 1.
Sin 6 is of like character.
But tan 0 is of quite a different character. It has an infinite
number of zero values corresponding to 6-mr; an infinite
number of infinite values corresponding to 6 = ^(2n+l)7r ; no
turning values ; and is susceptible of all real algebraical values
between — qo and + oo .
Cot 9 is of like character.
258
ADDITION FORMULA
CH. XXIX
Sec^ and cosec^ have again a distinct character. Each of
them has infinite and turning values, and is susceptible of all
real algebraical values not lying between - 1 and + 1. The
graphs of the functions y = sma!, y = cosiv, &c., are given in
Fig. 1. The curves lying wholly between the parallels KL,
K'L , belong to cos x and sin x, the cosine graph being dotted ;
all that lies wholly outside the parallels KL, K'L', belongs either
to sec X or to cosec x, the graph of the former being dotted. The
curves that lie partly between and partly outside the parallels
KL, K'L', belong either to tsmx or to cot;r, the graph of the
latter being dotted.
Again, from the geometrical definition combined with
elementary considerations regarding orthogonal projection are
deduced the following Addition Formulw : —
cos (0 ±(}>) = cos ^ cos <^ + sin ^ sin ^ ;
sin {6±<li) = sin ^ cos <^ ± cos ^ sin <^ ;
tan (^ ± ^) = (tan 6 ± tan <^)/(l + tan 6 tan 0).
As consequences of these, we have the following :
cos ^ + cos <^ = 2 cos |(^ + <ji) cos ^{0 - ^) ; '
cos <^-cos 6 - 2 sin ^{0 + <j>) sin ^{0-(f>) ;
sin ^ ± sin <;^ = 2 sin |(^ ± ^) cos ^{0 + ^).
cos ^ cos ^ = |cos (6 + <fi) + |cos (^ - <^) ; '
sin ^ sin ^ = ^cos (^ - ^) - ^cos (^ + <^) ;
sin 6 cos (ft - ^sin {$ + (fi) + |sin (6 - <^).
cos2^ = cos''e-sin'^^ = 2cos'^-l = l-2sin'^e '
= (l-tan^^)/(l+tan2^)
sin 2^ = 2 sin 6cosd = 2 tan 6/{l + tan'' $).
tan 2^ = 2 tan 9/(1 - tan' 6).
(5).
(C)
(7)
(8).
§ 3.] Inverse Circular Injunctions. When, for a continuum
(continuous stretch) of values of y, denoted by (y), we have a
relation
(t:-F{t/) (1),
§§ 2, S INVERSE CIRCULAR FUNCTIONS 259
which enables us to calculate a single value of x for each value
of y, and the resulting values of x form a continuum {x), then
the graph of F{y) is continuous; and we can use it either to
find X when y is given, or y when x is given. We thus see that
(1) not only determines a; as a continuous function of y, but also
^ as a continuous function of x. The two functions are said to
be inverse to each other ; and it is usual to denote the latter
function by F~^{x). So that the equation
y = F-Hx) (2)
is identically equivalent to (1).
It must be noticed, however, that, although F~^ (x) is con-
tinuous, it will not in general be single-valued, unless the values
in the continuum (x) do not recur. This condition, as the
student is already aware, is not fulfilled even in some of the
simplest cases. Thus, for example, if x = y^, for -oo <y< + oo,
the continuum (x) is given by 0:!f>x<+ oo ; and each value of x
occurs twice over. We have, in fact, y = ±x^', that is, the
inverse function is two-valued.
It is also important to notice that, even when the direct
function, F{y), is completely defined for all real values of y, the
inverse function, F~^ (x), may not be completely defined for all
values of x. F~^ (x) is, in fact, defined by (1) solely for the
values in the continuum (x). Take, for example, the relation
x=y^, for - cc <y< + (x>. The continuum (x) is given by
0 1^ iJ? < + 00 ; hence y is defined, by the above relation, as a
function of x for values of x between 0 and + qo and for no
others.
The application of the above ideas to the circular functions
leads to some important remarks. It is obvious from the
geometrical definition of sin^ that the equation
x = smy (3)
completely defines ir as a single-valued continuous function of
y, for — CO <y <+ cc. Hence, we may write
y = sin"' X (4),
17—2
260
MULTIPLE-VALUEDNESS
CH. XXIX
where the inverse function, sin~^a;*, is continuous, but neither
single-valued, nor completely defined for all real values of x.
Since, by the properties of sin y, x lies
between - 1 and + 1 for all real values
of y, sin"^ x is, in fact, defined by (3)
only for values of x lying between — 1
and + 1. For other values of x the
meaning of sin"^ x is at present arbitrary.
By looking graphically at the problem
"to determine y for any value of x lying
between -1 and +1," we see at once
that sin~^^ is multiple-valued to an
infinite extent.
If, however, we confine ourselves to
values of sin"^a; lying between -^tt and
+ |^7r, we see at once from the graph
(Fig. 2) that for any value of x lying
between — 1 and + 1 there is one, and
only one, value of sin~^:r. If we draw
parallels to the axis of x through the
points A, B, C, . . ., A', B', . . .,
whose ordinates are +-|-7r, +f tt, +|-7r, . . ., -I-t, -f tt, . . ., then
between every pair of consecutive parallels we find, for a given
value of X (- 11f^x'^+ 1), one, and only one, value of ^ = sin~^ar.
The values of y corresponding to points between the parallels
A' and A constitute what we may call the Principal Branch of
the function. Similarly, the part of the graph between A and B
represents the 1st positive branch ; the part between B and G
the 2nd positive branch ; the part between A' and B' the 1st
negative branch; and so on.
If, as is usual, we understand the symbol sin~^ x to give the
value of y corresponding to x, for the principal branch only, and
use yn or „ sin~^ x for the wth branch, then it is easy to see that
3/„ = „sin-^;r = TC7r + (-l)"sin-^a; (5),
Fig. 2.
* This may be read "angle whose sine is x" or " arc-sine x." In
Continental works the latter name is contracted into arc-sin x ; and thia is
used instead of sin"*x.
§ 3 BRANCHES DEFINED 261
where n is a positive or negative integer according as the branch
in question is positive or negative.
It is obviously to some extent arbitrary what portion of the
grapli shall be marked off as corresponding to the principal
branch of the function ; in other words, what part of the function
shall be called the principal branch. But it is clearly necessary,
if we are to avoid ambiguity — and this is the sole object of the
present procedure — that no value of y should recur within the
part selected ; and, to secure completeness, all the different values
of 3/ should, if possible, be represented. Attending to these con-
siderations, and drawing the corresponding figures, the reader
will easily understand the reasons for the following conventions
regarding cos'^a?, tan~^a?, cot"^a;, sec~^a;, cosec'^^r, wherein 3/
and the inverse functional symbols cos~^^, &c., relate to the
principal branch only, and i/n to the nth branch, positive or
negative.
1/ = cos"^ a?, y between 0 and + tt ;
I/n = (n + I- + i-T-'i)^ + i-T cos-' X.
y = tzxr' X, y between -\tt and + I^tt;
yn, = WTT + tau~^ X.
y = cot~^ X, y between 0 and ir ;
y^ = tm + cot~^ X.
y = sec~^ X, y between 0 and ir
2/ = sec ' i2?, y oetween u ana ir ;
2^» = (^ + |- + (-r-'^)T+(-)'*sec-^^.
y = cosec"^ x, y between -^tt and + ^^ 5
yn = n'7r+ (- )" cosec"^ x.
(6)
(7)
(8)
(9)
(10)
Since every function must, in practice, be unambiguously
defined, it is necessary, in any particular case, to specify what
branch of an inverse function is in question. If nothing is
specified, it is understood that the principal branch alone is in
question.
It is obvious that all the formulae relating to direct circular
functions could be translated into the notation of inverse circular
functions. In this translation, however, close attention must be
paid to the points just discussed. Thus
262 INVERSION or W = Z'^ CH. XXIX
If X be positive, the formula cos ^ = + ^(1 - sin^ B) becomes
sin~^ X = cos~^ ^(1 - of) ;
but, if X be negative, it becomes
sin~^ x = — cos~^ J(l - x^). ,
If 0<x<l/j2, 0<3/<l/^2, we deduce from the addition
formulae for the direct functions
sin~^ X + sin~^^ = cos~^ [\/{(l - ^^) (1 -y^)} - ^y] ',
if 0<a;<l, 0<2/<l,
tan~^ X + tan~^ y = tan~^ [(x + y)/(l - xy)].
If X and y be both positive, but such that xy>l, then
tan~^ X + tan~^ y = Tr ■\- tan~^ [{x + y)l{l — xy)] * ;
and, in general, it is easy to show that
flitan"^ X + „tan~^ y = {m + n+p)Tr + tan~^ {{x + y)/{l - xy)},
= r>i+n+pt&n-^{{x + y)l{l-xy)} (11),
where p=l, 0, or -1, according as tan"^ ^ + tan~^ ?/ is greater
than ^TT, lies between ^tt and -^tt, or is less than -^tt.
ON THE INVERSION OF W = Z'K
§ 4.] When the argument, and, consequently, in general,
the value of the function are not restricted to be real, the
discussion of the inverse function becomes more complicated,
but the fundamental notions are the same.
For the present it will be sufficient to confine ourselves to
the case of a binomial algebraical equation. Let us first consider
the case
w-^" (1),
where w is a positive integer, 2; is a complex number, say
z = x + yi, and, consequently, w also in general a complex
number, say w = u + vi.
To attain absolute clearness in our discussion it will be
* In English Text-Books equations of this kind are often loosely
stated; and the result has been some confusion in the higher branches
of mathematics, such as the integral calculus, where these inverse functions
play an important part.
§§ 3, 4 INVERSION OF W = z'^ ^6^
necessary to pursue a little farther the graphical method of
chap. XV., § 17.
It follows from what has there been laid down, and from the
fact that any integral function of x and 2/ is continuous for all
finite values of a; and 3/, that, if we form two Argand Diagrams,
one for a; + yi (the ;2;-plane), and one for u + vi (the tr-plane), then,
whenever the graphic point of s* describes a continuous curve, the
graphic point of w also describes a continuous curve. In this sense,
therefore, the equation (1) defines w as a continuous function of
z for all values, real or complex, of the latter. For simplicity in
what follows we shall suppose the curve described by z to be the
whole or part of a circle described about the origin of the ;^-plane.
We shall also represent z by the standard form r (cos ^ -^ * sin 6),
and w by the standard form s (cos (f> + i sin <^) ; but we shall, con-
trary to the practice followed in chap, xii., allow the amplitudes
6 and ^ to assume negative values. Thus, for example, if we
wish to give z all values corresponding to a given modulus r,
without repetition of the same value, we shall, in general, cause
6 to vary continuously from - ir to + tt, and not from 0 to 27r,
as heretofore. In either way we get a complete single revolution
of the graphic radius ; and it happens that the plan now adopted
is more convenient for our present purpose.
It is obvious that by varying the amplitude in this way, and
then giving all different values to r from 0 to + qo , we shall get
every possible complex value of z, once over ; and thus eff'ect a
complete exploration of any one- valued function of z.
Substituting in (1) the standard forms for w and z, and
taking, for simplicity, w = 3, we have
s (cos ^ + * sin ^) = r* (cos ^ + « sin df
= r" (cos 3d + i sill '66) (2)
by Demoivre's Theorem. Hence we deduce
s = r^y (t>^39 + 27i7r ;
* For shortness, in future, instead of "graphic point of z" we shall say
'• z " simply.
264
CmCITLO-SPmAL GRAPHS
CH. XXIX
or, if (as will be sufficient for our purpose) we confine ourselves
to a single complete revolution of the graphic radius of z,
s^r", cf> = Sd (3).
If, therefore, we give to r any particular value, s has the
fixed value r^ ; that is to say, w describes a circle about the
origin of the w-plane (Fig. 4). Also, if we suppose z to describe
its circle (Fig. 3) with uniform velocity, since ^ = 3^, w will
describe the corresponding circle with a uniform velocity three
times as great. To one complete revolution of z will therefore
Fm. 3.
Fig. 4.
correspond three complete revolutions of w. In other words, the
values in the (w)-continuum which correspond to those in the
(«)-continuum are each repeated three times over*.
The actual course of w is the circle of radius i^ taken
three times over. We may represent this multiple course
of w by drawing round its actual circular course the spiral
0', T, r, 0, 1', 1, 0', which re-enters into itself at 0' and 0'.
The actual course may then be imagined to be what this spiral
becomes when it is shrunk tight upon the circle.
* To indicate this peculiarity of w we shall occasionally use the term
"Eepeating Function." A repeating function need not, however, be periodic
as w=:a?i8.
H 4, 5 riemann's surface 265
If we now letter the corresponding points on the 2;-circle with
the same symbols we have the circle O'll' in the w-plane, cor-
responding to the circular arc O'll' in the z--p\a.ne, and so on, in
this sense that, when z describes the arc O'll', then w describes
the complete circle O'll', and so on.
It follows from this graphical discussion that tlie equation
W'=^, which defines w as a one-valued continuous function of z
for all values of z, defines z as a three-valued continuous function
of w for all values of w.
In other words, since, in accordance with a notation already
defined, (1) may be written
z = yw (1'),
we have shown that the cube root of wis a three-valued continuous
function of w for all values of w.
It is obvious that there is nothing in the above reasoning
peculiar to the case n = 3, except the fact that we have a triple
spiral in the w-plane, and a trisected circumference in the 2;-plane.
Hence, if we consider the equation
w = «" (4),
and its equivalent inverse form
z = ^w (4'),
all the alteration necessary is to replace the triple by an w-ple
spiral, returning into itself on the negative or positive part of
the w-axis, according as n is odd or even ; and the trisected
circumference by a circumference divided into n equal parts.
Thus we see that the equation (4), which defines w as a
continuous one-valued function of z for all values of z, defines z
{that is, the nth root of w) as a continuous n-valued function of w
for all values ofw.
% 5.] Riemann's Surface. It may be useful for those who are to pursue
their mathematical studies beyond the elements, to illustrate, by means of
the simple case tD=z^, a beautiful method for representing the continuous
variation of a repeating function which was devised by the German mathema-
tician Eiemann, who ranks, along with Cauchy, as a founder of that branch
of modern algebra whose fundamental conceptions we are now explaining.
266 BRANCHES OF ^Jw CH. XXIX
Instead of supposing all the spires of the w?-path in Fig. 4 to lie in one
plane, we may conceive each complete spire to lie in a separate plane super-
posed on the w-plane. Instead of the single it'-plane, we have thus three
separate planes, Pj, P^ , Pj , superposed upon each other. To secure continuity
between the planes, each of them is supposed to be slit along the M-axis from
0 to - 00 ; and the three joined together, so that the upper edge of the slit in
Po is joined to the lower edge of the slit in Pj ; the lower edge of the slit in
Pj to the upper edge of the slit in Pj ; the lower edge of the slit in Pj to the
upper edge of the slit in Pj, this last junction taking place across the two
intervening, now continuous, leaves. We have thus clothed the whole of the
w-plane with a three-leaved continuous flat helicoidal* surface, any continu-
ous path on which must, if it circulates about the origin at all, do so three
times before it can return into itself. This surface is called a Eiemann's
Surface. The origin, about which the surface winds three times before
returniug into itself, is called a Whiding Point, or Branch Point, of the
Third Order. Upon this three-leaved surface lo will describe a continuous
single path corresponding to any continuous single path of z, provided we
suppose that there is no continuity between the leaves except at the junctions
above described.
§ 6.] If we confine 0 to that part T'Ol' of its circle which
is bisected by OJC, and <f> to the corresponding spire I'Ol' of its
path, so that <^ lies between — tt and + ?r, and 6 between - ir/n
and +7r/n, then z becomes a one-valued function of w for all
values of w. We call this the principal branch of the n-\a\ued
function ^w; and, as we have the distinct notation w^^"' at our
disposal, we may restrict it to denote this particular branch of
the function z. In other words, if
w = s{GO&(f> + isiw (ji), — 77 < ^ < -t- TT,
we define w^'^ by the equation
and we also restrict (cos ^ + * sin <^)^''"' to mean cos . ^Jn + i sin . <f>ln.
Just as in § 4, we take the next spire after T'Ol' in the
positive direction (counter-clock) to represent the first positive
branch of IJw ; the next in the negative direction to represent the
first negative branch of ^w ; and so on, the last positive and the
last negative being full spires, or only half spires, according as n
is odd or even.
If, as is usual, we represent the actual analytical value of w
* Like a spiral staircase.
§§ 5, 6 PRINCIPAL VALUES 267
by the form s (cos «^ + / sin ^), where <^ is always taken between
- TT and + TT, then it is easy to find expressions for the values of z,
belonging to the w - 1 positive and negative branches of 'i/w and
corresponding to any given value of w, in terms of the value
belonging to the principal branch. We have, obviously, merely
to add or subtract multiples of 27r to represent the successive
positive and negative whole revolutions of the graphic radius of
w. Thus, if z, Zt, Z-t relate to the principal, tth. positive, and
^th negative branches oiz= l^w respectively, we have
z = s^/" {cos . ^/w + * sin . ^/w} ; \
Zt = s^'" {cos . (<)) + 2tTr)/n + i sin . (<^ + 2tT)ln} ; I (5).
Z-t = s''"{cos . (<^ - 2tTr)ln + i sin . (<^ - 2tiir)ln]. J
We have thus been led back by a purely graphical process to
results equivalent to those already found in chap, xii., § 18.
Cor. 1. Hence, \f z denote the principal value of the nth root
ofw, and <an = cos. 27r/w + i sin . 27r/n, then
t
n ;
that is, Zt = w^'"'u>n, Z-t = 'U)
Zt = zwn', z.t = zo)n-'; \ .„.
Cor. 2. The principal value of the nth root of a positive real
number r is the real positive nth root, that is, what has already
been denoted by r^'"' (see chap, x., § 2).
For, in this case, we have w = r (cos 0 + i sin 0), that is, ^ = 0.
Hence J!jw = r^'\
Cor. 3. There is continuity between the last values of any
branch of ^Jtv and the first values of the next in succession, and
between the last values of the last positive branch and the first
values of the last negative branch; but elsewhere two values of
'^w belonging to different branches, and cori'esponding to the
same value of w, differ by a finite amount.
It should be noticed as a consequence of the above that the principal
value of the nth root of a real negative number, such as - 1, is not definite
until its amplitude is assigned. For we may write -l = cos7r + isin7r or
= cos ( - tt) + i sin ( - ir) ; and the principal value in the former case is
cos.7r/n + tsin.ir/n, in the latter cos(-7r/w) + i Bin(-ir/n). This ambiguity
does not exist for complex numbers differing from - 1, even when they differ
infinitely little, as will be at once seen by referring to Figs. 3 and 4,
268 DISCUSSION OF W^ = Zl CH. XXIX
§ 7.] It should be observed that if, instead of restricting «^
in the expression 2; = 5^^"{cos. ^/w + « sin. <^/n} to lie between
— TT and +7r, we cause it to vary continuously from -wtt to
+ W7r, then s^^"{cos . <^/w + «' sin. ^/«} varies continuously and
passes once through every possible value of I^w, where | «r | is
given =5.
It follows also that, if w describe any continuous path
starting from P and returning thereto, the value of J^w will
vary continuously ; and will return to its original value, if w
have circulated round the origin of the w-plane pn times, where
/> is 0 or any integer ; and, in general, will return to its original
value multiplied by wj, where t is the algebraical value of
+ fi — r, fi and V being the number of times that w has circu-
lated round the origin in the positive and negative directions
respectively. On account of this property, the origin is called a
Branch Point of l^w.
§ 8.] Let us now consider briefly the equation
'uf = z'^ (1),
where p and q are positive integers. We shall suppose p and q
to be prime to each other, because that is the only case with
which we shall hereafter be concerned*.
Our symbols having the same meanings as before, we
derive from (1)
s^ {cosp(ji + i sill p<}>) = r^ (cos qO + i sin qd) (2).
Hence, taking the simplest correspondence that will give a
complete view of the variation of both sides of the equation
last written, we have
s^ = r^, p^ = q6 (3).
If, then, we fix r, and therefore s, the paths of z and w will
be circles about the origins of the z- and ^^-planes respectively ;
and, since p is prime to q, if z and w start from the positive part
* If ^ and q had the G.C.M. k, so that ^ = fc^', q = kq', where p' and q' are
mutually prime, then the equation (1) could be written («7P')*'=(z9')*, which
is equivalent to the k equations, wp'—z1', wV = uj^sfi' , ioP'=(,)f?zi', . . ., ujp*
= wj*~^2;9', where w^. is a primitive A;th root of + 1. Each of these k equations
falls under the case above discussed.
§§ 7, 8
DISCUSSION OF WP = Z'i
2G9
of the X- and w-axes simultaneously, they will not again be
simultaneously at the starting place before z has made p, and
w has made q revolutions.
To get a complete representation of the variation we must
therefore cause 6 to vary from —pir to +ptt, and ^ from — qir to
+ g''7r. The graphs of z and w will therefore be spirals having
p and q spires respectively. To each whole spire of the g-spiral
will correspond the pjqih. part of the j9-spiral. The case where
p = Z and g = 4 is illustrated by Figs. 5 and 6.
FiQ. 5.
Fig. 6.
It follows, therefore, that the equation (1) determines w as a
continuous p-valued function of z, and z as a continuous q-valued
function of w. Taking the latter view, and writing (1) in the
form
z = Uw^ (1'),
and (3) in the form
r = ^i% e=p^lq (3'),
we see that, if we cause ^ to vary continuously from -qir to
+ qtr, then s**'* ( cos - <^ + e sin- ^ j will vary continuously through
all the values which ^w^ can assume so long as 1 1^? | = s, and
will return to the same value from which it started. In fact, we
270 BRANCHES OF Ijw'P CH. XXIX
see in general that, if w start from any point and return to the
same point again after circulating /* times round the origin in
the positive direction, and v times in the negative direction,
then ^liP returns to its original value multiplied by cos . 2ptTrlq +
i sin . 2pt-nrlq where # = +/*- r; that is, by w/', where Wg denotes
a primitive q\h root of + 1.
If, as usual, we divide up the circular graph of w into whole
spires, counting forwards and backwards as before, and consider
the separate branches of the function ^iif corresponding to these,
then each of these branches is a single- valued function of 6.
The spire corresponding to -7r<^<+7r is taken as the
principal spire, and corresponding thereto we have the principal
branch of the function z = ^w^, namely.
\cos~<fi + ism-({)[, -'n-<(f)< +
For the (+ t)th. and (-r t)th. branches respectively, we have
Zt = s^'^{cos .p{(l> + 2tir)lq + i sin .p(<fi + 2t7r)/q},
= iofz;
z^t = s'^'^{cos.p{cf>-2t7r)/q + 2 sin .p{<fi- 2tTr)/q},
= o>f'"z.
As before, we may use w*'^^ to stand for the principal branch
of ^w^, and we observe, as before, that the principal value
of ^w^ when w is a real positive quantity is the real positive
value of the qth. root, that is, what we have, in chap, x,,
denoted by w^'^.
% 9.] It must be observed that, when p is not prime to q, the expressions
sPl'i {cos. p (0±2<ir)/g + i sin.p (^±2tir)/g} no longer furnish all the q values
of i^wP, but (as may be easily verified) only q/k of them, where k is the
G.C.M. of p and q. The appropriate expression in this case would be
«P/« {cos . (p^ ± 2(ir)/g + i sin . (p^ ± 2t7r)/g } .
This last expression gives in all cases the q different values of ^wp ; but
it has this great inconvenience, that, if we arrange the branches by taking
successively f = 0, t=l, « = 2, . . ., the end value of each branch is equal,
not to the initial value of the succeeding branch, but to the initial value of
a branch several orders farther on. There will therefore be more than one
crossing in the graphic spiral. The investigation from this point of view will
§§ 8-10 EXERCISES XIV 271
be a good exercise for the student. When p is prime to q, the two expres-
sions for ^wP are equivalent ; and we have preferred to use the one which
leads to the simpler graphic spiral.
If we adopt Riemann's method for the graphical representation of the
equation wP—zi, then we shall have to cover the z-plane with a ^-leaved
Eiemann's surface, having at the origin a winding point of the _pth order ;
and the w -plane with a g -leaved surface, having at the origin a winding
point of the gth order.
Exercises XIV.
(1.) Solve the equation
tan-i {(x-t- l)/(a; - 1)} -1- tan-i {(a: -f- 2)/(a; - 2)} = ^ TT,
and examine whether the solutions obtained really satisfy the equation when
tan~^ denotes the principal branch of the inverse function.
(2.) If 27r2<4g', show that the roots of the equation x^-qx-r=0 are
2(g/3)V2cosa, 2 (g/3)i/2cos (|7r-J-a), 2 ((7/3)1/2 cos (§7r- a), ^Yiexe a is deter-
mined by the equation co8 3a = ^r (3/g)=V2,
Show that the solution of any cubic equation, whose roots are all real,
can be effected in this way ; and work out the roots of a;^ - S-r + 3 = 0 to six
places of decimals. (See Lock's Higher Trigonometry, § 135, or Todhunter's
Trigonometry, 7th ed., § 2G0.)
Trace the graphs of the following, x being a real argument : —
(3.)
2/ = sina;-(-sin2a;.
(4.)
y = smx + cos 2x.
(5.)
y = sinx sin 2x.
(6.)
y = tanx + t&n2x.
(7-)
y=xamx.
(8.)
y = sin x/x.
(9.)
y = sin 8a;/cos x.
(10.)
y = am~^x^.
(11.)
2/2= sin- ix.
(12.)
sin y = tan x.
Discuss graphically the following functional equations connecting the
complex variables w and z. In particular, trace in each case the «;-paths
when the 2-paths are circles about the origin of the 2-plane, or parallels to
the real and to the imaginary axis.
(13.) w-=z^
(14.)
w = Ijz.
(15.) w^ljz^
(16.)
102 = 1/^3.
(17.) w^=(z-a){z-b).
(18.)
w^ = {z-af(z-h).
(19.) w3=(z-af.
(20.)
w^={z-af.
(21.) w = {az + b)l{cz + d).
(22.)
w^=ll{z-a)(z-b).
§ 10.] We can now extend to their utmost generality some
of the theorems regarding the summation of series already
established in previous chapters.
It is important to remark that the peculiar difficulties of this
272 GENERALISATION OF INTEGRO-OEOMETRIC SERIES CH. XXIX
part of the subject do not arise where we have to deal merely
with a finite summation ; that is to say, the summation of a
series to n terms. For any such summation involves merely a
statement of the identity of two chains of operations, each con-
taining a finite number of links, and any such identity rests
directly on the fundamental laws of algebra, which apply alike
to real and to complex quantities.
Even when the series is infinite, provided it be convergent,
and its sum be a one-valued function, the difficulty is merely one
that has already been fully settled in chap, xxvi.
The fresh difficulty arises when the sum depends upon a
multiple-valued function. We have then to determine which
branch of the function represents the series ; for the series, by
its nature, is always one-valued.
We commence with some cases where the last-mentioned
point does not arise.
GEOMETRIC AND INTEGRO-GEOMETRIC SERIES.
§ 11.] The summation
l+z + z"-^. . . +;^" = (l-z"*0/(l-2;) (1),
since it depends merely on a finite identity, holds for all values
of z. We may therefore suppose that z-x + yi^r (cos 6 + i sin 6),
and the equation (1) will still hold.
Also, since L 2;""*"^ ^ Lr'"-^'^ (cos n + l6 + i dnn + 1^) = 0,
n=at>
when r<\, we have, provided |2;|<1, the infinite summation
l+z + s'^+ . . . adoo = l/(l-;r) (2)
for complex as well as for real values of z.
In like manner, the finite summation of the integro-geometric
n
series 2^(w)^"> which we have seen can always be effected for
real values of z (see chap, xx., § 14), holds good for all values
of z ; and, since 2«^ (w) z^ is convergent provided 1 2; | < 1 , the
infinite summation deducible from the finite one wilj hold good
for all complex values oi z such that |;2;|<1.
§§ 10, 11 EXAMPLES 273
By substituting in (1) or (2), and in the corresponding
equations for 5^ (n) z^, the value r (cos 0 + i sin 0) for z, and then
equating the real and imaginary parts on both sides, we can
dedxice a large number of summations of series involving circular
functions of multiples of 6.
Example 1. To sum the series
Sn=l + rcosd + r^co82d + . . . + r^coane,
T^=r sin d + r"^ sin 20 + . . . + r"-sinn6,
i[7„=cosa+rcos(a + ^) + r''cos(a + 2») + . . . + r'*cos (o + n^),
Vn=ama + r8m{a + e)+r^sm{a + 2e) + . . . +r"sin (a + Ji^),
to n terms ; and to oo when r < 1.
Starting with equation (1), let us put z = r (cob 9 + i ain 6), and equate real
and imaginary parts on both sides. "We find
l + r{coB$ + iaind) + f^{cos2e + ism2d) + . . .+r^{cos7i0 + iamn0)
= { 1 - r™+i (cos {n + l)9+i sin (n + 1) ^)}/{ 1 - r (cos ^ + i sin ^)} (3) ;
whence*
S;„= {1 - r cos ^ - r"+i cos (n + 1) ^ + r"+2 cos n9}l{l - 2r cos O+r^} (4) ;
T„= {r sin 9 - r^+i sin [n + 1)9 + r'^+^ sin n9}l{l - 2r cos 9 + r^} (5).
Again, since U'„ = cos aS„- sin aT^,
Vn = sin a5f„ + cos aT^ ,
we deduce from (4) and (5) the following: —
f7„= {cos a - r cos (a - ^) - r"+i cos {n + 19 + a) + r"+2 cos {n9 + a)}/
{l-2rcos9 + r"} (6),
F„= {sin a - r sin (a -9)- r'^+i sin (n + 10 + a) +r'»+2 gin („,9 + a)}/
{l-2rcos^ + r2} (7).
From these results, by putting r=+l, or r=-l, we deduce several
important particular cases. For example, (6) and (7) give
C08a + C08(a + 5)+C0s(a + 2e) + . . . + cos(ft+7i^)
=cos ^ {o + (a+n^)} sin ^ (n + l)^/sin ^^ (6') ;
sina + sin(a + ^) + sin(a + 2^) + . . . + sin(a + ne)
= sin^{a + (a + n&)}sin^(n + l)^/sin4^ (7').
Finally, if r<l, we may make n infinite in (4), (5), (6), (7) ; and we thus
find
S„ = (l-rcos^)/(l-2rcos0 + r2) (4");
r„ = r sin 91(1 - 2r cos d + r^) (5") ;
U^ = {cos tt - r cos (a - ^) }/{l - 2r cos 9 + r^ (6") ;
F„ = {sin a -r sin (a - 9)}I{1 - 2r ooa9 + r^} (7").
* For brevity, and in order to keep the attention of the reader as closely
as possible to the essentials of the matter, we leave it to him, or to his teacher,
to supply the details of the analysis.
c. u. 18
274 EXAMPLES CH. XXIX
Example 2. Sum to infinity the series
S=l-2rcose + 3r^coa2e-4kr^co&3e + . . . ('■<1)-
If « = r (cos e + isinO), then S is the real part of the sum of the series
T=l-2z + dz^-iz^ + . . . .
Now, by chap, xx., § 14, Example 2,
T=l/(l + z)2.
Henee S=B{ll{l + rcoa0 + rism0)^}*
=R{{l + rcose-rism0)^j(l + rcoa0^ + r^sin^e)^},
= {l + 2rco3 0 + r^cos2e)l{l + 2rcose + r^)^.
Example 3. Exemplify the fact that every algebraical identity leads to
two trigonometrical identities in the particular case of the identity
- (b-c){c~a) (a - b) = bc (b - c) + ca {c - a) + ab {a ~ b).
In the given identity put a=cosa + tsina, &=cosj3 + tsin^, c = cos7 +
i sin y, and observe that
cos /3 + i sin /3 - cos 7 - i sin 7 = 2i sin i (/3 - 7) {cos i (/3 + 7) + i sin J (/3 + 7)}.
We thus get
4n sin i (i3 - 7) {cos 4 (i3 + 7) + i sin J (/3 + 7)}
= Ssin J(/3-7){cos|3 + isin/3} {0087 + 1 sin 7} {cos4(j3 + 7)
+ isini(^+7)},
whence
4cos(a + /3 + 7)nsini(/3-7) = Ssini(/3-7)cosf (^3 + 7);
4 sm (a + /3 + 7) n sin i (i3 - 7) = 2 sin i (/3 - 7) sin f ifi + y).
FORMULA CONNECTED WITH DEMOIVRE's THEOREM AND
THE BINOMIAL THEOREM FOR AN INTEGRAL INDEX.
§ 12.] By chap. xiL, § 15 (3), we have
cos (^1 + ^2 + . . . + ^n) + i sin (^1 + ^2 + • • • + ^n)
= (cos Oi + i sin 61) (cos 6^ + i sin 6^ . . . (cos 9^ + i sin ^n).
If we expand the right-hand side, and use Pr to denote
% cos 6I1 cos B^. . . cos Or sin ^r+i ... sin ^„, that is, the sum of all
the partial products that can be formed by taking the cosines
of r of the angles ^1, ^2, . • ., ^n and the sines of the rest, then
we find that
cos {6^ + 6i+ . . . + ^„) + » sin (^i + 6/2 + . . . + 0,,)
= Pn + iPn-l-Pn-2-iPn-i + Pn-4 + iPn-S-- - - •
* We use Rf(x + yi) and If(x + yi) to denote the real and imaginary parts
of / (a; + yi) respectively.
§ 12 EXPANSIONS OF COS (^i + ^2 + • • • + On), &C. 275
Hence
cos(^i + ^2+. . . +6'„) = P„-P„_2 + P„_4-P„-6+ . . . (1);
sin (6, + 6,+ . . . + ^„) = Pn-l - Pn-, + A-5 - Pn-7 + • • • (^)-
From these, or, more directly, from
cos {di + 6^+ . . . +0n) + i sin {6^ + 62+ . . . + ^n) = cos 6^ cos $2
... cos 6n (1 + i tan ^1) (1 + i tan 6^ . . . (1 + * tan ^„),
we derive
tan (^1+^2 + . .. + er,)={T,-T, + T,-. . .)/{l-T2+T,- . . .) (3),
where T'r = 2 tan ^1 tan ^2 • • • tan Or.
The formula) (1), (2), (3) are generalisations of the familiar
addition formulae for the cosine, sine, and tangent.
From the usual form of Demoivre's Theorem, namely,
cos n6 + i sin nd - (cos 6 + 1 sin &)"■,
we derive, by expansion of the right-hand side,
cos nd + i sin nd = cos'' 0 + i „(7i cos**"^ ^ sin ^ - ^Cg cos""^ 0 sin^ 6
- i nGs co8"-3 d sin' 6 + „C4 cos™"* d sin* 6> + . . . .
Hence
coswe = cos''e-„C2Cos"-'^sin2^ + „C4Cos"-''esin*^-. . . (4)*;
sin nO = nC^ cos"-^ ^ sin ^ - ^C^ cos"-=^ B sin=* 9
+ „aiCos"-''^sin^^-. . . (5);
„ ^(^itan^-^Cstan'^ + ^Cgtan''^-. . . ,^.
tan no= 77- — ttb — ?n — rz l^;.
l-rtC/atan-^ + nC^tan^^- . . .
These are generalisations of the formulae (8) of § 2.
The formulae (4) and (5) above at once suggest that cosw^
can always be expanded in a series of descending powers of cos ^;
that, when n is even, cos nd can be expanded in a series of even
powers of sin 0 or of cos 0 ; sin nO/sin ^ in a series of odd powers
of cos 0 ; and sin 7i^/cos 0 in a series of odd powers of sin 9 :
and, when n is odd, cos n6 in a series of odd powers of cos 9 ;
cos nO/cos ^ in a series of even powers of sin 9 ; sin n9 in a series
of odd powers of sin 9 ; sin nO/sin ^ in a series of even powers
of cos 9.
* The formulas (4), (5), (6), (8) were first given by John Bernoulli in 1701
(see 0^., t. I., p. 387).
18—3
276 EXPANSIONS IN POWERS OF SIN 0 AND COS 6 CH. XXIX
Knowing, a priori, that these series exist, we could in various
ways determine their coefficients ; or we could obtain certain
of them from (1) and (2) by direct transformation , and then
deduce the rest by writing ^w-O \n place of 6. (See Todhunter's
Trigonometry, §§ 286-288.)
We may, however, deduce the expansions in question from
the results of chap, xxvii., § 7. If in the equations (9), (10), (9'),
(9"), (10'), (10") there given we put a=^cos^ + »sin ^, ft = cos 6-
i sin 0, and therefore jt? = 2 cos ^, (/ = 1, we deduce
2 cos nO = (2 cos 0)'' - ^, (2 cos 6)''-^ + ^il^izi) (2 cos ey-' - . . .
^_yMn-r-l)(n-r-2)...(n-2r^l)^^^^^^^^^_^^^_ ^^^,.
sin «^/sin 6 = (2 cos 0)^-' - ^^^~ (2 cos 6^-' + ^^lll^^^ J^^Ilil
(2 cos er-^ -...(-)'• (n-r-l)(n-r- 2).^^n-2r)
(2 cos 6)''-2'--i + . . . (8);
cos ne = {- Y^ |l - ^' cos'^ e + '''^''^7 ^'^ cos" 6 - . . .
, ^y{n'-2'') . . . {ti'-2s~2-') „,. 1, . ^.
{-y — ^ (2s)\ ^ cos-' ^ + .. .Uw even) (9);
cos n9 = {- )(™-^''^ ] — cos 6 ^—-. — ' cos^ 0 + —^ ^ ^
11! o! 5!
cos^6-. . .{-y— — — .„ ix, -^cos^+^^ + . . .[
^ (2s +1)! J
(wodd) (10);
sin n6/sm 6 ^ {- fi'-^ i- cos 0 - ''^'''^~^'^ cos'' ^ + . . .
( - ) ^ {2i+i)\ ^°^ • 'P'* ^^^'^^ ^^ ^^ '
* The series (7), (9'), (10') were first given by James Bernoulli in 1702
(see Op., t. II., p. 926). He deduced them from the formula
2Bin^n^=|;(2Bin^)^-»'(";-^^)(2sin.)^4-"''^"'-^;j<"'-^^2sin^)e-....
which he established by an induction based on the previous results of Yieta
regarding the multisection of an angle.
§§ 12, 13 EXPANSIONS IN POWERS OF SIN 9 AND COS 9 277
sin ne/sin 6 = (-f-^V' U ~ ^-^^ cos^ 6 + (tl}^^t:3^ cos"^ - . . .
(-y^ '-^ 7^^^ cos^ 6+ . . .y{n odd) (12).
If in the above six formulae we put ^tt - ^ in place of 9, we
derive six more in which all the series contain sines instead of
cosines. In this way we get, inter alia, the following : —
cos n6=l- ^, sin^ 0 + ^ ^^ ^ sin* 9 - . . . (n even) (9') ;
sm n9 = .,, sm 9 ^—-t — '^ sin^ 9 + — — {~,^ ^- sin^ 9- . . .
1! 61 5!
(wodd) (10');
sin n9/ cos 9 = --^ sm 9 — ^—-, — - sm^ 6 + -^ ^ ^ sin^ 9-. . .
(weven) (11');
coswg/cos^=l-^^sin^^ + ^'''"^y~-^sin*g-. . .
(wodd) (12').
The formulae of this paragraph are generalisations of the
familiar expressions for cos 29, sin 29, cos 39, and sin 39, in terms
of cos 9 and sin 9.
§ 13.] The converse problem to express cos"^, sin'*^, and,
generally, sin™ 9 cos" ^ in a series of sines or cosines of multiples
of 9, can also be readily solved by means of Demoivre's Theorem.
If, for shortness, we denote cos 9 + i sin 9 by x, then we have,
by Demoivre's Theorem, the following results : —
£c = cos 9 + i sin 9, Ijx = cos ^ — z sin ^ ;
af = cos n9 + i sin n9, 1/af = cos nO - i sin nO :
cos9 = -{w+l/a;), sin9=-^{a;- l/ce);
cos n9 = I (.r" + l/.r"), sin n9 = ^. (^'^ - 1/^").
} (1).
278 EXPANSIONS IN COSINES AND SINES OF MULTIPLES OF 0
Hence
= r^^^{cos 2m^ + 2mOi cos (2w - 2)0 + ^^C.^ cos (2w - 4)^ +
' . . . + 22m^«i} (2).
Similarly,
cos^+i ^ = ^ {cos (2m + 1)^ + 2«+iCi cos (2m - 1)0
+ 2m+i^2Cos(2m-3)0+ . . . +2»i+iC'mCos0} (3);
sin'^™ 0 = ^^^ {cos 2m5 - ^C^ cos (2m - 2)0
+ 2^0.008 (2m -4)0+. . .{-Yl^raCm) (4);
sin-'«+^ 0 = ^-^ {sin (2m + 1)0 - am+aC; sin (2m - 1)0
+ 2^+, C^ sin (2m -3)0+. . . (-)'"2«+iatsin0} (5).
These formulae are generalisations of the ordinary trigonometrical
formula sin^ Q = -\ (cos 20 - 1), cos« 0 = ^ (cos 30+3 cos 0), &c.
In any particular case, especially when products, such as
sin™ 0 cos" 0, have to be expanded, the use of detached coefficients
after the manner of the following example will be found to con-
duce both to rapidity and to accuracy.
Example 1. To expand sin' Q cos^ ^ in a series of sines of multiples of 0.
sin' e cos' ^ = 08^5 (^ - 1/^)' (* + 1/^)*'
Starting with the coefficients of the highest power which happens to be
remembered, say the 4th, we proceed thus —
Coefficients of Multiplier.
Coefficients of Product.
1-1
1-4+ 6- 4 + 1
1-6 + 10-10 + 5-1
1 + 1
1 + 1
1 + 1
1-4+ 5+ 0-5 + 4-1
1-3+ 1+ 5-5-1 + 3-1
1-2- 2+ 6+0-6+2+2-1
The coeffi cients in the last line are those in the expansion of (a; - 1/x)' (x + l/x)'.
Hence, arranging together the terms at the beginning and end, and replacing
§ 13 EXERCISES XV 279
-: (a;8 - l/a;8) by sin 8^, - {x^ - l/x") by sin &d, and so on, we find
sin" e cos8 e=^ {sin 8^-2 sin 6(? - 2 sin 4^ + 6 sin 2^ + ^ . 0},
= fL {sin 80 - 2 sin 6» - 2 sin 4(? + 6 sin 20}.
The student will see that sin"* 6 cos" 6 can be expanded in a
series of sines or of cosines of multiples of 0, according as m is
odd or even. The highest multiple occurring will be (m + n) B.
Example 2. If 0 = 2vln, and a any angle whatever, and
,„[/■„= cos'" a + CDS'" (a + 0) + . . . + cos'"(a + ra-10),
^F„=8in'"a + 8in"'(a + 0) + . . . + 8in"*(a + n-10),
where m is any positive integer which is not of the form r + inji, then
2mf^«=2m^n=«-1.3. . .(2m-l)/2.4. . .2m;
am+l t/„ = 2m+l 'n ~ '^ •
This will be found to follow from a combination of the formulas of the
present paragraph with the summation formulae of § 11.
Exercises XV.
Sum the following series to n terms, and also, where admissible, to
infinity : —
(1.) cos a - cos (a + 0) + cos (a + 20)-. . .
(2.) sina-sin(a + 0) + sin(a + 20)-, . .
(3.) Ssin^n^. (4.) nco80 + (n- l)cos20 + (n-2) cos30 + . . . .
(5.) S sin 710 cos (n + 1)0. (6.) S sin Ji0 sin 2ra0 sin 3«0.
(7.) sin a - cos a sin (a + 0) + cos* o sin (a + 20)-. . . .
(8.) 1 + cos 0/cos 0 + cos 20/CO8* 0 + cos 30/cos^ 0 + ... to n terms, where
6=nv.
(9.) l-2rcos0 + 3r2cos20-4?-3cos30 + . . . .
(10.) 8in0 + 3sin20 + 5sin30 + 7sin40 + . . . .
(11.) Sn* cos (?i0 + a). (12.) Sn (n + 1) sin (2n + 1) 0.
(13.) sin 2n0 - ^JJi sin (2n -2)0 + j^Cj sin (2n - 4) 0 - . . . (n a positive
integer),
(14.) 8in(2n + l)0 + 2„+iCiSiu(2n-l)0 + 2„+iC2sin(27i-3)0 + . . . (n a
positive integer).
(15.) 2m(»i+l) . . . (ni + n-1) r"cos(a + ji0)/nl to infinity, m being a
positive integer.
(16.) Does the function
(sin2 0 + sin2 20 + . . .+ sin" n0)/(cos' 0 + 008*20 + . . . + cos*n0)
approach a definite limit when n = QO ?
(17.) Expand 1/(1 -2 cos 0,a; + a;*) in a series of ascending powers of x.
280 FUNDAMENTAL SERIES FOR COS 6 AND SIN 6 CH. XXIX
(18.) Expand 1/(1-2 cos ^ . x + x^Y ^^ * series of ascending powers of x.
(19.) Exjjand (l + 2x)/(l-x3) j^ ^ series of ascending powers of x ; and
show that
^ ^^^ ^ (3n - 1) (3» - 2) (3n - 2) (.8n - 3) (37i - 4) ^ ^ _ . = (-l)».
(20.) Show that l/(l+x+x2) = l-x + x3-x* + x8-x'' + x9-xio+ . . .;
and that, if the sum of the even terms of this expansion be ^ (x), and the
sum of the odd terms \j/ (x), then {0 (x)}^ - {^(x)}— ^ (a;-) + \p (x^).
Prove the following identities by means of Demoivre's Theorem, or
otherwise. S and 11 refer to the letters a, /3, 7: —
(21.) S sin a/(l + S cos a) = -Iltan Ja, where a+§ + y=0.
(22.) S sin {e - /3) sin {9 - 7)/sin (a - /3) sin (a - 7) = 1.
(23.) SsinJ(a + i3)sin|(o + 7)cosa/sin4(a-/3)sin|(o-7) = cos(a + /3 + 7).
(24.) cos <7 cos (o- - 2a) cos (tr - 2j3) cos (<r - 27) + sin <t sin (er - 2a) sin (o- - 2§)
sin (or - 27) = cos 2a cos 2;8 cos 27, where o- = a + j3 + 7.
Expand in series of cosines or sines of multiples of 6 : —
(25.) cosi<>6». (26.) sin7^. (27.) sin^ ^.
(28.) coss^sins^. (29.) cosStfsin'*^.
Expand in series of powers of sines or cosines : —
(30.) COSIO6'. (31.) sin 7^.
(32.) sin3&cos6tf. (38.) cosm^cosji^.
EXPANSION OF COS 6 AND SIN 6 IN POWERS OF 0.
§ 14.] We propose next to show that, for all finite real
values of 6,
cose = l-^/2! + 6V4!-^V6! + . . . ad co (1);
mi6 = e- eysi + ^751 -6'IV. + . . . ad a> (2).
These expansions* are of fundamental importance in the
part of algebraical analysis with which we are now concerned.
They may be derived by the method of limits either from the
formulae of § 12, or from two or more of the equivalent formulae
of § 13. We shall here choose the former course. It will appear,
however, afterwards that this is by no means the only way in
which these important expansions might be introduced into
algebra.
* First given by Newton in his tract Analysis per cBqttationes numero
terminorum infinitas, which was shown to Barrow in 1669. The leading idea
of the above demonstration was given by Euler (Introd. in Anal. Inf., t, i.,
§ 132), but his demonstration was not rigorous in its details.
§ 14 FUNDAMENTAL SERIES FOR COS 6 AND SIN 6 281
From (4) and (5) of § 12, writing, as is obviously permissible,
6/m in place of 6, and taking n = m, we deduce, after a little
rearrangement,
cos 6 - cos"
:{-
2!
tan — / — )
ml mj
^ (i-i/m)(i-2M(i-3M) ^, A e_i6\\
4! \ mf m)
0 .
= cos™ —{l-u^ + Ui-. . .], say,
and
sin
d = cos'"-?j^ftan^/-^)
m { \ ml mJ
3! \ m/
e lev
m)
= cos*" - {ui -Us + ' • • }, say,
.} (3),
(3');
(4),
(4').
Here, from the nature of the original formula, m must be a
positive integer; but nothing hinders our giving it as large a
value as we please, and we propose in fact ultimately to increase
it without limit. On the other hand, we take 6 to be a fixed
finite real quantity, positive or negative.
The series (3), as it stands, terminates ; and its terms alter-
nate in sign.
We have
U2n+2
Uo
^ (l-2nlm){l-2n + lfm) ^, A ^^ ^ /0_V
(2w + 1) (2w + 2) \ m/ mJ
Hence, so long as n is finite,
T ^2ra+2
(2n + 1) {2n + 2) '
If, therefore, we take 2n-hl>6*, we can always, by taking
m large enough, secure that, on and after the term u.2n, the
numerical value of the convergency-ratio of the series (3) shall
be less than unity.
Strictly speaking, it is sufficient if ^<^{(2« + l) (2n + 2)}.
282 FUNDAMENTAL SERIES FOR COS d AND SIN 6 CH. XXIX
From this it follows that, if 2n-\-l>B, and m be only taken
large enough, cos 6 will be intermediate in value between
and
cos'"-{l-W2 + ««4-. . •(-r«2«} (5),
m
cos*"- {1 - W2 + ^4 - . . .{-fu^n+i- )"+' «2«+2} (6).
m
Therefore cos^ will always lie between the limits of (5) and
(6) for m-oo.
]V[ow (see chap, xxv., § 23)
Zcos'»(^/7») = l, Lu2 = 6y2\, Lui=e*/4.\, . . ,
Im,^ = 6/^/(2?0! , i^^^2»+2 = e^+V(2w + 2)!.
Hence cos B lies between
1 _ ^72 ! + ^V4 ! - • • . ( - )" ^' V(2w) !
and
l-e72! + ^/4!-. . .{-)™6''V(2w)! + (-)"+^^"+V(2w + 2)!.
In other words, provided 2w + 1 > ^,
cos^= 1 - ^/2! + ^74! - • . . (-)'^^^/(2w)! + {-Y^^B^ (7),
Here 2^^ may be made as large as we please, therefore since
L 6^+y{2n + 2)1^0 (chap, xxv., § 15, Example 2), we may
n=oo
write
cos^ = l-^/2! + ^/4!-. . .adoo (7')-
By an identical process of reasoning, we may show that,
provided 2n + 2>6*, then
s{ne = e-6'/S\ + . . .(-fe^+y{2n + l)\ + {-Y+'M^+, (8),
where M^n+i < 6"*+V(2« + 3) ! ,
and therefore
smO = e- Oysi + $'151 - . . . ad 00 (8').
It has already been shown, in chap, xxvi., that the series (7')
and (8') are convergent for all real finite values of 6 ; they are
• More closely, if 9<:^{(2n + 2) (2n+3)}.
§ 14 EXAMPLES 283
therefore legitimately equivalent to the one-valued functions
cos 6 and sin 6 for all real values of 0, that is, for all values of
the argument for which these functions are as yet defined. From
this it follows that the two series must be periodic functions of
0 having the period 2rr. This conclusion may at first sight
startle the reader ; but he can readily verify it by arithmetical
calculation through a couple of periods at least.
When 0 is not very large, say ^l^^-n; which is the utmost
value of the argument we ineed use for the purposes of calcula-
tion* the series converge with great rapidity, five or six terms
being amply sufficient to secure accuracy to the 7 th decimal
place.
We shall not interrupt our exposition to dwell on the many
uses of these fundamental expansions. A few examples will be
sufficient, for the present, on that head.
Example 1. To calculate to seven places the cosine and sine of the
radian.
We have
cos 1 = 1 - 1/2! + 1/41 - 1/6! + 1/8! - 1/10! + R^q ,
iJio<l/12!,
= 1 - -500,000,0 + -041,666,7 - -001,388,9 + 000,024,8 - -000,000,3 + R^q ,
i?io< -000,000,003.
= -540,302,8.
Similarly,
Bin 1=1 - 1/31 + 1/5! - 1/7! + 1/9! - Eg,
Eg < 1/11! < -000,000,03,
= •841,471,0.
The error in each case does not exceed a unit in the 7th place.
Example 2. If ^ < 3, then 0 > sin ^ > 0 - ^e» ; 1 - ^^2 < cos 0 < 1 - ^e^+^^O*.
These inequalities follow at once from (7) and (8) above. They are
extensions of those previously deduced, in chap, xxv., § 21, from geometrical
considerations.
Example 3. Expand cos (a + ^) in powers of 0.
Result, cos (a + ^) = cos a cos ^ - sin a sin 0,
= cos a - sin a 0 - cos a ^2/2! + sin a fl*/3! + cos a ^*/4! - . . .
* Seeing that the cosine or sine of every angle between ^n- and ^tt is
the sine or cosine of an angle between 0 and lir.
284 EXERCISES XVI
CIT. XXIX
Example 4. Find the limit of
9(l-cose)l{ta,n0-e) when e = 0.
LO (1 - cos d)l(ta,n 0 -e) = Lsece Le(l- cos ^)/(sia 0-0 cos 0),
= lxLd(d''l2-e*l4l + . . .)l(d-d'i3l + . . .-0 + 6^12-. . .),
= L{0^l2-0^li\ + . . .)I{0^I3 + . . .),
=L(II2 + P0^- + . . .)I(1I3 + Q0"' + . . .),
= 3/2.
Exercises XVI.
(1.) Expand sin (a + 0) sin (j3 + 0) in powers of 9.
(2.) Calculate sin 45° 32' 30" to five places of decimals.
(3.) Given tan 6i/(9 = 1001/1000, calculate 0.
(4.) Expand cos^ 9, sin^ 0, and sin^ 9 cos 9 in powers of 0 ; and find the
general term in each case,
(5.) Show that cos'" 0 {m a positive integer) can be expanded in a con-
vergent series of even powers of 0 ; and that the coefficient of 0^* in this
expansion is
(-)»{m2«+^Ci(7n-2)2»+^C2 (771-4)2"+. . .}/2"»-i(2ji)!,
(6.) Show that, if m and n be positive integers, and 1<w<7b, then
j?i" - ^(7i (m - 2)» + „,C., (771 - 4)» - . . . = 0.
Examine how this result is modified when ?i = l, or n=m.
Evaluate the following limits : —
(7.) (sin2 7?7^-sin2 7z^)/(co3 2>^-cos(j'^), 0=0.
(8.) {sin27(a + (?) -sin^Ja}/^, ^ = 0.
(9.) {sinXa H-fi")- sin" j5a}/^, 0 = 0.
(10.) {sin"p(a + ^)cos(a + e)-sin"2'aco8a}/^, ^ = 0.
(11.) (a^ sin a^ - &^ sin 5(?)/(6^ tan a^-a^ tan 6^), 0=0.
(12.) 1/2x2 -ir/2a; tan TTX-- 1/(1 -a;2), r=l (Euler).
(13.) {sinxlxy/"", x = 0.
(14.) { (xja) sin (a/a;) }*", a; = oo , (7?i = > 2) .
(15.) Show, by employing the process used in chap, xxvn., § 2, that the
series for sin ji^/cos 9 in powers of sin 9 can be derived from the series for
cos 71^ in powers of sin^; and so on.
(16.) Show, by using the process of chap, xxvii., § 2, twice over, that, if
C0S7i^=l + Ji8in2^ + J2sin'*^ + . . . + ^^sin2'-^ + . . .,
then
-n^oosn9=2A^ + {3.iA2-2^Aj)sin^9 + . . .
+ {(2r + l)(2r + 2)J^i-(27-)2^r}8in2'-6> + . . . .
Hence determine the coefficients A^, A„, &o. ; and, by combining Exercise
15 with Exercise 16, deduce all the series (7) . . . (12') of § 12.
(17 ) Show (from § 13) that cos"0 and sin"0 can each be expanded in a
convergent series of powers of 9 ; and find an expression for the coefficient of
the general term in each case.
In particular, show that
sin»a;/3I=x3/8!-(l + 32)x'>/5I + (l + 82 + 3^)x7/7!-(l + aH3*+3«)x»/9! + . . . .
15 BINOMIAL THEOREM 285
BINOMIAL THEOREM FOR ANY COMMENSURABLE INDEX.
§ 15.] If, as in chap, xxvii., § 3, we write
/(m) = l + 2™(7„;z'' (10),
where m is any commensurable number as before, but z is now
a complex variable, then, so long as |2;|<1, "^mPnZ^ will (chap.
XXVI., § 3) be an absolutely convergent series ; and f{m) will be
a one-valued continuous function both of m and of z. Hence
the reasoning of chap, xxvii., § 3, which established the addition
theorem /(wzi)/(?»2) -f{mx + m^ will still hold good; and all the
immediate consequences of this theorem — for example, the
equations (4), (5), (6), (7), (8), (9) in the paragraph referred to —
will hold for the more general case now under consideration.
In particular, if p and q be any positive integers (which for
simplicity, we suppose prime to each other), then
= {\^-zY (11).
It follows i\\2Xf{plq) represents part of the g-valued function
^{1 + zY ; and it remains to determine what part.
Let z = r (cos ^ + * sin 9), then, since we have merely to ex-
plore the variation of the one-valued function fiplq), it will be
sufficient to cause 6 to vary between — ir and + ir.
Also, let
w-\ + z-l-if x + yi, \
= 1 + r cos 6 + ir sin 0, I {a),
— p (cos ^ + i sin ^), J
so that
p = {(1 + xf + y'^Y'^ - (1 + 2r cos 6 + t^'f' :
tan ^ - yl{l ■\-x) = r sin Ojil + r cos 0), J
If we draw the Argand diagram for w = l+x + yi, we see
that when r is given w describes a circle of radius r, whose centre
is the point (1, 0). Since r<\, this circle falls short of the
origin. Hence </>, the inclination to the a;-axis of the vector
drawn from the origin to the point w, is never greater than
286 EXPANSION OF {1 + X + yi)'" CH. XXIX
tan-' {r/(l - 7^)^}, and never less than - tan"' {r/(l - r*)'/'}.
Hence <f> lies in all cases between - l^r and + ^tt. Therefore,
since f{p/q) is continuous, only one branch of the function
^(1 + zy is in question. Now, if we denote the principal
branch by {l + z)^'^, so that
(1 + zy^ = p^'^ (cos . p4>lq + i sin . js^/g),
we have, by § 8,
^{l + zY = {l + zY"^oyf (12),
where ^ = 0, ±1, ±2, . . ., according to the branch of the
function which is in question. Hence we have
f{plq) = {l + zy"^i^f,
where t has to be determined.
Now, when «=0, we hayefip/q) = 1, hence we must have
1 = «>/'.
Hence t = 0, and we have
/(p/q) = (1 + '^y^ = P^'^ (cos .pi>/q + i sin. p<fi/q),
where -^7r<<f)< ^ir.
Next consider any negative commensurable quantity, say
-p/q. Then (by chap, xxvii., § 3 (9)),
f{-p/q)-/(o)/f(p/q),
= W{plq\
If, therefore, we define (1 + z)~^'^ to mean the reciprocal of
the principal value of {l + zY'^, we have
/( -piq) = (1 + ^Y"'^ = 1/(1 + ^Y'^
= p-^/« {cos ( -p<i>lq) + % sin ( -p4>lq)] (13).
To sum up : We have now estahlisJied the following expansion
for the principal valiie of(l + «)'"■, in all cases where m is any
commensurable number, and 1 2; | < 1 : —
{l + zy-=l + '^r.CnZ^ (14).
The theorem may also be written in the following forms : —
1 + 2,„(7„(;i; + yiy = {(1 + xf + y}'"^ [cos . m tan"' {yl(l + x)]
+ i sin . m tan"' {yj{l + x)]"] (15) ]
^15-17 GENERAL STATEMENT OF BINOMIAL THEOREM 287
1 + SmCnr" (cos nd + i sin nO)
= (1 + 2r cos 6 + r^)™'^ (cos m^ + i sin m<f>),
where -|7r<<^ = tan"^ {rsin ^/(l +rcos 6)}<+ ^tt (16).
§ 16.] The results of last paragraph were first definitely
established by Cauchy*. In a classical memoir on the present
subject!, Abel demonstrated the still more general theorem
l + 2m+fciC'n(a^ + yO"
= [(1 + a^f + ff"^ [cos {m tan-i {yl{l + a;)} + P log {(1 + a;f + f}}
+ i sin {m tan"^ {?//(l + x)} + \k log {(1 + xf + y^]}]
Exp [ - ^ tan-^ {yl{l + x)]\
Into the proof of this theorem we shall not enter, as the
theorem itself is not necessary for our present purpose.
§ 17.] The demonstration of § 15 fails when |;^| = 1. Here,
however, the second theorem of Abel, given in chap, xxvi., § 20,
comes to our aid. From it we see that the summation of, say,
(16) will hold, provided the series on the left hand remain con-
vergent when r = 1.
Now the series 1 + %mCn (cos n9 + i sin nB) will be convergent
if, and will not be convergent unless, each of the series
S=l + SmCiiCOSW^,
be convergent.
In the first place, we remark that, if m< — l, Lm'On = ±^
when « = CO , so that neither of the series 8, T can be convergent.
If w = - 1, then ^C„ = ( - 1)», >S = 1 + 2 ( - 1)" cos iiB,
T- 2 ( - 1)" sin nQ, neither of which is convergent (see chap.
XXVL, § 9).
If - 1 < ?w < 0, then L mGn = 0 ; and the coefiicients ulti-
mately alternate in sign. Hence, by chap, xxvi., § 9, both the
series S and T are convergent, provided 6^±ir. When 6 has
one or other of these excepted values, then S=l +%{-lYm,Gn,
which is divergent when m lies between -1 and 0 (see chap.
XXVL, § 6, Example 3).
* See his Analyse Algebrique
t (Euvres Completes (ed. by Sylow & Lie), 1. 1., p. 238.
288 GENERAL DEFINITION OF ExP z CH. XXIX
If wz>0, then, as we have already proved (see chap, xxvi.,
§ 6, Example 4), %m.Gn is absolutely convergent, and, a fortiori,
1+2to^jiCosw^ and %raGn^va.nQ are both absolutely convergent.
It follows, therefore, that the equation
will hold when \z\ = \, in all cases v)here m>0 ', and also when m
lies between — 1 and 0, provided that in this last case tlie imaginary
part ofz do not vanish, that is, provided the amplitude ofz is not ±7r.
In other cases where 1 2; | == 1, the theorem is not in question,
owing to the non-con vergency of S^C^^;''.
In all cases where |2;1>1, the series 2mC»^'' is divergent, and
the validity of the theorem is of course out of the question.
EXPONENTIAL AND LOGARITHMIC SERIES — GENERALISATION
OF THE EXPONENTIAL AND LOGARITHMIC FUNCTIONS.
§ 18.] The series
l+z + zy2\+2^/S\ + . . .
is absolutely convergent for all complex values of z having a
finite modulus (see chap, xxvl, § 10). Hence it defines a single-
valued continuous function of z for all values of z. We may
call this function the Exponential of z, or shortly Exp z* ; so
that Exp z is defined by the equation
'Expz=l+z + zy2l+2?/Sl + . . . (1).
The reasoning of chap, xxvin., § 5, presupposes nothing but the
absolute convergence of the Exponential Series, and is therefore
applicable when the variable is complex. We have therefore
the following addition them'em for the function Exp z : —
* When it is necessary to distinguish between the general function of a
complex variable z and the ordinary exponential function of a real variable x,
we shall use Exp (with a capital letter) for the former, and either e^ or exp x
for the latter. After the student fully understands the theory, he may of
course drop this distinction. It seems to be forgotten by some writers that
the e in c*^ is a mere nominis umbra — a contraction for the name of a function,
and not 2-71828 . . . Oblivion of this fact has led to some strange pieces of
mathematical logic.
§§ 17, 18 ADDITION THEOREM FOR ExP 5 28&
ExpziEx^Zz . . . ExTpZm = 'Exip{Zi + Z2+. . . + z^) (2),
where Zi, z^, . . ., z^. are any values of « whatever.
In particular, we have, if m be any positive integer,
(Exp 2;)'" = Exp (m;^) (3).
Also
Exp z Exp {-z) = Exp 0,
= 1;
and therefore
Exp(-c) = l/Exp2; (4).
We have, further,
Expl = l + 1 + 1/2! + 1/3! + . . .,
= e (5);
and, if w he any real commensurable number,
'Exp x= I + a; + 0^/21 + 0^/31 + . . .,
= ^ (G),
by chap, xxviii., where e' denotes, of course, the principal value
of any root involved if x be not integral.
It appears, therefore, that Exp iv coincides in meaning with
e*, so far as e^ is yet defined.
We may, therefore, for real values of ^ and for the corre-
sponding values of ^, take the graph oi y = Exp x to be identical
with the graph oi y = e^, already discussed in chap. xxi. Hence
the equation
y = Exp X (7)
defines ;c as a continuous one- valued function of y, for all positive
real values of y greater than 0. We might, in fact, write (7) in
the form
^ = Exp-> (8);
and it is obvious that Exp~^y may, for real values of y greater
than 0, he taken to he identical with hgy as previously defined.
If we consider the purely imaginary arguments + iy and — iy,
we have, by the definition of Exp z,
c. II. 19
290 MODULUS AND AMPLITUDE OF EXP {x + iy) CH. XXIX
Exp( + ^» = l+^;^/-?/V2!-^y/3! + ?/V4! + ^//5!-. . .,
= (l-2/72!+y/4!-. . .)
= cos2/ + «siny (0);
Exp(-«3/) = (l-y/2! + y/4!-. . .)
-^•(2/-//3!+//5!-. . . ),
= cosy -«' sin y (9'),
by § 14.
Finally, by the addition theorem,
Exp {x + yi) = Exp (x) Exp (i/i),
= e^ (cos 2/ + « sin y) (10).
The General Exponential Function is therefore always expressible
by means of the Elementary Transcendental Functions e", cos y,
sin y, already defined.
Inasmuch as the function Exp 2; possesses all the character-
istics which ^ has when z is real, and is identical with ^ in all
cases where e^ is already defined, it is usual to employ the nota-
tion ef for Exp z in all cases. This simply amounts to defining
d" in all cases by means of the equation
e'=l+z + z''/2\+ 2^/31 + . . .,
which, as we now see, will lead to no contradiction.
§ 19.] Graphic Discussion of the General Exponential Function
— Definition of the General Logarithmic Function. Let w be
defined as a function of z by the equation ;
w = Exp2; (1);
and let z = x + yi, and w = u + vi = s (cos (f> + i sin <^). Then, since
Exp {x + yi) = e" (cos y + i sin y), we have
s (cos <i> + i sin <j>) = e^ (cos y + i sin y) (2).
Hence
s = e^, <i> = y (3),
where we take the simplest relation between the amplitudes that
will suit our purpose.
Suppose now that in the 2;-plane (Fig. 7) we draw a straight
line 2'ri'2' parallel to the y-axis, and at a distance x from it.
18, 19 GKAPH OF Exp (x + yi)
291
Y
K
2'
2'
IK
D
B
r
C
■
A
X
0'
c
0
■A
0 K
D
B
i'
Tk
2
Fio. 8.
19-2
292 GRAPH OF Exp (cc + yi) ch. xxix
Then, if we cause z to describe this line, a? will remain constant, and
therefore e* will remain constant ; that is to say, the point w will
describe a circle (K) (Fig. 8) whose radius is e" about the origin
in the w-plane. If we draw parallels to the a;-axis in the ;2;-plane,
at distances O'l' = tt, 0'2' = Stt, . . . , above, and O'l' = tt, 0'2' = Stt,
. . . , below, then, as 1/ varies from - tt to + tt, z travels from 1'
to 1' ; as 2/ varies from + tt to + Stt, z travels from 1' to 2', and
so on ; and each of these pieces of the straight line corresponds
to the circumference of the circle K taken once over. To make
the correspondence clearer, we may, as heretofore, replace the
repeated circle ^ by a spiral supposed ultimately to coincide
with it. Then to the infinite number of pieces, each equal to
27r, on the line K corresponds an infinite number of spires of the
spiral K.
In like manner, to every parallel to the ^/-axis in the 2;-plane
corresponds a spiral circle in the w;-plane concentric with the
circle K. To the axis of 3/ itself corresponds the spiral circle
BAOAB of radius unity ; to the parallel DO"D to the left of
the 2/-axis the spiral circle DO"D ; and so on.
To the whole strip between the infinite parallels DB and
DB corresponds the whole of the «^;-plane taken once over ;
namely, to the right half of the infinite strip corresponds the
part of the ■i<;-plane outside the circle BAOAB; to the left
half of the strip the part of the w-plane inside the circle
BAOAB.
To each such parallel strip of the «-plane corresponds the
whole of the w-plane taken once over.
Hence the values of w are repeated infinitely often, and we
see that the equation (1) defines w as a continuous periodic
fu/nction of z having the period 27r«.
Conversely, the above graphic discussion shows that the equation
(1) defines z as a continuous <X)-ple valued function of w.
Taking the latter view, we might write the equation in the
form
z = Exp^' w (!')•
§19 Log w = LOG\w\ + i A^ip{w) 293
Instead of Exp~^ w we shall, for the most part, employ the
more usual notation Logw, using, however, for the present at
least, a capital letter to distinguish from the one-valued function
logy, which arises from the inversion oi y = e", when x and y are
both restricted to be real.
In accordance with the view we are now taking, we may
write (3) in the form
w = \ogs, y = <t>.
Hence z = Logw
gives w+yi = Log {s (cos ^ + ^ sin ^)},
where x = log s, and y = <i>.
In other words, we have
Log w = \og\w\-\-i amp {w) (2') ;
and, if we cause ^ (that is, amp {w)) to vary continuously through
all values between - co and + oo , then the left-hand side of the
equation (2') will vary continuously through all values which
Log'?*' can assume for a given value of \w\.
If we confine ^ to lie between -ir and +7r, then Logw
becomes one-valued ; and we have
Log 'W = \ogs + i4> (4),
where s = | w; | = ^(ic^ + v^), and cos <^ = u/J(u^ + v"^), sin ^ = v/J{u^ + v^),
— 7r;:|><^;^+ tt.
This is called the principal branch of Log w ; and we may
denote it by z.
It is obvious from the graphic discussion that, if Zt or Jjogw
denote the value of Log w in its t-th branch, z being the value in
the principal branch corresponding to the same value of w {that
is, a value of w whose amplitude differs by an integral multiple
oflir), then
tLog W^Zt^Z^ 2tTri,
= log s + i (ff) + 2tTr) (5),
where (f> is the amplitude {confined between the limits — ir and + tt)
ofw, and t is any integer positive or negative.
If w be a real positive quantity, =u say, then s = \w\ = u,
ffi ■- amp w = 0 ; and we have, for the principal value of Log u,
Log u = locr u.
294 ' DEFINITION OF ExP ^^ CH. XXIX
Hence^ for real positive values of the argument, log u is tJie
piincipal value of Log u. The other values are of course given
hy tLog u-logu + 2tiri, t being the order of the branch.
We have also the following particular principal values : — •
Log( + «) = ^W,
Log(-*) = -i7r*,
Log(— l) = ±7ri:
the principal value in the last case is not determinate until we
know the amplitude ; and the same applies to all purely real
negative arguments.
§ 20.] Definition of Exp aZ. The meaning of a^, or, as it is
sometimes written, Exp „«, has not as yet been defined for values
of z which are not real and commensurable.
We now define it to mean Exp (;2 . jLog a), where tLog a is
the ^-th branch of the inverse function Log a, and t may have
any positive or negative integral value including 0.
Thus defined, a^ is in general multiple-valued to an infinite
extent. In fact, since ^Log a - log 5 + z (^ + 2^7r), where s = |a|,
and <^ = amp a ( - tt < <^ < + tt), we have, \i z-x + yi,
a*+s^ = Exp [{x + yi) {log s + « (<^ + 2tir))\
= Exp Wx log s - (<^ + 2tTr) y] + i {y log s + (^ + 2tir) x]],
= exp {x log s - (^ + 2^7r) y] . [cos {y log s + («^ + 2tir) x)
+ i sin [y log s + (^ + ^tir) x}] (1).
If we put ^ = 0, that is, take the principal branch of Log a,
in the defining equation, then we get what may be called the
principal branch of a'"+'^, namely,
a'+yi = 'EiX^{zhoga),
= exp{a;logs-^y}.[cos{ylogs+^;r}+«sin{2f'log5+^a;}] (2).
The value given in (1) would then be called the ^-th branch,
and might for distinction be denoted by t«'"+'^ or by jExp a{^ + yi)-
It is important to notice that the above definition of a" agrees
with that already given for real commensurable values ofz provided
we take the corresponding branches. In fact, when y = 0, (1) gives
a"" ^ exp {x log s) . [cos (^ + lit-rr) X + i sin {<f> + 2tir) x'\ j
§§ 19-21 ADDITION THEOREM FOR LoG Z 295
that is, if X -piq,
[s (cos ^ + ^ sin <^)]P/'^
= s^i^ [cos . (<^ + 2tir) pIq + i sin . (<^ + 2tir) p/q] (3) ;
the right-hand side of which is the ^-th branch of the left as
ordinarily defined.
Cor. It follows from the above tluit when x is an incommen-
surable number the function a" has an infinite number of values
even when both a and x are real.
The principal value of a% however, when both a and x are
real and a is positive, is exp {x log a), which differs infinitely
little from the principal value of a^', if x be a commensurable
quantity differing infinitely little from x.
§ 21.] The Addition Theorem for Logz.
By the result of § 19 we have
^Log w, + nLog Wz
= log I Wi I + log 1 2^2 1 + 2 amp Wi + i amp w-i + 2{m + n) iri.
Now (chap. XII., § 15) | Wi 1 1 Wg 1 = I m;i «^2 1 . and, if amp (wi w^
were not restricted in any way, we should have amp Wi + amp w.^
= amp {wi w^. Since, however, amp {wi w^ is restricted in the
definition of Log {wi w.^ to lie between - tt and tt, we have
amp Wx + amp w^ = amp (wi w^ + 2p7r,
where p = + 1, 0, or - 1 according as amp Wx + amp w<2.>^tt, lies
between + ir and -tt, or <-tt. Hence we have
^Log 10^ + „Log ^2 = m+n+pLog {w^ W^) (l),
where p is as defined.
In like manner, it may be shown that
mLog U\ - „Log ^2 = m-n+pLog ( Wj/Wa) (2 ),
where p = + l, 0, or -1 according as amp Wi -amp ttJ2>+ tt,
between + rr and — tt, or < — tt.
Taking the definition of a^+^* given in § 20, and making use
of equation (1) of that paragraph, we have
^96 EXPANSION OF jLoG (1 + 2^) CH. XXIX
iLog ^'^^ = log I ta="+^* I + (amp <a^+^* + 2kiT) i,
= a? log S - (<^ + 2tTr) 1/+{l/\ogS+ (<f) + 2tTr) x} i + 2 {k + I) iri,
where / is an integer, positive or negative, chosen so that
-7r<7/logS+(^ + 2t7r)a; + 2liT<-\-Tr.
Hence
( fcLog tff^'"' =^{x + yi) {log s + (<^ + 2tTr) i] + 2{k + l) ni,
= {a; + yi) ^Log a ■¥ 2 {k + 1) -n-i (3).
The equations (1), (2), (3) are generalisations of formulae for
log X with which the reader is already familiar.
If we confine each of the multiple-valued functions ^Log and
jExpa to its principal branch, we have
Log w'+y' ={x + yi) Log a + 2hi (3'),
where I is so chosen that
-7r<?/logS + 4"^ + 2lir<+Tr.
§ 22.] Expansion of tLog (l + z) in powers of z.
Consider first the principal branch of the function Log (1 + z).
By the definition and discussion of § 20, we see that, when x is
any real quantity, the principal branch of (1 + z)^ has for its
value Exp {a? Log (1 + 2;)}. Hence we have
• (l + ;2f-l + {^Log(l + c)} + {a;Log(l + ;^)}"/2! + . . .;
and, since the series 1 + ^JJ^z"^ represents the principal branch
of (1 + zY, we have
l + 5^(7„2;'» = l+{.»Log(l + c;)} + . . . .
Now all the conditions involved in the reasoning of chap.
XXVIII., § 9, will be fulfilled here, provided the complex variable
z be so restricted that | /S' | < 1.
Hence, if |2;|<1, we must have, as before,
Log(l + ;^) = ;Z-;2V2 + ;2^/3-«V4 + . . . (1).
In other words, so long as\z\<l, the series z - z^l2 + «^/3 - . . .
represents the principal branch of Exp~^ (1 + ^)'
Cor. Since ^Log (1 + 2;) = Log (1 + 2;) + 2tTn, we have
t\jog{l-^z)^2tni + z-z'l2+z'l^-z*l'^ + . . . (2),
§§ 21-23 GENERALISED CIRCULAR FUNCTIONS 297
which gives us an expansion for the t-th. branch of Exp~^ (1 + z)
within the region of the 2:-plane for which | ;2 1 < 1.
It follows readily, from the principles of chap, xxvl, § 9, that
when I ;s 1 = 1 the series z - z^/2 + «^/3 - ... is convergent, pro-
vided amp z=¥±7r (other odd multiples of tt are not in question
here). Hence, by the theorem of Abel so often quoted already,
the expansion-formultB (1) and (2) will still hold when |«| = 1,
provided amp^;=#±ir.
GENERALISATION OF THE CIRCULAR FUNCTIONS — INTRO-
DUCTION OF THE HYPERBOLIC FUNCTIONS.
§ 23.] General definition of Cosz, Sinz, Tanz, Cotz, Secz,
Cosecz. Since the series l-z'^l2l + 2!*/Al -. . ., z- z^/3\+z^/5l
— . . .are convergent for all values of z having a finite modulus,
however large, they are each single-valued continuous functions
of z throughout the 2;-plane. Let us call the functions thus
defined Cos^; and Sin 2;, using capital initial letters, for the pre-
sent, to distinguish from the geometrically defined real functions
cos a? and sin x. We thus have
Cosz=l-z^l2\+zyA\-. . . (1),
^mz = z-2^/3\+ 1^/51-. . . (2).
We also define Tan z, Cot z, Sec z, Cosec z by the following
equations : —
Tan 2; = Sin 2;/Cos « ; Cotz=Cosz/^mz;]
Sec z = 1/Cos z ; Cosec z = 1/Sin z. J w- .
In the first place, we observe that when z is real, =a^ say,
we have, by § 14,
Cos x=l— x^l2 ! + a^l^\ — . . . = cos i??,
^ma; = x — af/Sl + af/5\ — . . .=^sin^;
so that, when the argument is real, the more general functions
Cos., Sin., Tan., Cot., Sec, Cosec. coincide with the functions
COS., sin., tan., cot., sec, cosec already geometrically defined
for real values of the argument.
298 euler's foemul^ ch. xxix
Since
z-z^/Sl+zy5\-. . .=l{Exp(^•^)-Exp(-^•^)},
it follows from (1) and (2) that we have for all values of z
(4)*
Cos z = - {Exp {iz) + Exp ( - iz)\
Sin ^ - 2^ {Exp (^2;) - Exp ( - e^)} ;
with corresponding expressions for Tan;?;, Cot 2;, Sec^:, and
Cosec z.
By (4) we have
Cos^2; + Sin^2;
= \ [{Exp {iz)f + {Exp ( - %z)f + 2 Exp {iz) Exp ( - iz)
- {Exp {iz)f - {Exp ( - %z)Y + 2 Exp {iz) Exp ( - iz)\
Hence, bearing in mind that we have, by the exponential
addition theorem,
Exp {iz) Exp ( - iz) = Exp {iz - iz) = Exp 0 = 1,
we see that
Cos^2 + Sin2;^=l (5),
from which we deduce at once, for the generalised functions, all
the algebraical relations which were formerly established for the
circular functions properly so called.
We also see, from (4), that Cos {-z)- Cos z and Sin {- z)
= - Sin z ; that is to say, Cos z is an even, and Sin z an odd
function of z.
Since, by (4), we have
Cos z-^i Sin z = Exp {iz),
Cos z-i Sin z = Exp ( — iz),
* These formulsB were first given by Euler. See Int. in Anal. Inf., t. i.,
§ 138. He gave, however, no sufficient justification for their usage, resting
merely on a bold analogy, as Bernoulli and Demoivre had done before him.
§ 23 PROPERTIES OF Cos z, &c. 299
it follows from the exponential addition theorem, namely,
Exp {izx + iz^ - Exp {iz^ Exp {iz.^,
that
Cos {zi + z^ + i Sin {Z]_ + z^ - (Cos Zx + % Sin 0i) (Cos z^ + * Sin z^
= (Cos ;j;i Cos «2 - Sin ^fj Sin z^ + « (Sin 2;i Cos z^ + Cos 2^1 Sin %)*.
Hence, changing the signs of ^^j and Za, and remembering that
Cos. is even and Sin. odd, we have
Cos {z-i, + z^ - i Sin {zx + z^ = (Cos Zx Cos ;2;2 - Sin Zx Sin ^ig)
- 1 (Sin 2;i Cos z<2, + Cos 2;j Sin z^.
Therefore, by addition and subtraction, we deduce
Cos {zx + z^ - Cos Zx Cos Z2 - Sin 2;i Sin z^ O . x
Sin (2^1 + ^jg) = Sin 2:1 Cos ^^2 + Cos 2^1 Sin 2^2 J
In other words, the addition theorem for Cos. and Sin. in
general is identical with that for cos. and sin.
By (6) we have
Cos (z + 2mr) = Cos z Cos 2mr - Sin z Sin 2mr,
that is, if n be any positive or negative integer, so that
Cos 2;i7r = cos 2?nr = 1, and Sin 2n-ir = sin 2mr = 0, then
Cos (z + 2«7r) = Cos z.
In like manner, Sin (z + 2mr) = Sin z ; Tan {z + mr) = Tan z ; &c.
That is to say, the Generalised Circular Functions have the same
real periods as the Circular Functions proper.
Just in the same way, we can establish all the relations for
half and quarter periods given in equations (3) of § 2. Thus, for
example,
Cos {it + z) = Cos TT Cos z - Sin tt Sin z,
= cos TT Cos z - sin tt Sin z,
= - Cos z.
Also all the equations (5), (6), (7) o/" § 2 will hold for the
generalised functions ; for they are merely deductions from the
addition theorem.
* We cannot here equate the coeflScient of i, Ac, on both sides, because
8'm{zx + z.^), &c., are no longer necessarily real.
300 DEFINITION OF HYPERBOLIC FUNCTIONS CH. XXIX
§ 24.] We proceed next to discuss briefly the variation of
the generalised circular functions.
Consider first the case where the argument is wholly-
imaginary, say z = iy. In this case we have
Cos iiy) = - {Exp {iiy) + Exp ( - iiy)],
-\{e-y + ^^) (1);
Bn{iy)='^.{e-y-^J),
-l(ey-e-y) (2).
"We are thus naturally led to introduce and discuss two new
functions, namely, ^ {e" + e'^) and ^ (e" - e'^), which are called
the Hyperbolic Cosine and the Hyperbolic Sine. These functions
are usually denoted by cosh y and sinh y ; so that, for real values
of y, coshy and sinh^ are defined by the equations
coshy = ^{ey + e-y), sinh y = ^ (e^ - e-«) (3).
In general, when y is complex, we define the more general
functions Cosh 2; and Sinh^ by the equations
Cosh z=^ {Exp (z) + Exp ( - z)},
Sinh2; = ^{Exp(;^)-Exp(-;^)}, (3').
We also introduce tanh y, coth y, sech y, and cosech y by the
definitions
tanh y = sinh y/cos\\ y, coth y = cosh y/sinh y ;
sech y = 1/cosh y, cosech y = 1/sinh y ;
and the more general functions Tanh 5;, Coth 2;, &c., in precisely
the same way.
From the equations (1) and (2) we have
Cos (iy) = cosh y, Sin (iy) = i sinh y ; "j
T}a,n(iy) = itsinhy, Cot (iy) = — i coth. y; > (4),
Sec (iy) = sech y, Cosec (iy) = — i cosech y ; J
and, of course, in general, Cos iz ^ Cosh z, &c.
t Y
V
s
1/
^^^
y^ C^^vV^
0 X
/ \r
/ T'\l
/s U
-
Fia. 9.
302 GRAPHS OF HYPERBOLIC FUNCTIONS CH. XXIX
The discussion of the variation of the circular functions for
purely imaginary arguments reduces, therefore, to the discussion
of the hyperbolic functions for purely real arguments.
§ 25.] Variation of the Hyperbolic Fimctionsfor real argu-
ments. The graphs of y^cosha;, 3/ = sinh^, &c., are given in
Fig. 9 as follows : —
cosher, CC\ sinha?, 80B\
coth^, TT'T'T\ tanha;, TTOTT\
secha?, CO' \ cosechiP, S'S'S'S'.
By studying these curves the reader will at once see the truth
of the following remarks regarding the direct and inverse hyper-
bolic functions of a real argument.
(1) coshj» is an even function of w, having two positive
infinite values corresponding to a; = ± qo , no zero value, and a
minimum value 1 corresponding to x = 0.
cosh~^2/ is a two-valued function of y, defined for the con-
tinuum 11f>i/^(x>, having a zero value corresponding to y=l,
and infinite values corresponding to y=<x) , but no turning value.
(2) sinh X is an odd function of x, having a zero value when
^ = 0, and positive and negative infinite values when a? = + oo and
x = - CO respectively.
sinh~^y is one-valued, and defined for all values of y ; it has
a zero value for y = 0, and positive and negative infinite values
when y = + cc and y = — cc respectively.
(3) tanhiP is an odd function, has a zero value for x = Q,
positive maximum + 1, and negative minimum - 1, corresponding
toa7=+oo anda; = — 00 respectively.
tanh"^y ^s a one- valued odd function, defined for -i:^'?/:|>+ 1 ;
has zero value for y = Q, positive and negative infinite values
corresponding to 3/ = + 1 and y = - 1.
(4) coth X is an odd function, having no zero value, but an
infinite value for x = 0, and minimum + 1, and maximum — 1, for
x — +<x> and a? = — 00 respectively.
cothr^y is a one-valued odd function, defined, except for the
continuum -i:^3^:j>+ 1, having positive and negative infinite
values corresponding to y=+l and y = -l respectively, and
a zero value for y= co.
§§ 24-27 INVERSE HYPERBOLIC FUNCTIONS 303
(5) sech X is an even function, having a maximum + 1 for
ir = 0, and a zero value for x---^±^,
sech"^^ is a two-valued function, defined for ^1s^y1s^\^ having
a zero value for y = 1, and infinite values for y = 0.
(6) cosech X is an odd function, having zero values for
a? = ± Qo , and an infinite value for x = 0.
cosech~^2/ is one-valued and defined for all values of y, having
zero values for y = + c» , and infinite values for 3/ = 0.
§ 26.] Logarithmic expressions for cosh~^y, sinh~'^i/, dx.
li x = cosh~^y, we have
1/ = coah. X = ^ (e" + e"'') (1).
Therefore
±x/(2/^-l) = |-(^^-0 (2).
From (1) and (2),
a! = log{y±sJ(f-l)};
that is, cosh-^2/ = log {y ± J(y^ - 1 )} (3),
the upper sign corresponding to the positive or principal branch
of cosh~^3^, the lower sign to the negative branch.
In like manner we can show that
sinh-^?/ = log {y + J(f + 1)} (4) ;
tanh-^2/ = ilog{(l+2/)/(l-2/)} - (5);
coth-^2' = ^log{(2/+l)/(2/-l)} (6);
8ech-^2/ = log[{l±V(l-3/^)}/y] (7);
cosech-^?/ - log [{1 + J{1 + f)]ly\ (8).
§ 27.] Properties of the General Hyperbolic Functions ana-
logous to thme of the Circular Functions.
We have already seen that the properties of the circular
functions, both for real and for complex values of the argument,
might be deduced from the equations of Euler, namely.
Cos z = - {Exp ( + iz) + Exp ( - iz)] ;
Sin ^; = -1 {Exp { + iz)~ Exp ( - iz)}
(A).
In like manner, the properties of the general hyperbolic
functions spring from the defining equations
304 PROPERTIES OF HYPERBOLIC FUNCTIONS CH. XXIX
Cosh z = ^ {Exp ( + z) + Exp (-z)};
Sinh^; = | {Exp { + z)- Exp ( - z)
} (B).
We should therefore expect a close analogy between the
functional relations in the two cases. In what follows we state
those properties of the hyperbolic functions which are analogous
to the properties of the circular functions tabulated in § 2. The
demonstrations are for the most part omitted ; they all depend
on the use of the equations (B), combined with the properties of
the general exponential function, already fully discussed.
The demonstrations might also be made to depend on the
relations connecting the general circular functions with the
general hyperbolic functions given in § 24*, namely,
Cosh z = Cos iz, i Sinh z = Sin iz ; '
+ i Tanh z = Tan iz, - i Coth z = Cot iz ; - (C).
Sech z = Sec iz, - i Cosech z = Cosec iz ; .
Algebraic Relations.
Go&\i^z-^mVz=l, Sech^;2 + Tanh='5;=l (1),
&c.
Periodicity. — All the hyperbolic functions have the period
27ri ; and Tanh z and Coth z have the smaller period -jri.
Thus
Cosh {z + ^rnri) = Cosh z\ &c. l . .
Tanh(;2 + w7r/) = Tanh2;; &c.J ^^^•
Also,
Cosh {iri ±z) = - Cosh z, Sinh {iri ±z) = + Sinh z ; \
Cosh (IW + 2;) = ± « Sinh «, ^m\i {^tri ± z) = i Go^Ax z ; [ (3).
Tanh {^iri ±z) = ± Coth z, Coth (|W ±z) = ± Tanh z ; )
Addition Formulce.
Cosh {Zi ± Zi) = Cosh Zi Cosh z^ ± Sinh Zi Sinh Z2 ; 1
Sinh (Zi ± Z2) = Sinh Zi Cosh Z2 ± Cosh Zi Sinh z^ ; \ (5).
Tanh (zi ± z.,) = (Tanh z, ± Tanh z.^l{l ± Tanh 0, Tanli z^). 1
* This connection furnishes the simplest metnoria technica for the hyper-
bolic formul89.
§§ 27, 28 GENERAL HYPERBOLIC FORMULA 305
Cosh Zi + Cosh Z2 = 2 Cosh J (zi + z^) Cosh J (zi — 2^2) 1)
Cosh Zi - Cosh Z2 = 2 Sinh | (^^j + Z2) Sinh | (^i — ^^2) ; r (6).
Sinh «i ± Sinh Z2 = 2 Sinh |- (^Jj ± Z2) Cosh J (^i + z.2). )
Cosh 2i Cosh Z2 = ^ Cosh (2^^ + SJg) + I Cosh (zi — ^^2) f
Sinh Zi Sinh 2^2 = y Cosh (2^1 + Zz)-^ Cosh (2^1 — 2^2) ; ■ (7).
Sinh Zi Cosh 2:3 = ^ Sinh {zi + 2:2) + ^ Sinh (zi — z^). ,
Cosh 2z = Cosh^ z + Sinh' z = 2 Cosh' 2; - 1,
= 1 + 2 Sinh' z=(l + Tanh' 2)/(l - Tanh' 2).
Siuh 22 = 2 Sinh 2; Cosh z = 2 Tanh 2;/(l - Tanh' z).
Tanh 22; = 2 Tanh zl{l + Tanh' 2;).
h (8).
Inverse Functions. — Regarding the inverse functions Cosh~\
Sinh~\ &c., it is sufficient to remark that we can always express
them by means of the functions Cos~\ Sin~\ &c. Thus, for
example, if we have Cosh"-^2; = w, say, then
z = Cosh w = Cos iw.
Hence *w = Cos"^2;;
that is, W-- i Cos~^2;.
So that Cosh-^2; = -i Cos~^2; ;
and so on.
In the practical use of such formulae, however, we must
attend to the multiple-valuedness of Cosh~^ and Cos"-'. If, for
example, in the above equation, the two branches are taken at
random in the two inverse functions, then the equation will take
the form
Cosh~^2; = 2nnri ± i Cos~^2;,
where m is some positive or negative integer, whose value and
the choice of sign in the ambiguity ± both depend on circum-
stances.
§ 28.] Formulce for the Hyperbolic Functions analogous to
Demoivre's Theorem and its consequences.
We have at once, from the definition of Cosh z and Sinh 2;,
c. II. 20
306 ANALOGUE TO DEMOIVRE's THEOREM CH. XXIX
Cosh (Z1 + Z2 + . . . + ^^n) ± Sinh (Z1 + Z2 + . . . + Zn)
= Exp ±{Zi + Z2+ . . . + Zn),
= Exp + Zi Exp ±Z2 . . . Exp + Zn,
= (Cosh Zi ± Sinh Zi) (Cosh z^ ± Sinh Z2)
. . . (Cosh Zn ± Sinh 2;„) (A) ;
and, in particular, if n be any positive integer,
Cosh nz ± Sinh nz = (Cosh z ± Sinh z^ (B).
These correspond to the Demoivre-formulse, with which the
reader is already familiar*.
We can deduce from (A) and (B) a series of formulae for the
hyperbolic functions analogous to those established in § 12 for
the circular functions.
Thus, in particular, we have
G0H}\{Zi + Z2+ . . . +Zn)=Pn + Pn-2 + P7i-i + - ' • (l')>
Vfhere Pr = ^Gos]iZiGoshz2 . . . Cosh 2?^ Sinh 2r+i . . . Sinh2;„.
Tanh {Z1 + Z2 + . . . + Zn)
=^{T,+ Ts+T,+ . . .)/{l + T2+T, + . . .) (3'),
where TV = 2 Tanh ^^i Tanh ^^g . . . Tanh 2;,..
Cosh nz = Cosh»;s + ^O. Cosh''-^^; Sinh^z
+ „C4Cosh"-'';sSinh*;2; + . . . (4').
Sinh nz = nCx Gq^V^'-^z Sinh z + nCz Cosh»-«2; Sinh« z
+ „C;Cosh»-«;2;Sinh'';2; + . . . (5').
Cosh nz = {- fi' |l - 'I cosh'^ ;g + ""' ^'"'~ ^'^ cosh^ z - . . .
(_)._J (2^)! ^'cosh"z + . . .j (9),
{n even) ;
* As a matter of history, Demoivre first found (B) in the form
J/ = Hl/v'W(l + ''^)-«}-'C^W(l+^'^)-'^}]' where y is the ordinate of P in
Fig. 10 below, and v the ordinate of Q, Q corresponding to a vector OQ such
that the area AOQ is n times AOP, and OA is taken to be 1. He then
dechiced the corresponding formula for the circle by an imaginary trans-
formation. (See Miscellanea Analytica, Lib. II,, cap. i.)
^ 28, 29 HYPERBOLIC INEQUALITIES AND LIMITS 307
^-^ (2^Tl)! cosh-+^^+. . .| (11),
(n even) ;
and so on.
We may also deduce formulae analogous to those of § 13,
such as
(-)"'2m+iCmSinh«}.
§ 29.] Fundamental Inequality and Limit Theorems for the
Hyperbolic Ftmctions of a real a/rgument.
Ifu he any positive real quantity, then
tanhM<w<sinhw<cosh?« (1).
By the definitions of § 24 we have
sinh II = \ {exp {u) - exp ( - ii)\ ;
= m + wV3! + mV5! + . . • (2);
coshw=l + MV2! + wV4! + . . . (3);
whence it appears at once that sinhM>?^.
Again, cosh ?« = +>/(! + sinh^ «), so that cosh m> sinh u.
Finally, since
tanh u = sinh w/cosh u
= w(l+t*V3! + wV5! + . . .)/(1+m72! + mV4!- • •)>
and mV3!<mV2!, u''lb\<u'l^\, &c.,
we see that tanhM<«<.
Cor. When u = 0, Lm\\\ulu = \, and LtSin\iu/u=l. This
may either be deduced from (1) or established directly by means
of the series (2) and (3).
If a be a quantity which is either finite and independent of n
or else has a finite limit when w = oo , then^ when w = oo ,
i(c„sh^)^l. X(siuh^/^y=, X(unh^/^)%1.
20—2
308 GEOMETRICAL ANALOGIES CH. XXIX
We have
Hence, if we put 1 + e"-"'" ^2-2;:;, so that z-d coiTesponds
to vi = CO , then we have
L ('cosh-y = e» L {(i-;^)-v«}»2«/iog(i-2*0.
Now, X (1 - 2;)-^/* = e, and i/22;/log (1 -2z) = - 1. Hence, by
chap. XXV., § 13,
ifcosh-j =e»e~"
= 1.
We leave the demonstration of the second limit as an exer-
cise for the reader. The third is obviously deducible from the
other two.
A very simple proof of these theorems may also be obtained
by using the convergent series for cosh . a/w and sinh . a/w.
§ 30.] Geometrical Analogies between the Circular and Hyper-
bolic Functions.
If 6 be continuously varied from — tt to + tt, and we connect
a and y with 6 by the equations
a? = a cos ^, y = asm9 (1),
then we have
ar' + y^ = a''{cos'6 + siu'e) = a'' (2).
Hence, if (a?, y) be the co-ordinates of a point P, as 6 varies con-
tinuously from — TT to + TT, P will describe continuously the
circle A'AA" (of radius a) in the direction indicated by the
arrow-heads (Fig. 10).
Let P be the point corresponding to 6 ; and let 0 denote the
area AOP, to be taken with the sign + or — according as ^ is
positive or negative. Then 0 is obviously a function of 0. We
can determine the form of this function as follows : —
Divide 8 into n equal parts, and let P^, P^, . . ., Pr, . ■ . P
be the points corresponding to 6/w, 29 jn, . . . , rOjn, . . . ndjn
respectively. Then we have, by the lemmas of Newton,
Area^OP= L T'p^OP^+i.
m=oo r—Q
29, 30
AREA OF CIRCULAR SECTOR
fY
S09
— X
Fm. 10.
Now
= ^a^ {cos . r^/w sin . (r + l)e/n - sin . rO/n cos . (r + 1)^/;?},
= |a^sin. ^/w.
Hence
® = ^a'Ln sin. 6/n,
= la' OL (sin. d/7i)/(e/n),
Hence, if 6=^2@/a\ we have cos 6 = a/a, sinO^yJa, ts^nO ^yjx,
cot ^ = xjy, &c.
let
Then
Next, let u be continuously varied from - oo to + oo ; and
oe = a cosh u, y=^a sinh ?/, ( i ').
.r^ -y^^a^ (cosh^ w - sinh' z«) == a^ (2').
310
AREA OF HYPERBOLIC RECTOR
CH. XXIX
Hence, if {x, y) be the co-ordinates of P, as w* varies con-
tinuously from - CO to + CO , P will describe continuously the
right-hand branch AAA" of the rectangular hyperbola, whose
Fig. 11.
semi-axis-major is OA = a, in the direction indicated by the
arrow-heads in Fig. 11.
If P be the point corresponding to u, P^ Pr+i the points
corresponding to ru/n and (r+l)w/w, and U the area AOP
agreeing in sign with u, then, exactly as before,
* Adopting an astronomical term, we may call u the hyperbolic excentric
anomaly of P. The quantity u plays in the theory of the hyperbola, in
general, the same part as the excentric angle in the theoiy of the ellipse.
§§ 30, 31 GUDERMANNIAN 311
r=n—l
71=00 r=0
and
= a^ {cosh . rti/n sinh . (r + 1) u/n - sinh , ru/n cosh . (r + 1) m/w},
= a^ sinh . w/??.
Therefore 17= \a?Ln sinh . w/w,
= ^a^uL (sinh . u/n) /(u/n),
= ^a\ by § 29, (3').
Hence, if the area A OP = U, and u-2 U/a^, then, a; and
y being the co-ordinates of P, we might give the following
geometric definitions of cosh u, sinh u, &c, : —
cosh w = x/a , sinh u = i//a,
tanh « = 2//^> coth w = a?/y, &c.
It will now be apparent that the hyperbolic functions are
connected in the same way with one half of a rectangular
hyperbola, as the circular functions are with the circle. It is
from this relation that they get their name.
We know, from elementary geometrical considerations, that the area 9 is
the product of ^a^ into the number of radians in the angle AOP. It there-
fore follows from (3) that the variable 6 introduced above is simply the
number of radians in the angle A OP. Our demonstration did not, however,
rest upon this fact, but merely on the functional equation cos^S + sin^^^l.
This is an interesting point, because it shows us that we might have intro-
duced the functions cosd and Bin 6 by the definitions cos 0 = ^ {Exp (i^)
4-Exp(-ie)}, Bin^ = — . {Exp(2^)-Exp(-i^)}; and then, by means of the
above reasoning, have deduced the property which is made the basis for their
geometrical definition. When this point of view is taken, the theory of the
circular and hyperbolic functions attains great analytical symmetry ; for it
becomes merely a branch of the general theory of the exponential function as
defined in § 18.
When we attempt to get for u a connection with the arc AF, like that
which subsists in the case of the circle, the parallel ceases to run on the same
elementary line. To understand its nature in this respect we must resort to
the theory of Elliptic Integrals.
§ 31.] Expression of Real Hyperbolic Functions in terms of
Real Circular Functions,
312 GUDERMANNIAN CH. XXIX
Since the range of the variation of coshw when u varies from
- 00 to + 00 is the same as the range of sec 0 when 6 varies
from - |-7r to + -Itt, it follows that, if we restrict 0 and u to have
the same sign, there is always one and only one value of u
between - oo and + oo and of 6 between - ^tt and + ^rr such that
cosh M = sec ^ (1).
If we determine 6 in this way, we have
sinh u = ± /^(cosh^ ««-!)>
= ±1/(660^6-1);
hence, bearing in mind the understanding as to sign, we have
sinh u = tan 6 (2).
From these we deduce
e" = cosh u + sinh u,
= sec ^ + tan 6 ;
u - log (sec 6 + tan 6\
= logtan(i7r+i^) (3).
Also, as may be easily verified,
tanh \u = tan ^9 (4).
When 6 is connected with u by any of the four equivalent
equations just given, it is called the Gudermannian* of u, and we
write 6 = gd w.
* This name was invented by Cayley in honour of the German mathe-
matician Gudermann (1798-1852), to whom the introduction of the hyperbolic
functions into modern analytical practice is largely due. The origin of the
functions goes back to Mercator's discovery of the logarithmic quadrature of
the hyperbola, and Demoivre's deduction therefrom (see p. 306). According
to Houel, F. C. Mayer, a contemporary of Demoivre's, was the first to give
Bhape to the analogy between the hyperbolic and the circular functions. The
notation cosh. sinh. seems to be a contraction of coshyp. and sinhyp., pro-
posed by Lambert, who worked out the hyperbolic trigonometry in consider-
able detail, and gave a short numerical table. Many of the hyperbolic
formulae were independently deduced by William Wallace (Professor of
Mathematics in Edinburgh from 1819 to 1838) from the geometrical pro-
perties of the rectangular hyperbola, in a little-knowu memoir entitled New
Series for the Quadrature of Conic Sections and the Computation of Logarithms
(TruTis. R.S.E., vol. vi., 1812). For further historical information, see
Giinther, Die Lehre von den gew'dhnlichen vnd verallgevieinerten Hyperbel-
funktionen (Halle, 1881) ; also, Beitriige zur Geschichte derNeueren Mathematik
{Programmschrift, Ansbach, 1881).
§ 31 EXERCISES XVII 313
It is easy to give a geometrical form to the relation between 6 and u. If,
in Fig. 11, a circle be described about 0 with a as radius, and from M a
tangent be drawn to touch this circle in Q (above or below OX according as u
is positive or negative), then, since MQ'^=OM^- OQ:'^ = x'^-a^=y^, we have
ocoshti=a;=asec(30ilf. Therefore QOM=0, and we have y = 3IQ = a tan 6.
From this relation many interesting geometrical results arise which it would
be out of place to pursue here. We may refer the reader who desires further
information regarding this and other parts of the theory of the hyperbolic
functions to the following authorities : — Greenhill, Differential and Integral
Calculus (Macmillan, 1886), and also an important tract entitled A Chapter
in the Integral Calculus (Hodgson, London, 1888); Laisant, "Essai sur les
Fonctions hyperboliques," MSm. de la Soc. Phys. et Nat. de Bordeaux, 1875 ;
Heis, Die Hyperbolischen Functionen (Halle, 1875). Tables of the functions
have been calculated by Gudermann, Theorie der Potential- oder Cyclisch-
hyperbolischen Functionen (Berlin, 1833); and by Gronau (Dantzig, 1863).
See also Cayley, Quarterly Journal of Mathematics, vol. xx. ; and Glaisher,
Art. Tables, Encyclopedia Britannica, 9th Ed.
Exercises XVII.
(1.) Write down the values of the six hyperbolic functions corresponding
to the arguments ^iri, iri, 2iri.
Draw the graphs of the following, x and y being real: —
(2.) y=s,mh.xlx. (3.) y = xc,oihx.
(4.) y=gdix. (5.) t/ = sinh-Ml/(«-l)}.
(6.) Express Sinh""^«, Tanh-^2, Sech~^2, Cosech~iz, by means of Sin"^^,
Cos~^^, &c.
(7.) Show that cosh*«-8inh^M = l + 3sinh''Mcosh2M.
(8.) Show that
4 cosh'M - 3 cosh u — cosh 3w = 0 ;
4 sinh^M + 3 sinh u - sinh 3m = 0.
(9.) Show that any cubic equation which has only one real root can be
numerically solved by means of the equations of last exercise. In particular,
show that the roots of x'^-qx-r=0 are ^{ql3) coshu, 2J{qlS){Gos^Tr
cosh «±i sin Itt sinh tt), u being determined by cosh 3h = 3r/,y3/2,^g^
(10. ) Solve by the method of last exercise the equation a;^ + 6a; + 7 = 0.
Express
(11.) tanh~^ar + tanh-iy in the form tanh-^z.
(12.) cosh~i a; + cosh-i ^ in the form cosh^^^.
(13.) sinh'^x- sinh-iy in the form cosh~i2.
Expand in a series of hyperbolic sines or cosines of multiples of m: —
(14.) Coshi««. (16.) sinh^M. (16.) cosh^u sinhSy.
314 EXERCISES XVII CH. XXIX
Expand in a series of powers of hyperbolic sines or cosines of u : —
(17.) Cosh 10m. (18.) sinhTw.
(19.) cosh 6m sinh 3w. (20.) sinh nm cosh tim.
Establish the following identities : —
(21. ) tanh l{u + v)- tanh ^ (m - v) = 2 sinh u/(cosh u + cosh v),
,„„ , sinh (m - 1;) + sinh M + sinh (m + v) , ,
(22.) — ~ {— V- — ; ^ ' = tanhM.
^ ' cosh (m - v) + cosh 7i + cosh (m + v)
(23.) tanh u + tanh (^iri + u) + tanh (firt + m) = 3 tanh Su,
cosh 2m + cosh 2v + cosh 2m? + cosh 2 (m + ?; + «;) = 411 cosh {v + w).
(24.) Tan |(m + iv) = (sin u + i sinh v)/(cos u + cosh v).
(25. ) Express Cosh^ (u + it') + Sinh* {u + iv) in terms of functions of u and v.
EHminate u and v from the following equations: —
(26.) a; = acosh (m + X), y = b sinh [u + im).
(27.) y coshw-x sinh M=a cosh 2m,
y sinh u + x cosh « = a sinh 2m.
(28.) X =: tanh M + tanh v, y = cothM + cothv, u + v = c.
(29.) Expand sinh (u + h) in powers of h.
(30.) Expand tanh-^x in powers of x; and deduce the expansions of
cosh-^x and sinh~ix. Discuss the limits within which your expansions are
valid,
(31.) Given sinh m/m = 1001/1000, calculate u.
00 1 /a;V-"~^-l\
(32.) Show that the series S — ( — ^^ ) is convergent, and that its
1 ^ \x^/2 _f.l/
sum is (x2+l)/(x2-l)-l/loga; (Wallace, I.e.).
(33.) Prove that the infinite product cosh —^ cosh — cosh — ... is con-
vergent, and that its value is sinh m/w.
(34.) Show that
_x-x-^ 2 2 2
log X~ 2 • ^1/2 + x-y'' XV* + .T-y* ' xl/« + X-V8 • • • ad 00 .
(Wallace, I.e.)
(35) If i. = *-ZL2^\_j^_^-^ . . . ^j^^iT^n. «how that P„ differs
from 1/logx (in defect) by less than
{1 + i (x'/2"+' + x-V-!"+') }/3 . 4»+iP„.
Evaluate the following limits : —
(36.) (sinh x - sin x)jx^, x=0.
(37.) (sinh^ mx - sinh^ 7ja;)/(co8hpx - cosh qx), x = 0.
(38.) (tan'^ x - tanh* x)/(coa x - cosh a;) , x = 0.
§ 31 EXERCISES XVII 315
Show that, when h=0,
(39.) L {cosh a(x+ h) - cosh ax} lh = a ainh ax.
(40.) L {sinh a (x + h) -sinh ax} lh = a coah ax.
(41.) L {tanh a{x + h) - tanh ax} lh = a sech- ax.
(42.) L {coth o (x + 7i) - coth ax}lh= - a cosech^ ax.
(43.) Show that
2^. «°*h ^i = «o*^ " ~ f 2^. **"^ ^' '
- = coth M - S TTT tanh—, ,
Ji 12" 2" '
and state the corresponding formulsa for the circular functions (Wallace,
Trans. E.S.E., vol. vi.).
(44.) From the formulffi of last exercise, derive, by the process of chap.
XXVII., § 2, the following: —
24«oth'^|.-«otl^'^«-2.itanh='|..,
-„ = coth2 u - S j^ tanh" -^^ .
u^ I 2-^ 2"
(Wallace, I.e.)
In the following, 0 is the centre of the hyperbola x^la^-y^lb^=l', A one
of its vertices ; F the corresponding focus ; P and P' any two points on the
curve, whose excentric anomalies are u and u', and whose co-ordinates are
(x, y){x', y'), so that a; = a cosh m, y = b sinh«, &c. ; and N is the projection
of P on the axis a. Show that
(45.) Area ANP=iab (sinh2u- 2m).
(46.) Area of the right segment cut off by the double ordinate of P
= -x J(x^ - a^) - ah cosh~^ - ,
a ^ ^ ' a
= - x ^(x^ - a^) - ablog — ^^-^ ' .
(47.) Area of the segment cut off by PP' = ^ab {sinh («' -u)-{u'-u)}.
Express this in terms of x, y, x', y'.
(48.) If R be the middle point of PP', and OR meet the hyperbola in S,
the co-ordinates of S are {a cosh^ (w-i-m'), b sinh^{u+u')}.
(49.) OS bisects the hyperbolic area POP'.
(50.) If PP" move parallel to itself, the locus of E is a straight line passing
through 0.
(51.) If PP' cut off a segment of constant area, the locus of i? is a
hyperbola.
316
GRAPH OF Cos {x + yi)
CH. XXIX
GRAPHICAL DISCUSSION OF THE GENERALISED CIRCULAR
v^ FUNCTIONS.
§ 32.] Let US now consider the general functional equation
w - Cos z, or, as we may write it,
u + w = Cos (w + yi) (1),
where u, v, x, y are all real.
Since Cos {x + yi) = Cos x Cos yi - Sin x Sin yi = cos x cosh y -
i sin X sinh y, we have
M = cos ^ cosh y, t) = - sin ;r sinh y (2);
and therefore
u^/cofi' X - v^/sin^ x = l (3) ,
Y
u
d
, TH.
M
L
K
K
L
M
N U
U
M
L K
K
l-1>
BR.
PRIN.
B«.
(+1)
BR.
c
5
s
D
R
B
B
R
D
S 0
c
s
D
R B
B
G
G
Q
A
P
F
F
P
A
Q G
Q
Q
A
P F
G X
c
C
S
D
R
B
B
n
D
S C
C
S
D
R B
B
D
U
N
M
L
K
K
L
M
N U
U
N
M
L K
K
Fig. 12.
In order to avoid repetition of the values u and v, arising
from the periodicity of cos x and sin x, we confine z, in the first
instance, to lie between the axis of y and a parallel UCGCU to
this axis at a distance from it equal to tt (Fig. 12).
If we draw a series of parallels to the y-axis within this strip,
we see, from equation (3), that to each of these will belong half
§32
GRAPH OF C0S(x + yi)
317
of a hyperbola in the w-plane (Fig. 13), having its foci at the
fixed points i^and G, which are such that 0F= 0G=1. Thus,
for example, if in the 2;-plane FF ^ ^tt and FQ = |7r, then to the
parallels LPL, NQN correspond the two halves LPL, NQN of
a hyperbola whose transverse axis is PQ = jj2.
^
V
M
5
/
/
JK^
A
XJ
ef
1
/
f
B
K
u
cl G
Jq a
M
X
B
/
\l
K
Fig. 13.
K
B
K
B
L
R
M
BR.
D
N U
S C
u
5
PR IN.
S
Y
M
BR.
D
L K
R B
K
B
L
R
M
BR.
D
N U
S C
a
c
F
F
P
A
Q G
G
Q (
|A
P F
F
P
A
Q G
FX
B
K
B
K
D
S C
N U
0
U
s
N
D
M
J
R B
L K
B
K
R
L
D
M
S C
N D
C
U
TiQ. 14.
318 GRAPH OF Cos (x + yi) CH. XXIX
To the parallel MAM, which bisects the strip, corresponds
the axis of v (which may be regarded as that hyperbola of the
confocal system which has its transverse axis equal to 0) ; and
to the parallels KFK and UG U, which bound the strip, corre-
spond the parts KFK axvd. UGU oi the w-axis, each regarded as
a double line (flat hyperbola).
Again, if we draw parallels to the ^r-axis across the strip, to
each of these will correspond one of the halves of an ellipse
belonging to a confocal system having F and G for common foci.
Thus to BRDSG and BRDSC equidistant from the a^-axis corre-
spond the two halves BRDSC and BRDSC of the same ellipse
whose semi-axes are coshy and sinh^/. In particular, to FPAQG
on the ir-axis itself corresponds the double line (flat ellipse)
FPAQG. __
Thus, to the whole of the first parallel strip between KOK
and UU corresponds uniquely the whole of the tc-plane. Hence,
if we confine ourselves to this strip, (1) defines w and z each as
a continuous one- valued function of the other. To each succeed-
ing or preceding strip corresponds the w-plane again taken once
over, alternately one way or the opposite, as indicated by the
lettering in Fig. 12. w is therefore a periodic function of z,
having the real period 27r ; and 2; is a multiple-valued function
of w of infinite multiplicity, having two branches for each period
of w.
The value of z corresponding to the first strip on the right
of the y-axis is called the principal branch of Cos~^ w, and the
others are numbered as usual. We therefore have for the ^-th
branch
«Cos-'w = ;^t = (^ + | + (-)'-U)7r + (-)*Cos-^w; (5),
where Cos"^ w is the principal value as heretofore ; and Cos~^ w
= a; + yi, x and y being determined by (3) and (4), when u and v
are given.
It should be noticed that for the same branch of z there is
continuity fi'om B to B not directly across the w-axis, but only
by the route BFB\ whereas there is continuity from B io B
§§32-34 GRAPH OF Sin (a; + 2/0 319
directly, if we pass from one branch to the next. This may be
represented to the eye by slitting the w-axis from i'^ to + co and
from G to — oo , as indicated in Fig. 13. If we were to con-
struct a Riemann's surface for the w-plane, so as to secure unique
correspondence between every w-point and its ^^-point, then the
junctions of the leaves of this surface would be along these slits.
The reader will find no difficulty in constructing the model.
Since to the line KFPA QG U (the whole of the w-axis) corre-
sponds in the 2;-plane the three lines KF, FPA QG, G U taken
in succession, we see that as w varies first from + co to 1, then
from 1 to —1, and finally from -1 to -oo, Cos'^m? varies first
from CO i to 0, then from 0 to tt, and finally from tt to tt + oo 2 ;
so that an angle whose cosine is greater than 1 is either wholly
or partly imaginary.
§ 33.] If w = Sin 2;, say
u + iv^ Sin (x + yi) ( 1 ),
then, as in last paragraph,
u - sin a; cosh y, v = cos a; sinh y (2) ;
u^/siii^ X - v^lco^^ x=l (3) ;
?*7cosh^ y + 'jr^/sinh'^ y = ^ (4).
The graphical representation is, as the student may easily
verify, obtained by taking Fig. 13 for the i<;-plane and Fig. 14
for the 2;-pIane.
We have also, for the ^-th branch of the inverse function,
tSin~^ w = Zt = tTr + {-Y Sin~^ w,
where Sin~^ w^x + yi, x and y being determined by equations
(3) and (4), under the restrictions proper to the principal branch
of the function.
§ 34.] If w = Tan z, say
M + «v = Tan (^ + ^?') (1),
then {u + iv) Cos {x + yi) = Sin {x + yi),
that is,
{u cos X cosh y + v&iux sinh y) + i{~u&mx sinh y + vco^x cosh y)
= sin X cosh y + i cos x sinh y.
320
GRAPH OF Tan (x + yi)
CH. XXIX
Therefore
u cos X cosh y + v&mx sinh y=mix cosh y,
— u sin X sinh y + v cos x cosh ?/ = cos x sinh ^.
From the last pair of equations it is easy, if we bear in mind
the formulse of § 27, to deduce the following : —
u - sin 2xl{coB 2x + cosh 2y), v = sinh 2?//(cos 2x + cosh 2y) (2) ;
u'' + iP + 2ucot2x-l = 0 (3) ;
u' + v'- 2v coth 22/ + 1 = 0 (4).
The graphical representation of these results is given by
Figs. 15 and 16.
|«o<.
«,
Y
Ij+OO
i-
I^'
B
R,
PR
N.
B
R.
(*
TH.
1)
B
R.
<
"
/3
-l
a
a
c
r\
(^.
'^
/3
7
5
e
C
"
<*,
"
/3
7
5
e
f
n
f ,
"
T
K
L
U
N
p
Q
R
ST
K
L
M
N
0
p
<s
R
S T
K
L
M
N
P
Q
R
81
K
-'
A
B
0
D
e
F
Q
HJ
A
B
0
D
E
F
Q
H J
A
B
C
0
E
F
Q
HJ
A ^
T
K
L
M
N
p
Q
R
ST
K
L
M
N
P
Q
R
S T
K
L
M
N
P
Q
R
8 T
K
4
s
7
i
e
K
1
O'i
;?
r
6
e
c
U
3:
O
/3
7
a
e
C
1
tf'l
a
Ir
■CO
1,
-«
1,
-OS
Fig. 15.
When X is kept constant, the equation to the path of w is
given by (3), which evidently represents a series of circles passing
through the points (0, +1) and (0, - 1).
When y is constant, the equation to the path of w is (4),
which represents a circle having its centre on the ■y-axis ; and it
is easy to verify that the square of the distance between the
centres of the circles (3) and (4) is equal to the sum of the
squares of their radii, from which it appears that they are
orthotomic.
If we consider a parallel strip of the 0-plane bounded by
a; = - |7r, a; = + |7r, we find that to this corresponds the whole
§34
GRAPH OF Tan {w + yi)
321
t<;-plane taken once over. The corresponding values of z are
said to belong to the principal branch of the function Tan~^ w.
To the vertical parallels in the 2;-plane correspond the circles
passing through / and I in the w-plane, and to the horizontal
parallels correspond the circles in the w-plane which cut the
former orthogonally.
It should be noticed that / and / in the «<;-plane correspond
to + 00 and - co in the direction of the i^-axis in the ;2-plane, and
Y^
V
A
cl
D E
' F
/
H
7
1^
%
Vyj^
Y
/
L\
K
T
J
ys
A
4- 09
J V
Fig. 16.
that to A and J in the 2;-plane correspond the points at oo on
the u- and v-axes in the t<;-plane ; also that there is no continuity
directly across IKcc or IKco in the w-plane, except in passing
from one branch of Tan"^ w to the next.
For the ^-th branch of the inverse function we have
{Tan~^ 'w = Zt = tTr + Tan~^ w (5),
where the principal value Tan~^ w is given by ^dixr''- w = x + yi,
X and y being determined, under the restrictions proper to the
principal branch, by means of (3) and (4).
c. II. 21
322 GRAPHS OF f(x+yi) and llf{a) + yi) CH. xxix
§ 35.] It will be a useful exercise for the student to discuss
directly the graphical representation of w = Secz, w = Cosecz,
and w = Cotz. The figures in the w-plane for these functions
may, however, be derived from those already given, by means of
the following interesting general principle.
If Zbe any z-path, W and W the corresponding w-pathsfor
w =f{x + yi) and w = \lf{x + yi), then W is the image with respect
to tJie lb-axis of the inverse of W, the centre of inversion being the
origin of the w-plane and the radius of inversion being unity.
This is easily proved; for, if (p, ^), (p', <^') be the polar
co-ordinates of points on W and W corresponding to the point
{x, y) on Z, then we have
p (cos ^ + «■ sin 4>) =f (x + yi),
P (cos tf>' + i sin ^') = l//(^ + yi).
Hence p (cos ^ + * sin ^) = l/p'(cos ^' + * sin <i>),
= {l/p')(co8(-cf>') + imi(-<l>')).
Therefore p =^ 1/p', ^ = - <^' + 2^7r, which is the analytical ex-
pression of the principle just stated.
From this it appears at once that, if we choose for our standard z-paths
a double system of orthotomic parallels to the x- and y-axes, then the ic-paths
for w=Cotz will be a double system of orthotomic circles, and the w-paths
for 'w=Seoz andw=Cosec2 a double system of orthotomic Bicircular Quartics.
Example 1. If u + vi = Sec (x + yi), show that
w = 2 cos a; cosh yj {cos 2x + cosh 2y) ;
t) = 2 sin a; sinh j//(cos 2x + cosh 2y) ;
{u^ + v^)^=u^Igo8!^ X - v^Jsin^ x ;
(«a 4- ^2^2 _ u^JQosifi y + u^ysinh^ y.
Discuss the graphical representation of the functional equation, and show
how to deduce the t-th. branch from the principal branch of the function.
The curves represented by the last two equations are most easily traced
from their polar equations, which are
/32= 2 (cos 20 - cos 2x)/sin2 2a;,
p^=2 (cosh 2y - cos 2<f>)lsmh^ 2y,
respectively.
Example 2. The same problem for u+vi = Cosec (a; + yi).
Example 3. The same problem for u + ri = Cot {x + yi).
§§ 35, 36 ORTHOMORPHIC TRANSFORMATION 323
§ 36.] Before leaving the present part of our subject, it will
be well to point out the general theorem which underlies the
fact that to the orthogonal parallels in the 2;-plane in the six
cases just discussed correspond a system of orthogonal paths in
the w-plane.
Let us suppose that f{z) is a continuous function of the
complex variable z^ such that for a finite area round every
point z-a within a certain region in the 5;-plane f{z) can
always be expanded in a convergent series of powers Qi z- a,
so that we have
f{z)=f{a)^-A^{z-a) + A^{z-af-^. . . (1),
where Ai, A2, . . . are functions of a and not of z.
Then we have the following general theorem, which is funda-
mental in the present subject.
^Ai + 0, the angle between any two z-paths emanating from
a is the same as the angle between the corresponding w-paths
emanating from the point in t/ie w-plane which corresponds
to a.
Proof. — Let z be any point on any path emanating from a,
(r, 6) the polar co-ordinates of z with respect to a as origin, the
prime radius being parallel to the a^-axis. Let w and b be the
w-points corresponding to z and a, (p, ^) the polar co-ordinates
of w with respect to b. Then we have
p (cos <!> + i sin <f>)
= w-b=f(z)-/(a),
= Ai(z-a)+A2(z-aY + . . ., by (1),
= Air (cos 6 + i 8va 0) + A2r^ {cos 0 + i sin 6y + . . . (2).
Let now Ai = n (cos a^ + i sin aj), A^ = rg (cos a^ + i sin aj), . . . ,
then (2) may be written
p (cos i> + i sin <^) = r^r {cos (a^ + 6) + i sin (a^ + 6)}
+ r^r^ {cos (ag + 2^) + i sin {o^ + 2^)} + . . . (3).
Whence
P cos ^=rir cos (aj + ^) + r-^i^ cos (ag + 2^) + . . . (4) ;
p sin ^ =- rir sin (a^ + ^) + r^r^ sin (a.^ + 2(9) + . . . (5).
21—2
324 ORTHOMORPHIC TRANSFORMATION CH. XXIX
In the limit, when r and consequently p are made infinitely
small, (4) and (5) reduce to
{plr) cos ^ = ri cos (a^ + &), {pjr) sin ^ = rj sin (aj + 6) (6).
Since p and r are both positive, these equations lead to
pjr = /-i, and ^ = 2kTr + ai + ^ (7).
Hence, if we take any two paths emanating from a in directions
determined by 6 and 0\ we should have (fi- cj/ = 6-6', which
proves our theorem.
We see also, from the first of the equations in (7), that if we
construct any infinitely small triangle in the ;2;-plane, having its
vertex at a, to it will correspond an infinitely small similar
triangle in the w-plane having its vertex at b.
Hence, if we establish a unique correspondence between points
{uyv) and {x, y) in any two planes by means of the relation
u + vi =f{x + yi) = X {x, y) + # {x, y),
then to any diagram D in the one plane corresponds a diagram
D' in the other which is similar to D in its infinitesimal detail.
The propositions just stated show that, if we have in the
z-plane any two families of curves A and B such that each curve
of A cuts each curve of B at a constant angle a, then to these
correspond respectively in the w-plane families A' and B' such
that each curve of A' cuts each curve of B' at an angle a.
Since the six circular functions satisfy the preliminary condition
regarding the function f(x + yi), the theorem regarding the
u-v-cuTves for these functions which correspond to a; = const.,
?^ = const, follows at once.
If J.i = 0, J.2 = 0, . . ., An-i = 0, Jn + 0, then the above con-
clusions fail. In fact, the equations (7) then become
p/r^ = rn, <ty = 2kir + a^ + n6 (T);
and we have <}} - <f>' = n (6 - 6').
In this case, as the point z circulates once round a, the point
w circulates n times round b. That is to say, & is a winding
point of the wth order for z ; and the Riemann's surface for the
w-plane has an w-fold winding point at b. We have a simple
example of this in the case of «; = «*, already discussed, for which
§ 36 EXERCISES XVIII 325
w = 0 is B, winding point of the third order. The points w = +l
and z = ±0 are corresponding points of a similar character for
w = cos z.
The theorem of the present paragraph is of great importance in many parts
of mathematics. From one point of view it may be regarded as the geomet-
rical condition that ^(x,y) + ix(x,y) may be, according to a certain definition,
a function of x + yi. In this way it first made its appearance in the famous
memoir entitled Grundlagen fiir eine allgemeine Theorie der Functionen einer
veranderlichen complexen Grosse, in which Riemann laid the foundations of
the modern theory of functions, which has borne fruit in so many of the
higher branches of mathematics.
From another point of view the theorem is of great importance in
geometry. When the points in one plane are connected with those in
another in the manner above described, so that corresponding figures have
infinitesimal similarity, the one plane is said by German mathematicians to
be conform abgebildet, that is, conformably represented (Cayley has used the
phrase " orthomorphically transformed") upon the other; and there is a cor-
responding theory for surfaces in general. Many of the ordinary geometrical
transformations are particular cases of this ; for example, the student will
readily verify that the equation w^a^Jz corresponds to inversion.
Lastly, the theory of conjugate functions, as expounded by Clerk-
Maxwell in his work on electricity (vol. i. chap, xii.), depends entirely on the
theorem which we have just established. In fact, the curves in Figs. 12,
13, 15, and 16 may be taken to represent lines of force and lines of equal
potential; so that every particular case of the equ&tion u + vi=f {x+yi) gives
the solution of one or more physical problems.
Exercises XVIII.
(1.) Discuss the variation of sin~^M and Birr^iv, where u and v are real,
and vary from - oo to + oo .
Draw the Argand diagrams for the following, giving in each case, where
they have not been given above, the ic-paths when the ^-paths are circles
about the origin and parallels to the real and imaginary axes : —
(2.) w = logz. (3.) w=:exp2.
(4.) 10 = cosh. z. (5.) w = t&nh.z.
(6.) Show that cos~^(« + iu) = cos-^{7-icosh~i F";
sin~i (u + iv) = sin-i U+i eoshr^ V,
where 2U^^{(u+l)^ + v^} -J{{u-l)'^ + v'^},
2V=J{(u+l)^ + v^} + ^{{u-l)^ + v%
the principal branch of each function being alone in question.
326 EXERCISES XVIII CH. XXIX
(7.) Show that the principal branch oi ta,n~^ {u + iv) is given by x + yi,
where y=^t&nh.-^{2ul(u^ + v^+l)};
and a;=^tan-i{2M/(l-M2_r2)}, itu^ + v^<l;
= ± Iff + i tan-i {2m/(1 - u^ - v^)}, if u2 + i;2> i^
the upper or lower sign being taken according as u is positive or negative.
(8.) U u + vi = cot{x + yi), show that
M=sin 2a;/ (cosh 2y - cos 2x), v= - sinh 22//(cosh 2y - cos 2a;);
«2^^2_2ucot2a;-l = 0, u2 + i;2 + 2i; coth 22/ + l=0.
(9.) If M + Tt = cosec (a; + yi) , show that
M=2sina;coBhj//(cosh22/-cos2a;), v= - 2 cos a; sinh ?//(cosh 2y - cos 2a;) ;
{u^ + v^Y — w'^/cos^a; - v^JBin^y, (u^ + v'^)^ = w^/cosh^j/ + v^'/sinh^y.
Express the following in the form u + vi, giving both the principal branch
and the general branch when the function is multiple-valued : —
(10.) Cosh-i(x + 2/i). (11.) Tanh-i(x + 2/i).
(12.) iLog{(a;-(-yi)/(a;-i/i)}. (13.) hog Sin {x + yi).
(14.) (cos^ + isin^)*. (15.) hog a+ip {x + yi).
(16.) Show that the general value of Sin-^ (cosec ^) is {t+^)T+ilog
cotl{tTr + 6), where t is any integer.
(17. ) Show that the real part of Exp^ {Log (1 + i)} is e-"^/8 cos (Itt log 2).
(18.) Prove, by means of the series for Cos ^ and Sin 0, that Sin 2tf = 2 Sin 6
Cos 61.
(19.) Deduce Abel's generalised form of the binomial theorem from
§§ 20, 22.
(20.) Show that
^ + m+ni(^l^ + m+ni(^2^^+ • • • ad oo
= (1 + x)^ [cos {n log (1 + x)} + i sin {n log (1 + x)}].
(21.) Show that the families of curves represented by
sin a; cosh 7/ = X, cos x sinh ?/=/*
are orthotomic.
(22.) Find the equation to the family of curves orthogonal to r"
cosn^=X.
(23.) Find the condition that the two families
Ax^ + 2Bxy+Cy^ = \, A'x^ + 2B'xy + CY=fi
be orthotomic.
(24.) If tan {x + hj) — sin (m + iv), prove that coth v sinh 2y = cot u sin 2ar.
SPECIAL APPLICATIONS OF THE FOREGOING THEORY TO
THE CIRCULAR FUNCTIONS.
§ 37.] In order to avoid breaking our exposition of the
general theory of the elementary transcendents, we did not stop
§§ 87, 38 APPLICATIONS TO CIRCULAR FUNCTIONS 327
to deduce consequences from the various fundamental theorems.
To this part of the subject we now proceed ; and we shall find
that many of the ordinary theorems regarding series involving
the circular functions are simple corollaries from what has gone
before.
Let us take, in the first place, the generalised form of the
binomial theorem given in § 15. So long as l + %mCaZ^ is
convergent, we have seen that it represents the principal value
of (1 + 2;)™. Hence, if z = r (cos ^ + * sin 6), where r is positive,
and -7r::)>^:|> +7r, we have
1 + 2^(7„r" (cos nO + i sin n6)
= {1 + 2rcos6 + T^Y"'^ (cos w^ + i sin m4>\
where - ^tt :|> ^ = tan~^ {r sin ^/( 1 + r cos ^) } :|> + f tt.
Hence, equating real and imaginary parts, we must have
1 + :S™C„r" cos ne = {l + 2r cos 0 + r')'"/^ cos mcf> (1) ;
2wCn^" sin nO = (l + 2r cos 6 + r^)™^ sin m4> (2).
These formulae will hold for all real commensurable values of
m, provided r<l.
When r = 1, we have
<^ = tan-^ {sin 6/(1 + cos 6)} = 1$,
and (1) and (2) become
1 + 2 ™(7„ cos ne = 2"^ cos'"i^ cos ^mO (1'),
-$mGn sin nO = 2™ cos'^^O sin ^mO (2').
These formulae hold for all values of 6 between — -rr and + tt*,
when m>—l; and also for the limiting values — tt and + n
themselves, when m>0.
§ 38.] Series for cos m(j> and sin m^, when m is not integral.
If in (1) and (2) of last paragraph we put 9 = ^tt, and
r = tan ^, so that <^ must lie between - ^tt and + \rr, then
(1 + 2r cos 6 + r^yi'^ ^ sec"*^ ; and we find
cos7w^ = cos"*^(l-„(72tan^^ + w(74tan^<^- . . .) (3),
sin7»^ = cos"'^(TOCitan^-mC3tan^<^ + . . .) (4).
* Since the left-hand sides of (!') and (2') are periodic, it is easy to
see that, for 2p-n -ir>d>2pTr-\-ir, the right-hand sides will be 2"* cos™ .J ^
cos \m {e - 2p7r) and 2™ cos"*^^ sin \m [0 - 2/37r) respectively, where 2"* cos"*^^,
being the value of a modulus, must be made real and positive.
328 SERIES FOR COS m(f) AND SIN mcf) CH. XXIX
WhencG
tan mq> =- 77— — 5-7—; — tti — n v''/*
These formulae are the generalisations of formulae (4), (5), (6)
of § 12. They will hold even when <j> has either of the limiting
values ± jTT, provided m>-l ; so that we have
2™^'cos|m7r=l-„C; + ^C4-. . .;
2'^'^smlmTr = mCi-mC-i + . ...
Since
cos'"-^<^ = (1 - sin^</.)('"-^'-)/^ = 1 + 2 ( - )\rr,-,r),2Cs sin^«^,
and the terms of this series are ultimately all positive, it follows
that the double series deducible from (3), that is to say, from
S ( - Ym02r cos™"^'"^ sin^*"^ by substituting expansions for the
cosines, satisfies Cauchy's conditions (chap, xxvi., § 34), for
there is obviously absolute convergency everywhere under our
present restriction that — ^irlfx})^ + ^tt.
Hence we may arrange this double series according to powers
of sin (f>.
The coefficient of ( - )'' sin^''^ is
«=r
•^ {m~2s)/2^r~s m^28
_m (771 — 2) . . . (m-2r + 2) ^ r* ri
~ r~3 (2r—l) ■^(»n-l)/2^«(2r-l)/2t/r-«.
Now, by chap, xxiii., § 8, Cor. 5,
^(m-i)/2^« {2r-i)/2^r-8 = (m+2r-2)/2^r'
Hence the coefficient of (-/sin^'"^ is
m{m-2) . . . {m-2r + 2)(m+2r-2) . . . (m + 2)m
1.3 .. . (2r-l)2 . . . (2r-2)2r
_m' {7)1^-2^) . . . (^"-2^^")
(2r)!
Hence
cosOT^ = l---rSur<;6 + — ^—r-\ ^sm^<^-. . . (6).
A 1 4*
§ 38 <^ IN POWERS OF SIN <f) 329
In like manner, we can show that
, m . , m (m^ -V) . „ ,
sin 7w^ = — , sm ^ ^-^. ^sin^0
+ — ^^ ~ -^sm'«A-. . . (7).
o!
Also
cos w^ = cos <^ U ^^ m^<i>
+ ^^ ^j ^sin*«^-...| (8);
, {m . , m (m^ - 2^) . , ,
sm m4> = cos ^ j — i ^^^ *P ~~^\ sin^^
+ — ^ rj^^ ^sin'<^-. . . V (9).
The demonstration above given establishes these formulae
under the restriction -\Trl!f<f>l!^\Tr. It can, however, be shown
that they hold so long as -■k'^'^^'^l,'^', that is to say, so long
as the series involved are convergent.
Cauchy, from whom the above is taken, shows that by
expanding both sides in powers of m and equating coefficients
we obtain expansions for <^, </)^, ^^, &c., in powers of sin ^.
Thus, for example, we deduce
^ . ^ lsin^«^ 1.3sin^</> 1 . 3 . 5 sin^<A
<i = sm </> H + — + + . . .
^^232.45 2.4.6 7
If we put X = sin <J!), this gives
. 1 la? 1.3^ 1.3.5^' ,,^.
sm-^ = ^ + --+^^- + 2-^g-+... (10).
In particular, if we put x = \, we obtain
-g
1 1.3 1 ,..
2. 3. 2^ "^2. 4. 5. 2«'^" " 'I ^ ^'
from which the value of tt might be calculated with tolerable
rapidity to a moderate number of places. The result to 10
places is 77-31415926536 ....
330 ' EXAMPLES CH. XXTX
The important series (10) for expanding 8in~i x is here demonstrated for
values of x lying between - 1/\/2 and + 1/\/2. It can be shown that it is
valid between the limits x= -1 and x= +1.
The series was discovered by Newton, who gives it along with the series
for sin a; and cos a; in powers of a; in a small tract entitled Analysis per
jEquationes Numero Terminorum Ivfinitas. Since this tract was shown by
Newton to Barrow in 1669, the series (10) is one of the oldest examples of an
infinite series applicable to the quadrature of the circle.
Example 1. If ni>0, and
C = 2-"* S ^C„cos(m-2n)a:,
n=0
S = 2-'" S ^C„ sin (m - 2n) x,
n=0
C'=2-'" 2 (-)"-! ^C„ COS (m-2n) a;,
n=0
S'=2-^ -Z {- )"- VCn sin (m - 2n) x,
n=0
then, p being any integer,
1°. C = (cos x)"* COS 2mp7r, S = (cos a;)"* sin 2mp7r,
from a; = (2p - ^) TT to a; = (2p + 1) tt.
2°. C=(-cosa;)'"cos77i(2/) + l)7r, S=:(-cosa;)'"sinm(2p + l) ir,
from X = (2/D + 1) TT to a; = (2p + 1) TT.
3°. C' = (8ina;)"'cosm(2/) + 4)7r, -S' = (sina;)»'sinm(2/) + i) ir,
from X = 2pir to x = (2p + l)ir.
4°. C'= (- sin a;)™ cos m(2/) + f) IT, S' = ( - sin x)™ sin m (2p + f ) ir,
from x — [2p+l)ir io x = (2p + 2) t.
These formulae will also hold when m lies between - 1 and 0, only that
the extreme values of x in the various stretches must be excluded. (Abel,
(Euvres, t. i., p. 249.)
If we multiply (1') and (2') above by cos a and sin a respectively, and add,
we obtain the formulae
cos a + S,„C„ cos (o - nd) = 2"* cos^^^ cos (a - ^mO + vipv),
wherein it must be observed that cos"*^^ is the modulus of (l + 2rcos^+r2)'"/2
when r~l, and must therefore be always so adjusted as to have a real positive
value.
From the equation just written, Abel's formulae can at once be deduced
by a series of substitutions.
Example 2. Show, by taking the limit when m=0 on both sides of
(1) and (2) above, that the series (1) and (2) of § 40 can be deduced from the
generalised form of the binomial theorem.
Example 3. Sum to infinity the series Sw3„C„ sin" 6 cos nO. This series
is the real part of Sn'^m^^n sin" 6 (cos ne + i sin nO). Hence
S=R['En^,n(^n sin" d (cos d + ism 0)"],
= J? [{ni3 sin3 e (cos e + i sin e)^+m (3m - 1) sin^ e (cos ^+i sin oy
+mBmd(cosd + isind)}{l+sm6{cose + iBind)}^-%
FORMULAE FROM EXPONENTIAL & LOGARITHMIC SERIES 331
by Example 5 of chap, xxvii., § 5,
= [m^ sin3 e cos {36* + (m -3)<f>} + m (3m - 1) sin^ d cos {2^ + (m - 3) (p}
+ m sin e COB {^ + (m - 3) ^}] (1 + 2 sin 6* cos ^ + sin^ ^)(»*-3)/2,
where ^ = tan-^ {sin2^/(l + sin 6 cos 6)}.
§ 39.] Formulw deduced from the Exponential Series.
From the equation
^ (cos y + imiy) = l-\-'%{x + yifln\ ,
putting x = r cos 6, y = r sin ^, we deduce
gr eosfl |(.Qg (^ gin ^) 4. ^- gii^ (^ gin ^)} = 1 + ^yw (cos n9 + i sin n6)Jn\.
Hence
gT- cos e cos (r sin ^) = 1 + 2 r"cos w^/w! (1) ;
gr cos 0 gin (^ gin ^) = :S r"sin w^/»! (2) ;
which hold for all values of r and 9.
In like manner, many summations of series involving cosines
and sines of multiples of 6 may be deduced from series related
to the exponential series in the way explained in chap, xxviii.,
§8.
Thus, for instance, from the result of Example 3, in the paragraph just
quoted, we deduce
S(13 + 23+. . .+n3)a;«/nI = e^''°s^{rcos((9 + rsin^) + Jr2cos(25 + rsin5)
^ +2r3cos(3e+rsine)+Jcos(4& + rsin0)}.
§ 40.] FormulcB deduced from the Loga/rithmic Series. Since
the principal value of Log(l+^;) is given by Log(l +2;) = log
|l + 2;| + 2amp(l+2;), and since the series z-z'^/2+s^/S- . . .
represents the principal value of Log (1 + z), if we put z = r (cos 6
+ i sin 0), we have
log (1 + 2r cos 9 + t^y^ + i tan"^ {r sin 6/(1 + r cos 9)}
= %{- )"-' r" (cos n9 + i sin n9)ln,
where -|-7r::^tan~^ {r sin6/(l + r cos 6)}:|>|7r, that is, the prin-
cipal value of the function tan~^ is to be taken.
Hence we have the following : —
I log (1 + 2/- cos 6 + r^) = 2 ( - )"-^ r" cos w6/w (1) ;
tan-i [r sin djil + r cob 6)} = 2 ( - )"-' r" sin nOln (2).
332 sin^-^sin2^ + ^sin8^- . . . =J^ ch. xxix
Although, strictly speaking, we have established these results
for* values of 6 between - tt and + tt both inclusive, yet, since
both sides are periodic functions of 0, they will obviously hold
for all values of 0, provided r<l.
If r = l, (1) and (2) will still hold, provided ^4=±7r; for the
series in (1) and (2) are both convergent, and we have, by
Abel's Theorem,
cos^-^cos2^ + ^cos3^-. . .= L ilog(l + 2rcos^ + r^),
r=l
= log (2 cos i^) (3);
sin ^ - ^ sin 2^ + 1 sin 3^ - . . . = tan"^ {sin 6/(1 + cos 0)},
= tan-^ {tan ^{0 + 2^7r)},
=^e + u (4),
where k must be so chosen that ^6 + krr lies between - ^tt
and + ^TT. Thus, if 6 lie between - tt and + tt, ^ = 0, and we
have simply
sin^-|sin2^ + ;^sin3^-. . . =1^ (4').
In particular, if we put 0 = ^tt, we get
i- = l-i+i-^ + ^-TV + A + . . . (5),
which is Gregory's quadrature ; see § 41.
When 6= ±(2j3 + l)7r, the series in (3) diverges to -oo, and the right-
hand side becomes log 0, that is - oo , so that (3) still holds in a certain
sense.
The behaviour of the series in (4) when ^ = ± (2^) + 1) tt is very curious.
Let us take, for Bimplicity, the case 6= ^ir. With this value of 6 we have
for values of r as near unity as we please tan-^ {rsin^/(l+r cos ^)} = 0.
Hence, by Abel's Theorem, when ^=±7r, sin ^-^sin 2^+ . . .=0, as is
otherwise sufficiently obvious.
On the other hand, for any value of 0 differing from ±7r by however little,
we have L tan-^{rsin^/(l + rcose)}=J^. Hence, again, by Abel's Theorem,
for 6=^ir={:<p, where (f> is infinitely small, we have
sin ^-^ sin 2^+ . . . = ±^7r=Fi0.
The series y = sind — ^sm2d + . . .is therefore discontinuous in the neigh-
bourhood of 0= ±7r; for, when fl= ±7r, 2/ = 0, and when d differs infinitely
little from ± tt, ?/ differs infinitely little from ± ir/2. This discontinuity is
accompanied by the phenomenon of infinitely slow convergence in the
neighbourhood of r=l, ^= ±7r; and the sudden alteration of the value of
the sum is associated with the fact that the values of the double limits
§§ 40, 41 Gregory's series 333
L L tan-i {rsine/(l + rcos^)} and L L t&n-^ {r sin 61(1 + r cos 0)}
r=16 = ±iT e=±7r r=l
are not alike.
When 6 lies between v and Sir, we may put 9 = 2Tr + d', where 6' lies
between -ir and +7r, then, for such values of 0, we have
?/ = sin^-Jsin2^'+ . . .,
= ^6', as we have already shown,
Hence, however small <p may be, we have, for 6 = ir + <f>, y=^<f>-^ir. But,
as we have just seen, for ^ = 7r-^ we have ?/= -^0 + ^7r. Hence, as ^ varies
from TT - </> to TT + 0, y varies abruptly from -^cp + ^ir to i<p - ^ir. In other
words, as 6 passes through the value tt, y suffers an abrupt decrease
amounting to tt*.
We have discussed this case so fully because it is probably the first
instance that the student has met with of a function having the kind of
discontinuity figured in chap, xv., Fig. 5. It ought to be a good lesson
regarding the necessity for care in handling limiting cases in the theory of
infinite series.
§ 41.] Gregory s Series. If in equation (2) of last paragraph
we put 6 = |ir, we deduce the expansion
tan-V=r-^r3+|r^-. . . (6),
where tan"^/* represents, as usual, the principal value of the
inverse function, and —lli^rl^l.
In particular, if r= 1, we have
7r=.4(l-^ + ^-. . .).
The series (6), which is famous in the history of the quadrature of the
circle, was first published by James Gregory in 1670 ; and independently,
a few years later, by Leibnitz. About the beginning of the 18th century, two
English calculators, Abraham Sharp and John Machin (Professor of Astronomy
at Gresham College), used the series to calculate tt to a large number of places.
Sharp, using the formulas ^7r=tan-il/v'3 = (iy3){l- 1/3.3 + 1/5.32- . . .},
suggested by Halley, carried the calculation to 71 places ; that is, about
twice as far as Ludolph van Ceulen had gone. Machin, using a formula
of his own, for long the best that was known, namely, Jtt = 4 tan"^ 1/5
- tan~^ 1/239, went to 100 places. Euler, apparently unaware of what
the English calculators had done, used the far less effective formula
J7r=tan-i^+tan-i J. Gauss {Werke, Bd. ii., p. 501) found, by means
of the theory of numbers, two remarkable formul89 of this kind, namely : —
47r= 12 tan-i 1/18 + 8 tan-i 1/57 - 5 tan-i 1/239,
= 12 tan-i 1/38 + 20 tan-i 1/57 + 7 tan-i 1/239 + 24 tan-i 1/268,
* The reader should now draw the graph of the function y, for all real
values of 6.
334 EXERCISES XTX CH. XXIX
by means of which ir could be calculated with great rapidity should its value
ever be required beyond the 707th place, which was reached by Mr Shanks
in 1873 1*
Exercises XIX.
Sum the following series to infinity, pointing out in each case the limits
within which the summation is valid: —
(1.) l-^cosd + i^5cos2^-^-^cos30+. . . .
cos 5 1 „ cos 35 1 • 3 . cos 55
(2-) «'-i- + 2-^-^ + 2T4^-^+--- •
,„ , cos 5 1 COS 35 1.3 cos 55
<^-) -T+2-3- + 2:4-^+---'
result \ cos~i (1-2 sin 5).
(4.) S(27i-l)(2/i-3)cos7i5/nl (5.) Ssinn5/(7i + 2)7i!
(6.) e-«sin5-^e~3"sin35 + ^e-5'»sin55-. . . .
(7.) sin5-2— 3sin25 + g— ^sin35-. . . .
(8.) sin2 5-isin2 25 + ^sin2 35-. . .;
result \ log sec B.
(9.) Scos27i5/n(n-l). (10.) S sin n5/(n2 _ 1).
(11.) ^sin5sin5-^sin25sin25 + -^sin35sin35- . . . .
(12.) cos(a + j3)-cos(a + 3^)/3I+cos(a + 5/3)/5!-. . . .
(13.) cos5--|cos25 + ^cos35-. . .;
result \ log (2 + 2 cos 5), except when 5 = (2;j + 1) v.
(14.) cos5 + ^cos25 + Jcos35+. . .;
result - \ log (2-2 cos 6), except when 5 — 2pir.
(15.) sin 5+ I sin 25 + ^ sin 35+ . . .;
result =0, if 5 = 0; =^ (7r-5), if 0<5>7r; &o.
(16.) sin5-^sin35 + ^sin55- . . . .
(17.) X cos 5 - |x3 cos 35 + J^x^ cos 55 - . . . ;
result i tan-i {2x cos 5/(1 -x^)}.
(18.) cos5cos^-i cos25cos20 + ^cos35cos 30- . . .;
result J log {4 cos ^ (5 + 0) cos i{9-<p)}-
(19.) ar cos 5 cos ^-^x^ cos 35 cos 30 + ^a;'' cos 55 cos 50- • . .;
result i tan-i [ix (1 - x^) cos 5 cos 0/{ (1 + x")^ - 4x2 (cos^ 5 - cos^ <p)}].
(20.) Show that log (1 + x + x^) = 2S ( - )"-i cos inir x»/n, provided | x | < 1,
and examine whether the result holds when |x| = l.
* For the history of this subject see Ency. Brit., art. "Squaring the
Circle," by Muir.
§ 41 EXERCISES XX 335
(21.) Show that, under certain restrictions upon 9,
log (1 + 2 cos ^) = - 2S cos ^nir cos ndjn\
^ = - 2 cos ^nj- sin nOjn.
(22.) Show that
2^2~ "^3 5 7 '''9 "''11 12 13"^"" * * *
(Newton, Second Letter to Oldenburg, 1676.)
Exercises XX.
(1.) Calculate ir to 10 places by means of Machin's formula.
(2.) Show that, if a; <1,
(tan-ia:)2
=a;2-(l + l/3)a;4/2 + . . .(-)'»-i {1 + 1/3 + . . . + l/(2ra- l)}a;2»/w . . . .
Does the formula hold when x=l?
(3.) Expand tan~i (x + cot a) in powers of x.
(4.) Deduce the series for.sin"^* from Gregory's series by means of the
addition theorem for the binomial coefficients.
(5.) If X lie between 1/^/2 and 1, show that
sin~^x=7r-
s/{l-x')
.-x^) ( 1 1-x^ 1 (l-a;2)2 1
X \ 3 x^ '^ 5 X* ■ • I
(6.) Show that § 38 (10) is merely a particular case of (7).
(7.) Show that
e . , 2 . ,, 2.4 . ,, 2.4.6 . .,
cos 6 3 3.5 3.5.7
(Pfaff.)
,„, 1,„ sin^^ 2 sin*^ 2.4sin«6l ,^,, . .„ ,
(^•^ 2^'=-^ + 3-^ + 3-:5-^6 +••• • (btamvme.)
(9.) e^^sinH + ^.^(l + ^^am^e + . . .
3.5.. .(2ft-l) 3 /, 1 1 \ . .,„x,.
4.6...2n 2n+l \ 6^ (27i-l)^J
(10.) e*=sm*d + ^.^{l + ^^Bm^e + . . .
4.6. . . (2w-2) 2 /, 1 1 \ . o „
5.7 ... (2/1-1) 71 V 2-' (71-1)2/
(11.) Deduce from § 38 (6) and (7) an expression for ^"'/sin"*^ in powers
of sin 6.
(12 .) If sin ^ = X sin (^ + a) , show that d + rir = Sa:" sin 7ia/7i.
(13.) If c2=a2- 2a 6cosC+6^then
logc = loga-(6/a)cos(7-^(&/a)2cos2C-i(6/a)3cos3C-. . . .
(14.) Show that
1 - 'LZ^ 4. (w - 4) (w - 5) _ («-5)(7i-6)(7i-7) _ 1+ (-)"+! 2 cos|7i7r
2"^ 2.3 2.3.4 +•••- ^ .
336 EXERCISES XX CH. XXIX
Show that
(15.)* 52=sin2^ + 22sm*^ + 2«sin*|^ + 2«sm*^ + . . . .
23
22-" sin. 23
(16.)* u'^=Bmh^u - 22 sinh*| - 2* sinh* J - 2« sinh*^ - .
(17.)* |^ = sm(9 + 3sin3| + 32sin3|5 + . . . .
.S 1 ml
(18.)* j8in^= ^--3^^,sin3-0 + S 3;^,8in33--i^.
(19.)* tCos(?= S ^ ' , cos3 3'"-^^.
* See Laisant, " Essai sur les Fonctions hyperboliques," M^m. de la Soc.
de Bordeaux, 1875.
CHAPTER XXX.
General Theorems regarding the Expansion of
Functions in Infinite Forms.
EXPANSION IN INFINITE SERIES.
§ 1.] Cauchys Theorem regarding the Expansion of a Function
of a Function.
u
y^a^^^a^cf- (1),
the series being convergent so long as \a;\<B, and if
z = b, + ^bny'' (2),
this series being convergent so long as \y\<S, then from (1) a7id
(2) we can derive the expansion
provided x be such that \x\<Ry and also
\ao\ + %\an\\x\'<8.
This theorem follows readily from chap, xxvi., §§ 14 and 34.
We have already used particular cases of it in previous chapters.
§ 2.] Expansion of an Infinite Product in the form of an
Infinite Series.
If %Un be an absolutely convergent series, and ^Smi, „2miW2,
. . . , «2mi ^2 . • -Ur, . . . denote the sums of the products of its
first n terms taken one, two, . . ., r, . . ., at a time, then
L „2Wi=2\, L n'^^hU-i= T^, . . ., L ^U-^U^. . .Ur=Tr, . . .
n=a> n=oo n=oo
where Ti, T^, . . ., Tr, . . . are all finite.
Also the infinite series l + %Tn is convergent ; and converges
to the same limit as the infinite product n (1 + itn).
c. II. 22
338 INFINITE PRODUCT REDUCED TO A SERIES CH. XXX
After what has been laid down in chap, xxvi., it will
obviously be sufficient if we prove the above theorem on the
assumption that all the symbols «i, «2, • • •> %, • • • represent
positive quantities. In the more general case where these are
complex numbers the moduli alone would be involved in the
statements of inequality, and the statements of equality would
be true as under.
Since Ml, w^, . , ., w„, . . . are all positive, we see, by the
Multinomial Theorem (chap, xxiii., § 12), that
< {ill + U2+ . . . + Un+. , . ad CO y/r\
<S^/rl, (1),
where S is the finite limit of the convergent series 2m„ ; and the
inequality (1) obviously holds for all values of r up to r-n,
however great n may be.
Therefore n^UxU2 . . . Ur has always a finite limit, Tr say,
such that
0>>7;:j>/S^/r! (2).
By (2), we have
0<l + Ti+r2 + . . . adco<l+/S'/l! + /S'72! + . . . adoo,
that is,
0<H-i7;<e^ (3).
Hence 1 + S7^„ is a convergent series, whose limit cannot
exceed e^.
Again, since Lr^Ui u^. . . Ur=Tr when w = oo , we may write
n^UiUi. . .Ur = {l+rAn)Tr ^ (4),
where LrA^ = 0 when n=<x>.
Hence, An being a mean among i^„, ^An, . . ., «^», and
therefore such that X^„ = 0 when w=qo, we have
n(H-M„)sl+„S?fi + „2MiM2+. . .+«22^e«a. • 'Un
= l + (l + ^«)2Tn (5).
§ 2 INFINITE PRODUCT AND SERIES 339
If in (5) we put w = oo , we get
= l + ^Tn ' (6),
1
since LAn = 0, and 2r„ is finite.
This completes the proof of our proposition.
Cor. 1. If 2un be absolutely convergent, then, Tn having the
above meaning, 1 + 2^"7^« will be convergent for all finite values
of x; and we shall have
Jl{l + xUn) = l + ^x''Tn (7).
1 1
This follows at once by the above, and by chap, xxvi., § 27.
Cor. 2. Let
Urt = nVo + nViX + nV2X'^ + . . . (8),
where n%, nVi, &c., are independent of x, and the series on the
right of (8) may either terminate or not ; and let
Un=\nVo\ + \nVi\\x\+\nV^\\x\^ + . . . (9).
Then, if %Un be convergent far all values of x such that
\x\<py it follows that for all such values n (1 + m„) is convergent,
and can be expanded in a convergent series of ascending powers of x.
For, if Tn have the meaning above assigned to it, then it will
obviously be possible to arrange T^ as an ascending series of
powers of x. Moreover, if we consider the double series that
thus arises from \-v'%Tn, we see that all Cauchy's conditions
(see chap, xxvi., § 35) for the absolute convergence of this
double series are satisfied. Hence we may arrange l+57^„ as
a convergent series of ascending powers of x.
Example 1. To expand (1 + a;) (I + x^) (1 + x^j (i + ^^Sj , . .in an ascending
series of powers of x. (Euler, Introd. in Anal. Inf., § 328.)
The series 2 1 a; P" is obviously convergent so long as | x | < 1. Hence, so
long as |x|<:l, we may write
(l + x)(l + x2)(l + x'')(l + a;8). . . = l + (7ia; + C2a;2 + . . . + C„a:" + . . . (10).
To determine the coefficients C^, G^, C„, we observe that, if we multiply
both sides of (10) by l~x, the left-hand side becomes L (l-x^"), that is,
n=oo
1, since jx|<l. We must therefore have
l/(l-a;) = l + Cia; + C2a;2+. . .+C„a;»+. . .,
22—2
340 PRODUCTS OF EULER AND CAUCHY CH. XXX
that is,
l+x+x'^+. . .+a;'»+. . .=l + C-iX + C2x'^+ . . . + (7„x"+. . .,
therefore Ci=C.2=. . . = C7„=. . .=1.
Another way is to put x^ for x on both sides of (10), and then multiply by
(1 + x). We thus get
l + SC„a;™=l+a; + Cia;2+. . . +C7„a;2» + C„x2»+i+ . . .;
whence C^^ = C^n+i = C„ , ^i = 1,
from which it is easy to prove that all the coefficients are unity.
Example 2. To show that
{l+xz){l + x'^z) . . . (l + a;"»z)
„=1 (l-x)(l-x^) . . . (l-x») ^ '
(Cauchy, Comptes Rendus, 1840.)
Let
{l + xz){l + x^z) . . . (l+a;"*2)
= 1 + ^13 + ^2^2+. . .+^„2«+. . .+^^2"* (2),
where ^j, Jj, . . . are functions of x which have to be determined.
Put xz in place of z on both sides of (2), then multiply on both sides by
{1 + xz)l{l+ x'^+^z), and we get
{l + xz){l + x^z) . . . (1 + x"*^)
= {l + (l+A^)xz + {AT^ + A^)x^z^+... + {An.i + An)x'^z'' + ...-i-A^x'^+'^z^+^},
X { 1 - X'^+^Z + x2(™+l)32 + ..,(- )na.n(,«+l)2« + . , . J (3).
Hence, arranging the right-hand side of (3) according to powers of «,
replacing the left-hand side by its equivalent according to (2), and then
equating the coefficients of a™ on the two sides, we get
A„= (J„ + J„_j)a;«- x»>+i (^„_i + 4n_2)x»-i
+x2(»'+i)(^„_2+^„_3)a:»-a
(-)»-ia;(»-i)(™+i)(4i + l)a;
(_)«a.n(m+l).
whence
1 — a;"
A =A ,—A ^x^ + A „3;2»»_ ( -\n-lT{n-l}m /A\
Putting n - 1 in place of n in (4), we have
X»-i"(l - X"') ^n-l = -^n-2 - ^n-sa:"* + A^_,X^ - . . . (_ )n-2^(n-2)m (5).
If we multiply (5) by x"» and add (4), we derive, after an obvious
reduction,
(l-x")^„=(x»-x'»+i)^„_i (61).
In like manner,
(1 - X»-l) ^„_i = (X»-1 - X'»+l)^„_2 (62),
(1 - X»-2) 4„.2= (x"-2 - X'»+l)^„_3 (63),
(l-x)^i = (x-x'»+i) (6„).
2, 3 EXPANSION OF SeCH X AND SEC X 341
Multiplying (6i), (62), . . . , (6,j) together, we derive
_(x- x'»+^) {x^ - a;'»+i) . . . (x" - a:"^^)
"~ (l-a;)(l-a;2). . .(1-a;")
_(l-a;"')(l-a;"^^)...(l-a:'n-n+i) ^
(l-a;){l-a;--').. .(l-x»)
which establishes our result.
(7).
(8),
If |a;|<l, the product {iy-\-xz){\-vx'^z) . . . will be convergent when
continued to infinity, and will, by the theorem of the present paragraph, be
expansible in a series of powers of z. The series in question will be obtained
by putting m = Qo in (1), We thus get
to ».n(n+l)/2
(1 + ..)(1 + .^.). . . ad 00 =1+^1, (i.,)(i_..)... (!_,„) ^- (9).
an important theorem of Euler's {Introd. in Anal. Inf., § 306).
§ 3.] Expansion of Seek x and Sec x.
We have, by the definition of Exp x,
2/(Exp ^ + Exp - ar) = 1/(1 + tx^l{2n) !) (1).
Hence, if y = %x^l{2n)\ (2),
2/(Exp a; + Exp - a?) = 1/(1 + y),
= l + 2(-r2/" (3).
The expansion (3) will be valid provided | ?/ 1 < 1 ; and the
series (2) is absolutely convergent for all finite values of x.
Hence, if ^=|a;|, it follows from § 1 that the series (3) can
be converted into a series of ascending powers of x provided
i ^V(2w)!<l (4).
n=l
This last condition involves that
l{^ + e-^)-\<l',
that is, that ^<log (2 + ^3).
This condition can obviously be satisfied ; and we conclude
that 2/(Exp X + Exp - x) can be expanded in a series of ascending
powers of x provided | a; | do not exceed a certain finite limit.
Since the function in question is obviously an even function
of X, only even powers of x will occur in the expansion. We
may therefore assume
2/(Exp ^ + Exp - ip) = 1 + S ( - fEnX^I{2n)\ (5).
To determine Ei, E^, . . ., ^Q multiply one side of (5) by
342 euler's numbers ch. xxx
J (Exp X + Exp - w), and the other by its equivalent 1 + %!c^l{2n)\ ;
we thus have
1 = {1 + :S ( - YEnX^^'li^ny] {1 + :Sa^/(2;i)!} (6).
El, Eit . . . must be so determined that (6) becomes an
identity. We must therefore have
_J ^ + ^' (-Y ^" =0 r7V
{2n)\Q\ (2w-2)!2! (2«-4)!4! '''^ ^0!(2w)! " '^'^ '
or,
•^n = iriPiEn-\ - inG^En-^ + • • • ( — )~'^l'nP'2.n-'iEx + ( — l)""-^ (8).
The last equation enables us to calculate E^, E^, E^, . . .
successively. We have, in fact,
Ei = l; E,= 6Ei-l; Es=15E,-15Ei + l;
E,^28E3-70E2 + 28Ei-l; &c.
whence
Es= 2702765,
E-,= 199360981,
Es= 19391512145,
^9 = 2404879675441,
E,= 1,
•£2= 5,
Es^ 61,
Ei= 1385,
^5-50521,
These numbers were first introduced into analysis by Euler* ,
and the above table contains their values so far as he calculated
them.
Since the constants E^, E2, . . .are determined so as to make
(6) an identity, (6), and therefore also (5), will be valid for all
values of w, real or complex, which render all the series involved
convergent. Hence, since 1 -^ %ar^'^/(2n)\ is convergent for all
values of a;, (5) will be valid for all values of cc which render the
series l + %{-)"'EnaP"'/{2n)\ convergent. We shall determine
the radius of convergency of this series presently. Meantime,
we observe that (5) as it stands may be written
Sech a; = l + :^{-T Ena^/{2n)\ (9) ;
and, if we put ix in place of ic, it gives
Sec ^ - 1 + %EnX^''l{2n)\ (10).
• See Inst. Gale. Biff., § 224 : the last five digits of Eg are incorrectly
given by Euler as 61671.
For a number of curious properties of the Eulerian numbers see Sylvester,
Comptes Bendus, t. 52 ; and Stern, Crelle^s Jour., Bd. izxix.
§§ 3, 4 EXPANSION OF Tanh X, &c. 343
Cor. SechP'x and Seo^x can each he expanded in a series of
even powers of x.
The possibility of such an expansion follows at once from the
above. . The coefficients may be expressed in terms of Euler's
numbers. We may also use the identity 1 = (1 + %AnX^'^l{2ny)
cos** a?; expand cos" a; first as a series of cosines of multiples of a;;
finally in powers of x ; and thus obtain a recurrence formula for
calculating J.1, ^2> . • • The convergency of any expansion thus
obtained will obviously be co-extensive with the convergency of
(10).
§ 4.] Expansion of Tanh x, x Coth x, Cosech x ; Tan x,
xQiotx, Cosec^*.
We have already shown, in chap, xxviii., § 6, for real values
of X, that
xl{l - e-'') = l+lx+%{- f-^ ^„^»/(2w)!,
the expansion being valid so long as the series on the right is
convergent. In exactly the same way we can show, for any
value of X real or complex, that
xl{l - Exp - a;) = 1 + 1^ + S ( - )"-^ Bnx'^l{2n)\ (1),
where Exp — a? is defined as in chap, xxix,, and x is such that
I a? I is less than the radius of convergency of the series in (1).
From (1) we derive the following, all of which will be valid so
long as the series involved are convergent :
X (Exp X - Exp - i»)/(Exp X + Exp - x)
= 4^/(1 - Exp - Ax) - 2x1 {I - Exp - 2x) - x,
= 2(-r-^2^"(22'^-l)^„^/(2w)! (2);
X (Exp X + Exp - .T)/(Exp X - Exp — x)
= x/{l - Exp - 2x) - xl(l - Exp 2x),
= ! + %{- )"-^ 2^ BnX^I(2n)l (3) ;
2a;/(Exp X - Exp -x) = 2a?/(l - Exp -x)- 2^/(1 - Exp - 2x),
= 1 + 2S ( - )" (2^-^- 1) BnX^''l(2n)\ (4).
From these equations, we have at once
Tanha7 = 2(-)"-'2^"(2"'-l)^«a^"-V(2»)! (5);
a?Coth;» = l + 2(-)"-i2-'*i?„a?V(2w)! (6);
X Cosech x^l + 2^{-f (2^"-^ - 1) B^a?''\{2n)\ (7).
* Euler, I.e.
344 EXERCISES XXI CH. XXX
If in (2), (3), and (4), we replace x by ix, we deduce
Tan^ = 22^'^(2^-1)^„^-V(2w)! (8);
a; Cot a; = 1 - 22^" B^a^''\{'ln)\ (9) ;
X Cosec a; = 1 + 22 {2^-^ - 1) B^c^''l{^n)\ (10).
Cor. Each of the functions ( Tanh ccf, {x Coth x)''\ (x Cosech x)^,
{TanxY, (x CotxY, {x Cosec xY can be expanded in an ascending
series of powers of x.
Exercises XXI.
(1.) If tf=gdM (see chap, xxix., § 31), show that
B=a-^u-a^x{? ■'ra^w'- . . .,
where a^n+i = EJ{2n + 1) I.
(2.) Find expressions for the coefficients in the expansions of Sin" a; and
Cos^x.
(3.) Find recurrence-formulas for calculating the coefficients in the
expansions of (x cosec x)" and (sec a;)".
In particular, show that
^ „=«" ' 'mi 'm'
where S,. denotes the sum of the products r at a time of 1^, 3^, 5^, . . . , (2p - 1)^.
(Ely, American Jour, Math., 1882.)
(4.) If |x|<l, show that
{l + x^){l + x*)(l + x^) ... ad 00 =l + Sx"2+«/(l-x2)(l-a:4) . . . (l-a;^").
(5.) If I a; I > 1, and p be a positive integer, show that
00 ^n{n+l— 2p)/2
(6.) Show that the Binomial Theorem for positive integral exponents is
a particular case of § 2, Example 2.
(7.) Show that
{l + xz){l + x^z) ...{1 + x^-^z)
_ rn (1 - x-^^) (1 - X^^-^ ... (1 - x^-2»+2) „
~ n=r {l-X^)(l-X*)...{l~X^r.) ^^^
(Cauchy, Comptes Rendus, 1840.)
(8.) Show that
(l-xa)(l-a;*2).. .(l-x"»2)~ "^ (l-x)(l-x^) . . .(1-x^)
also that, if | x | < 1, | zx | < 1,
ll{l-xz)(l-x'^z). . .adoo =l + 2a;»2'V(l-a;)(l-x«). . .(l-x»).
(Euler, Int. in Anal. Inf., § 313.)
§ 4 EXERCISES XXI 345
(9.) If m be a positive integer (1 - x'"*) (1 - a;»*-i) ... (1 - x^-^+i) is exactly
divisible by (1 -x)(l-x^) . . . (1 - x™).
(Gauss, Summatio quarumdam serierum singularium,
Werke, Bd. ii., p. 16.)
(10.) Uf(x,m) = l + ^i-)^^' (l-l)(l-.j;V.(l-.>») '''''''' 1^1
>1, show that
/ (a;, m)=f{x, m - 2\) (1 - x""-^) (1 - a;'"-^) ... (1 - ai'^-sA+i)
_l-x'"-i 1-x^-'^ l-x»»-5
- l-x-i • l-x-s • l-a;-5 • • • ^<i =^-
Hence show that, if |a;|<l, then
1 - r2 l-T* 1 - r6
l + Sa;"(»+i)/2=i-^ .^,.i-^ ... ad 00.
l-x l-a;-* l-x"
(Gauss, 16.)
(11.) Show that, if m be a positive integer,
(l + x)(l + x^) . . . (l-|-.^) = l-l-Z.-( (l-xVl-xV-- (1-X-) •
(Gauss, I&.)
(12.) Show that
1
{l-xz)(l-a^z) . . . (l-a;2"»-i2)
_ (l-X^-^){l-X^rn+2) . . . (1 _ ^2m+2n-2)
(1-X^)(1-X*) . . . (1-X2»)
Also that, if | aj | < 1, and | za; | < 1,
ll(l-xz)(l-x»z) ... ad oo=l + Sa;»2"/(l-x2)(l-a;'') , . . (l-a;2»).
(13.) Show that, if |a;|<l,
1/(1 - X) (1 - x3) (1 - x5) ... ad 00 =(l + a;)(l + a;2)(l + a;3) . . . ad oo .
(Euler, i.e., §325.)
(14.) If |xl<l,
+00
(l-a;)(l-a;2)(l-a;3) . . . ad oo = 2 ( - )»a;(3'''^»)/2.
— 00
(Euler, Nov. Comm. Pet., 1760.)
(15.) If |a;|<l,
log{(l-a;)(l-x2)(l-a;») ... ad oo }= -Sj(n)a;»/n,
where j[n) denotes the sum of all the divisors of the positive integer ra; for
example, J(4) = l + 2 + 4.
Hence show that
oo ■JJT™ "^
1 1 - X" 1 ■' ^ '
(Euler, lb.)
(16.) If d{n) denote the number of the different divisors of the positive
integer n, and lx|<l, show that
Sd(n)x»=S
1 l-x»'
(Lambert, Ensai d^Architectonique, p. 507.)
346 EXPANSION IN INFINITE PRODUCT CH. XXX
Also that
00 00 /I 4- t'»'\
(Clausen, Crelle's Jour., 1827.)
(17.) If |a;|<l, show that
, + ,-4-1 + ^^— «+• • . .
1-x 1-x^ l-x' 1 + x^ 1 + x* 1 + x^
(18.) Sx'*+V(l - a;2™+i)2=Sna;"/(l - sc^n).
S ( - )'»-ina;»/(l + a;») = S ( - )'»-ia;»/(l + a;«)2.
(19.) The sum of the products r at a time of x, x^, . . . , a;" ia
a.r(r+i)/2(a;'-+i-l)(a;'-+2-l) . . . {x^ - 1) I (x - 1) {x"^ - 1) . . . (a;"-'--l).
(20.) If Sj. be the sum of the products r at a time of 1, a;, . . ., a;"-!, then
(21. ) Show that, if x lie between certain limits, and the roots of ax^ +bx + c
be real, then {px + q)l(ax^ + bx + c) can be expanded in the form Ug +
2 {u^x'^+VnX'^) ; and that, if the roots be imaginary, no. expansion of this
kind is possible for any value of x.
ON THE EXPRESSION OF CERTAIN FUNCTIONS IN THE FORM
OF FINITE AND INFINITE PRODUCTS.
§ 5.] The following General Theorem covers a variety of
cases in which it is possible to express a given function in the
form of an infinite product ; and will be of use to the student
because it accentuates certain points in this delicate operation
which are often left obscure if not misunderstood.
Let f{n, p) be a function {with real or imaginary coefficients)
of the integral variables n and p, such that L f{n, p) is finite for
P=QO
all finite values of n, say L f{n, p) =f{n) ; and let us suppose
that for all values of n and p {n<p), however great, which exceed
a certain finite value, \f{n, p) \l\f{n) \ is not infinite.
Thsn L n {l+f{n, p)} = U{l+f{n)] (1),
p=oo n=l 1
provided 2 \f{n) \ be convergent {that is, provided n {1 +f{n)} be
absolutely convergent).
Let us denote n {1 +f{n, p)} by Pp ; L U {1 +f{n, j))] by
n=l p=» n=l
P ; !/(«, P) 1 by ^i {n, p) ; and \f{n) \ by/, {n).
§ 5 GENERAL THEOREM 347
We may write
m p
P,-=U{l+f{n,p)} IT {l+f(n,p)},
= PmQm, say, (2).
Just as in chap, xxvi., § 26, we have
\Qm-i\> n {1+A(n,p)}-1.
n=m+l
Now, by one of our conditions, if m, and therefore p, exceed
a certain finite value, we may put fi (n, p)/fi (n) = An, where An
is not infinite. If, therefore, A be an upper limit to An, and
therefore finite and positive, we have/i {n, p)1f>Afi (n). Hence
>> n {l + AMn)}-l, (3).
TO+l
Let us now put p- <x> in (2). Since m is finite, and
L f{n, p)'^f{n), we have
m
LPm=ll{l+f{n)}.
p=w 1
■m
Therefore F = U{1 +f{n)} Q,, (4),
where Qm is subject to the restriction (3),
Let us, finally, consider the effect of increasing m.
Since n {1 +fi (n)} is absolutely convergent, II {1 + Afi (n)} is
absolutely convergent. It therefore follows that, by sufficiently
increasing m, we can make TI {1 + Afi (n)} - 1, and, a fortiori,
m+l
\Qm-l\ as small as we please. Hence, by taking m sufficiently
great, we can cause Qm to approach 1 as nearly as we please.
In other words, it follows from (4) that
P = n{l+/(n)} (5).
In applying this theorem it is necessary to be very careful to see that both
the conditions in the first part of the enunciation regarding the value of
f(n,p) are satisfied. Thus, for example, it is not sufficient that L f{n, p)
have a finite definite value / (n) for all finite values of n, and that S/^ (n) be
348 INFINITE PRODUCTS FOR SINH pu, SINH U CH. XXX
absolutely convergent. This seems to be taken for granted by many mathe-
matical writers ; but, as will be seen from a striking example given below,
such an assumption may easily lead to fallacious results.
§ 6.] Factorisation of sink pu, sink u, sinpO, and sin 6*.
From the result of chap, xii., § 20, we have, p being any
positive integer,
a;'P-l={x'-l)uL''-2a;cos''^ + l\ (1).
From this we have
-^ — - = n (or -2a; cos — + 1 ) ;
^-1 »=iV p /'
whence, putting w = l, and remembering tha,t Lia^-l)/{a^-l)=p,
we have
p-i
p = 2P-^ n (1 - cos . mr/p) (2) ;
= 4P-^'usm\'mr/2p (3);
and, since sin . Tr/2p, sin . 27r/2/>, . . . , sin . (j9 - 1) 7r/2p are
obviously all positive.
p-i
sjp = 2^-^ n sin . n7rl2p (4).
If we divide both sides of (1) by a?^, we deduce
aF-x-P^{x- x-^) 'n.{x + x-^-2 cos . mrip) (5),
where for brevity we omit the limits for the product, which are
as before.
If in (5) we put x - e", we get at once
sinhpu = 2^-^ sinh u U (cosh w - cos . mr/p) (6),
= 4^-^ sinh u U (sin^ «7r/2^ + sinh^ w/2) (7).
Using (3), we can throw (7) into the following form : —
sinh j9M =p sinh uU {1 + sinh^ M/2/sin^ mr/2p} (8).
Finally, since (8) holds for all values of u, we may replace u
by u/p, and thus derive
* The results in §§ 6-9 were all given in one form or another by Euler in
his Introductio in Analysin Infinitorum. His demonstrations of the funda-
mental theorems were not satisfactory, although they are still to be found
unaltered in many of our elementary text-books.
§§ 5, 6 INFINITE PRODUCTS FOR SINK pu, SINK V, 349
smh u=p sinh - IT U + ^^r '-.-/-} (9).
i^ n=i I snr.mr/2p) ^ '
We shall next apply to (9) the general theorem of § 5.
Before doing so, we must; however, satisfy ourselves that the
requisite conditions are fulfilled.
In the first place, so long as n is a finite integer, we have
(10).
^ sinh^ . u/2p _ W
p=« sin^ . mrl2p n^-n^
This can be deduced at once, for complex values of u, from
the series for siuh.u/2p and sm.mr/2p. When u is real it
follows readily from chap, xxv., § 22.
The product n (l+u^/irir^) is obviously absolutely convergent.
We have, therefore, merely to show that, for all values of n and p
exceeding a certain finite limit,
sinh^ . u/2p I u^
sm'^.mr/2p/ nrir^
<A
(11),
where Aha. finite positive constant. That is to say, we have
to show that
remains finite.
Now
sinh . u/2p
ul2p
sin . mr/2p"^
mr/2p >
sinh . ul2p
u/2p
N 1 /««V
>l +
3! \2p/
3! \2p) "^
(12).
Since the series within the bracket is absolutely convergent,
its modulus can be made as small as we please by taking p
sufficiently great.
Again we know, from chap, xxix., § 14, that, if 0:!f^J{6 x 7)
:l>6*48, and, a fortiori, if 6:lf>2rr, then
that is, if 6 be positive,
sinH^-i^',
350 INFINITE PRODUCTS FOR SIN p6, SIN 6 CH. XXX
Now, since n'i^p - 1, mrj^pl^ hir. Therefore
~m^]2p~ ^ '^ \2p)
H:i-^<t:-58 (13).
From (12) and (13) it is abundantly evident that the con-
dition (11) will be satisfied if only j3 be taken large enough ; and
it would be easy, if for any purpose it were necessary, to assign
a numerical estimate for A. All the conditions for the applica-
bility of the General Limit Theorem being fulfilled, we may make
p infinite in (9). Remembering that Lp sinh . ujp = u, we thus get
sinh M = w n (1 + u^ln'^tr') (14).
To get the corresponding formulae for mipO and sin^, we
have simply to put in (5) x = exp id. The steps of the reasoning
are, with a few trifling, modifications, the same as before. It will
therefore be sufficient to write down the main results with a
corresponding numbering for the equations.
sin 2)(^ - 2^-1 sin ^ n (cos ^ - cos . mrip) (6') ;
= 4^-^ sin en (sin^. mrj^p - sin^. ^/2) (7').
sin j3^ =p sin dU (1 - sin^. ^/2/ sin^ mtj^p) (8').
' = ^sin^ njl-^^^^n (9').
p n=\\. ain^.nirj2p}
smO=eu{i-eyn'Tr'] (W).
It should be noticed that, inasmuch as (6), (7), (8), (9), and
(14) were proved for all values of u, real and complex, we might
have derived (6'), (7'), (8'), (9'), and (14') at once, by putting
u = iO.
Cor. 1. The following finite products for sinpO and sinhpu
should be noticed : —
smi
§^ 6, 7 WALLIS'S THEOREM 851
sin pO =^ 2^-^ sin 0 sin (O + tt/p) am {d + 27r Ip) ...
sin{0 + p-l7rlp) (15);
sinh pu = {- 2^y "^ sinli u sinh {u + tV/p) sinh (m + 2i7r/p) . . .
sinh {u+p — l i-rr/p) (16).
The first of these may be deduced from (6'), as follows : —
sin pO - 2^~^ sin ^n (cos 6 — cos.mr/p),
= 2^-^ sin ^n {2 sin {mr/2p + 0/2) sin (w7r/2/j - 6/2)},
= 2^-^ sin eu {2 sin {mr/2p + 6/2) cos {p-nir/2p + 6/2)}.
Hence, rearranging the factors, we get
sin p6 = 2P-^ sin 6U {2 sin {mr/2p + 6/2) cos {mr/2p + 6/2)},
= 2^-^ sin 6 n sin {6 + mr/p).
n=l
We may deduce (16) from (15) by putting 6 = -iu.
Cor. 2. Wallis's Theorem.
If in (14') we put 6 = ^tt, we deduce
l = |,rn(l-l/2V) (17);
1
whence 2 = 173.375. • . ^2n-l){2n^l)' ' ' ^^°°'
2 2 4 4 2n 2n , /,_x
= Vrrl"-2^^-2^i--'^^'^ ^^^^'
This formula was given by Wallis in his Arithmetica In-
finitorum, 1656. It is remarkable as the earliest expression
of IT by means of an infinite series of rational operations. Its
publication probably led to the investigations of Brouncker,
Newton, Gregory, and others, on the same subject.
§ 7.] Factorisation of cosp6, cos 6, cosJipu, coshu. Following
the method of chap, xii., § 20, and using the roots of — 1, we
can readily establish the following identity : —
x^ + \= Jl(x'-2xcos^^^^^^^ + l) (1).
n=l \ Zp J
Putting herein x = \, we get
2 = 2^n(l-cos.(2;2-l)7r/2;?) (2);
= 4Pn sin^ (2;^ - 1) iv/4,p (3).
352 INFINITE PRODUCTS FOR COS p6, COS 6 CH. XXX
Hence, since all the sines are positive,
J2 = 2^ n sin . (2?i - 1) irjip (4).
From (1),
oF + x-^= II {cc + x-^-2 cos . (2w - 1) 7rj2p) (5) ;
whence, putting x - Exp iO, we deduce
cos^^ = ^.2^U (cos ^ - cos . (271 - 1) 7r/2p) (6) ;
= I . -l^n (sin^. {2n - 1) 7r/4^ - sin^ ^/2) (7).
From (7), by means of (3), we derive
cosj9^ = n (1 - sin^ ^/2/sin^ {2n - 1) 7r/4p)
From (8), putting djp in place of 0, we get
sin^^/2;?
cos
„=i I sin^. {2n - 1) Tr/Ap)
(8).
(9).
For any finite value of n we have
^ sm\e/2p
^^
(10).
p=oo sin^ (2w - 1) 7r/4j9 (2w - If tt^
Also the product 11 (1 + 4.6^l{2n - IYtt^) is absolutely con-
vergent.
Moreover,
sin.^/2/?
ei2p
S\\2p) "*■ • •
>l +
<2p
\^l\2pj
(12);
so that I sin . 0l2pl6l2p \ can be brought as near to 1 as we please
by sufficiently increasing p.
Also, since (2n - 1) Tr/^p1f>^^, we have, exactly as in last
paragraph,
&m^{2n-l)7rl4p
i2n-l)^lip ^^^
We may, therefore, put jt?= oo in (9) ; and we thus get
cos e = n {1 - 4^V(2w - 1)^71^}
(13).
t
(14).
§§7,8 INFINITE PRODUCTS FOR COSB. pu, COSH M 353
In like manner, putting a; = e"' in (5), we get
p
cos\i pu = 1 . 2^ n (cosh w - cos . (2n - 1) 7r/2/?) (6') ;
n=l
- i . 4^n (sin'^ . (2n - 1) n/Ap + sinh^ . w/2) (7').
cosh pu = n (1 + sinh^ . w/2/sin . (2n — 1) Tr/Ap) (8').
, PC sm\iKul2p ] ,^,.
coshM= U U+ . rrTTf (9)-
„=i (. sin^(2w- l)7r/4^jj ^ ^
cosh M = 3 {1 + 4m7(27j - 1)' TT^} (14').
We might, of course, derive the hyperbolic from the circular
formulae by putting $ = iu.
It is also important to observe that we might deduce (14)
from the corresponding result of last paragraph, as follows : —
From (14') and (17) of last paragraph, we have
^"^^=^°ii^:i7k)4'
ll{2nr
_2^ /2w7r-2^ 2mr + 2$ 1
~ TT t(2w-l)7r'(2w+l)7rj *
Hence, putting ^tt - ^ in place of 0, we deduce
cos(^- ^ n| (2n-l)7r •"~(2^n^l)^J'
= (1 - 2e/7r) n {(1 + 2ej(2}i - 1) tt) (1 - 2^/(2w + 1) tt)},
= (1 - 2^/7r) (1 + 26/7r) (1 - 2^/37r) (1 + 2^/37r) ....
Written in this last form the infinite product is only semi-
convergent, and the order of its terms may not be altered
without risk of changing its value ; we may, however, associate
them as they stand in groups of any finite number. Taking
them in pairs, we have
cos ^ = (l-4^/7r2) (1-4^73^772) . . .,
= n{l-467(2;i-l)V-=}.
§ 8.] From the above results we can deduce several others
which will be useful presently.
c. II. 23
354 VARIOUS INFINITE PRODUCTS CH. XXX
We have, since all the products involved are absolutely
convergent,
sin {6 + <^) ^ 6 + i^ n{l-(g + <^)7y^V^}
sine ~ 6 n {1 - ^/wV^} '
provided O + mr.
Hence, provided O^nir,
cos*H-sin*cote=(l4)n{l_^^:} (1).
In like manner, starting with cos (6 + ^)/cos 6, we deduce
cos*-s!n*tan<»=n{l-4(-^^^-^,} (2),
provided 9^^ (2n - 1) tt.
Also, from the identity
sin ^ + sin e _ sin l((f> + 0) cos |(<^ - (9)
sin^ sin ^e cos I e '
we derive
1 + cosec 0 sin <^
^ = (-!)"{- ^4^} (3).
provided 6=^mr.
A great variety of other results of a similar character could
be deduced ; but these will suffice for our purpose.
§ 9.] Before leaving the present subject, it will be instructive
to discuss an example which brings into prominence the neces-
sity for one of the least obvious of the conditions for the applica-
bility of the General Theorem of § 5.
We have, 6 being neither 0 nor a multiple of tt,
apP - 2xP cose + l = {af- (cos ^ + i sin $)} {of - (cos 0 - i sin 6)].
The joth roots of cos 6 + i sin 6 are given by
co&.{2mr->f6)lp-\-i%m.i27nr + 6)lp, n^O, 1, . , ., p-l (1).
The joth roots of cos^-zsin^, that is, of cos {-0) + i
sm{-0), by
cos . {2mr - $)Ip + i sin . {2mr - 6)lp, w = 0, 1, . . ., p-l (2),
§§ 8, 9 PRODUCT FOR COS (f) - COS 0 366
Since cos . (2n-ir - 6)/p - cos . {2 (ja - w) tt + 6]lp,
sin . (2w7r - 6)Ip = - sin . {2 {p -n)7r + 6]lp,
(2) may be replaced by
co&.{2mr + 6)lp-i&m.{2mr + 6)lp, n-0, 1, . . ., p-1 (2').
We have, therefore,
ar?P-2A-Pcos^ + l
p-i
= {a^-2w cos . ^/jt) + 1) n {af - 2x cos . (2«7r + ^)/j9 + 1} (3).
Since cos . (2w7r + ^)/p = cos .\1{p-n)Tt- 6]lp, we may, if ^ be
odd, arrange all the factors of the product on the right of (3)
in pairs. Thus, if j9 = 2^* + 1, we have
;B*a+2 -2^23+1 cos^ + l =
r;r'-2^cos— +l\h[ (^-2^cos.(2;^7r + ^)/(2g+l)+l)|
V 2^+1^ V«=ilx(^-2a;cos.(2n7r-^)/(2^+ 1)4- 1)1
(4).
If we now put x=\, we get
4 sm^ - = 4^«+^ sin^ , n \ %m^ . sin^ . — \ (5).
2 4g + 2m=il 4g' + 2 4g' + 2j ^ '
If we divide both sides of (4) by x'^^'^, and put x = Exp i4>,
we deduce
2(cos(2g+l)<^-cos6)
= 2^+' {cos <^ - cos . 6l{2q + 1)} n {cos «^ - cos . (^Imr ± 0)1 {2q + 1)}
(6),
where the double sign indicates that there are two factors to be
taken.
Transforming (6), and using (5), &c., just as in the previous
paragraphs, we get, finally,
cos ^ - cos Q
-28inH^ll -'-^5!iM4i±2)| fi fi _ sin^«^/(4g + 2) |
(7).
Since nli^q, (2w7r+6')/(4g'+ 2):}>(2g7r±^)/(4^ + 2) ; and the
limit of this last when g'= go is \-k. Hence, by taking q large
enough we can secure that i^mt ± 0)l{4.q + 2) shall have for its
23—2
356 PRODUCT FOR COSH U — COS 6 CH. XXX
upper limit a quantity which differs from ^-rr by as Uttle as
we please ; and therefore (see § 6) that sin . {2mr + 6)l{iq + 2)/
(2w7r + 0)/(4g + 2) shall have for its lower limit a quantity not
less than "58.
We may, therefore, put q= <x> , &c., in (7). We then get
cos <^ - cos ^ = 2 sin^ | ^ ( 1 - <^7^') S {1 - </.7{2w7r ± Of} (8),
n=l
that is,
cos ^ - cos ^
.,3i„'i.{i-i]{i-^-^}{:-^}... .
Putting ffi = iu in (8), we deduce
cosh w -cos ^ = 2 sin^^^ (1 + u^jO'') S {1 + u^l{2mT±6)"] (9).
The formula (8) might have been readily derived from those
of previous paragraphs . by using the identity cos <^ — cos 6
= 2 sin |(^ + ^) sin ^{0-4) and proceeding as in the latter part
of §7.
Remark. — At first sight, it seems as if we might have dis-
pensed with the transformation (4) and reasoned directly from
(3), thus—
From (3) we deduce
p-i
2 (cosptji - cos 0) = 2^ (cos ^ - cos . 0/p) U {cos ^ - cos . (2mr + 6)/p}.
71=1
Hence
cos (f>-cos6
Jsm ,t/|l g|j^.^/2^|^l^A^|l sm\{2n7r + e)l2pi'
Put now jt? = oo , &c., and we get
cos <^ - cos ^ = 2 sin'' I ^ ( 1 - <t>y6^) fi { 1 - <f>y(2mr + df},
1
This result is manifestly in contradiction with (8), although
the reasoning by which it is established is the same as that often
considered sufficient in such cases.
§ 9 INSTANCE OF FALLACY 357
In point of fact, however, the condition of § 5, that
M=fi{n, p)/fi{n) must remain finite when w and p exceed certain
limits, is not satisfied.
In the present case the upper limit of {2mT + 0)/2p, namely,
{2(p—l)-ir + 0}l2p, can be made to approach as near to tt as we
please. Hence in this case M may become infinite. We have,
in fact,
sm.{<l>l2p}l(<t>/2p)
M-
sin . {2mr + Q)\2p\{2mT + ^)/2/>
hence, if we give n its extreme value p — \, and put p= co , M
becomes infinite. No finite upper limit to the modulus M can
therefore be assigned ; and the General Theorem of § 5 cannot be
applied.
This is an instructive example of the danger of reasoning
rashly concerning the limits of infinite products.
Exercises XXII.
(1.) If (1 + ixja) (1 + ixlb) (1 + ixjc) . . . = A + iB, then
2 tan-i (xja) = tan-i (B/^).
Hence show that S tan"' (2/n2) = dirji.
1
(Glaisher, Quart. Jour. Math., 1878.)
(2.) Find the n roots of
^ ^^n(7i-r-l){n-r-2)...{n-2r + l)^_^^_ . . = 0.
(3.) If n be an odd integer, find the n roots of the equation
x+ 3, x+ g, x+ —j^ x+...-a.
(4.) Solve completely
a;" + „CjCOsaa;"-i + „C2C08 2aa;"-2+ . . .+cosna = 0.
(Math. Trip., 1882.)
(5.) The roots of
.T™ sin nd - nOia;"-! sin {nd + <j>) +JJ^x'^-'^ sin (n^ + 20) - . . . = 0
are given by a;=sin(0 + 0- /i;7r/n)cosec(^- fe7r/n), where k = Q, 1, . . ., or
Oi-1).
If a = 7r/2p, prove the following relations : —
(6.) jj = 2P-isin2asin4a. . .sin(2jj-2)a;
1 = 2^-1 sin a sin 3o . . . sin {2p - 1) a.
ass EXERCISES XXII CH. XXX
(7.) v'j» = 2P-'co8acos2a. . .cos (p-l)a.
(8.) l = 2P-isin.a/2sin.3o/2. . .sm.(22)-l)a/2;
= 2P-1 cos . a/2 cos . 3a/2 ... COS . (2p - 1) a/2.
(0.) sini)» = 2P-isin^sin(2a + e)sin(4o + e). . . sin (2p-2a + 0);
cospe = 2P-1 sin (a + ^) sin (3a + 6) sin {5a + 0). . . sin (2^) - ] a + 0).
(10.) tanp^ = tan0tan(tf + 2a). . .tan(e + (2j)-2)a), where p is odd.
(11.) tan e tan (^+ 2a).. . tan (^ + (2^ - 2) a) = { - 1)^/2, where;) is even.
(12.) Sliow that the modulus of
cos (^ + i(p) cos (61 + i<p + ttIp) . . .coB(0 + i(p + (p-l) irfp)
is {coshj)0-cos(2)7r + 22)e)}/2P-J.
(13.) If n he even, show that
. „^ , > «^ ^ 9 d + 2ir « + 47r 0 + (2)i-2)7r
8in2 - = ( - )»/22»-2 cos - cos cos ... cos — .
2 ^ ' n n n n
(14.) Show that n(l + sec2"^) = tan2»^/tan»;
and evaluate n < ^ V .
(15.) Show that
Sr /, 4 . „ 0\ sine
flU-4sin''-J=cos6i;
and write down the corresponding formulae for the hyperbolic functions.
(Laisant.)
Prove the following results (Euler, Int. in Anal. Inf., chap, ix.): —
na\ «^*+e'^'' „ j, ^ i(b-c)x + ix^ ] .
(16-) -iFiT^ = n |1 + ^2^^ _ ^^, ^2 ^ (^ _ ^)2)- .
e6+g-ec-a: _ / 2x \ j 4(?>-c)a; + 4a;n
c^-e" ~V "^b-cj^ I ■^(2n)2^2+(6-c)--iJ •
coshy + co8hc_ ( ±2cjy + y'' ) .
^ '"' 1 + cosh c - " "[^ "^ (2n - l)-* TT^* + c'l '
cosh y- cosh c_ /i _y^\ ^ ii ^^cy + y^ \
1-coshc V cV Y (2n)' t'+ c'i '
Binhy + sinhc^/^^y\^L^(-);2c.V + y^) ;
siuhc \ c/ ( 7i-7r'* + C'' J
sinhy-Binhc^_/^_y\„L^(-)"-^2cy + y2|
sinhc \ c/ ( n^ir^+c^ )
Write down the corresponding formulaa for the circular functions, and deduce
them by transformation from § 9.
§10
EXERCISES XXII
359
cos 0 + cos B
{
^^^■^ 1 + cos^ ~"r ((27i-l)7r±&)2p
(19.) cos^ + tani^Bin^ = n{(l + ^-4^— ) (^l-^^^-)|.
cos(g-0)_ [/ 2^ W 20 \1
^''"•^ cose -^M\ (2«-l)7r-2e;V^ (2n-l)x + 2e;f '
sine ~V 5;/ IV 2«7r-eyV 2)i7r + eyj"
(21.) Show that
cosh 2v - cos 2m = 2 (!<2 + ^2) n j(!i![±Jf)!+l'}
^ ^„ (((2«-l)7r±2M)2 + 4j;2}
cosh 2v + cos 2m = 2n ^^ t^^^ — rr^'-^ \ ;
< (2j1 - 1)- TT^ )
( 4m^ )
cosh2(i-cos2M = 4M^n U+ . — .\ ;
^ n* IT*)
( 2''u* )
cosh 2m + cos 2m = 211 {1 + y^z 5Ti -^^ •
( (2n - 1)* ir*)
(Schlomilch, Handh. d. Alg. Anal., chap, xi.)
(22.) Evaluate n ( %^-\ ] •
^ ' 1 \4n2-4tt + l/
(23.) If OT = log (1 + ^2), show that
EXPANSION OF THE CIRCULAR AND HYPERBOLIC FUNCTIONS
IN AN INFINITE SERIES OF PARTIAL FRACTIONS.
§ 10.] By § 8 we have, provided ^ + 1 (2w - 1) tt,
cos.#.-sin.#,tan« = n{l-4^-^^|^±^,} (1).
Now, referring to § 2, Cor. 2, we have here
Un =
{2n - l)'7r2 - 46=
<^l + 4
>
W\^' '^
{2n - 1)^2 - AO'
A
w,,
{2n - ly-r^ - 4.0'^ r I {2n - 1)'-^^ - 46''
where 0' = \6\, ^' = | ^ | . It follows, therefore, that the product
in (1) may be expanded as an ascending series of powers of ^.
360 INFINITE SERIES OF PARTIAL FRACTIONS CH. XXX
Expanding also on the left of (1), we have
l-|J + ...-tan^(<^-|-;4-...)
{{2m - IfTi^ - 4e^}{(2« - 1)^^ - 4^
. . . . . . . . (2).
Since the two series in (2) must be identical, we have, by
comparing the coefficients of <f>,
This series, which is analogous to the expansion of a rational
function in partial fractions obtained in chap, viii., is absolutely
convergent for all values of 0 except ^tt, f7r, |7r, ... It should
be observed, however, that when 0 lies between | (2w - 1) tt and
^ (2n + 1) TT, the most important terms of the series are those in
the neighbourhood of the wth term, so that the convergence
diminishes as 0 increases.
We may, if we please, decompose 80/{{2n— l)V^-4^} into
2/{(2ra - 1) TT - 2^} - 2/{{2n - 1) tt + 20}, and write the series (3)
in the semi-convergent form
tan«= 2 2 2 2
TT - 2^ Tr + 26 37r-26 Sir + 2$
/o'\
"^ 57r - 2^ ~ 57r + 2^ "^ ■ • • ^'^''
In exactly the same way, we deduce from (1) and (3) of § 8
the following : —
Scot 6 = 1 -26'%^^-^^ (4),
or
6 cot 6=1 ^ +
-6 17 + 6 2Tr-6 27r + e
^^irrh-' • • (4),
Zir-6 37r + ^
§ 10 INFINITE SERIES OF PARTIAL FRACTIONS 361
provided ^ 4= tt, 27r, Stt, . . . ;
and
e cosec 6 = 1 + 26'- S-^7p"^! (5),
or
a a ^ ^ e e 6
6 cosec p = 1 + Ti 7i - t: 7. +
-^ 7r + ^ 2ir-6 2Tr + 0
provided 6^7r, 27r, Stt, . . . .
"^ 377 - ^ Stt + ^ • • • ^^'^'
We might derive (4) from (3) by writing (Jtt-^) for 6 on
both sides, multiplying by 6, decomposing into a semi-convergent
form like (3'), and then reassociating the terms in pairs ; also
(5) might be deduced from (3) and (4) by using the identity
2 cosec 6 = tan ^6 + cot 1-9.
When we attempt to get a corresponding result for sec 6,
the method employed above ceases to work so easily ; and the
result obtained is essentially different. We can reach it most
readily by transformation from (5'). If we put (5') into the form
.111 1 1
cosec 6=y:+ 7, „ - ^ +
-6 TT + e 2ir-6 2Tr + 0
1
3ir-e 377 + 6 • • •'
which we may do, provided 6=¥0, and then put ^ir-O in place
of 6, we get
.22 2 2
sec 6 -^ +
ir-26 T7+26 377-26 377+26
2 2 _ , ,.
"^ 577 - 26 "^ 577 + 26 • • • ^^'''
or, if we combine the terms in pairs,
where 6 4= ^77, §77, |77, . . . .
The series (6), unlike its congeners (3), (4), and (5), is only
362 INFINITE SERIES OP PARTIAL FRACTIONS CH. XXX
semi-convergent ; for, when n is very large, its wth term is com-
parable with the «th term of the series 2l/(2w - 1).
We might, by pairing the terms differently, obtain an abso-
lutely convergent series for sec 6, namely,
but this is essentially different in form from (3), (4), and (5).
Cor. 1. The sum of all the products two and two of the terms
of the series 21/{(2w - 1)271^ - 4^^! ^g (tan 6 -6)11280^; and the
like sum fw the series 21/{wV^ - 6''} is (3-0^- 36 cot 6)/S6\
This may be readily established by comparing the coefficients
of 4*^ in (2) above, and in the corresponding formula derived from
§ 8 (1).
Cor. 2. The series Sl/{(27Z- If 7r2-4^'P converges to th^
value (0 tan' 6 -tan 6 + 6)164.0'; and 21/(^^^-67 to tlie value
{e^cose(f6 + 6cot6-2)/A6\
Since the above series have been established for all values of
6, real and imaginary, subject merely to the restriction that 6
shall not have a value which makes the function to be expanded
infinite, we may, if we choose, put 6 = ui. We thus get, inter alia,
tanh u = 8m21/{(2w - IfTt" + 4m=^} . (8)
WCOthM=l + 2w'2l/K7r2 + w2| (9^
u cosech u=l- 2u^% ( - l)"-V{«-7r2 + u'] (10)
sech w = 42 (-)""' (2w - 1) 7r/{(2w - If-jr'' + 4m^} (1 1).
EXPRESSIONS FOR THE NUMBERS OF BERNOULLI AND EULER.
RADIUS OF CONVERGENCY FOR THE EXPANSIONS OF
TAN 6, COT 6, COSEC 6, AND SEC 6.
§ 11.] If |6|<7r, then every term of the infinite series
^ff^Kn^TT^ - 6^) can be expanded in an absolutely convergent series
of ascending powers of 6. Also, when all the powers of 6 are
replaced by their moduli, the series arising from \l{n^tr^ - 6^)
will simply become l/lw^ir'— \6\^}, which is positive, since |6|<'''".
The double scries
§§10-12 EXPRESSION FOR BERNOULLI S NUMBERS 363
„?i XTi^'^^iiF^'^ ' ' ' '"IF^'' + • • •/
therefore satisfies Caiichy's criterion, and may be arranged
according to powers of 0. Hence, if
0-2^ = l/P"* + 1/2^'" + 1/3'"* + . . . (1),
we have, by § 10 (4),
6cot 6 = 1-22^7(72^2-6^), ' •
= l-2^(r2,„6''"/7r2™ (2).
Since o-2,„(<o-2) is certainly finite*, the series (2) will be
convergent so long as, and no longer than, 6<7r,
Now, by § 4 (9), we have
6cot6=l-22=""5«6''"/(2w)! (3),
provided Q be small enough.
The two series (2) and (3) must be identical. Hence we
have
_2(2m)!(r2».^2(2m)! fill \
§ 12.] If, instead of using the expansion for 6 cot 6, we had
used in a similar way the expansion for tan 0, we should have
arrived at the formula
Bm =
2 (2w)!
(1 - 1/2^'") (27r)2
1 1^"* ^ 3-'" ^ 5-"' ■*■•••! v^/"
This last result may be deduced very readily from (4) ; it is,
indeed, merely the first step in a remarkable transformation of
the formula (4), which depends on a transformation of the series
o-„ due to Eulert. We observe that the result of multiplying
the convergent series (r^m by 1 - 1/2'"* is to deprive the series of
all terms whose denominators are multiples of 2. Thus
(1 - 1/2^'") a-2^ = 1 + 1/3'™ + 1/5"™ + . . . .
* It may, in fact, be easily shown that L<x^^=l when m=Qo; for, by
chap. XXV., § 25, we have the inequality l/(2m-l)>l/22'"-fl/32'« + l/42^
+ . . .>l/(2m-l)22»»-i, which shows that L(l/22»t + 1/32'" + 1/42'"+. . .)=0,
when m = CO .
t See liUrod. in Anal, luj., g 283.
864 PROPERTIES OF BERNOULLI'S NUMBERS CH. XXX
If we take the next prime, namely 3, and multiply
(1 - 1/2-"0 o-2» by 1-1/3^'", we shall deprive the series of all
terms involving multiples of 3 ; and so on. Thus we shall at
last arrive at the equation
(1 - 1/2^-) (1 - 1/3^-) (1 - 1/5^'") ... (1 - llp'^) a^
= l + l/g^'»+. . . (6),
where 2, 3, 5, . . . , p are the succession of natural primes up to
p, and q is the next prime to p. We may, of course, make q
as large as we please, and therefore l/g^'" + . . . (which is less
than the residue after the q — 1th term of the convergent series
o-2m) as small as we please. Hence
cr^™ = 1/(1 - 1/2^-) (1 - 1/3^'«) (1 - 1/5^'") . . . (7),
where the succession of primes continues to infinity. Hence
Bm = 2 (2m)!/(27r)^'» (1 - 1/2^"') (1 - 1/32"") (1 - 1/5^™) . . . (8).
§ 13.] Bernoulli's Numbers are all positive; they increase
after B^; and have oo for an upper limit.
That the numbers are all positive is at once apparent from
§ 11 (4). The latter part of the corollary may also be deduced
from (4) by means of the inequality of chap, xxv,, § 25. For
we have
l/(2w-l)>l/2-'" + l/3''"+l/42'"+ . . . > 1/(2ot - 1) 2-'"-^ (9).
Hence
■g^+i ^ {2m + 2) (2m + 1) o-g^+a
Br, (27r)V^
(2m + 2) (2m + 1) {1 + l/(2w + 1) 2'^'"+^}
^ (27r)^{l+l/(2m-l)}
(2m)'' -1
^ 47r2 •
Hence Bm+i/Bm>l, provided m>^(Tr^+\), that is, if
w>3-16. Now Bi>Bi, hence Bi<Bi<Bc< . . . .
Again, it follows from (9) that Lar^m = 1 when m = co, and
i/(2m)!/(27r)^'" is obviously infinite; hence LBm is infinite.
Cor. Bm/{2m)l ultimately decreases in a geometrical pro-
gression having for its common ratio l/Ait^. From which it follows
§§ 12-14 CONVERGENCE OF SERIES FOR TAN 0, &C, 365
that the series for tan 9, 6 cot 6, and 6 cosec 9, given in § 4, have
for their radii of convergence 9 = Jtt, tt and -n- respectively.
§ 14.] Turning now to the secant series, we observe that
42 ( - )"-' (2w - 1) 7r/{(2« - Ifn'' - A9^} does not, if treated in the
above way as it stands, give a double series satisfying Cauchy's
criterion, for, although when | ^ | < ^tt the horizontal series are
absolutely convergent after we replace 9 by \9\, yet the sum
of the sums of the horizontal series, namely, 42 (-)"~^ (2n - 1) tt/
{(2w - 1)V- - 4 1^1^}, is only semi-convergent. We can, however,
pair the positive and negative terms together, and deal with the
series in the form
(4n-3)7r {An-l)'n-
42
.(4» - 3)^2 - 4^2 (An - If-n-^ - 4.9
} (10),
,, ,. ^ ^ (47^-3)(4;z-l)7r^ + 4^^ .^^,
tnat IS, «7r2i 1(4^ _ 3)2^2 _ 4^2| 1(4,^ _ ^^^2^2 _ 4^2| (H).
Since (11) remains convergent when for 9 we substitute
1^1, it is clear that we may expand each term of (10) in as-
cending powers of 9, and rearrange the resulting double series
according to powers of 9. In this way we get
sec 6 = 4 J^ Ln?ii(4w-3)^™+^ " (l^T^lT+^J J 1^^^ '
= 2 2^'»+V2«+,e2"'/7r2'"+i (12),
7n=0
where r^m+i=^l/l^+'-l/S''^+'+ 1/5"-"'+'-. . . (13).
Comparing (12) with the series
sec9^1 + :S,Er„9'"'/{2m)l,
obtained in § 3, we see that
= 2 (2w)! y |p^ - 32^1 + 52^1+1 - .. .j (14),
which may be transformed into
£„ = 2(2».)! (?)""/(! + 3^-.) (1 - ^,1^) (1 . ^L-.) . . .
in the same way as before. (15)*.
See again Euler, Introd. in Anal. Inf., % 284.
366 PROPERTIES OF EULER's NUMBERS CH. XXX
Cor. 1. Eulers numbers are all positive; they continually
increase in magnitude, and have infinity for their tqyper limit.
For we have
l>T2„m>l-l/3'"'+' (16).
Hence
Ern+i ^ (2m + 2) (2m- + 1) 4t„„+3
■Em. TT^T,
>
2m+l
(2m + 2)(2m + 1) 4 (1 - 1/3'"'+')
But this last constantly increases with m, and is already
greater than 1, when m = l. Hence Ei<E2<Ei<. . . Also,
from (16), we see that Xram+i = 1 when W2 = (», and
L (2m)\ (2/7r)2'»+i = 00 , hence LEr^ = « .
Cor. 2. E,n/(2m)\ ultimately decreases in a geometrical
progression whose common ratio is A/tt^. Hence the radius of
convergence of the secant series is 0 = ^tt.
§ 15.] "We have, by § 11 (4),
— _L 4. _!_ 4. _— + = "^ TT^w n\*'
^2m— ,2m "^ 2^™ 3^™ T . . . (2m)[ '
and hence
o- on, = — „™ + -— + Tsr; + . . .—11 — -^ I — 7:z — r;- ^r ,
2/» j2m - 3-2«. - 52»» - • • • \^- 2''V (2w)!
(2'»--l)^^^,^
2 (2?»)!
2m J2TO 2^ 3'"* ' * ■ V 2'"/ (2w)!
(2'""-^-!)^^
(2m)!
(2);
(3).
Again, from (14) of last paragraph
nim+i • • • o2m+2/o,w,\f V /
■I2HI+1 Q2m+1 K2»H-1 • • • 2'^"'''^(27W)!
* The remarkable summations involved in the formula (1), (2), (3) were
discovered independently by John Bernoulli (see Op., t. iv., p. 10), and by
Euler (Comm. Ac. Petrop., 1740).
§§ 14-16
SUMS OF CERTAIN SERIES
367
Inasmuch as we have independent means of calculating the
numbers Bm and Em, the above formulae enable us to sum the
various series involved. It does not appear that the series 0-2^+1
can be expressed by means of Bm or Em', but Euler has cal-
culated (to 16 decimal places) the numerical values of 0-2^+1 in a
number of cases, by means of Maclaurin's formula for approxi-
mate summation*. As the values of o-^ are often useful for
purposes of verification, we give here a few of Euler's results.
It must not be forgotten that the formulae involving tt for o-^
are accurate when m is even j but only approximations when
m is odd.
0-2= 1-6449340668
0-3= 1-2020569031
0-4= 1-0823232337
0-5=1-0369277551
^6= 1-0173430620
0-^=1-0083492774
cr8= 1-0040773562
(r9= 1-0020083928
= 7rV6.
= 7rY25-79436
= 7rV90.
= 7r7295-1215
= 7r7945.
= 7rV2995-286
= 7r79450.
-'n-729749'35
EXPANSIONS OF THE LOGARITHMS OF THE
CIRCULAR FUNCTIONS.
§ 16.] From the formulae of §§ 6 and 7, we get, by taking
logarithms,
log sin ^ = log ^ + i log (1 - ^YwV^),
= \og6- 2 o-g^e^^/wTT^
(1),
since the double series arising from the expansions of the
logarithms is obviously convergent, provided |<9|<7r.
If we express 0-2^ by means of Bernoulli's numbers, (1) may
be written
log sin ^ = log 6 - 5 22'»-i5™62'7m (2m)! (1').
* Inst. Gale. Diff., chap. vi.
368 STIRLING'S THEOREM CH. XXX
The corresponding formulae for cos 6 are
log cos 6* = - 2 (2='" - 1) cr2^^'"/»?7r2'» (2) ;
= - 22^™-^ (2^'" - 1) Br^e^"'/m (2m)\ (2').
The like formulse for log tan 9, log cot 6, log sinh u, log cosh u,
&c. , can be derived at once from the above.
If a table of the values of a-om or of B^. be not at hand, the
first few may be obtained by expanding log (sin 0/0), that is,
log(l-^73! + ^Y^' ~ • • •)> ^Dtd comparing with the series
-2<r2m^^'"/w7r'^"*. For example, we thus find at once that
Stirling's theorem.
§ 17.] Before leaving this part of the subject, we shall give
an elementary proof of a theorem of great practical importance
which was originally given by Stirling in his Methodus Differen-
tialis (1730).
When n is very great, n\ approaches equality with J(2mr) (n/e)'^;
or, more accurately, when n is a large number, we have
n\ = J(27rn) {njeY exp [IjUn + 0} (1),
where - l/24»'< ^ < l/24» {n-\).
Since log {w/(w - 1)} = - log (1 - Ijn), we have
, w 1 1 1 1 , 1
^ n-\ n 2n^ Zri^ 4;i'* ' ' ' wzw'"
We can deprive this expansion of its second term by multi-
plying by w - 1. We thus get
/ 1 M w ^ 1 1 m - 1
In - h) log = 1 + -—— 2 + -.Tn + . . . + ^ — 7 — , iv m + • • • •
V ^/ ^n-1 I2n^ 12?r 2m {m + 1) li"'
Hence, taking the exponential of both sides, and writing suc-
cessively n, n-1, n-2, . . ., 2 in the resulting equation, we
deduce
/ n V-i /, 1 1
ijrri) ='''^V-'i2^^-'m'-"-
m-l \
"*■ 2w (tw + 1) «"* "^ ' ' 7'
§§ 16, 17 Stirling's theorem 369
/n-iy-'^-^ _ f 1 1
\n-2) -e^^Pi^l '^l2{n-lY'^12in-iy'^- ' '
m-1 _ _
2m'(m+l)(n^T)"' "^ '
\n-s) ~ ^""P V "*" 12{n-2y ^ 12 {n-2y ^ ' ' *
m — 1
2m(m+l){n-2y
/3\=^-* /, 1 1
(2) =^"p('"'i^^'-i2:
33+...
m— 1
+ -
2m {m + 1) 3''
(1) =^^p(^'"n:2-^^i2:2^^---
m—1
+
2m {m + 1) 2™
By multiplying all these together, we get
w^~i f 11
m — 1
S'm +
•} (2),
2m (m + 1)
where S'n,= 1/2™ + l/S*" + 1/4™ + . . . + 1/w™
Now
S'm = S^- l/{n + ir - 1/(71 + 2)'» - . . . (3),
where /Sf^= 1/2™ + 1/3™+ . . . + 1/w™ + . . . ad 00.
By the inequality (6) of chap, xxv., § 25, we have
l/{m - 1) M*"-^ > 1/(71 + 1)™ + l/(w + 2)™ + . . . > l/(m - 1) (w + l)'»-i.
Hence
>S'™ - l/(7n - 1) (71 + 1)™-^ > /S",„ > S,r, - l/{m - 1) n^-\
Therefore
12'^^''l2'^^''- • •'"2^(^+1)'^™'"- • •
• ^i^im-llSrr, ,g 1
^2m{m+l) ^2 fn{m + l)n"'-^ ^^'
^jJ^(m-l)Sr, ,g 1
^2 m{m + l) ^2 m {m + 1 ) (w + 1)'"-^ ^^^•
c. II. 24
370 STIRLING'S THEOREM CII. XXX
Since Sm<lKm - 1), the series 2 {m - 1) Snlm (m + 1) con-
verges to a finite limit which is independent both of m and of n.
Again,
i ^
2 7n{m + 1)m™~^
111 /a\
2.3n 3. 4:9V' 4.5n^
1 1 f, 1 1 1
(in 12n^ I w ?r J
^Qn'^12n(n-1) ^ ^'
Also, by (6),
1
.5/^1 l_^ L__
7Vw2 ?» + 1/(^ + 1)™-'*
00 ]^ °° 1
^ '2^(^ + 1)'" 2 (w + i)(w+ ir^^
= (^^l){-^-log(l-^0}
1 111 1^11^
1
2.3»
3
1
.in"
^T
1
. 5n^
1
^6n
1
12^2
(8).
Combining (2), (4), (5), (7), and (8), we have
-^>^^pf-i+^?mi^rri)~r27r24«(«-i)l (^>'
<exp ]w - 1 + |2^ . ^ tT- - tit- + o^T^^h (10).
^ t "* 2 ^ (^ + 1) 12» 24wv
§ 17 STIRLING'S THEOREM 371
Hence, putting
C=exp{l-iS<??,:ii^f"| (11).
so that C is a finite numerical constant, we have
«!>&-,.-» exp(i-^,) (12),
<ft-VHexp(i.^-j^) (13);
or, since the exponential function is continuous,
w! = (7^-^^«Hexp(^^ + ^) (14),
vfhere -l/2Aii^<e<l/24tn{n-l).
Hence, putting w = go on both sides of (14), we have
L7il = CLe'''n''+i (15).
The constant C may be calculated numerically by means of
the equation (11). Its value is, in fact, sJ{2Tr), as may be easily
shown by using Wallis's Theorem, § 6 (18).
Thus we have, when w = oo ,
TT ^ 2"'^(w!)^(2w+l) ^ J 2^"(w!)*(2?^ + l)
2~ P3^ . . . {2n+lf~' {(2w+l)!P *
Hence, using (15), we get
'^^_P2T 2'''e-*^n*^+^2n + 1)
2 e-^™-2(2w+l)^''+^ '
^C^j. ^
4 {(l + l/2»)^'»}2{l + l/2w}2'
4 e''*
Therefore, since C is obviously positive,
C=V(27r) (16).
Using this value of C in (14), we get finally
n\ = J{2Trn) {n/efexi^ {l/Un + e}* (17),
where -l/24w'<^<l/247i(w-l).
* An elementary proof that Ln\ = Ls/(2irn){nle)^ was given by Glaisher
(QiMrt. Jour. Math., 1878). In an addition by Cayley a demonstration of
the approximation (17) is also given ; but inasmuch as it assumes that series
24—2
372 EXERCISES XXIII CH. XXX
Cor. By combining (11) and (16), we deduce that
l-^S^^'"^^f'" = |log2 + Hog7r (18),
■^ 2 ^^(W+ 1)
where >S',„ = 1/2™ + 1/3™ + 1/4"' + . . . ad oo .
Exercises XXIII.
(1.) Show that, when |a;|>ir, a; cot a; can be expanded in the form
^o + S{B„a;~"+C„a;"); and determine the coefficients in the particular case
where Tr<.x<2ir.
(2.) Show that the sum of the products r at a time of the squares of the
reciprocals of all the integral numbers is 7r2''/(2r + l)!; and find the like sum
when the odd integers alone are considered.
(3.) Sum to n terms
tan ^ + tan (^ + tt/ji) + tan (^ + 27r/H) + . . .;
tan2^ + tan2(e + 7r/K) + tan2(^ + 27r/?j) + . . . .
Sum the following : —
(4.) 1/(12 + a;2) + 1/(22 + .t2) + 1/(32 + a;2) ....
(5.) l/a;2-l/(a;2-7r-^) + l/(x2-227r2)-. . . .
(6.) l/x + l/(x-l)+l/(a; + l) + l/(a;-2) + l/(.r + 2) + . . . .
(7.) 1/(1 - e) + 1/(4 -e) + 1/(9 -e) + . . . + l/(«2-^) + . . ,
(8.) 1/1. 2 + 1/2. 4 + 1/3. 6 + 1/4. 8 + . . . .
Show that
(9.) (a-2-6)/6 = l/12. 2 + 1/22. 3 + 1/32. 4 + . . . .
(10.) 7r/8-l/3 = l/1.3.5-l/3.5.7 + l/5.7.9-. . . .
(11.) If fr (m) be an integral function of n whose degree is r, show that
2/r ('*)/(2« - 1)^™ can be expressed in terms of Bernoulli's numbers, provided
r t> 2m - 2 ; and 2 ( - )"~^/r (n)/(2« - l)2"»+i in terms of Euler's numbers, pro-
vided r > 2m - 1.
In particular, show that
1 1 + 2 1 + 2 + 3 _^/^i_:^\
3;+ 54 + 74 +• • '-64 1, 12y*
(12.) Show that
Sl/(?i7r + ^)2 = cosec2tf;
00
S Hinir + ^)*= cosec*^ - ^ cosec2^,
n=0 being included among the values to be given to n. (Wolstenholme.)
of the form of 1/2"* + 1/3"* + . . . can be expanded in powers of 1/m, it cannot
be said to be elementary. The proofs usually given by means of the Mac-
laurin-sum-formula are unsatisfactory, for they depend on the use of a series
which does not in general converge when continued to infinity, and which can
only be used in conjunction with its residue. See Raabe, Crelle's Jour., xxv.
§§ 17, 18 EXERCISES XXIII 373
,j„ , g 1 _ VsJ'i sinh . vx^2 + sin . ■irx^2 _ 1
^ " ' xn* + x*~ 4x^ cosh . ■trXiJ2 - cos . irx^2 2x* '
(Math. Trip., 1888.)
(14.) Show that
I 1 ^ TT^ 1^
„=i{(2n)2-(2m-l)2}2 16 (2m -1)2 2(2»i-l)*'
„=i { (2n - 1)2 - (2m)2} « 64m2 *
Also that the sum of the reciprocals of the squares of all possible differ-
ences between the square of any even and the square of any odd number is
ir*/384.
(15.) If2?<n, show that
cosP^
1 "^1 sin ■ (2r + 1) 7r/2n . cos . P(2r + 1) 7r/2rt
COS nd n r=o cos 0 - cos . (2r + 1) 7r/2n
(16.) Show that
V "^ i V v \
tan-1 — S -^tan-i tau-^ V = tan-i (tanh v cot u) :
u n=i I nir-u nir + uj ^ ''
S -Itan-i j^ ^Y W - 1^"~^ 7;i r; ^ Y = ta^^"^ (tai^li '^ tan «)•
n=i I (2n-l)7r-2u (2n-l)7r + 2M) '
(Schlomilch, Handb. d, Alg. Anal., cap. xi.)
(17.) If X(x) = xn{l-(x/7ia)2}, /i(a;) = n{l-(2x/27i-la)2}, express
1 1
X(a5+o/2) in terms ot n(x), and also iJi,{x + al2) in terms of X(a;).
Hence evaluate i 1.3.5 .. . (2m-l)V(2ni + l)/2™m!.
(Math. Trip., 1882.)
(18. ) Show that, if r be a positive integer,
.£('-T(-r---('-'-^)"'"'='-«-
(19.) Show that
/ X x X \ — '^
(20. ) If n, p, X be all integers, prove
{n + x){n-^x + \) . . . (n+p + x-1)
L
(l + a:)(2 + a;) . . . (p + x)
REVERSION OF SERIES — EXPANSION OF AN ALGEBRAIC
FUNCTION.
§ 18.] The subject which we propose to discuss in this and
the following paragraphs originated, like so many other branches
of modern analysis, in the works of Newton, more especially in his
tract De Anahjsi per ^quationes Numero Terminwum Infinitas.
374 STATEMENT OF EXPANSION PROBLEM CH. XXX
Let us consider the function
%{m, n)af^y'^^{l, Q)x+{0, l)y + (2, 0)^^+(l, \)xy+{Q, 2)f+. . .,
where the indices m and n are positive integers, and we use tl\e
symbol (m, n) to denote the coefficient of ofy^, so that {m, n) is
a constant. We suppose the absolute term (0, 0) to be zero ;
but the coefficients (1, 0) (0, 1) are to be different from zero.
The rest of the coefficients may or may not be zero ; but, if the
number of terms be infinite, we suppose the double series to be
absolutely convergent when | ^r | = | ^^ | = l*. From this it follows
that the coefficient (w, n) must become infinitely small when m
and n become infinitely great ; so that a positive quantity \ can
in all cases be assigned such that | {m, n)\1^X whatever values we
assign to m and n. It also follows (see chap, xxvi., § 37) that
1 (m, n) x^'if' is absolutely convergent for all values of a; and 3/
such that I ic 1 ;t> 1, 1 2/ 1 ;:}> 1.
We propose to show tJiat one value, and only one value, of y as
a function of x can be found which has the following properties: —
1°. y is expansible in a convergent series of integral powers of
X for all values of x lying within limits which are not infinitely
narrow.
2°. y has the initial value 0 when x = Q.
3°. y mahes the equation
:^{m, n)x'^y'^ = 0 (1)
an intelligible identity.
Let us assume for a moment that a convergent series for y
of the kind demanded can be found. Its absolute term must
vanish by condition 2°. Hence the series will be of the form
y = biX + bnar -^-b-iX^ + . . . (2).
In order that this value of y may make (1) an intelligible
identity, it must be possible to find a value oi x<\ such that
(2) gives a value of y<l. The series (1), when transformed by
means of (2), will then satisfy Cauchy's criterion, and may be
arranged according to powers of x. All that is further necessary
* The more general case, when the series is convergent so long as | x | >■ o
and \y\>p, can easily be brought under the above by a simple transforma-
tion.
§18
GENERAL EXPANSION THEOREM
375
to satisfy condition 3° is simply that the coefficients of all the
powers of x shall vanish.
It will be convenient for what follows to assume that
(0, 1) = - 1 (which we may obviously do without loss of
generality), and then put (1) into the form —
y = {(1, 0)x + {2,0)x' + {S,0)a^ + . . . }
+ {(1, l)a; + (2, l)aP + {S, 1)^ + .
+ {(0, 2) + (1, 2)a; + (2, 2) x" + {3, 2) or" + .
+ {(0, n) + (1, n)x + (2, n) a^ + (3, n)x^ + .
(3).
Using (2), we get
= {(1, 0)^ + (2, 0)ir^ + (3, 0)V + ,
+ {{l,l)x+{2,l)a^ + {3, l)a? + .
+ {(0, 2) + (1, 2)x + (2, 2)a^ + (3, 2)a? +
,\{hi+hiX+h3x'^+. . .}V
+ {(0,w)+ {l,n)x + {2,n)iu^ + {3,n)x'-^. . .]{h^+h^x+b.iX'+. . .fx""
(4).
Hence, equating coefficients, we have
&i=(l,0),
&2=(2, 0) + (l, 1)6, +(0, 2)^>l^
«>3=(3, 0) + (l, 1)62 +(2, l)&i +(l,2)^>i»+2(0,2)JA + (0, 3)6x«,
^'^ = {n, 0) + (1, l)&„-i + (2, 1) hn-^ + . . . + (0, w) &i"
(5).
Here it is important to notice that each equation assigns one
of the coefficients as an integral function of all the preceding
coefficients. Hence, since the first equation gives one and only
one value for &i, all the coefficients are uniquely determined.
There is therefore only one value of y, if any.
In order to show that (5) really affi)rds a solution, we have to
show that for a value of x whose modulus is small enough, but
not infinitely small, the conditions for the absolute convergency
of (2) and (4) are satisfied when hi, 62, • • • have the values
assigned by (5).
376 GENERAL EXPANSION THEOREM CH. XXX
This, following a method invented by Cauchy, we may show
by considering a particular case. The moduli of the coefficients
of the series (3) have, as we have seen, a finite upper limit X.
Suppose that in (3) all the coefficients are replaced by A, and
that a^ has a positive real value <1. Then we have
+ X.{x + aP + a^ + . . . ]y
+ X{1 + x + a^ + x^ + . . . ]y^
(6).
This series is convergent so long as a;<l and |2/|<1. It
can, in fact, be summed ; for, adding k + \y to both sides, we get
(l+X)y + X=^Xl{l-x){l-y),
that is, (1 + X)y'^ -y + Xa;/(l -ic) = 0.
Hence, remembering that the value of y with which we are
concerned vanishes when a; = 0, we have
y=[l-J{l- U (1 + X) w/{l - x)}]l2 (X + 1) (7).
Now, provided 4X (1 + X) a;/(l -a;)<l, that is, a;<l/{2X + If,
the right-hand side of (7) can be expanded in an absolutely con-
vergent series of integral powers of a;, the absolute term in which
vanishes. Also, when ^<l/(2X+l)^ the value of y given by
(7) is positive and < 1, therefore the absolute convergency of (6)
is assured.
It follows that the problem we are considering can be solved
in the present particular case. If we denote the series for y in
this case by
y = G-^x + C^a^ + C^x^ + . . . (8),
then the equations for determining (7i, Co, Cs, . . . will be
found by putting (1, 0) = (2, 0) = (1, 1) = . . . = A in (5), namely,
c,=x,
Cn = X(l + Cn., + Cn-. + . . . + C^%
(9);
from which it is seen that Ci, C^, C^, . . . are all real and
positive.
§ 18 GENERAL EXPANSION THEOREM 377
Returning now to the system (5), and denoting moduli by
attaching dashes, we have, since (1, 0)', (2, 0)', &c., are all less
than X,
&2>(2, 0)' + (l, l)V + (0, 2)'W'<X{1 + C^ + C^')<C„
h'XS, 0)'+(l, l)'6,'+(2, 1)'^'/+(1, 2)V+2(0, 2yWW+{0, 3)V,
. . . . . . ■ _■ y . . (10).
Hence the moduli of the coefficients in (2) are less than the
moduli in the series (8), which is known to be absolutely con-
vergent. It therefore follows that the series (2) will certainly be
absolutely convergent, provided \a;\< 1/(2A + 1)^.
It only remains to show that x may be so chosen (and yet
not infinitely small) that y as given by (2) shall be such that
y'<l. We have
i/'<bia;' + b2a;"^ + b3'a;'^ + . . .,
<Cia;'+C2x'^+Csa:'' + . . .,
<[\-J{l-AX{l+X)x'l{l-x')]]l2{X+l) (11).
Now the right-hand side of (11) is less than 1, provided
x'<ll{2k + l)\ If, therefore, | ^ [ < 1/(2X -t- 1)"^ the absolute
convergency of the double series (3) or (4) will be assured ;
and (2) will convert (1) into an intelligible identity.
We have thus completely established that one and only one
value of y expansible within certain limits as a convergent series
of integral powers of w can be found to satisfy the equation (1) ;
and the like follows for w as regards y. The functions of x and y
thus determined, being representable by power-series, are of course
continuous. The limits assigned in the course of the demonstra-
tion for the admissibility of the solution are merely lower limits ;
and it is easy to see that the solution is valid so long as (2) itself
and the double series into which it converts the left-hand side of
(1) remain absolutely convergent.
It should be remarked that we have not shown that no other
power-series whose absolute term does not vanish can be found to
satisfy (1) ; nor have we shown that no other function having
zero initial value, but not expansible in integral powers of x, can
378 REVERSION OF SERIES CH. XXX
be found to satisfy (1). We shall settle these questions presently
in the case where the series 2 {m, n) x^y^ terminates.
§ 19.] The problem of the Reversion of Series properly so
called is as follows : —
Given the equation
x^aa + aray'^ + ara+iy'^^^ + . . . (1),
where ttm + O, hut Uq may or may not be zero, and tJie series
«m y^ + (f"m-¥\ 'iT"^^ + . . . is absolutely convergent so long as
\y\^ a fixed positive quantity p, to find a convergent expansion,
or convergent expansions, for y in ascending powers of x-a^.
Let t denote {{x-a^jar,^'^'^, that is, the principal value of
the mth root of {x-a^jam, and w^ a primitive mi\x root of
unity, then (1) is equivalent to m equations of which the
following is a type : —
-mn=y(i-^f^y^'^f^..T (2).
Now, the series inside the bracket in (2) being absolutely
convergent for all values of y such that l^/l^^p, it follows from
the binomial theorem combined with § 1 that we can, by taking y
within certain limits, expand the right-hand side of (2) in an
ascending series of powers of y. We thus get, say,
-i>>ra'^+y+C,f+G,f + , . .=0 (3).
It follows, therefore, from the general theorem of last para-
graph that we have, provided | $ \ does not exceed a certain
limit,
y=b^<oji+ho>^^e+bs<^j'-e+. . . (4).
We have, of course, m such results, in which the coefficients
bi, bz, bs, . . . will be tJie same, hut r will have the different
values 0, 1, 2, . , . , (w - 1).
Each of these solutions is, by chap, xxvi., § 19, a continuous
function of x. If we cause x to circulate about a^ in Argand's
Diagram, the m branches of y will pass continuously into each
other; and after m revolutions the branches will recur. The
point tto is therefore a Branch Point of the wth order for the
function y, just as the point 0 is for the function w^-"* in
chap. XXIX., ^ 5, 6.
§§ 18-20 EXAMPLES OF EEVERSION 379
Cor. In the particular case where ao = Oj m — l, we get the
single solution
y-=hiX + h2CC^ + ha^ + . . . (5).
Example. To reverse the series
2 = l + 2//l!+2/2/21 + 2/3/3! + . . . (6).
Let z = l + x, then we have
Hence, provided \x\ lie within certain limits, we must have by the
general theorem
y = bj^x + b^x'^ + bgx^ + . . . (8).
Knowing the existence of the convergent expansion (8), we may determine
the coeflBcients as follows.
Give y a small increment k, and let the corresponding increment of a; be ft;
then, from (7), we have
{y + k)-y {y + kf-y^ , (y + lcf-y^ ,
n- jj + 2! + 3! +• • • •
Hence, since L{{y + k)'^-y^}lk = ny"'-'^ when k = 0, and since, owing to
the continuity of the series as a function of y, h = 0 when fc = 0, we have
^fc-^ + l] + 2'! + - • ••
= l + x (9).
Again, from (8), we have, in like manner,
k
Z,^ = 6i + 262a; + 3&3a;2+. . . (10).
Combining (9) and (10), we have
61 + 262X + 363x2 + . . , = l/(l + x),
= l-x + x'^ - . . . .
We must therefore have
61 = 1, &2=-l/2, 63 = 1/3
Therefore
X x' x*
1 2 3
- 1 2 + 3 ■ ■ • ("^
It must be remembered that (11) gives only that branch of the function y
which is expansible in powers oi z-1 and which vanishes when 2 = 1. We
have, in fact, merely given another investigation of the expansion of the
principal value of log 2.
§ 20.] Expansions of the various branches of an Algebraic
Function.
The equation
5 {m, n) ^'"v/" + (0, 0) = 0 (1),
380 DEFINITION OF ALGEBRAICAL FUNCTION CH. XXX
where tJie series cm the left terminates, gives for any assigned value
of a; a finite number of values of y. If the highest power of y
involved be the wth, we might, in fact, write the equation in the
form
A^y' + J.„-,y^-^ + . . . + ^12/ + ^0 = 0 (2),
where A^, A^, . . ., An are all integral functions of ^. If, then,
we give to x any particular value, a, real or complex, it follows
from chap, xii., § 23, that we get from (2) n corresponding values
of y, say bi, h, . . ., bn. If we change a; into a value a + h
differing slightly from a, then bi, h, . . ., b^ will change into
bi + h, b2 + k2, . . .,bn + kn; that is to say, we shall get n values
of y which will in general be ditferent from the former set. We
may therefore say that (2) defines y as an w-valued function of
X ; and we call y when so determined an algebraic function of x.
Since every equation of the form y = F{x), where F{x) is an
ordinary synthetic irrational algebraic function (as defined in
chap. XIV., § 1), can be rationalised, it follows that every ordinary
irrational algebraic function is a branch of an algebraic function
as now defined. Since, however, integral equations whose degree
is above the 4th cannot in general be formally solved by means
of radicals, it does not follow, conversely, that every algebraic
function is expressible as an ordinary synthetic irrational alge-
braic function.
In what follows we assume that the equation (2) contains (so
long as X and y are not specialised) no factor involving x alone
or y alone. We also suppose that, so long as x is not assigned,
the equation is Irreducible, that is to say, that it has not a
root in common with an integral equation of lower degree in y
whose coefficients are integral functions of x. If this were so, a
factor could (by the process for obtaining the G.C.M. of two
integral functions) be found having for its coefficients integral
functions of x, and the roots of the equation formed by equating
this factor to 0 would be the common root or roots in question.
Therefore the equation (2) could be broken up into two integral
equations in 3/ whose coefficients would be integral functions of a?;
and each of these would define a separate algebraic function of x.
The condition of irreducibility involves that (2) cannot have
§ 20 SINGULAR POINTS 381
two or more of its roots equal for all values of x. For, if (2)
had, say, r equal roots, then, denoting all the roots by
2/i, ^2, ' • ', Vn, the equation
'^iy-yi){y-y-^ • • • {y-?/s-i)(y-?/s+i) . • • (y-i/n)^o (3)
would have r-1 roots in common with (2), for r - 1 equal
factors would occur in each of the terms comprehended by 2.
Now the coefficients of (3) are symmetric functions of the roots
of (2) ; therefore (3) could be exhibited as an equation whose
coefficients are integral functions of-4o, ^i, • . -, An, and there-
fore integral functions of w*. Hence (2) would be reducible,
which is supposed not to be the case.
It must, however, be carefully noticed that irreducibility in
general (that is, so long as a; is not specialised) does not exclude
reducibility or multiplicity of roots for particular values of x. In
fact, we can in general determine a number of particular values
of X for which (2) and (3) may have a root in common t. In
other words, it may happen that the n branches of y have points
in common ; but it cannot happen that any two of the n branches
wholly coincide.
When, for x = a, the n values bi, b2, . . ., 6^ are all different,
a (or its representative point in an Argand-diagram) is called an
ordinary point of the function i/, and 6i, 62, . . ., 6» single values.
If fti = ^2 = • ■ . = br, each =b, say, then a is called an r-ple point
of the function, and b an r-ple value.
For every value of a; (zero point) which makes Ao = 0, one
branch of y has a zero value ; for every value of a; (double zero
point) which makes ^o = 0 and -4i = 0, two branches have a zero
value ; and so on. These are called single, double, . . . , zero
values.
For every value of x (pole) which makes ^^ = 0, one branch
of y has an infinite value ; for every value of x (double pole)
which makes An = 0 and An-i. = 0, two branches have an infinite
* See chap, xviii. , § 4.
t These are the values of x for which
and n^„2/«-i + (w-l)^„_i2/"-- + . . .+^i = 0
have a root iu common.
382 EXPANSION AT AN ORDINARY POINT CH. XXX
value; and so on. These may be called single, double, . . .,
wfinities of the function.
The main object of what follows is to show that every branch
of an algebraic function is {within certain limits), in the neigh-
bourhood of every point, expansible in an ascending or descending
power series of a particular kind; and thus to show that every
branch is, except at a pole, continuous for all finite values of x.
§ 21.] If , at the point x = a, the algebraic function y has a
single value y = b, then y-b is, within certain limits, expansible
in an absolutely convergent series of the form
y-b=^Ci{x-a) + C2{x-aY+Cs{x-ay + . . . (4).
Let x = a + $, y = b + i], then the equation (1) becomes, after
rearrangement,
(0, 0) + (l, 0)^+(0, l)77 + (2, 0)^^ + &c. = 0 (5).
Since y = b is a single root of (1) corresponding to x = a, it
follows that when ^=0 (5) must give one and only one zero
value for r]. Therefore we must have (0, 0) = 0 and (0, 1) + 0.
It follows, from the general theorem of § 18, that within
certain limits the following convergent expansion,
V = C^$+C,e+Cd' + . . .,
and no other of the kind will satisfy the equation (5) ; that is, .
y = b + Ci{x-a) + C2{x-af+Cs{x-ay + . . . (6)
will satisfy (1).
The function y determined by (6) is continuous so long as
I a? — a 1 is less than the radius of convergency of the series
involved ; and it has the value y = b when x = a.
If we suppose all the values of y, say bi,b2, • . ., bn, corre-
sponding to « = a to be single, then we shall get in this way for
each one of them a value of the function y of the form (6).
Hence we infer that
Cor. So long as no two of the branches of an algebraic function
have a point in common, each branch is a continuous function of x ;
and the increment of y at any point of a particular branch is ex-
§§ 20-22 EXPANSION AT A MULTIPLE POINT 383
pansible in an ascending series of positive integral powers of the
increment of a; so long as the modulus of the increment of x does
not exceed a certain finite value.
§ 22.] We proceed to discuss the modification to which the
conclusions of last paragraph are subject when ^ = a is a multiple
point of the function y.
We shall prove that for every multiple point of the qth order, to
which corresponds a q-ple value y = h, we can find q different con-
vergent expansions for y of the form y = h-^%Cr{x- aY, where the
exponents rform a series of increasing positive rational numbers.
It will probably help the reader to keep the thread of the
somewhat delicate analysis that follows if we premise the follow-
ing remarks regarding expansibility in ascending power-series
in general : —
If t) be expansible in an absolutely convergent ascending
series of positive powers of ^, of the form
^ = C^^o., + C;^a.+a, + C^i'^,^''^-'-, + . . . (A),
where oi, aj, . . . are all positive, then we can establish a series
of transformations of the following kind : —
■n^^^'iCi + yii), rj^ = t^{C, + rj2), v^^i^'iCs + Vs), . . .,
Vn-l^^'^HC'n + Vn) (B),
where i/i, v^, ' ' •, Vn all vanish when ^=0; Ci, C2, . . ., Cn
are all independent of $, and all different from zero ; and
C^L-nli'^^, C^^Lr^.li'^^, . . ., C„ = i:%_,/^' when ^ = 0.
Conversely, if we can establish a series of transformations of
the form (B), and if we can show that yjn is expansible in a series
of ascending positive powers of ^, it will obviously follow that r]
is expansible in the form (A).
Let now y = 6 be a g-ple value of y corresponding to x = a,
and put as before x = a + ^, y = b + y], then the equation (1)
becomes
% {m, n) e't = 0 (7).
384 EFFECTIVE GROUP OF TERMS CH. XXX
Since q values of y become b when x-a, q values of 17 must
become 0 when ^ = 0. Hence the lowest power of 77 in (7)
which is not multiplied by a power of ^ must be yf. There
must also be a power of $ which is not multiplied by a power of
7], otherwise (7) would be divisible in general by some power of
17, which is impossible since (1) is irreducible. Let the lowest
such power of t be |^.
Put now
^ = IMCi + 770 = ^"^ (8),
and let us seek to determine a positive value of A. such that
Ci=-Lv = Lr]/i^ is finite both ways* when ^ = 0.
The equation (7) gives
^(ot, w)^™+^"'»'' = 0 (9).
Now (9) will furnish values of v which are finite both ways when
1=0, provided we can so determine A that at least two terms of
(9) are of the same positive degree in $, and lower in degree
than all the other terms.
Assume for the present that we can find a value of A for
which a group of r terms has the character in question, so that
8 = mi + Xni = m2 + ^n2 = . . .^mr + ^rir (10),
where iii^n^^ . . . :|> w^ ;
and X = (twi - mr)/{nr - Wi) = g/h, say, (11),
where g is prime to h,
8 - {mji + nxg)lh. •
Then, putting ^i = ^^'\t so that 4 = 0 when ^ = 0, and dividing
out |^% ''+«!«', we deduce an equation of the form
</> (4 , V) 4 + (^r, nr) V'r + {mr-i, Wr-l) ^"'-^ + . • . + (w2i , Wj) V"l = 0
(12),
where <^ (li, v) is an integral function of ^ and v.
For our present purpose we are concerned only with those
* That is, neither zero nor infinite — a useful phrase of De Morgan's,
t It is sufficient for our purpose to take the principal value merely of the
hth root of ^.
§ 22 EFFECTIVE GROUP OF TERMS 385
roots of (12) whose initial values are finite both ways. There are
evidently rir - n^ such roots, and their initial values are given by
(wr , rir) ®"'--"i + {rrir-u nr-i) v'^r-i-H + . . . + (7^^J, m) = 0
(13).
If the roots of (13) are all different, then we get rir — ni trans-
formations of the form (8) ; and the corresponding values of v,
that is, of Ci + 771, are given by the algebraical equation (12).
Moreover, since all the values of v are single, we shall get for
each value of -rji an expansion of the form
= d^^'i'' + d^e^^ + . . . (14);
and each of these will give for r] a corresponding expansion of
the form
^ = Ci^«'/* + <?i^(''+i)/'' + (?2^(''+2)/n. . . (14').
If a group of the roots of (13) be equal, then we must
proceed by means of a second transformation,
Vl = ^l''{C2 + V2) (15),
to separate those roots of (12) which have equal values. If the
next step succeeds in finally separating aU the initial values,
then we have for each of the group of equal roots of (13) two
transformations (8) and (15), and finally an expansion like (14'),
the result being the final separation of all the iir-ni roots of
(12), with convergent expansions for each of them.
Moreover, we must in every case be able, by means of a
finite number of transformations like (8) and (15), to separate
the initial values, otherwise we should have two branches of y
coincident up to any order of approximation, which is impossible,
since (1) is irreducible.
The indices in the series (14') may be all integral or else
partly or wholly fractional (see Examples 2 and 1 below).
In the former case the corresponding branch of the function
7] is single-valued in the neighbourhood of tlie point ^ = 0 ; that
is to say, if we cause i to circulate about the point $ = 0 and
c. II. 25
386 NEWTON'S PARALLELOGRAM CH. XXX
return to its original position, 17 returns to the value with which
we started.
If some or all of the indices be fractional, the series will take
the form
rj = c,t'^ + c^m + Csiy'^ + . . . (14"),
where one at least of the fractions a/q, ^/q, . . . , is at its lowest
terms. The function 77 is then ^'-valued and the series (14")
will as in § 19 lead to a c^/cle, as it is called, of q branches
which pass continuously into each other when i is made to
circulate q times round ^ = 0. At any multiple point there
may be one or more such cycles ; and for each of them the
point is said to be a branch point of the qth order, q being the
number of branches belonging to the cycle.
All that now remains is to show that we can in all cases
select a number of groups of terms satisfying the conditions (10)
sufficient to give us q expansions corresponding to the q branches
which meet at the q-p\e point a; = a.
The best way, both in theory and in practice, of settling this
point is to use Newton's Parallelogram, which is constructed as
follows : — Let OX and OF (Fig. 1) be a pair of rectangular axes,
the first quadrant of which is ruled into squares (or rectangles)
for convenience in plotting points whose co-ordinates are positive
integers. For each term {m, n) i'^rf in equation (7) we plot a
point K {degree-point) whose co-ordinates are 0M= m, MK- n.
We observe that, if KF be drawn so that cot KPO = k, then
OP = 0M+ MP = m + nX. Hence OP is the degree in i of the
term in (9) which corresponds to (m, n) i'^r]\ If, therefore, we
select any group of terms whose degree-points lie on a straight
line A, these will all have the same degree in ^, namely, the
intercept of A on OJT.
The necessary and sufficient conditions, therefore, that a
group of two or more terms furnish the initial values of a group
of expansions, let us say be an effective group, are ; —
1°. That the line A containing the degree-points shall cut
OX to the right of 0, and 0 Y above 0. This secures that X be
positive.
§22
NEWTON S PARALLELOGRAM
387
2°. That all the other degree-points shall lie on the opposite
side of A to the origin. This secures that all the other terms in
(9) be of higher degree in i than those of the selected group.
Y
Y,
X,
>t
^
^
f.
■^
V,
■K
\
x,i
/
F
^
s
1
/
H
\,
^
J
K
\,
\
s
\
V
s
V
\
N
V
\
s
1
\
p
s
/
^.
L
vj
S
s
/
\
N
c
/
\
^
■>^i
B
/
s
c
1
-~
■^1
' .
k
>
fs
r.
1
^
h
p
X 1
1
Yl
v^
Fig. 1.
We have thus the following rule for selecting the effective
groups : —
Let A and E be the degree-points of the terms that contain
i and rj alone, so that OA^p, OE=q. Let a radius vector,
coinciding originally with AX^, turn clock-wise about A as
centre until it passes through another of the degree-points B.
If it passes through others at the same time as B, let the last of
them taken in order from A be 0. Next, let the radius turn
about C as centre in the same direction as before, until it passes
through another point or points, and let the last of this group
taken in order from G be D. Then let the radius turn about D ;
25—2
388 EXAMPLE OF EXPANSION CH. XXX
and so on, until at last it passes through E, or through a group
of which E is the last.
We thus form a broken line convex towards 0, beginning at
A and ending at E, every part of which contains a group of
degree-points the terms corresponding to which satisfy the
conditions (10).
Now the degree of the equation (13) corresponding to any
group CD is the difference between the degrees of -q in the first
and last terms C and D ; but this difference is the projection of
CD on OY. The sum of all the projections oi AG, CD, &c., on
0 F is OE, that is to say, q. Hence we shall get, by taking all
the groups AC, CD, &c., q different expansions for 3/ correspond-
ing to the q different branches that meet at the multiple point
a; = a. Each one of these has the same initial value b, and each
is represented by a separate expansion in positive ascending
rational powers of a; -a.
Example 1. To separate the branches of the function 17 at the point ^=0,
7] being determined by
+ Hp7?^2=o. (16).
The lowest term in r/ alone is y, so that ^=0 is a multiple point of the
10th order. Plotting the degrees of the terms in Newton's diagram, and
naming the points by affixing the coefficients, we find (see Fig. 1) that the
effective groups are ABC, CD, DE. Taking, for simplicity of illustration,
A=+2, B=-3, C=+l, D=-l, E=+l,
we get from the group ABC
X= 6/2 = 3/1, so that h=l, and t)2-3i; + 2=0 gives the initial values of w,
that is, v = l, or 2, the corresponding expansions being
V=^^l + d,^ + d^^^+. . .),
From the group CD, we get
X = 4/3, u* - 1 = 0 gives the initial values of v,
that is, r = l, u, u^, where w is a primitive imaginary cube root of 1, the
corresponding expansions being
§ 22 EXAMPLE OF EXPANSION 389
In like manner, DE gives five expansions of the type
where a is any one of the five 5th roots of 1.
All the ten branches are thus accounted for ; and they fall into cycles of
the orders 1, 1, 3, 5.
Example 2. To separate the branches of 97 at the point ^=0, 7; being
determined by
4f5-3$*-4,t2(^_^) + 4(^-|)2 = 0 (17).
The effective group for (17) at the point f =0 corresponding to branches
which have the initial value 7; = 0 is 4(?;-$)2; as will be readily seen from
Newton's diagram.
X=l, h=l and, it ■n=H^i + '^i)=^'"> '^^ ^^^°
4^-3^-4f(v-l) + 4(v-l)2=0 (18).
Hence two branches have the same initial value for v, viz. v = l. For
each of these »; = ^ (1 + iji) ; and we have for 17^ the equation
4^3_3^2_4^^^ + 4^^2^0 (18').
If we draw Newton's diagram for (18'), we find that the effective group is
47;i3 - 4f 77i - 3^ ; and that X = l. Fnt now 7]^=^ {0^ + 712) = ^Vi', and we get
4^ + (2i;i-3)(2vi + l) = 0 (19).
The initial values of v^ are given by (2vj-3) (2vi + l) = 0, which give the
single values ^^ = 3/2, v^=: - 1/2. Hence for the two branches we have
'7i=?(3/2 + %); Vi=H-h + V-2');
and the farther procedure will lead to integral power series for 772 and 173 .
We have therefore for the two branches
V = ^ + 3e'l2 + C\e+. . .;
and the double point is not a branch point on either.
It should be observed that, if we form an integral equation
by selecting from any given one a series of terms which form an
effective group, the new equation gives an algebraic function.
Those branches of this function that have zero initial values
coincide to a first approximation (that is, as far as the first term
of the expansion) with certain of the branches of the algebraic
function determined by the original equation which have initial
zero values. Thus, reverting to Example 1 just discussed,
from the group ABC vfe have
Ae' + Bt^'-q + cerf = 0.
This gives, when we drop out the irrelevant factor ^,
Cf + Bev + Ae = 0,
390 ALGEBRAIC FUNCTION ALWAYS EXPANSIBLE CH. XXX
which breaks up into two equations,
and thus determines two functions, each of which has a branch
coincident to a first approximation with a branch of -q (as deter-
mined by (16)) which has zero initial value.
In like manner, CB gives Gi^^Drf = ^\ and BE gives
B^+Erf=^.
We thus get a number of binomial equations, each of which
gives an approximation for a group of branches of the function
i\ determined by (16). We shall return to this view of the
matter in § 24.
§ 23.] Before leaving the general theory just established, we
ought to point out that Newton's Parallelogram enables us to
obtain, at every point {singular or non-singular), convergent
expansions for every branch of an algebraic function in ascending
or descending power-series, as the case may be.
To establish this completely, we have merely to consider the
remaining cases where x or y or both become infinite.
1st. Let us suppose that the value of the function y tends
towards a finite limit b when x tends towards oo . Then, if we
put r]=y — b,x = ^,we shall get an equation of the form
^(m,n)i^rj^ = 0 (17),
which gives i; = 0 when ^= oo .
Let us suppose that Fig. 1, as originally constructed, is the
Newton-diagram for (17), and let i" be the highest power of i
that occurs in (17) so that 003 = k. Now in (17) put i=l/i',
and multiply the equation by $'^ ; we then get the equation
S(«i, w)P-™>7'*=0 (18),
which is obviously equivalent to (17).
But the Newton-diagram for (18) is obviously still Fig. 1,
provided OsXg and O3F3 be taken, instead of OJC and OY, as
the positive parts of the axes.
Hence, if we make a boundary convex towards O3 in the
same way as we did for 0, we shall obtain a series of branches
of rj all of which are expansible in ascending powers of $', that
§§ 22, 23 EXPANSION AT POLES, &C. 391
is, in descending powers of i, and all of which give »; = 0 when
^ = CO . For each such branch we have
that is,
(i/-b)a^ = c + d/iif + e/a;^ + . . . (19), ,
where A, a, y8, . . . are all positive, and c is finite both ways.
2nd. Suppose that w = a is a pole of y, so that 2/ = qo when
a; = a; and put r] = i/, ^ = x-a, so that we derive an equation
%{m,n)e^7f = Q (20),
for which Fig. 1 is the Newton-diagram with OX and 0 F as
axes. Then, putting t] - l/rj', we get an equation of the form
2 (m, n) e'r,''-'' = 0 (21),
I being the highest exponent of t] in (20).
The Newton-diagram for (21) is then Fig. 1 with O^X^
and OiFi as axes; and we construct, as before, a boundary,
EFG say, convex towards Ox, every part of which gives a series
of branches of rf, that is, of !/>?, expansible in ascending powers
of ^. For every such branch we shall have
■qe'=ll{c + di'' + e^^ + . . .),
where \, a, /3, . . . are all positive, and c is finite both ways.
Hence also, by the binomial theorem combined with § 1,
rii''=llc + d'i'' + ei^' + . . .,
that is,
i/{a^-a)^--l/G + d'{x-af + e'(a;-ay+. . . (22),
where \, a, /3', . . . are all positive, and c is finite both ways.
3rd. Suppose that 3/ has an infinite value corresponding to
^ = 00 (pole at infinity). Then, if we put x = ^ = 1/$', 2/ = '^ = i/v,
we shall get, by exactly the same kind of reasoning as before, a
boundary GHI convex to O2, each part of which will give a
group of expansions of the form
r)' = i'^{o + di"' + ee^ + . . .}.
Whence, as before, for every such branch
if/a;^ = l/(c + d/x"^ + e/afi + . . . ),
= l/c + d'/w'' + 6'/afi' + . . . (23),
where X, a, /3', . . . are all positive, and c is finite both ways.
392 ALGEBRAIC ZEROS AND INFINITIES CH. XXX
If we combine the results of the present with those of the
foregoing paragraphs, we arrive at the following important
general theorem regarding any algebraic function y : —
Ify = ^ when x^a (a+co), tJien L yl{x-aY is finite both
ways.
Xfy = 0 when x=cc , then L y\x~^ is finite both ways.
a;=oo
Ify = CO when x-a(a^cc), then L yl{x - a)~^ is finite both
ways.
Jfy= CO when x— cc , then L yjx^ is finite both ways.
X=to
X is in all cases a finite positive commensurable number
which may be called the ordee of the particular zero or infinite
value of y.
This theorem leads us naturally to speak of algebraical zero- or infinity-
values of functions in general, meaning such as have the property just
stated. Thus sin .r = 0 when a; = 0 ; but L sin a;/a;=l when x = 0; therefore
we say that sin a; has an algebraic zero of the first order when x=0. Again,
tana; = QO when x = ^tr; but Ltanxl{x-^ir)~^ is finite when x = ^ir ; the
infinity of tanx is therefore algebraical of the first order. On the other
hand, e^=oo when x = oo ; but this is not an algebraical infinity, since no
finite value of \ can be found such that Le^jx^ is finite when a; = Qo . (See
chap. XXV., § 15.)
§ 24.] Application of the method of successive approxima-
tion to the expansion of functions. This method, when applied
in conjunction with Newton's diagram, greatly increases the
practical usefulness of the general theorems which have just
been established. The method is, moreover, of great historical
interest, because it appears from the scanty records left to us
that it was in this form that the general theorems which we have
been discussing originated in the mind of Newton.
Let us suppose that the terms of an equation (which may be
an infinite series) have been plotted in Newton's diagram, and
that an effective group of terms has been found lying on a line
A ; and let rT - 1" (the coefficients are taken to be unity for
simpHcity of illustration) be a factor in the group thus selected,
repeated, say, a times, so that the whole group is (f)i($, ri){rf—^Y.
Let A be moved parallel to itself, until it meets a term or group
§§ 23, 24 SUCCESSIVE approximation 393
of terms <^a {$, rj) ; then again until it meets a group ^3 (i, -q) ;
and so on.
The complete equation may now be arranged thus —
<t>i(i.v){r-er + M^,v)+<f>s{i,v)+. . .=0,
or thus —
say, {v'^-ty + r., + Ts+ . . .=0.
Now, by the properties of the diagram, when -q = ^"^™,
^2 (i, v), ^3 (^> v), • ■ ' a.re in ascending or descending order as
regards degree in ^, and the same is true of xg, T3, . . . Let
us suppose that ^ and rj are small, so that t^, tj, . . . are in
ascending order.
As we have seen, yp^i^, that is, 7; = ^"^'", gives a first
approximation. To obtain a second, we may neglect tj, T4, . . ,,
and substitute in xg the value of 77 as determined by the first
approximation. To get a third approximation, neglect t^, . . .,
substitute in xg the value of rj as given by the second approxima-
tion, and in xj the value of rj as given by the first approximation.
We may proceed thus by successive steps to any degree of
approximation ; the only points to be attended to are never to
neglect any terms of lower degree than the highest retained,
and not to waste labour in calculating at any stage the co-
efficients of terms of higher degree than those already neglected.
There is a special case in which this process of successive
substitution requires modification. We have supposed above
that the substitution of the first approximation, rj — i'^'^, in Xg
does not cause Xg to vanish, which will happen, for example,
when <f>2{^, v) contains rj'^-i"' as a factor. In such a case the
beginner might be tempted to put xg = 0 and go on to substitute
the first approximation in xj. This would probably lead to error.
For, if we were to substitute the complete value of rj in Xg, it
would not in general vanish, but simply become of higher order
than is indicated in Newton's diagram, of the same order
possibly as xj. The best course to follow in such cases may be
learned from Example 5 below, which deals with a case in point,
394 EXAMPLES CH. XXX
Example 1. Taking the equation (16), to find a third approximation to
one of the branches of the group CD.
Next in order to G and D a parallel to CD meets successively B and A.
Hence, putting, for simplicity, D=+l, C=:Bz=A=-l, the equation (16)
may be written
Whence 17' - ?* - f /t? - ^^^h^ + . . . = 0 (25) .
The first approximation is 7] = ^*l^; hence, neglecting f^'/'?' i^ (25), we get
for the second
Whence ,7 = ^4/3(1 + ^5/3)1/3=^4/3(1 + 1^5/3) (26).
If we use this second approximation in ^^/ij, and the first approximation
in ^^"Ir}^ now to be retained, (25) gives for the third approximation
^3 _ ^4 _ ^7/^4/3 (1 + 1 ^5/3) _ ^10/ 18/3 = 0.
Whence, if all terms higher than the last retained be neglected,
^3_|4_p/3_|p/3 = 0,
which gives
,, = f4/3(l + ^5/3 + |p/3)i,
=^'IH1 + W^+U^'^') (27),
which is the required third approximation.
This might of course be obtained by successive applications of the method
of transformation employed in the demonstration of § 22, or by the method
of indeterminate coefficients, but the calculations would be laborious. It
will be observed on comparing (27) with the theoretical result in § 22 that
^1=^2=^3=^4=^6=^7 = ^8=^9 = 0 ; a fact wMch, in itself, shows the advan-
tages of the present method.
The other branches of the cycle to which (27) belongs are given by
V = H''')' {i+l i^e^')' + h H'''T} >
where w is any imaginary cube root of unity.
Example 2. To find a second approximation for the branches corre-
sponding to ABC in equation (16), in the special case where A= +1, B= -2,
C= + 1,D=-1.
The terms concerned in this approximation are {ABC) and (D). We
therefore write
or {v-^r'-v'l^' = 0.
The first approximation is ?; = |^ ; hence the second is given by
{v-ef-e'i^'=o,
that is, ('?-^y-i"=0.
Whence ,-^3 ±^11/2=0,
which gives the two second approximations corresponding to the group.
There are two, because to a first approximation the branches are coincident.
This, therefore, is a case where a, second approximation is necessary to
distinguish the branches.
§ 24 EXAMPLES ' S95
Example 3. To find a second approximation, for large values both of
f and 7], to the branch corresponding to HI in equation (16).
Beferring to Fig. 1, we see that, if HI move parallel to itself towards 0,
the next point which it will meet is G, Hence, so far as the approximation
in question is concerned, we may replace (16) by
(JJ ^10^12 + J^U^7) + G ^s^n = 0.
For simplicity, let us put H = l, 1= G= —1, and write the above equation
in the form
Confining ourselves to one of the five first approximations, say rj = ^^/', we
get for the second approximation
^5 _ ^4 _ ^8/5^0,
which gives rj = ^'^l^ (1 + i^~^-/').
The other branches of the cycle are given by
where w is any imaginary fifth root of unity.
Example 4. Given
a;=2/ + 2/2/2! + 2/3/31 + 2/4/4!+ . . .,
to find 2/ to a fourth approximation. We have
j/ = a;-2/2/2!-2/='/3!-2/V4!- ....
Hence 1st approx. 2/ = ^.
2nd „ y = x-ix\
3rd „ 2/=^-4(^-i^T-|^^
4th „ y = x-^(x-ix^ + ix^)^-^{x-l^X-')^-^\x\
= x- |x^ + ^x^ - ^x*.
Example 5. To separate the branches of 17 at f =0, where
4«-3^*-4^2(^_^) + 4(^_|)2^0.
If we plot the terms in Newton's diagram, and arrange them in groups
corresponding to their order of magnitude, we find
where the suffixes attached to the brackets indicate the orders of the groups.
The first approximation 77 = $ is common to the two branches.
Since 7;-^ is a factor in { }2, we cannot obtain a second approximation
by neglecting { jg and putting 77=^ in { jg. In obtaining the second
approximation we therefore retain { ]g, treating rj-i as a variable to be
found. We thus get
4(,-^)2_4f(„-|) = 344;
whence {2 (1? - |) - ^2}^ = 4t4,
which gives v = ^ + 3^^/2 ;
or v'=^-i'l2.
The branches are thus separated.
396 HISTORICAL NOTE CH. XXX
If a third approximation were required, we should now retain { }^, and
•write
|2(^-f)-r}2 = 4|4_4,5.
Hence 2(7;-^)-^2= ±2^2(1- t)i^
We thus get
^ = $+3^/2-^3/2;
Sistoncal Note. — As has already been remarked, the fundamental idea of the
reversion of series, and of the expansion of the roots of algebraical or other equa-
tions in power-series, originated with Newton. His famous " Parallelogram " is
first mentioned in the second letter to Oldenburg; but is more fuUy explained
in the Geometria Analytica (see Horsley's edition of Newton's Worlcs, t. i.,
p. 398). The method was well understood by Newton's followers, Stirling and
Taylor ; but seems to have been lost sight of in England after their time. It was
much used (in a modified form of De Gua's) by Cramer in his well-known Analyse
des Lignes Oourbes Algdbriques (1750). Lagrange gave a complete analytical form
to Newton's method in his "Me'moire sur I'Usage des Fractions Continues," Nouv.
Mem. d. VAc. roy. d. Sciences d. Berlin (1776). (See CEuvres de Lagrange, t. iv.)
Notwithstanding its great utility, the method was everywhere all but forgotten
in the early part of this century, as has been pointed out by De Morgan in an
interesting account of it given in the Cambridge Philosophical Transactions,
vol. IX. (1855).
The idea of demonstrating, a priori, the possibility of expansions such as the
reversion-formulae of § 18 originated with Cauchy ; and to him, in effect, are due
the methods employed in §§ 18 and 19. See his memoirs on the Integration of
Partial Differential Equations, on the Calculus of Limits, and on the Nature and
Properties of the Eoots of an Equation which contains a Variable Parameter,
Exercices d' Analyse et de Physique Ilatkematique, t. i. (1840), p. 327 ; t. rt.
(1841), pp. 41, 109. The fonn of the demonstrations given in §§ 18, 19 has
been borrowed partly from Thomae, El. Theorie der Analytischen Functionen
einer Complexen Verdnderlichen (Halle, 1880), p. 107 ; partly from Stolz, Allge-
meine Arithmetih, I. Th. (Leipzig, 1885), p. 296.
The Parallelogram of Newton was used for the theoretical purpose of estabhsh-
ing the expansibility of the branches of an algebraic function by Puiseux in
his Classical Memoir on the Algebraic Functions {Liouv. Math. Jour., 1850).
Puiseux and Briot and Bouquet {Theorie des Fonctions Elliptiques (1875), p. 19)
use Cauchy's Theorem regarding the number of the roots of an algebraic equation
in a given contour ; and thus infer the continuity of the roots. The demonstra-
tion given in § 21 depends upon the proof, a priori, of the possibihty of an
expansion in a power- series ; and in this respect follows the original idea of
Newton.
The reader who desires to pursue the subject further may consult Durbge,
Elemente der Theorie der Functionen einer Complexen Verdnderlichen Grosse, for
a good introduction to this great branch of modern function-theory.
The English student has now at his disposal the two treatises of Harkness and
Morley, and the work of Forsyth, which deal with function-theory from varioua
points of view.
The appUcations are very numerous, for example, to the finding of curvatures
and curves of closest contact, and to curve-tracing generally. A number of
beautiful examples will be found in that much-to-be-recommended text-book,
Frost's Curve Tracing.
§ 24 EXERCISES XXIV - 397
Exercises XXIV.
Revert the following series and find, so far as you can, expressions for
the coefficient of the general term in the Eeverse Series : —
, nx n(w-l) „ n(n-l) (n-2) , ,
(2.) y = x-ix» + ^x^-}x'^+ ....
X* x^ x'^
(3.) 2/=^- 37 + 51-71+ •• • ■
(4.) y = a; + x2/22 + a;3/32 + a;V42+ ....
(5.) If 2/ = sin .-c/sin (x + a), expand x iu powers of?/.
a; and y being determined as functions of each other by the following
equations, find first and second approximations to those branches, real or
imaginary, for which | a; | or | j/ 1 , or both, become either infinitely small or
infinitely great : —
(6.) ?/2-2y = a;*-a;2.
(7.) a3{2/ + a;)-2a2x(2/ + a;) + a;'»=0, (F. 69*).
(8.) {x-yf-{x-y)x-'-\x^-\y^^Q, (F. 82).
(9.) a(2/2-x2)(2/-2x)-2/^=0, (F. 88).
(10.) aa;(2/-x)2-?/=:0, (F. 96),
(11.) a;(y-x)2-a»=0, (F. 115).
(12.) a;y-2a2x2?/ + a%-fc» = 0, (F. 121),
(13.) 2/(j/-a;)2(2/ + 2a;) = 9cx3, (F. 131).
(14.) \x{y-x)-a?-Yy^=a'', (F. 140),
(15.) x''-a;V + aV-«^2/'' = 0, (F. 143).
(16.) a(x'> + 2/=)-a2x32/+a;V = 0, (F. 143).
(17.) x^y* + ax^3/* + &x^i/ + ex + dz/"^ = 0, where a, 6, c, d are all positive,
(F, 155).
(18.) If e„ be any constant whatever when n is a prime number, and
such that e^=-epeqe^ . . . when n is composite and has for its prime factors
p, q, r, . . . , then show that
If a, &, c, . , , be a given succession of primes finite or infinite in number,
s any integer of the form a^-h^cl . . ., t any integer of the forms a, ah,
abc, . , . (where none of the prime factors are powers), and if
F(x) = ZeJ{x%
then /(x) = S(-)«e4F(a^),
where u is the number of factors in t.
(This remarkable theorem was given by Mobius, Crelle's Jour., ix. p. 105.
For an elegant proof and many interesting consequences, see an article by
J. W, L. Glaisher, Phil. Mag., ser. 5, xvni., p. 518 (1884).)
* F. 69 means that a discussion of the real branches of this function,
with the corresponding graph, wiU be found in Frost's Curve Tracing, § 69.
CHAPTER XXXI.
Summation and Transformation of Series
in General.
THE METHOD OF FINITE DIFFERENCES.
§ 1.] We have already touched in various connections upon
the summation of series. We propose in the present chapter to
bring together a few general propositions of an elementary
character which will still further help to guide the student in
this somewhat intricate branch of algebra.
It will be convenient, although for our immediate purposes it
is not absolutely necessary, to introduce a few of the elementary
conceptions of the Calculus of Finite Differences. We shall thus
gain clearness and conciseness without any sacrifice of simplicity ;
and the student will have the additional advantage of an intro-
duction to such works as Boole's Finite Differences, where he
must look for any further information that he may require
regarding the present subject.
Let, as heretofore, «„ be the wth term of any series ; in other
words, let Un be any one-valued function of the integral variable
n ; w„_i, Un-2, . . ., «i the same functions of w- 1, w-2, . . .,1
respectively.
Farther, let A?/„, Am„_i, . . ., Awj
denote w„+i-Wn, w„-m„_i, . . ,, u^-Ui)
also A(A?f„), A(Am„_i), . . ., A(Ami),
which we may write, for shortness,
DIFFERENCE NOTATION
A2w„, A2^^„_l, . . .,
A2«^,
§ 1 DIFFERENCE NOTATION 399
denote
A^f„+^-A^f„, A?^„-Aw„_j, . . ., ^u^-^Ui',
and so on. Thus we have the successive series,
Ui, Ma, Us, . . ., Un, ... (1)
Aui, Au2, Aw3, . . ., Aun, ... (2)
A^Mi, A^U2, A^Wg, . . ., A^Un, ... (3)
A^Wi, A^Ma, A^Ws, . . ., A='?«„, ... (4)
where each term in any series is obtained by subtracting the one
immediately above it from the one immediately above and to the
right of it.
The series (2), (3), (4), . . . are spoken of as the series of
1st, 2nd, 3rd, . . . differences corresponding to the primary
series (1).
Example 1. If Uj^—iv^, the series in question are
1, 4, 9, 16, . . . n2, . . . ;
3, 5, 7, 9, . . . 271 + 1, . . .;
2, 2, 2, 2, ... 2, ... ;
0, 0, 0, 0, ... 0, ... ;
where, as it happens, the second differences are all equal, and the third and
all higher differences all vanish.
Cor. If we take for the primary series
A''mi, A^'Mg, ^''^3, . . ., A''m„, . . .,
then the series of 1st, 2nd, 3rd, .
A'-'+'u,, A'-+^W2, A'-+^M3,
A^'+'u,, A^'+'u,, A^+^Us,
A^+^Uu A^'+^Ma, A'^+Swa,
differences will be
. ., A'-+iw„ . . .
. ., A'+hi,, . . .
. ., A'-+^w„ . . .
In other words, we have, in general, A'^A^Un = A'^+^Un. This is
sometimes expressed by saying that the difference operator A
obeys the associative law for multiplication.
Although we shall only use it for stating formulae in concise
and easily-remembered forms, we may also introduce at this
stage the operator E, which has for its office to increase by unity
the variable in any function to which it is prefixed. Thus
400 EXAMPLES CH, XXXI
E(t> (n) = fl> (n + 1) ; Eun = Un+i', Eui = Uz\
and so on.
In accordance with this definition we have E{Eu^, which we
contract into E'^Un, = Eun+i = w„+2 ; and, in general, i/™w„ = Un+m-
We have also, as with A, E''E^Un = E'''^*Un, for each of these is
obviously equal to Un+r+a-
Example 2. E^n^ = {n + r)^.
Example 3. The with differeuce of an integral function of n of the rth
degree is an integral function of the (r - m) th degree if vi<r, a constant if
r=m, zero if m>r.
Let
^y(n) = an'" + &n'"~^ + cn'"-2+. . .;
then
A0r(n) = a(ra + l)'-+6(n + l)'-i+c(n + l)'-2 + . . .
-anr- hn^-^- cn^-^ + . . .,
=rarf-^ + {^r{r-l)a+{r-l)h}'nT~^ + . . .,
= 0r-l(«),
say, where 0^_i [n) is an integral function of n of the (r - l)th degree. Then,
in like manner, we have ^<pr-i {n) = <Pr-2 (")• S'lt ^4'r-i i^) — ^'^<f>r'"' 5 lience
A^(pj.{"') = <Pr-2{n)' Similarly, A^r (w) = <^r-s (") 5 and, in general, A'^tpr{n)
= 0y_^(n). We see also that A''^^ (n) will reduce to a constant, namely, rla;
and that all differences whose order exceeds r will be zero.
The product of a series of factors in arithmetical progression, such as
a{a + b) . . .(a + {m-l)b), plays a considerable part in the summation of series.
Such a product was called by Kramp a Faculty, and he introduced for it the
notation a"*'*, calling a the base, m the exponent, and 6 the difference of the
faculty. This notation we shall occasionally use in the slightly modified
form a""i^, which is clearer, especially when the exponent is compound.
Since
a(a + b) . . . (a + (m-l)6) = 6™(o/6)(a/6 + l) . . . (a/6 + m-l),
any faculty can always be reduced to a multiple of another whose difference
is unity, that is, to another of the form c""'^, which, omitting the 1, we may
write c '"*! . In this notation the ordinary factorial /n! would be written 1 1"*' .
The reader should carefully verify and note the following properties of
the differences of Faculties and Factorials. In all cases A operates as ubual
with respect to n.
Example 4.
A(a + 6n)i'"i6=m6{a + 6(n + l)}i"*-ii&.
Example 5.
A{l/(a+6w)i'»i6}=-m6/{a + 6w)i"'+il6.
Example 6.
_a-c(g-b)'"+^i«>
6 c^n+ub •
§§ 1, 2 FUNDAMENTAL DIFFERENCE THEOREMS 401
Example 7.
Acos(o + /3«)= -2sin J/3sin(a+ij8+/3n);
A sin (a + /3w) = + 2 sin i/3 cos (a + J/3 + /3n).
§ 2.] Fundamental Theorems. The following pair of
theorems* form the foundation of the methods of differences,
both direct and inverse : —
To prove I. we observe that
- Un+i + Un,
hence
and so on.
— Un+2 ~ 2W»4.x + Un ]
A^lln = Un+3-2Un+2+ Un+i
= Un+3 ~ OUn+z + oUji+i — Un J
Here the numerical values of the coefficients are obviously
being formed according to the addition rule for the binomial
coefficients (see chap, iv., § 14) ; and the signs obviously alter-
nate. Hence the first theorem follows at once.
To prove II. we observe that we have, by the definition of
Aum, Um+i = 'Um + Aum. Heuce, siuce the difference of a sum of
functions is obviously the sum of their differences, we have, in
like manner, u^+i = «m+i + ^u^+i = «» + Am™ + A (u^ + Au„^) =
Urn + AWto + AUm + A^Um. We therefore have in succession
* The second of these was given by Newton, Principia, lib. in., lemma v.
(1687) ; and is sometimes spoken of as Newton's Interpolation Formula. See
his tract, Methodus Differentialis (1711) ; also Demoivre, Miscellanea Analytica,
p. 152 (1730), and Stirling, Methodus Differentialis, &c., p. 97 (1730).
C. II. '2(j
402 SUMMATION BY DIFFERENCES CH. XXXI
Um+2 — ^m + ^Ujn
«m+3 = Mm + 2AW„ + A^M^
+ AWto+2A'^W;„ + A^M,„,
Um + S^Um + SA^M^ + A^Um',
and so on.
The second theorem is therefore established by exactly the
same reasoning as the first, the only difference being that the
signs of the coefficients are now all positive.
If we use the symbol E, and separate the symbols of opera-
tion from the subjects on which they operate, the above theorems
may be written in the following easily-remembered symbolical
forms : —
A^Ur, = (E-l)'^Un (L); «™+„ = (l + A)"M« (IL).
§ 3.] The following theorem enables us to reduce the sum-
mation of any series to an inverse problem in the calculus of
finite differences.
I/vn be any function of n such that A'y„ = M„, then
n
n=«
This is at once obvious, if we add the equations
lis =AVg =Vs+i-Vs.
The difficulty of the summation of any series thus consists
entirely in finding a solution (any solution will do) of the finite
difference equation Av„ = w„, or Vn+i -Vn = w». This solution can
be effected in finite terms in only a limited number of cases,
some of the more important of which are exemplified below.
On the other hand, the above theorem enables us to con-
§§ 2, 3 EXAMPLES 403
struct an infinite number of finitely summable series. All we
have to do is to take any function of n whatever and find its
first diflference ; then this first difference is the wth term of a
summable series. It was in this way that many of the ordinary
summable series were first obtained by Leibnitz, James and John
Bernoulli, Demoivre, and others.
n
Example 1. i; {a + j(6}{a + (w + l) 6} . . . {a + (n + m-l) &}.
n=s
Using Kramp's notation, we have here to solve the equation
Av„={a + n6}i'»i& (2).
Now we easily find, by direct verification, or by putting m + 1 for m and
n - 1 forn in § 1, Example 4, that
A[{a + (n-l)6}i'»+Jl6/(m + l)6]={a + ni!)}i"»l6.
Hence 1^^= {a + (?i-l) 6}""+i'*/(m + l) 6 is a value of «;„ such as we
require.
Therefore
S{a + nZ,}.^'^^^^ + "^>""^^''-^^ + ^^-^)^^'"'^^'^ (3)
g ^ (m + 1) 6 ^ ''
Hence the tuell-known rule
l,{a + nh}{a + (n+l)b} . . . {a+{n + m-l)b}
= C+{a + nb}{a+{n + l)b} . . , {a + (n + m-l)b} {a + {n + m)b}/(m + l)b
(4),
where C is independent of n, and may be found in practice by making the two
sides of (4) agree for a particular value of n.
Example 2. To sum any series whose jith term is an integral function of
n, sa.yf{n).
By the method of chap, v., § 22 (2nd ed.), we can express f{n) in
the form a + bn + cn{n + l) + dn(n + l) {n + 2) + . . . Hence
7:,f(n) = C + an + ^bn{n + l) + lcn{n + l){n + 2) + ldn{n + l)(n + 2){n + 3) + . . .
(5),
where the constant G can be determined by giving n any particular value
in (5).
Examples. Sl/{a + 6n}"»'6.
Proceeding exactly as in Example 1, and using § 1, Example 5, we deduce
» 1 _ l/{a + 68}i'»-ii»-l/{a + fe(n + l)}i'»-ii&
, {a + 6n}i«i6"" (m-l)6 ^''
Hence a rule for this class of series like that given in Example 1.
Example 4. To sum the series S/(n)/{a + 6n}i''*i'>, /(n) being an integral
function of n,
26—2
404 EXAMPLES CH. XXXI
Decompose /(n), as in Example 2, into
a+i3(a + 6re)iii6 + ^(a + 6„)i2ib + 5(a + 6ra)i3i6 + . . . (7).
Then we have to evaluate
aSl/{a + 6M}l'»l6 + /3Sl/{a + 6(n + l)}i'"-ilH. • • (8),
which can at once be done by the rule of Example 3*.
Example 5.
na\n\b _ a ((a + 6)l»l6 _ (o + fcp^l /q\
This can be deduced at once from § 1, Example 6, by writing a + 6 for 6
and n - 1 for n.
Example 6. To sum the series whose terms are the Figurate Numbers of
the mth rank.
The figurate numbers of the 1st, 2nd, 3rd, . . . ranks are the numbers
in the 1st, 2nd, 3rd, . . . vertical columns of the table (II.) in chap, rv.,
§25. Hence the (ra + l)th figurate number of the mth rank is „+^_jC^_i
=„+^_iC„=m(??i + l) . • . (m + 7i-l)/?t!. Hence we have to sum the series
"m(m + l) . . . (m + n-1)
l + Zi — .
1 1.2 ... 71
Now if in (9), Example 5, we put a=m, 6 = 1, c = l, we get
n »n,l'»l_ (m + l)i"l m + 1
llnl llnl 1
Hence
TO(m + l) m{m+l) . . . (m + n-1)
l + ^ + _L_J+.. .+ 1.2... n
(>ra + l)(m + 2) . . . (m + l + 7i-l)
' 1.2 ... n
(10);
that is to say, the sum of the first n figurate numbers of the tnthrank is the nth
figurate number of the (m + l)th rank.
This theorem is, however, merely the property of the function ^JBT^, which
we have already established in chap, xxni., § 10, Cor. 4. The present
demonstration of (10) is of course not restricted to the case where m is a
positive integer.
Many other well-known results are included in the formula of Example 6,
some of which will be found among the exercises below.
* The methods of Examples 1 to 4 are all to be found in Stirling's Methodus
Bifferentialis. He applies them in a very remarkable way to the approxi-
mate evaluation of series which cannot be summed. (See Exercises
xxvii., 17.)
§§ 3, 4 DIFFERENCE SUMMATION FORMULA 405
Example 7. To sum the series
5„ = cosa + co8(a + j3) + . . . + cos(a + (n-l) j3) ;
r„=sina + sin(a + j3) + . . . + sin (a + (n-l)i9).
From § 1, Example 7, we have cos(a + j3n) = A {sin(o-i|8 + i3rt)/2sin^^}.
Hence
S„ = {sin (a- i/3 + ^n) - sin (a- ii3)}/2 sin ^p,
= ^-Yo cos {a + iiS (ra - 1 }.
Similarly,
§ 4.] Expression for the sum of n terms of a series in terms
of the first term and its successive differences.
Let the series be Wi + Wa + • • . + w™ ; and let us add to the
beginning an arbitrary term Uq. Then if we form the quantities
So = Uq, Si = Uo + Ui, S^^Uo + Ui + Uz,
. . ., 8n = Uo + tli + U.i + . . , +Un, . . .,
we have
Hence, putting n = 0,
ASo = u„ /i.'So = Au„ . . ., A^So = A'^-'u^, . . . (1).
Now, by Newton's formula (§ 2, 11. ),
Sn^So + nOi^So + nC^A'K + . . . + A"/^o (2).
If, therefore, we replace Sq, ASq, A^/S'q, ... by their values
according to (1), we have
n
%Un = Uo + nCiUi + nO^^Ui + nCA'^Ui + . . . + A'^-^Mj (3) ;
0
or, if we subtract Uq from both sides,
2W« = nCiUi + nCAUl + nOs^^Ui + . . . + A^-^W^ (4)*
1
The formula (4) is simply an algebraical identity which may
be employed to transform any series whatsoever; for example,
in the case of the geometric series ^af^ it gives
* This formula, which, as Demoivre (Miscell. An., p. 153) pointed out, is
an immediate consequence of Newton's rule, seems to have been first explicitly
stated by Montmort, Journ. d. Savans (1711). It was probably independently
found by James Bernoulli, for it is given in the Ars Conjectandi, p. 98 (171S).
MONTMORTS THEOREM
CH. XXXI
+ ir"
2! ^ ^ 3!
'-^a;{x-iy + . . .
+ a;{x-l)'*-\
406
^' + ^ + . . .
= nx +
which can be easily verified independently, by transforming the
right-hand side. The transformation (4) will, however, lead to
the sum of the series, in the proper sense of the word sum, only
when the m\h differences of the terms become zero, m being a
finite integer. The sum of the series will in that case be given
by (4) as an integral function of n of the mth degree. Since the
wth term of the series is the first difference of its finite sum, we
see conversely that any series whose sum to n terms is an
integral function of n of the m\h degree must have for its «th
term an integral function of n of the (m — l)th degTee. We have
thus reproduced firom a more general point of view the results of
chap. XX., § 10.
Example. Sum the series
S(n + l)(n + 2)(?i + 3).
1
If we tabulate the first few terms and the successive differences, we get
1, 2, 3, 4, 5
•"'n
24,
60,
120,
210,
336,
Aw„,
36,
60,
90,
126,
A'««
24,
30,
36,
A3«,
6,
c.
A^"„
0.
Hence, by (4),
S(n + l)(n + 2)(n + 3)
= „ . 24 + "("-^) . 36 + >^ ("-!)(» -2) _ 24 ^ «(n-l)(7i-2)(n-3) ^ ^^
2 6 24
= |(«J + 10w» + 35?i2 + 50/1).
§ 5.] Montmort's Them'em regarding the summation ofZunOf^.
An elegant formula for the transformation of the power-
series "ZunX^ may be obtained as follows. Let us in the first
place consider S^ 2M„a?", which we suppose to be convergent when
|a7l<l; and let us further suppose that |a;l<|l-a?|. Put
X = 2//(l + 2/) ; so that
4,5
montmort's theorem
407
i2^/(i+y)l = kl<i,
and \i/\ = \w/(l-x)\<l.
Then, since
we liave
1
= u^y-Ui'if+ Uiy^— Uiy* +
u^f
+ u^y^ - iCiU^if + 3^2^22/* - iG^u^f +
+ Usf - sCi Usy* + 4C2 M3/ - .
+ u^y* - iOiihf + .
+ u^f-
This double series evidently satisfies Cauchy's criterion, for
both 1^1 <1 and |y/(l +2/) | < 1. Hence we may rearrange it
according to powers of y. If we bear in mind § 2, L, we find
at once
>S^=Mi2^ + Ami2/^ + A2wi2/^ + A^t/i2/* + A^?^i/ + . • • •
Hence, replacing y by its value, namely, a?/(l - x), we get
UxX
1 i
+ ,. + 7^ — ^^, +
(ly
X {\-xf {\-xy
When the differences of a finite order m vanish, Montmort's
formula gives a closed expression for the sum to infinity ; and,
if the differences diminish rapidly, it gives in certain cases a
convenient formula for numerical approximation.
Cor. 1. We ham for the finite sum
** X 0^
\ JL Ou (X OCj
a?
For, if we start with the series e<„+ia7"+^ + i^,i+2^""'"^ +
proceed as before, we get
From (1) and (3) we get (2) at once by subtraction.
(2).
., and
(3).
* First given by Montmort, Fhil. Trans. R.S.L. (1717). Demoivre gave
in his Miscellanea a demonstration very much like the above.
408 EULER's theorem CII. XXXI
The formula (2) Avill furnish a sum in the proper sense only
when the differences vanish after a certain order. The summa-
tion of the integro-geometric series, already discussed in chap.
XX., §§ 13 and 14, may be effected in this way. It should be
observed that, inasmuch as (2) is an algebraic identity between
a finite number of terms, its truth does not depend on the con-
vergency of ^UnOf', although that supposition was made in the
above demonstration.
Cor. 2. If Un be a real positive quantity which constantly
diminishes as n increases, and if Lu^ = 0, then
U1-U2 + U3-. . .=-^ih- ^^Ui+ ^.^■''ui-. . . (4)*.
This is merely a particular case of (1) ; for, if in (1) we put
-a; for x, we get
S(-rz.„^» = i(-)«A-^e.,.(^y (5).
Since the differences must ultimately remain finite, the right-
hand side of (5) will be convergent when a:=l. Also, by Abel's
Theorem (chap, xxvi., § 20), since 2 ( - )"w„ is convergent, the
CO
limit of the left-hand side of (5) when iz: = 1 is ^ ( - )"«„. Hence
1
the theorem follows.
The transformation in formula (4) in general increases the
convergency of the series, and it may of course, in particular
cases, lead to a finite expression for the sum.
Cor. 3. We get, by subtraction, the following formula : —
«i - Wa + • . . ( - )""'«« = 2 (^'1 - ( ~ )"«n+i) - 22 (^«i - ( - )"^«n+l)
+ |(^'«'i-(-r^'w„+,)-. . . (6),
in which the restrictions on m„ will be unnecessary if the right-
hand side be a closed expression, which it will be if the differences
oiun vanish after a certain order.
♦ Euler, hist. Biff. Calc., Part IL, cap. i. (1787).
§ 5 EXERCISES XXV 409
Example 1. We have (Gregory's Series)
IT , 1 1 1 /_«
4=1-3 + 5-7 + - •• <^>-
If we apply (4), we have m„= l/(2« - 1). Hence
A*- «„=(-)'• 2. 4 . . . 2j7(2ra-l)(2n + l)(2n + 3) . . . (2n + 2r-l);
A'-Mi = (-)'-2.4 . . . 2r/1.3.5 . . . (2;- + l),
= (-)'-2'-.1.2 . . . r/1.8.5 . . . (2r + l).
«,, , IT , 1 1.2 1.2.3 ...
Therefore 2 =1+ 3 + 375 + 37577 + - ' * ^^^'
Example 2. To sum the series
S„=P- 22 + 32-. . . (-)«-in2.
Since AUn+i = 2n + 3, Aiii = 3,
Ahi„+, = 2, A2m,=2,
A3u„+i=0, A3mi=0,
we have, by (6),
Sn=Ml-(-)"(« + iP}-i{3-(-)M2n + 3)}+i{2-(-)"2},
= (-)»-4w(n + l).
Exercises XXV,
(1.) Sum to n terms the series whose nth term is the nth r-gonal
number*.
Sum the following series to n terms, and, where possible, also to
infinity : —
(2.) S»i(n + 2)(n + 4). (3.) Sl/(ri2-l),
(4.) 1/3.8 + 1/8.13 + 1/13. 18 + . . . .
(5.) 1/1. 3. 5 + 1/3. 5. 7 + 1/5. 7.9 + . . . .
(6.) 1/1. 2. 3. 4 + 1/2. 3. 4. 5 + 1/3. 4. 5. 6 + . . . .
(7.) 2(an + 6)/n(n + l)(n + 2).
(8.) 1/1.3.5 + 2/3. 5. 7 + 3/5. 7.9 + . . . .
(9.) 1/1.2.4 + 1/2. 3.5 + 1/3. 4.6 + . . . .
(10.) 1/1. 3. 7 + 1/3. 5. 9 + 1/5. 7. 11 + . . . .
(11.) S(n+l)2/n(n + 2).
(12.) 4/1. 3. 5. 7 + 9/2. 4. 6. 8 + 16/3. 5. 7.9 + . . . .
(13.) Ssecn^sec(n + 1)^. (14.) S tan (^/2»)/2".
(15.) Stan-i{(7ia-n+l)a»-V(l + «(w-l)a-»-0}-
(16.) Stan-i{2/ri2}.
(17.) ml + (m + l)!/l! + (7/i + 2)!/2! + . . . .
(18.) lI/ml + 2!/(m+l)! + 3!/(m + 2)l + . . . .
* The sums to n terms of arithmetical progressions whose first terms are
all unity, and whose common diSerences are 0, 1, 2, . . ., (r-1), . . . respec-
tively, are called the nth polygonal numbers of the 1st, 2nd, 3rd, . . . , rth, . . .
order. The numbers of the first, second, third, fourth, . . . orders are spoken
of as linear, triangular, square, pentagonal, . . . numbers.
410 EXERCISES XXV CH. XXXI
(19.) l-^Cj + ^C^-. . .(-)VC„.
(20.) Show that the figurate numbers of a given rank can be summed by
the formula of § 3, Example 1.
. 1 1.2 1.2.3
^ '' m vi(m + l) m (m + 1) (m + 2)
a{a + l) . . . (g + r) a{a + l) . . . (a + r + 1)
^ '■' c c(c + l)
. a a(a + l)
^ ■' c^{c + l) . . . (c+r)'*'c(c + l) . . . (c + r + 1)"^' ' ' '
(24.) S(a + n)""-2i/(c + w)"»l.
1.3 1.3.5 1.3.5.7
^ ' 1.2.3.4'^1.2.3.4.5'^1.2.3.4.5.6''"* " * *
/9fi^ (l + r)(l + 2r) (l + r)(l + 2r)(l + 3r)
^ ' 1.2,3.4.5'^ 1.2.3.4.5.6 *
2 2^ 2*
(27.) jm-j-^m(m-l) + j— g— gm(m-l)(m-2)-. . . .
(28.) Show that
\'M • • • ("+!) -ri/M • • • ("-^)+^/M • • • ("-I)-- ••
(Glaisher.)
(29.) Show that
l + 2(l-a) + 3(l-a)(l-2a) + . . . + n(l-a) (l-2a) . . . (l-(n-l)a)
= a-i{l-(l-a)(l-2a) . . . (1-na)}.
(30.) • =^-,,,,^„. '>
a; + l~a;-l (x-l) (x-2) ^ (a;- 1) (a;-2)(a;-3) "*•
(-)"+^w! / n + l\
(x-l){x-2) . . . (x-n)\ x + lj'
(31.) If a + & + 2 = c + d, then
n olnlftlnl _ a& ^(g + 1) '"I (6 + 1) !»' _ (g+ 1) "-n (6 + 1) I'-n^
(32.)
1 g-y . g(g-i)-y(^-i)
{p-q + l).{p + r-l) (p-q + l)(p-q+2).(p+r-l)(p + r-2)
p.{p-q + r) '
{Educational Times Reprint, vol. xli., p. 98.)
(33.) Transform the equation
log2 = l-4 + i-i + . . .
by § 5, Cor. 2.
(34.) Show, by means of § 2, I., that, if m be a positive integer, then
1 r "^^ r "fcll_ r «(a-l)J^) ,
^-m^ll+rn^H(b-l) "'^H (& - 1) (6 - 2) "*"' * '
(^-|)(-.^)---(-.-r^)
§ G DEFINITION OF RECURRING SERIES 41 1
RECURRING SERIES.
§ 6.] We have already seen that any proper rational fraction
such as {a + bx + ca^)l{l +px + qcc^ + rar^)* can always be expanded
in an ascending series of powers of x. In fact, \i\x\ be less than
the modulus of that root of ra? + qx^ +px +1 = 0 which has tlie
least modulus, we have (see chap, xxvii., §§ 6 and 7)
a + bx + cx^ ^ „ . ,
:; ^ ^ = Uo + UiX + U.yX" + . . .+UnX^+. . . (1).
1+px + qx^ + ra^ ^ ^
We propose now to study for a little the properties of the
series (1).
If we multiply both sides of the equation (1) by 1 +px
+ qa^ + ra^, we have
a + bx + cx'^ = (1 +px + qaf + rx^) (uq + UiX + u.^x^ + . . . +UnX^ + . . . )
(2)-
Hence, equating coeJB&cients of powers of x, we must have
Mo = « (3i);
Ui+puo = b (Sa);
Ui+pu^ + qua = c (Ss);
«^3 + pu2. + QUi + rwo = 0 (84) ;
Un + pUn-1 + qUn-2 + ^%-3 = 0 (3„+i).
Any power-series which has the property indicated by the
equation (3«+i) is called a Becurring Power-Series^; and the
equation (3„+i) is spoken of as its Scale of Relation, or, briefly,
its Scale. The quantities p, q, r, which are independent of n,
may be called the Constants of the Scale. According as the scale
has 1, 2, 3, . . .,/•,. . . constants, the recurring series is said to
be of the 1st, 2nd, 3rd, . . ., rth, . . . order. When x=l, so
that we have simply the series Mq + Wj + Wg + . • . + w„ + . . . ,
with a relation such as (3„+i) connecting its terms, we speak of
* For simplicity, we confine our exposition to the case where the
denominator is of the 3rd degree; but all our statements can at once be
t The theory of Eecuning Series was originated and largely developed
by Demoivre.
412 MANIFOLDNESS OF RECURRING SERIES CH. XXXI
the series as a recurring series simply * ; so that every recurring
series may be regarded as a particular case of a recurring power-
series.
It is obvious from our definition that all the coefficients of a
recurring power-series of the Hh order can be calculated when
the values of the first r are given and also the constants of its scale.
Hence a recurring series of the rth order depends upon 2r constants;
namely, the r constants of its scale, and r others.
From this it follows that if the first 2r terms of a series (and
these only) be given, it can in general be continued as a recurring
series of the rth order, and that in one way only ; as a recurring
series of the (r + l)th order in a two-fold infinity of ways ; and
so on.
On the other hand, if the first 2r terms of the series be
given, two conditions must be satisfied in order that it may be a
recurring series of the (r - l)th order ; four in order that it may
be a recurring series of the (r - 2)th order ; and so on.
Example. Show that
is a recurring series of the 2nd order. Let the scale be M„+i'M„-i + ?Wn_2=0.
Then we must have
3 + 22) + 2 = 0, 4 + 3p + 2g=0, 6 + 4i3 + 33=0, 6 + 52) + 4g=0.
The first two of these equations give ^= -2, q=+\; and these values
are consistent with the remaining two equations. Hence the theorem.
§ 7.] The rational fraction (a + 6ic + ca^)/(l +J3a? + qn^ + ro^),
of which the recurring power-series u^ + u^x + u^oc^ + ... is the
development when | a; | is less than a certain value, is called the
Generating Function of the series. We may think of the series
and its generating function without regarding the fact that tiie
one is the equivalent of the other under certain restrictions. If
we take this view, we must look at the denominator of the
function as furnishing the scale, and consider the coefficients
* We might of course regard a recurring power- series as a particular case
of a recurring series in general. Thus, if we put JJ^—UnX^, we might regard
the series in (1) as a recurring series whose scale is
§§ 6-8 GENERATING FUNCTION 413
as determined by the equations (3i), (82), . . ., (3„+])*. No
question then arises regarding the convergence of the series.
Given the scale and the first r terms of a recurring power-
series of the rth order ^ we can always find its generating function.
Taking the case r = 3, we see, in fact, from the equations (3i),
(82),. . .,(8„+i), . . .of§6, that
{mq + (^z +i3Wo) X -•- (w2 + J9^*l + g'Mo) ^^}/{l + p-^? + (10^ + ra?}
is the generating function of the series u^-^ UyX -v u-^oc^ -^ . . .,
whose scale is u^ +pun-i + qUn-2 + rUns = 0.
Cor. 1. Every recurring power-series may, if \x\ be small
enough, be regarded as the expansion of a rational fraction.
Cor. 2. The general term of any recurring series can always
be found when its scale is given and a sufficient number of its
initial terms.
For we can find the generating function of the series itself
or of a corresponding power-series ; decompose the generating
function into partial fractions of the form A(x-a)-^; expand
each of these in ascending powers of x ; and finally collect the
coefficient of x^ from the several expansions.
Example. Find the general term of the recurring series whose scale is
Uj^-4m„_i + 5w„_2— 2m„_3=0, and whose first three terms are 1 + 0-5. Con-
sider the corresponding power-series. Here 2?= -4, g = 5, r= - 2; so that
a = Mo = l, 6 = Mi+j)u„= -4, c = U2+pUi + quQ = 0.
The generating function is therefore
1 - 4:c _ 1 - 4a;
l-4a; + 5a;2-2a;» ""' (l-xP(l-2a;) '
_ 2 3 4
~l-x'^ (1-x)^ (l-2a;)*
Expanding, we have
1 -4t
l-4a; + 5x^-2x3 = 2{l + S^n + 3{l + 2(n + l).xn-4{l + S2^a;"}.
= l + S(3n + 5-2"+2)a;™,
The general term in question is therefore 3w + 5 - 2"+^,
§ 8.] If Un be any function of an integral variable n which
satisfies an equation of the form
Un +pUn-l + qUn-2 + rUn-3 = 0,
or, what comes to the same thing,
Un+3 +pUn+2 + qUn+1 +rUn^O (l),
* We might also regard the series as deduced from the generating
function by the process of ascending continued division (see chap, v., § 20).
414) LINEAR DIFFERENCE- EQUATION CH. XXXI
we see from the reasoning of last paragraph that w„ is uniquely
determined by the equation (1), provided its three initial values
Mo, Ml, % are given ; and we have found a process for actually
determining Un-
it is not difficult to see that we might assign any three
values of w„ whatever, say Ua, up, Uy, and the solution would
still be determinate. We should, in fact, by the process § 7,
determine Un as a function of n linearly involving three arbitrary
constants u^, Va, ii^, say/(Mo, «*i, Ma, w) ; and Uq, ii^, ti^ would be
uniquely determined by the three linear equations
f(uo,Ui,U2,a) = Ua, /(uo,Uj,U2,^) = up, /{u^, th, th, y) = Uy (2).
An equation such as (1) is called a Linear Difference- Equation
of the 3rd m-der with constant coefficients ; and we see generally
that a linear difference-equation of the rth order with constant
coefficients has a unique solution when the values of the function
involved are given for r different values of its integral argument.
Example. Find a function «„ such that w»+3 - 4m„+2 + •''"n+i - 2u„ = 0 ;
and Mo = l» ^1 = 0, u^=-5.
We have simply to repeat the work of the example in § 7.
§ 9.] To sum a recurring series to n + 1 terms, and {when
convergent) to infinity.
Taking the case of a power-series of the 3rd order, let
Sn-UQ + U-i^X + UiCC^ + . . .+UnX''\
then
pxSn =pUoX + pUiOp +... +pUn-iaf'+ pUnX^"^^,
qa^Sn = qu^a^ + . . .■^-qUn-^x^+qUn-iX^'^'^ + qUriX""-^^,
ra?8n = . . . + riin-z x"" + rUn-i. a?"+^ + ru^_^ x^^"^ + ru^ o^^'^
Hence adding, and remembering that u„, + jt?M«_i + qu^-^
+ run-z - 0 for all values of n which exceed 2, we have
(1 +px + qaP + ra^) Sn = UQ+ (ui +pu^ x-¥{Ui-\-pui + qu^) o^
+ {pUn + qun-x + run-^ a;™+^ + {quy, + rUn-^ 37"+^ + rM„;r"+' (1) ;
whence ^» can in general be at once determined by dividing by
1 +^,r + qa^ + ra?.
The only exceptional case is that where for the particular
value of w m question, say x = a, it happens that
1 +pa + qa? + ru' = 0.
§§ 8, 9 SUMMATION OF RECURRING SERIES 415
In this case the right hand of (1) must, of course, also
vanish, and Sn takes the indeterminate form 0/0. S^ may in
cases of this kind be found by evaluating the indeterminate form
by means of the principles of chap, xxv. This, however, is often
much more troublesome than some more special process applicable
to the particular case.
If the series ^Univ^ be convergent, then LunX^ = 0 when
w = Qo ; therefore the last three terms on the right of (1) will
become infinitely small when 7i = go . We therefore have for
the sum to infinity in any case where the series is convergent
1 +px + qa? + ra? ^ ''
The particular cases
Wo + Ml + ^2 + . . . + M„ + . . . (3),
Mo-'?*i + W2-. . .+(-)"%„+. . . (4),
are of course deducible from (1) and (2) by putting x=-\-\
and x = -\. Exceptional cases will arise if l+j9 + 2' + r = 0, or
if 1 —p + q-r = 0.
It is needless to give an example of the above process, for
Examples 1 and 2, chap, xx., § 14, are particular instances,
2w^^" and 1 + iS ( — y~'^2nx^ being, in fact, recurring series whose
scales are w„ - 3w„-i + 3m„_2 - Wn-s = 0 and u^ + 2m„_i + Un^^ = 0
respectively.
Exercises XXVI.
Sum the following recurring series to n + 1 terms, and, where admissible,
to infinity : —
(1.) 2 + 5 + 13 + 35 + 97+ . . . .
(2.) 2 + 10 + 12-24 + 2 + 10 + 12+. . . .
(3.) 2 + 17X + 95x2 +461x3+. . . .
(4.) 5 + 12x + 30a;2 + 78x3 + 210x4+. . . .
(5.) 1 + 4x + 17x2 + 76x3 + 353x*+. . , ,
(6.) l + 4x + 10x2 + 22x3 + 46x4+. . . .
(7.) If a series has for its rth term the sum of r terms of a recurring
series, it will itself be a recurring series with one more term in the scale of
relation.
Find the sum of the series whose rth term is the sum of r terms of the
recurring series 1 + 6 + 40 + 288 + . . . .
416 EXERCISES XXVI CH. XXXI
(8.) If T„, T„+i, r„+2 te consecutive terms of the recurring series
whose scale is Tn+2=aTn+i-hT„, then
(9.) Form and sum to n terms the series each term in which is half the
difference of the two preceding terms.
(10.) Show that every integral series (chap, xx., § 4) is a recurring series;
and show how to find its scale.
(11.) If M„=M„_i+M„_2, and M2=arti, show that
V-«n+i«™-i=(-)"(«'-«-l)V.
(12.) If the series iij, ?<2, "3> • • •> w„, . . . be such that in every four
consecutive terms the sum of the extremes exceeds the sum of the means by
a constant quantity c, find the law of the series ; and show that the sum of
2m terms is
\m (m - 1) (4ffi - 5) c - m (m - 2) iti + mMj + m (m- 1) Wg.
(13.) If tt„+2=«n+i + "n> Mi=l, W2 = l. sum the series
1.2 1.3 ■ ■ * M„+iWn+3*
(14.) By French law an illegitimate child receives one-third of the portion
of the inheritance that he would have received had he been legitimate. If
there be I legitimate and n illegitimate children, show that the portion of
inheritance 1 due to a legitimate child is
1 n n (ra - 1) . w (w - 1) . . . 2 . 1
l~3Z(J + l)"*'3'-'i(i + l)(i + 2) • ■ • ^ '3»J(i + l) . . . (i + 7i)"
(Catalan, 'biouv. Ann., ser. ii., t. 2.)
Simpson's method for summing the series formed by
TAKING every IcTB. TERM FROM ANY POWER -SERIES
WHOSE SUM IS KNOWN.
§ 10.] This method depends on the theorem that the sum of
the pth powers of the kth roots of unity is k if p be a multiple
ofk, but otherwise zero.
This is easily seen to be true ; for, if w be a primitive ^th
root of 1, then the k roots are w°, w\ a?, . . ., w*-\ If p-fik,
then (a)y = w«'^*' = (a>*)'*' = l. If p be not a multiple of k, then
we have
((0°)^ + (o>i)p + . . . + (iJ'-'y = 1 + (o>py + (i^y + . . . + {<^^f-\
={i-Kmi--^x
= 0,
for (o)^)* = {J'Y = 1, and w^ 4= 1.
§ 10 Simpson's theorem 417
Let us suppose now that f{x) is the sum of n terms of the
power-series u^ + 2M,ia?", n being finite, or, it may be, if the series
is convergent, infinite.
Consider the expression
k
... , (1)'
where m is 0 or any positive integer <k.
The coefficient of x^ in the equivalent series is
w,.{(o)*')^-'«+'-+(o)0*-'"+'" + (o)'^)'=-'"+'" + . . . + {J'-'f -'"'+'-] I Jc (2).
Now, by the above theorem regarding the ^th roots of unity,
the quantity within the crooked brackets vanishes if Jc-m + r
be not a multiple of k, and has the value ^if^-w+rbea
multiple of k. Therefore we have
U^ = W^^'» + W»+fc«™+* + Ura+.T.X^^'^ + . . . (3),
where the series extends until the last power of x is just not
higher than the nih., and, in particular, to infinity if /(;r) be a
sum to infinity*^.
If we put w = 0, we get
{f{x)+f{o>'x)+f{oy^x) + . . .+f{J^-'x)}lk
= Uq + UkO^ + u^a?^ 4- u^^x^^ + . . . (4).
Example 1.
l + ar+a;2+. . . +a;"=(l -a;»+i)/(l-a;).
Hence, if w be a primitive cube root of 1, we have
\ \-x 1-wx 1- la^x J
where 3s is the greatest multiple of 3 which does not exceed n.
Example 2. To sum the series
x^ x'' x^^ ,
* This method was given by Thomas Simpson, Phil. Trans. R. S. L.
Nov. 16, 1758 (see De Morgan's Trigonometry and Double Algebra (1849),
p. 159). It was used apparently independently by Waring (see Phil. Trans.
B. S. L. 1784).
c. II. 27
418 MISCELLANEOUS METHODS CH. XXXI
We have
e*=l + x + 2j + 'o7+ • • • ad 00.
Hence, if w be a primitive 4th root of unity, say u=i, then, since here
A;=4, ?ft=3, k-m=l, w2=-l, ca^=~i, we get
4(e»=+ie»»-e-»=-ie-i^)=|j + |j+|jj + . . ..
that is, ^(sinhx-sina;) = gT + ^ + jj|+. . . .
MISCELLANEOUS METHODS.
§ 11.] When the wth term of a series is a rational fraction,
the finite summation may often be effected by merely breaking
up this term into its constituent partial fractions ; and even
when summation cannot be effected, many useful transformations
can be thus obtained. In dealing with infinite series by this
method, close attention must be paid to the principles laid down
in chap, xxvi., especially § 13 ; otherwise the tyro may easily
fall into mistakes. As an instance of this method of working,
see chap, xxviil, § 14, Examples 1 and 2.
Example 1. Show that
( 1 1 1 \
f 1 1 1 1 _ 1
"^ ((x + l)(a; + 2)2'*'(x + 2)(x + 3)2"^(a; + 3)(a; + 4)2+' ' -[-(aj + l)"-
Denote the sums of n terms of the two given series by S„ and T„
respectively, and their nth terms by m„ and v„ respectively. Then
«„= -l/(x + n) + l/(x + n)2 + l/(x + n + l);
Vj,=ll{x + n)-ll{x + n + l)^-ll{x + n + l).
Whence we get at once
5„ + r„=l/(x + l)2-l/(x + n + l)2.
Therefore S^ + T^ = ll(x + 1)K
Example 2. Resolution into partial fractions will always effect the
summation of the series
n
S/{n)/{7H-o)(7i + 6) . . . {n + k),
where a, b, . . ., k are positive or negative integers, and /{ii) is an integral
function of n whose degree is less by two at least thau the degree of
{n + a){n-i-b) . . . {n + k).
§§ 10-12 euler's identity 419
For we have
f{n)l{n + a){n + b) . . . (n + fc) = S^/(n + a),
and
/(n) = S4(n + 6)(n + c) . . . (n + k).
Since the degree of f(n) is less by one at least than the degree of the
right-hand side of this last identity, we must have
A + B+. . .+K=0.
But, since a, b, . . ., k are all integral, any partial fraction whose
denominator p is neither too small nor too great will occur with all the
numerators A, B, . , ., K, bo that we shall have Alp + Bjp+ . . . +Klp=0.
On collecting all the fractions belonging to all the terms of the series we
shall be left with a certain number at the beginning and a certain number at
the end ; so that the sum will be reduced to a closed function of n.
§ 12.] Euler's Identity. The following obvious identity*
1 - «! + «! (1 - a^ + aifta (1 - «3) + • • • + ai«2 ... a™ (1 - a„+i)
= 1 -aia.2 . . . a„+i (1)
is often useful in the summation of series. It contains, in fact,
as particular cases a good many of the results already obtained
above.
If in (1) we put
a; _^+pi _x+p.2 _x+pn
fti — , Qi^— , ff 3 — , .... fl^n+1 — ,
y y^Vv y+P2 y+p,^
and multiply on both sides by yl{y - x), we get
1 + ^ + ^(^+.Pi) ^ ^ xix+p^) . . . {x+pn-^)
y^Pi (y+Pi)(y+P2) ''' {y+Pi){y+p-2) • . • (y+Pn)
(2).
(3),
= -K— _ ^ {^+Pi)(^+Pi) • • ■ i^+Pn)
y-x y-x'\y +^i) {y +p^) . . . (y +pn)
If the quantities involved be such that
^ {x+pi){x+p2). . . (x+pn) _Q
«=« (y +pi) (y+P2)- . . (y +Pn)
then
- X x(x + pi) J y ...
1 + ■ + 7 -^r^^^^—< + . . . ad 00 = -^— (4).
y+Pi {y+Pi){y+P2) y-^
* Used in the slightly different form,
(l + a^)(l + a2)(l + as){l + a^) . . .
= l + ai + 02(l+ai) + a8(l+Oi)(l-l-a2) + a4(l + ai)(l + a2) (1 + 03) + . .
by Euler, Nov. Comm. Petrop. (1760).
27—2
420 EXERCISES XXVll CH. XXXI
If in (2) we put y = 0, we get
Pi P^Pz ' ' ' P1P2 • ' ■ Pn
(-!)(- !)•••(-#.) («'•
From (5) a variety of particular cases may be derived by
putting 72 = CO , and giving special values to pi, p^, • ■ • Thus,
for instance, if the infinite series ^l/pn diverge to + oo, then (see
chap. XXVI. , § 24) we have
1 + ^^ ^-^- . . . ad 00 - 0 (6).
Pi PlP-2
CO
In general, if the continued product 11(1 +xjpn) converge to any
1
definite limit, then the series l + '^x{x+pi) . . . {a;+pn-i)/piP2 • - -Pa
1
converges to the same limit.
Example. Find when the infinite series
y+p {y+p){y+^p) (y+p)(y+'b){y+^p)
converges, and the limit to which it converges.
If in (2) above we put Pi=p, P2=^P, &c., . . ., we have
^__y x_ ^ (x^-'p)(x + 2p) . . . (X+7>J>) .g.
y-x y-x n=<»\y-^v){y + ^p). . .\y + np)
Now the limit in question may be written
fi fi^(^-y)M
1 I l + ylnpj'
but this diverges to oo if (x - y)lp be positive, and converges to 0 if (a; - y)lp
be negative (chap, xxvi., § 24).
- Hence, if ^j denote in all cases a positive quantity, we see that
X x(x + v) , , y
1 + — + , w n >+ • • . ad 00 = -^^ ,
y+p {y+p)(y+^p) y-^
if y>x; and
x x(x-p) . J y
1 + + \ , '-. + ... ad 00 = — =^— ,
y-p {y-p)(y-^p) y-^
if y<x.
Exercises XXVII.
(1.) Given ll(l-x)'^=l + 2x + 3x^ + 'h^+ . . .,
sum l + 'ix^ + 7x« + 10x^+ . . , .
(2.) Sum the series
l + ar'/4 + x«/7+. . .;
l + x3/3I+a:«/6!+. . . .
I 12 EXERCISES XXVII 421
(3.) If f{x)=iiQ + UiX + u,2.v"+. . ., and a, p, y, . . . be the Jith roots
of - 1, show that
^{a2»-»»/(ax)+/3«»-»'/(/3x)+. . .} = M,„x™-M„+^a;'"+» + M„,+2„a;'«+2»- . . .
where m<n. (De Morgan, Diff. Calc, p. 319 (1839).)
Sum the following series, and point out the condition for convergency
when the summation extends to infinity : —
(4.) l-x3/4 + x«/7- . . . adoo;
x-x*li\ + x'l7l- . . . adoo.
(5.) l + m03 + mG6+mC9+- . . adoo;
l-mC3 + mC8-wC'9+' • • ad 00 .
(6.) 1/1.3 + 1/1.2.4 + 1/1,2.3.5+. . . to n terms.
(7.) 1/1. 2. 3+^Ci/2. 3. 4 + ^(72/3. 4.5+. . . ad oo .
(8.) l-2j;/l + 3a;2/2-4a;3/3+. . . ad oo .
(9.) cose/1.2.3 + cos2S/2.3.4 + cos3e/3.4.5+ . . . ad oo .
(10.) l/12.22 + 7/2^32+. . . + (2tt2 + 4n + l)/(ji + l)2(,i + 2)3.
(11.) 1/12.22- 1/22. 3«+. . . (-)«-il/n2(n + l)H. . . adoo.
(12.) If n be a positive integer, show that
n 1 n{n-l) 1 n(n-l){n-2)
m + n 2 (wi + n) (m + n-1) 'd{m + n) (m + n-1) {m + n-2)
_ n 1 n{n-l) 1 n(n-l) {n-2)
~m + l 2(?/i + l)(m + 2) 3 (?» + 1) (nt + 2) (?« + 3)
(13.) Show that
n^l n<^2 , flA " .
1-x/l (l-.'c/l)(l-x/2)'^(l-x/l)(l-a;/2)(l-a;/3) **• n-x'
and hence show that
where er,. = 1/1 + 1/2 + . . . + 1/r.
(14.) Sum the series
1 - p + —12:22- ^^-12:22:32— ' + • • • ad CO ;
m2 m2 (m2 + P) n^jjn^+ V) (^2 + 3^)
^ + F + —1X3^" ■*" r^^3i:5"2 + ... ad CO .
(15.) Show that
fli . yi«2 ^ ;PiP£3 ^
ai+Pi (aa+2'i)(«2+F2) (ai+i'i)(«2+i'2)(«3+i'3)
iJlPa • • . Pn-l^n _-, P1P2 ' • 'Pn
= 1--
(Oj+Pi) (a2+i>2) . . . ian-^Pn) («1 +2^1) (a2+i'2) ■ • • («n+Fn) '
(It).) Show that
tan2^r-l'-(l''-^T, 34-(32-x2)2
2 ~ (12-x2)2 ■^(12-a;-'')2(,S2-a;2)-''^' • • *
(Glaisher, Math. Mess., 1873, p. 138.)
422 EXEECISES XXVII CH. XXXI
(17.) Show that
11.1^ 1.2
n^ n{n+l) n{n + l)(n + 2) n (n + 1) {n + 2) (n+3) " ''
and apply this result to the approximate calculation of ir^ by means of the
formula
7r2/6 = 1/12 + 1/22+1/32+. . . .
(Stirling, Methodus Differentialis, p. 28.)
(18.) Show that Sl/(m»-l) = l and 21/(a«- l) = log2, where m and n
have all possible positive integral values differing from unity, a is any even
positive integer, and each distinct fraction is counted only once.
(Goldbach's Theorem, see Liouv. Math. Jour., 1842.)
(19.) If n have any positive integral value except unity, and r be any
positive integer which is not a perfect power, show that S (n - l)/(r" - 1)
= 7r-/6 ; and, if d(ii) denote the number of divisors of n, that S (d(n) - l)/r"
= 1; also that S(n-l)/r = 2;i/(?-- 1)2. (lb.)
CHAPTER XXXII.
Simple Continued Fractions.
NATURE AND ORIGIN OF CONTINUED FRACTIONS.
§ 1.] By a continued fraction is meant a function of the form
. a3+^--. (1);
the primary interpretation of which is that bz is the ante-
cedent of a quotient whose consequent is all that lies under the
line immediately beneath b^, and so on.
There may be either a finite or an infinite number of links in
the chain of operations ; that is to say, we may have either a
terminating or non-terminating continued fraction,
h h
In the most general case the component fractions —, — ,
— , . . . , as they are sometimes called, may have either positive or
a^
negative numerators and denominators, and succeed each other
without recurrence according to any law whatever. If they do
recur, we have what is called a recurring or periodic continued
fraction.
For shortness, the following abbreviative notation is often
used instead of (1),
a, + AAA_... (2),
^2 + as + 054 +
the signs + being written below the lines, to prevent confusion
with
424 SIMPLE CONTINUED FRACTIONS CH. XXXII
^2 ^3 bi ^
^2 «3 «4
Examples have already been given (see chap, iii., Exercises
III., 15) of the reduction of terminating continued fractions ;
and from these examples it is obvious that evert/ tertninating
continued fraction whose constituents ai, a2, . . ., b^, bs, . . . are
commensurable numbers reduces to a commensurable number.
§ 2.] In the present chapter we shall confine ourselves
mainly to the most interesting and the most important kind
of continued fraction, that, namely, in which each of the nume-
rators of the component fractions is +1, and egxjh of the
denominators a positive integer. When distinction is necessary,
this kind of continued fraction, namely,
111 ,,,
«i + . . . (1),
tta + «3 + «4 +
may be called a simple continued fraction. Unless it is otherwise
stated, we suppose the continued fraction to terminate.
In this case, for a reason that will be understood by and by,
the numbers ai,a2,as,. . . are called the first, second, third, . . .
partial quotients of the continued fraction.
§ 3.] Every number, commensurable or incommensurable, may
be expressed uniquely as a simple continued fraction, which may
or may not terminate.
For, let X be the number in question, and «i the greatest
integer which does not exceed X; then we may write
X=«i+^ (1),
where Xi>l, but is not necessarily integral, or even commensur-
able.
Again, let a^ be the greatest integer in Xi, so that a^^l',
then we have
where X2>\, as before.
Xi = aa + ^ (2),
* The notation a, + — + — + -* + . . .is frequently used by Continental
•* a« a^ a^
writers.
§§ 1-3 CONVERSION OF ANY NUMBER INTO S.C.F. 425
Again, let a-i be the greatest integer in X2 ; tlien
X.==a3 + ^^ (3);
and so on.
This process will terminate if one of the quantities ^Y", say
jr„_i, is an integer ; for we should then have
Now, using (2), we get from (1)
X=ai + Y
Thence, using (3), we get
A = «! + -
and so on.
Finally, then.
«2 + Y
X=ax+ . . . — (a),
flf2 + % + «n
It may happen that none of the quantities X comes out
integral. In this case, the quotients «i, «2, • • • either recur, or
go on continually without recurrence ; and we then obtain in
place of (a) a non-terminating continued fraction, which may be
periodic or not according to circumstances.
To prove that the development is unique, we have to show
that, if
11 ,11 ,^.
a2 + a3+ «2 +«3 +
then ffli = a/, ^2 = 0^2', ^s = eta, &c.
Now, since a^ and a^ are positive integers, and . . . and
el's "^
-—, — . . . are both positive, it follows that ... and — -. —
a-i + tta + «3 + ^2 +
— — . . . are both proper fractions. Hence, by chap, iii., § 12,
426 CONVERSION UNIQUE CH. XXXII
(r),
- (8).
we must h
ave
and
1
a2 +
1
•«3 +
(h
1
ai->r
1
, . ,
Again,
from
(S), we have
0^2
1
■f —
«3 +
1
a^+ ' '
= a^ +
1
1
From (c), by the same reasoning as before, we have
, 111 111 , .
and . . . = —7— —7 7— . • • ('?)•
^3+ ^4+ a5+ as + «4 + ^5 +
Proceeding in this way, we can show that each partial
quotient in the one continued fraction is equal to the partial
quotient of the same order in the other*.
This demonstration is clearly applicable even when the
continued fraction does not terminate, provided we are sure
that the fractions in ()8), (S), {rj), &c. have always a definite
meaning. This point will be settled when we come to discuss
the question of the convergency of an infinite continued fraction.
Cor. If «!, a2) • • •, «re, h, bz, . . ., bn be all positive
integers, Xn+i and yn+i any positive quantities rational or irra-
tional each of which is greater than unity, and if
1 J_J_^^,4.J_ 1 1
^ ^2 + ' ' " a» + ^«+i ~ ^ h+ ' ' ' bn+ yn+l '
then must
ai = bi, a.2 = b2, . . ., an -= bn, and also Xn+i = yn+\-
§ 4,] As an example of the general proposition of § 3, we
may show that every commensurable number may be converted
into a terminating continued fraction.
Let the number in question be A/B, where A and B are
integers prime to each other. Let Oi be the quotient and C the
remainder when A is divided by j9 ; Og the quotient and D the
* We suppose, as is clearly allowable, that, if the fraction terminates, the
last quotient is > 1. It should also be noticed that the first partial quotient
may be zero, but that none of the others can be zero, as the process is
arranged above.
%S,4i CASE OF COMMENSURABLE NUMBER 427
remainder when B is divided by C; a^ the quotient and E the
remainder when C is divided by D ; and so on, just as in the
arithmetical process for finding the G.C.M. of A and B. Since
A and B are prime to each other, the last divisor will be 1, the
last quotient a,j, say, and the last remainder 0. We then have
A
B^
= «i
G
= «!
1
'^ BIG'
B
C~
■■a.2,
D
= a^
1
^ GjD'
Hence ^''^
B ^ Oa + % + " * " a» '
It should be noticed that, if ^ <5, the first quotient ai will be zero.
Example 1.
To convert 167/81 into a continued fraction.
Going through the process of finding the G.C.M. of 167 and 81, we have
81)167(2
162
5)81(16
80
1)5(5
6
0
Hence
Example 2.
Consider -23 = 23/100.
We have
81~ 16+ 5*
100)23(0
0
Hence
23)100(4
92
8)23(2
16
7)8(1
7
1)7(7
7
0
428 CASE OF INCOMMENSURABLE NUMBER CH. XXXII
Cor. If we remove the restriction that tJie last partial quotient
shall be greater than unity, we may develop any commensurable
number as a continued fraction which has, at our pleasure, an
even or an odd number of partial quotients.
For example, 2 + -^^ — - has an odd number of partial quotients ; but we
lo -|- o
may write it 2+ - . — -; — -, which has an even number.
16+4+1
§ 5.] Any single surd, and, in fact, any simple surd number,
such as A + Bp^'"' + Cp'^'"' + . . . + Aj»<"~^"", can be converted into
a continued fraction, although not, of course, into a terminating
continued fraction.
The process consists in finding the greatest integer in a series
of surd numbers, and in rationalising the denominator of the
reciprocal of the residue. Methods for effecting both these
steps are known (see chap, x.), but both, in any but the
simplest cases, are very laborious. It will be sufficient to give
two simple examples, in each of which the result happens to
be a periodic continued fraction.
Example 1.
To convert Ji^ into a continued fraction.
We have, 3 being the greatest integer < JlS,
Vl3 = 3 + (Vl3-3) = 3+ — ^
1/(^13-3)'
= 3+--J: (1).
(Vl3 + 3)/4
Again, since the greatest integer in (,^13 + 3)/4 is 1, we have
4 4 4/(^/l3-l)'
= 1+— 7= (2).
(Vl3 + l)/3 ^ '
Vl3 + 1_ jYi-2 _ 1
3 "" ■^3/(Vl3-2)'
= 1 1
'*'(s/l3 + 2)/3 ^^^'
§§ 4, 5 EXAMPLES 429
Vl3 + 2_ Vl3-1^
3 a
3/(Vl3-
1
-1)'
1
4/(n/13-
1
-3)'
^13 + 3
1
1
-3)'
= 1 + ;^=^— (4);
n/13 + 1^.^^ VT3-3^^^
4 4
=1+—^— (5);
Vl3 + 3 = 6+ Vl3-3 = 6 +
(Vl3 + 3)/4
after which the process repeats itself.
From the equations (1)...(6) we derive
^13 = 3+ — -_____...,
♦ * .
where the * * indicate the beginning and end of the cycle of partial quotients.
Example 2.
Jb-1
To convert ^^-^ — into a continued fraction.
We have
^3-1
=0 +
2/(V3-l)
0+ '
V3 + l'
^3 + 1 = 2 + ^^3-1 = 2 + ^ ,
1/(^3-1) •
=2+— yJ^ — ;
(V3 + l)/2
v/3 + l_ J3-l_ 1
2 - + 2 -' + 2/(V3-l)'
V3+1
after which the qaotients recur. We have, therefore,
Vizi.o+J^JL... .
2 ^2+ 1 +
* *
It will be proved in chap, xxxiii. that every positive number of the form
(J^ + Q)IR, where P is a positive integer which is not a perfect square, and
Q and R are positive or negative integers, can be converted into a periodic
continued fraction ; and that every periodic continued fraction represents an
irrational number of this form.
430 • EXERCISES XXVIII CH. XXXII
Exercises XXVIII.
Express the following as simple continued fractions, terminating or
periodic as the case may be: —
/I \ 15 to\ 5^2 n \ 39293 76
(1-) 73- (^-^ il93- (^-^ 36932- ^^'^ ^^i^-
(5.) 2-718281. (6.) -0079. (7.) ^2. (8.) ^5. (9.) ^/(ll).
(10.) ^(10). (11.) ^/(12). (12.) Vf. (13.) V3 + 1.
(14.) ^-±1^^
(15.) Show that H-ig = l + ^^^-L... .
* «
(16.) A line AB is divided in C, so that AB.AC=BC^. Express the
ratios ACjAB, BCIAB as simple continued fractions.
(17.) Express sj(a^ + a) and ij{a^-a) as simple continued fractions, a
being a positive integer.
(18.) If a be a positive integer, show that
2V(l4-a^) = 2a + ^jl^... .
« ♦
(19.) If a be a positive integer > 1, show that
« *
(20.) Show that
(21.) Show that every rational algebraical function of x can be expanded,
and that in one way only, as a terminating continued fraction of the form
_1 1_ J^
where Qi, Q^^ • • • » Qn ^^^ rational integral functions of x.
Exemplify with (a?+x^+x+l)l(x*+3x^ + 2!>i^ + x + l).
(22.) If x=^ A . . ..
, b a
and j, = __...,
* «
show that x-y = a-b.
§ 6 COMPLETE QUOTIENTS AND CONVERGENTS 431
PROPERTIES OF THE CONVERGENTS TO A CONTINUED FRACTION.
§ 6.] Let US denote the complete continued fraction by x^, so
that
and let
,111
^2 + «3 + »4 +
1
'a.
0);
1 1 1
Xn — Ctln + ...
(2);
1 1
a?3 = ^3 + . . . —
(3);
and so on.
Then x<2,Xz, . . . are called the complete quotients corresponding to
a^, <h, • • -J Of) simply, the second, third, . . . complete quotients.
The fraction itself, or x^, may be called the first complete quo-
tient. It will be observed that ai, a.2, a^, . . . are the integral
parts of ^1, X2, X3, . . . .
Let us consider, on the other hand, the fractions which we
obtain by first retaining only the first partial quotient, second by
retaining only the first and second, and so on ; and let us denote
the fractions thus obtained, when reduced (without simplifica-
tion, as under) so that their numerators and denominators are
integral numbers, by pilqi, p-ilq^, . . . Then we have
ax Px / X
«x =r =q, (")'
(7),
(8).
where
a.2 a^
^2
1 1 _ ai^a^s + ai + Os
fta + 0,% CL-^z + 1
2'3
11 1 .
ai + ... — = &c.
^2 + «3 + «»
and so on,
Px = ax, 9'i = l
Pi - a^a^ +1, g'2 = Oa
_P3 = aia2«3 + ai + «3, g'3 = a2«3
+ 1
and so on.
in
(7)>
432 RECURRENCE-FORMULA FOR CONVERGENTS CH. XXXII
The fractions p^jqi , Pijq^ , ■ • ■ are called the Jirst, second, . . .
convergents to the continued fraction.
Cor. If the continued fraction terminates, the last convergent
is, hy its definition, the continued fraction itself.
§ 7.] It will be seen, from the expressions for pi, p.^, pa and
qi, q'l, qs in § 6 (a), i/3'), (y), that we have
Pi = a3P2+Pi (1);
q3 = a;q2 + qi (2).
This suggests the following general formulw for calculating the
numerator and denominator of any convergent when the numerators
and denominators of tlie two preceding convergents are known,
namely,
Pn = anPn--l+PH-2 (3);
qn = dnqn-l + qn-2 (4).
Let us suppose that this formula is true for the nth. con-
vergent. We observe, from the definitions (a), (J3), . . ., (8) of
§ 6, that the n + lth convergent, pn+i/qn+i, is derived from the
wth if we replace a„ by a„ + l/a„+i. Hence, since pn-i, qn-i,
p„_2, qn-2 do not contain a„, and since, by hypothesis,
Pn ^anPn-l+Pn-2
2'n ^nQn-i + qn-2
it follows that
Pn+1^ {Cl'n + llan+\)Pn-\ + Pn-2
qn+1 {(^n + l/<^Ji+l) 9'ji-l + qn-2
or, after reduction,
Pn+l _ an+i {anPn-l+Pn-2) +Pn-1
qii+1 Cin+l y^nqn-l + qn-i) + 3'n-l
^ dn+lPn -^Pn-l
Ctn+iqn + qn-1
by (3) and (4).
Hence it is sufficient if we take
Pn+l = (f-n+lPn +^n-l \
qn+\ — (tn+\qn + qn-1'
In other words, if the rule hold for the wth convergent, it holds
for the wTlth. Now, by (1) and (2), it holds for the third;
hence, by what has just been proved, it holds for the fourth ;
hence for the fifth; and so on. That is to say, the rule is
general.
§§6,7
PROPERTIES OF CONVERGENTS
433
Cor. 1. Since a„ is a positive integral number, it follows from
(3) and (4) that the numerators of the successive convergents form
an increasing series of integral numbers, and that the same is true
of the denominators.
Cor. 2. From (3) and (4) it follows that
' ' ' (5);
and
- — = a» +
qn 111
qn~i " an-1 + an-1 + ' ■ 'as
(6).
For, dividing (3) by j9„_i, and writing successively w-1, w-2,
. . ., 3 in place of w, we have
•PnlPn-l = «n + " /" ;
/'»-]//'»l-2
= tts +
aa + «i '
From these equations, by successive substitution, we derive (5) ;
and (6) may be proved in like manner.
Example 1.
The continued fraction which represents the ratio of the circumference
of a circle to the diameter is 3 + ^r— t^— ^j — :r-- — :; — :; — . . , . It is
7+ 15+ 1+ 2U2+ 1+ 1 +
required to calculate the successive convergents.
1 3 22
The first two convergents are 3 and 3 + = , that is, - , — .
Hence, using the formulae (3) and (4), we have the following table : —
n
a
P
q
1
3
3
1
2
7
22
7
3
15
333
106
4
1
355
113
5
292
103993
33102
6
1
104348
33215
7
1
208341
66317
where p4 = 355, for example, is obtained by multiplying the number over it,
namely 333, by 1, and adding to the product the number one place higher
still, namely 22.
28
II.
434 EXAMPLES CH. XXXII
The successive convergents are therefore
3 22 333 355 103993
1' 7 ' 106' 113' 33102 » ' ■ • *
Example 2.
If Pi/^i. Ihl<l2> ... be the convergents tol + s— ^— -r— ...— - ...
ad 00 , show that
Pn={n-l)Pn-i + in-l)Pn-2 + {n-2)p^_s+. . .+3p2 + 2pi + 2.
By the recurrence-formula we have
Pn=fiPn-l+Pn-2;
Pn-l = (n-'^)Pn-2+Pn-3'
JPn-2 = («-2)i?„_3+i'„-4;
Ps=^P2+Pi;
and (since Pi = l, p^=3)
p.,^2pi + l.
Adding all these equations, and observing that Pn-^, Pn-Sf • • -t Ps
each occur three times, once on the left multiplied by 1, once on the right
multiplied by 1, and again on the right multiplied by n-1, n-2 3
respectively, we have
2J„=(n-l)i)„_i + (w-l)i)„_2 + («-2)j>„_3+. . .+dp2 + 2pi + {p^ + l),
■which gives the required result since i5i = l.
Example 3.
In the case of the continued fraction a, h . . • prove
a2+ ai+ a2+ 0^ +
that P2n = q2,i+l . ^^2™-! = «l92n/«2 •
By the definition of a convergent, we have
P?n+i=a,+^...^ (a),
32,1+1 * a2+ «!
since every odd partial quotient is a^ .
Again, by Cor. 2 above,
Hence
•which gives
Pin ' 02+ «1
Pin+\ _ Pin-¥\
3211+1 Pin
2'2»=9'2n+l (t).
Also, since l'2» = «2P2n-l+i'2n-2»
92n+] = '^iS'Zn + 92n-l i
(7) leads to
«h!P2»-l +i'2n-2 = «l9'2n + 5'2n-l (5).
Now, if we write n-1 for n in (7), we have i'27»-a=3'2»-i*» hence (5) gives
«2P2n-l = ai5'2»-
Therefore
i'2H-i=r'32» (0-
"2
§ 8 PROPERTIES OF CONVERGENTS 436
§ 8.] From equations (3) and (4) of last section we can prove
the following important property of any two consecutive con-
fer gents : —
For, by § 7 (3) and (4),
Pn^-X^r, -Pnqn+\ = ifln+\Pn + Pn-^ qn' Pn («»i+i9'>i + qn-i),
= -(Pnqn-i-Pn-iqn)-
Hence, if (1) hold, we have
Pn+iqn -Pnqn+l = - ( - 1)",
In other words, if the property be true for any integer n, it
holds for the next integer n + 1. Now
P^qi -Piq2 = (aia2 +1)1- fti^s,
= 1,
that is to say, the property in question holds for n = 2, hence it
holds for w = 3 ; hence for w = 4 ; and so on.
Cor. 1. The convergents, as calculated by the rule of %7, are
fractions at their lowest terms.
For, if pn and qn, for example, had any common factor, that
factor would, by § 8 (1), divide (-1)" exactly. Hence pn is
prime to qn', and Pn/qn is at its lowest terms.
Cor. 2.
Pn Pn-l ^ ( - I)'*
qn 2'n-i qnqn-i
(2).
Cor. 3.
'n qi \q2 qJ \q3 qJ ' ' ' Vg™ qn-J'
1 1 (-1)"
+ . . . + ^ — '-
= «! + ■ +. . . + ^ '- (3).
qiq^ q^qz gn-iqn ^
Cor. 4.
Pnqn-2-Pn-2qn={-Y~^an (4).
For
Pnqn-i -Pn-^qn = {^nPn-X ^Pn-^ g'n-2 - Pn-1 {flnqn-X + g'n-2),
= {Pn-\qn-1 -Pn-^qn-l) ^n,
= (-r-^a™,byCor. 1.
28—2
436 EXAMPLE CH. XXXII
Cor. 5.
Piikn -Pn-llgn-Z = ( " Y'%Jqnqn-i (5).
Cor. 6. The odd conver gents continually increase in value, the
even conver gents continually decrease; every even convergent is
greater than every odd convergent; and every odd convergent is less
than, and every even convergent greater than, any following con-
vergent.
These conclusions follow at once from the equations (2) and (5).
Cor. 7. Given two positive integers p and q which are prime
to each other, we can always find two positive integers p and q
such that pq -p'q=+ 1 or = - 1, as we please.
For, by § 4, Cor., we can always convert piq into a continued
fraction having an even or an odd number of partial quotients,
as we please. If p'lq' be the penultimate convergent to this
continued fraction, we have in the former case pq' -p'q- + 1, in
the latter pq^ —p'q = — 1.
Example. If jp„/g'„ be the nih. convergent to a^H . . . — , and
s^nlsQn *^^ convergent to Ug-i ... — which corresponds to the
"s+l + ^P
partial quotient a„, show that
Pn9n-r -Pn-r<ln={ " l)""''*"^n-J+lQn«
We have, by our data,
— = a, 4 ... — (a),
ffn «2+ «n
Q.n-r ^2+ ^n-r
heno ^' = a^ + . . . ^-^ -^ (7).
Now
Pn-r _ ^n-rVn-r-\ +Fn-r-
9n-r ^n-r 3n-r-i + 1n-r-2
Hence, by (a) and (7),
Pn ^ i<^n-r + n-r+lQnln-r+lPn)Pn-r-l +J'n-r-3
ffii (^n-r + n-r+l^n/n-r+l^n) 9n-r-X + 3n-r-2
_ Pn-r"^ n-r+T.QnPn-r-lln-r+\-^ n
3n-r + n-r+lQn 9n-r-lln-r+\"n
n—r+l"n Pn—r ' n-r+l VnPn— ^ — 1
n-r+1 °n3n— r + n-r+lQn 3n-r-l
Now it la easy to see that the numerator and denominator of the fraction
last written are mutually prime; therefore
(8).
Pn -~ n-r+l "nPri-r + n-r+l '■
9n — n-r+l "n 2 n-r + n-r+l Q
^nPn-r-l' I /^v
'n3n-r-i.i ^ '"
§§ 8, 9 APPROXIMATION TO CONTINUED FRACTION 437
From (e) we derive
PnQn-r ~Pii-r1ii~ ~ (Pn-rQn-r-l ~ Pn-r-1 Qn-rJn-r+lQuy
= (-l)(-l)"-Vr+l<?n.
by (1) above,
=(-ir-'-+VrH<3„;
as was to be shown.
§ 9.] The convergents of odd order are each less than the
whole continued fraction, and the convergents of even order are
each greater; and each convergent is nearer in value to the whole
continued fraction than the preceding.
We have, by § 7,
Pn^-x ^ (in+^Pn -^ Pn-\ ,
3'n+i <^n+i^n + ^n-l
and the whole continued fraction a?i is derived from pn+^/qn+i by
replacing the partial quotient an+i by the complete quotient Xn+i.
Hence
_ iVn+iPn + Pn-i
W\ — .
From this value of x^ we obtain
„ Pn _ ^n+}Pn+Pn-i Pn
OOi — ,
Pn-\qn-Pnqn-i
qn(a!n+iqn + qn-i)
(1).
Similarly
_Pn-\ ^ ^n+l(Pnqn-l-Pn-iq») ,^.
qn-1 2'w-l (^re+l5're + 3'?i-l/
From (1) and (2) we deduce
Pn
Xi--
q-n, qn-1
Pn~\ qn^n+l
(3).
qn-i
Now qn-\, qn ar^> positive integers ; a?„+i -^ 1 ; and, by § 7,
Cor. 1, qn-i<qn- It follows, therefore, from (3) that Xi-pnjqn
is opposite in sign to, and numerically less than, Xi-pn^j/qn-i.
In other words, pjqn differs from Xi by less than pn-i/qn-i does ;
and if the one be less than Xi , the other is greater, and vice versa.
438 APPROXIMATION TO CONTINUED FRACTION CII. XXXII
Now the first convergent is obviously less than a^i, hence the
second is greater, the third less, and so on ; and the difference
between Xi and the successive convergents continually decreases.
Cor. 1. The difference between the continued fraction and
the nth convergent is less than 1/qnQn+i, and greater than
Cl'n+2/(J!nqn+2'
For, by what has just been proved,
Pn Pn+2 ^ Pn+l
Qn Qn+a 9'n+l
are, in order of magnitude, either ascending or descending.
Hence
Pn ^ ^Pn Pn+l
9'n Qn 9'n+l
<r^,by§8(2).
Again,
Pn ^ ^Pn Pn+a
Qn 9'n Qn+i
>-^,by§8(5).
Since gn+i>qn, and since qn+ijan+i = («n+2g'n+i + gn)/an+2
= qn+i + qn/an+2<qn+i + qn (ci'n+2 being <|:l), it follows that the
upper and lower limits of the error committed by taking the nth
convergent instead of the whole continued fraction may be
taken to be l/q^ and l/qn(qn + qn+i)- These, of course, are not
so close as those given above, but they are simpler, and in many
cases they will be found sufiicient.
Cor. 2. In order to obtain a good approximation to a
continued fraction, it is advisable to take that convergent wJiose
corresponding partial quotient immediately precedes a very much
larger partial quotient.
For, if the next quotient be large, tliere is a sudden increase
in qn+i , so that l/qnqn+i is a very small fraction.
The same thing appears from the consideration that, in
taking pn/qn instead of the whole fraction, we take a» instead of
§ 9 CONDITION THAT Pn/g'n BE A CONVERGENT TO X^ 439
ttn + ■ . . . , that is, we neglect the part ... of the
complete quotient. Now, if a„+i be very large, this neglected
part will of course be very small.
Cor. 3. The odd convergents form an increasing series of
rational fractions continually approaching to the value of the
whole continued fraction; and the even convergents form a
decreasing series having the same property*.
Cor. 4. If Pnlc[n-i^x<'^l<in{<ln + qn-\), where qn-x is the de-
nominator of the penultimate convergent to Pn/^n when converted
into a simple continued fraction having an even number of
quotients, then Pnlqn is one of the convergents to the simple
continued fraction which represents Xi; and the like holds if
3^i-Pnlqn<'i-/^n(qn + Qn-i), whore qn-i is the denominator of the
penultimate convergent to pn/qn when converted into a simple
continued fraction having an odd number of quotients.
Let «!, ttaj •..>«» be the n partial quotients of pnjqn
when converted into a simple continued fraction having an
even number of quotients, and let Pn-\lqn-\ be the penultimate
convergent. Then pnqn-\ -Pn-iqn = 1-
Let Xn+i be determined by the equation
1 1 1
^1 = «i + . . . .
^2 + an + iy„+i
Then we have
^1 = (-^^n+li^/i +P.i-l)liXn+iqH + qn-l).
whence
^71+1 = (iViqu-i -Pn-i)/(Pn - a^iqn),
* The value of every simple continued fraction lies, of course, between
0 and 00 ; and we may, in fact, regard these as the first and second con-
vergents respectively to every continued fraction. If we write 0 = f , and
00 = i , and denote these by -^ and — , so that we understand »_, to be 0,
Pg to be 1, q_i to be 1, and q^^ to be 0, then p_j and p^ will be found to fall
into the series p^, p^, p.^, &c., and g_j and Qq into the series q^, q^, q^, &c.
It will be found, for example, that l>i = a^pQ+p_-^^, <j'j = ajg'o + ?-i > Pol-i -^-i?o
= ^ - l)** =r 1, and so on,
440 CONDITION TRAT pn/qn BE A CONVERGENT TO CC^ CH. XXXII
or, if we put ^ =pn/qn - ^i,
^»+l = {{Pnqn-l -Pn-\ qr^hn - qn-X i]\q%^,
= i^/qn - qn-1 ^)/qni'
Hence the necessary and sufficient condition that iCn+i > 1 is that
i/qn-qn-ii>qJ,
that is,
^<'i-/qn(qn+qn--d,
which is fulfilled by the condition in the first of our two
theorems.
Let now bi, h^, . . ., bn be the first n partial quotients in the
simple continued fraction that represents ^j. Then we have
,1 11
Xi-bi + y . . . 7 ,
02+ bn +2/n+l
where ^»+i > 1.
Hence
1 J ]_^j ^ J_ 1 1
^ 02+ ' ' ' an + iCn+1 ^ b.2+ ' ' ' bn+1/n+i'
Therefore, by § 3, Cor., we must have
Cl'i — bi, ^2 = 62, . . ., ttn — bn, ^n+i—^n+i-
1 1 . p .
Hence «i + . . . + — , that is, — is the nth convergent to
Cti + G'n qn
The second theorem is proved in precisely the same way.
Since qn-\<qni the conditions above are a fortiori fulfilled if
X^~Pn\qn<^\'^qn-
§ 10.] The propositions and corollaries of last section show
that the method of continued fractions possesses the two most
important advantages that any system of numerical calculation
can have, namely, 1st, it furnishes a regular series of rational
approximations to the quantity to be evaluated, which increase
step by step in complexity, but also in exactness ; 2nd, the error
committed by arresting the approximation at any step can at
once be estimated. The student should compare it in these
respects with the decimal system of notation.
§§ 9-11 CONVERGENCE OF S.C.F. 441
§ 11.] It should be observed that the formation of the suc-
cessive convergents virtually determines the meaning we attach
to the chain of operations in a continued fraction.
If the continued fraction terminate, we might of course pro-
ceed to reduce it by beginning at the lower end and taking in
the partial quotients one by one in the reverse order. The
reader may, as an exercise, work out this treatment of jfinite con-
tinued fractions, and he will find that, from the arithmetical
point of view, it presents few or none of the advantages of the
ordinary plan developed above.
In the case of non-terminating continued fractions, no such
alternative course is, strictly speaking, open to us. Indeed, the
further difficulty arises that, a priori, we have no certainty that
such a continued fraction has any definite meaning at all. The
point of view to be taken is the following : — If we arrest the
continued fraction at any partial quotient, say the sth, tlien, in
the case of a simple continued fraction, however great s may be,
we have seen that the two convergents, p2n-i/Q2n-i, P-^l^^n, in-
clude the fraction psjqa between them. Hence, if we can show
that p^-i/q2n~i and Pznlq2n, each approach the same finite value
when n is increased without limit, it will follow that as s is
increased without limit, that is, as more and more of the partial
quotients of the continued fraction are taken into account,' pa/qa
approaches a certain definite value, which we may call the value
of the whole continued fraction. Now, by § 8, Cor. 5, p2n-i/q2n-i
continually increases with n, and p-2n/q2n continually decreases,
and P2n/q2n>p^-ilq2n-i- Hence, since both are positive, each of
the two must approach a certain finite limit. Also the two
limits must be the same ; for by § 8, Cor. 2, P2nlq^-p^-\lq2n-i
= l/q^q^n-i, and by the recurrence formula for q^ it follows that
q2n and g'sn-i increase without limit with n ; therefore p-aijqon
—p^n-ilq-in-i may be made as small as we please by sufficiently
increasing n.
It appears, therefore, that everi/ simple contimied fraction has
a definite finite value.
Example.
To obtain a good commensurable approximation to the ratio of the
&0.
442 EXERCISES XXIX CH. XXXII
circumference of a circle to the diameter. Referring to Example 1, § 7,
■we have the following approximations in defect: —
3 333 103993
1' 106' 38102 • '
and the following in excess : —
22 355 104348
7 ' 113 ' 33215 '
Two of these*, namely 22/7 and 355/113, are distinguished heyond the others
by preceding large partial quotients, namely, 15 and 292.
The latter of these is exceedingly accurate, for in this case l/g„?„+i
= 1/113 X 33102 = -0000002673, and a„+2/g'„q'„+2= 1/113 x 33215 = •0000002665.
The error therefore lies between -000000266 and -000000267 ; that is to say,
355/113 is accurate to the 6th decimal place. In point of fact, we have
7r = 3-14159265358 . . .
355/113 = 3-14159292035 . . .
Differences -00000026677 ....
Exercises XXIX.
769
(1.) Calculate the various convergenta to yprj, and estimate the errors
committed by taking the first, second, third, &c., instead of the fraction.
(2. ) Find a convergent to the infinite continued fraction :j — ;r — - — ...
which shall represent its value within a millionth.
(3.) Find a commensurable approximation to /v/(17) which shall be
accurate within 1/100000, and such that no nearer fraction can be found
not having a greater denominator.
(4.) The sidereal period of Venus is 224-7 days, that of the earth 365-25
days ; calculate the various cycles in which transits of Venus may be expected
to occur. Calculate the number of degrees in each case by which Venus is
displaced from the node, when the earth is there, at the end of the first cycle
after a former central transit.
(5.) Work out the same problem for Mercury, whose sidereal period ia
87-97 days.
(6.) According to the Northampton table of mortality, out of 3635
persons who reach the age of 40, 3559 reach the age of 41. Show that
this is expressed very accurately by saying that 47 out of 48 survive.
* The first of them, 22/7, was given by Archimedes (212 b.c). The
second, 355/113, was given by Adrian Metius (published by his son, 1640
A.D.) : it is in great favour, not only on account of its accuracy, but because
it can be easily remembered as consisting of the first three odd numbers
each repeated twice in a certain succession.
§11
EXERCISES XXIX 443
(7.) Find a good rational approximation to sj(19) which shall differ from
it by less than 1/100000 ; and compare this with the rational approximation
obtained by expressing ij{19) as a decimal fraction correct to the 6th place.
(8.) If a be any incommensurable quantity whatever, show that two
integers, m and n, can always be found, so that 0<an-m<K, however small
K may be,
(9.) Show that the numerators and also the denominators of any two con-
secutive convergents to a simple continued fraction are prime to each other ;
also that if p^ and p^-^ have any common factor it must divide a„ exactly.
(10.) Show that the difference between any two consecutive odd convergents
to ij(a^ + l) is a fraction whose numerator, when at its lowest terms, is 2a.
(11.) Prove directly, from the recursive relation connecting the numera-
tors and denominators, that every convergent to a simple continued fraction
is intermediate in value to the two preceding.
(12.) Prove that
3n%-i'«= (-1)"+V^2^3 • • • ^n+l-
Show that pjqn differs from x^ by less than Ija^a^ . . . a„+i(|'„. Is this a
better estimate of the error than llgn^n+i '
(13.) If the integers x and y be prime to each other, show that an integer
u can always be found such that
where z is an integer.
(14.) Prove that
(PJ - in) d'n-l" - 3n-l^) = {PnPn-1 " 9n ^n-lY ' 1 '.
Pn+9n ^ {PnPn-l + QnQn-^)^+'^
Pn-i + in-i {Pn-lPn-i + in-\ in-^f + 1 '
(15.) Prove that p^-iPn - 1n-\1n^i ^^ positive or negative according as n
is even or odd.
(16.) If P/Q, P'/Q', P"IQ" be the nth, n-ltb, Ji^^th convergents of
1
1
02 +
1
03 +
1
04 +
1
02 +
1
03 +
1
«4 +
1
a3 +
1
O4 +
respectively, show that
P = a2P' + P", (3 = (aia2 + l)P' +«!?".
(17.) If the partial quotients of Xi=pjq^ form a reciprocal series (that is,
a series in which the first and last terms are equal, the second and second
last equal, and so on), then Pn-i = qn^ ^^^ {ln'^^)IPn i^ ^^ integer; and,
conversely, if these conditions be satisfied, the quotients will form a
reciprocal series.
(18.) Show, from last exercise, that every integer which divides the sum
of two integral squares that are prime to each other is itself the sum of two
squares. (See Serret, Alg. Sup., 4'"« ed., t. i., p. 29.)
444 EXERCISES XXIX CH. XXXII
(19.) Show that
11 ' '
1 1,1 1
ajH- a„-i an-i+ «a
(20.) If ^1 = rv rz ;7T • • • ' ^^°^^ *^^* 2'™=3»-i-
1111
(21.) The successive convergents of 2a -\ — ; ^ — - . . . are
^ ' a+ 4a+ a+ 4a +
always double those of a + 2— - 0^-77 • • • •
(22.) If the reduced form of the nth complete quotient, x^, in
0,+ ... be ^Jvn, show that
fn = '^n fn+1 "•" fn+2 '
(23.) Find the numerically least value of ax -by for positive integral
values of x and y, a and & being positive integers, which may or may not be
prime to each other.
CLOSEST COMMENSURABLE APPROXIMATIONS OF GIVEN
COMPLEXITY.
§ 12.] One commensurable approximation to a number
(commensurable or incommensurable) is said to be more complex
than another when the denominator of the representative frac-
tion is greater in the one case than in the other. The problem
which we put before ourselves here is to find the fraction, whose
denominator does not exceed a given integer D, which shall most
closely approximate {by excess or by defect, as may he assigned)
to a given number commensurable or incommensurable. The
solution of this problem is one of the most important uses of
continued fractions. It depends on a principle of great interest
in the theory of numbers, which we proceed to prove.
Lemma. — Ifplq andp'jq' he two fractions such thatpq'-p'q=l,
then no fraction can lie between them unless its denominator is
greater than the denominator of eitJier of them.
Proof — Let ajb be a fraction intermediate in magnitude to
pjq and p'jq. Then
q h^q q' ^^' '
?-^<^-< (2).
o q q q ^
^ 12-14 SIMPLICITY OF APPROXIMATION 44)5
From(l), PkzM^PlJLS^.
^ ' qb qq
pb — qa 1
qb qq '
Hence qb > qq (pb - qa) ;
and b>(pb- qa) q.
Now p\q — ajb is positive, hence pb - ga is a positive integer.
It follows, therefore, that bxq.
Similarly it follows from (2) that b>q.
Hence no fraction can lie between p\q and p\(l unless its
denominator is greater than both q and q. In other words, if
pq' -p'<l = ^, no commensurable number can lie between pjq and
p'l<][ which is not more complex than either of them.
§ 13.] The nth convergent to a continued fraction is a nearer
approximation to the value of the complete fraction than any
fraction whose denominator is not greater than that of the con-
vergent. For any fraction ajb which is nearer in value to the
continued fraction than Pnlqn must, a fortiori, be nearer than
Pn-il^n-i' Hence, since Pn/gn and Pn-il^n-i include the value of
the continued fraction between them, it follows that a/b must
lie between these two fractions. Now we have, by § 8, either
Pnqn-l-Pn-iqn^'i-, Or Pn-iqn- Pnqn-l= i- HcnCC, by § 12, b
must be greater than qn, which proves our proposition.
Example.
Consider the continued fraction a>, = 3 + - — — - ■-— -— -.
^ 1+ 3+ 4+ 2+ 5
mt, • ^ 3 4 15 64 148 779 ,,
The successive convergents ^'^^ 3; » j » 4" » pf ' "og > on? ' ^^ ^^ ^^''^
any one of these, say 64/17, the statement is, that no fraction whose
denominator does not exceed 17 can be nearer in value to x^ than 64/17.
§ 14.] The result of last section is a step towards the solution
of the general problem of § 12; but something more is required.
Consider, for example, the successive convergents Pn-2lqn-2,
Pn-i/qn-i, Pnlqn to o^i, and let n be odd, say. Then
Pn-2 Pn ^ Pn-1
> > "^1 >
qn-2 qn qn-1
are in increasing order of magnitude. We know, by last
446 INTERMEDIATE CONVERGENTS CH. XXXII
section, that no fraction whose denominator is less than g^-i can
lie in the mtevwaX pn-ijqn-i, Pn-il^n-i, and also that no fraction
whose denominator is less than q^ can lie in the interval
Pnlqn, Pn-i/qn-i'} but we havG no assurance that a fraction
whose denominator is less than qn, may not lie in the interval
Pn-2/qn-2, Pnlqn, ^^'^ Pnqn-i- P%-iq% = a,,, where «„ may be>l.
This lacuna is filled by the following proposition : —
1°. The series of fractions
Pn-2 Pn-1 +Pn-1 Pn-2 + '^Pn-l
qn-i qn-2 + qn-1 9'n-2 + ^qn-l
Pn-i + an- IPn-l Pn-I + (^nPn-X f ^ PA /i\
qn-2 + an- ^qn-i' 9'«-2 + «re2'«-i V qJ
form (according as n is odd or even) an increasing or a decreasing
series.
2°. Each of them is at its lowest terms; and each consecutive
pair, say P/Q, P'/Q', satisfies the condition PQ — P'Q = ± 1 ; so
that no commensurable quantity less complex than the mm'e complex
of the two can he inserted between them.
The first and last of these fractions (formerly called Con-
vergents merely) we now call, for the sake of distinction, Principal
Convergents ; the others are called Intermediate Conmrgents to
the continued fraction. To prove the above properties, let us
consider any two consecutive fractions of the series (1), say PjQ,
P'lQ'; then
P_P^ Pn-2 + rpn-i _ Pn-2 + r+ 1 jP„_i
Q Q' qn-2 + rqn-i q^.^ + V+lq^-x
(where r = 0, or 1, or 2, . . ., or a„- 1),
^ - {Pn-l gn-2 -Pn-1 g^-i)
(g'„_2 + rqn-^ {qn-t + r + 1 q^-i) '
+ 1
{qn-2 + rqn--,) {qn-2 + r+l q^-i) '
~ OO ^ ** be odd,
+ 1
= ^^ if n be even.
(2).
§§ 14, 15 COMPLETE SERIES OF CONVERGENTS 447
Hghcb
Fq-P'Q=-li(nheodd, ] ,^.
= + 1 if w be even. J
(2) and (3) are sufficient to establish 1° and 2".
3°. Since P/Q-pn-i/qn-i = ±l/qn-i{qn~2 + rqn~i), and since
iVi obviously lies between P/Q and Pn-i/qn-i, it follows that the
intermediate convergent PjQ differs from the continued fraction
by less than l/q,i-i Q, a fortiori by less than Ijqn-i*
§ 15.] If we take all the principal conver gents of odd order
with their intermediates wherever the partial quotients differ from
unity, and form the series
9. Pi P^ Pl^^ Pn /A\
1 J • • •» J • • -J > • • •> ,...,,... {IV},
^ yi </3 qn-2 qn
and likewise all the principal convergents of even order with
their intermediates, and form the series
1 P2 Pi Pn-3 Pn-l /T>x
'J q-2 q* qn-s qn~i
then (A) is a series of commensurable quantities, increasing in com-
plexity and increasing in magnitude, which continually approach
the continued fraction; and (B) is a series of commensurable
quantities, increasing in complexity and decreasing in magnitude,
which continually approach the same; and it is impossible between
any consecutive pair of either series to insert a commensurable
quantity which shall be less complex than the more complex of the
two.
If the continued fraction be non-terminating, each of the two
series (A) and (B) is non-terminating.
If the continued fraction terminates, one of the series will
terminate, since the last member of one of them will be the last
convergent to Xi ; that is to say, Xi itself. The other series may,
however, be prolonged as far as we please; for, if Pn-i/qn-i c^nd
pjqn be the last two convergents, the series of fractions
Pn-l Pn-l +Pn Pn-1 + 2pn
qn-1 qn-i + qn g'„-i + 2g'„'
* For a rule for estimating the errors of principal and intermediate
convergents to a continued fraction, see Hargreaves, Mesa. Math., Feb. 1898.
448 CLOSEST RATIONAL APPROXIMATION CH. XXXII
forms either a continually increasing or a continvxilly decreasing
series, in which no principal convergent occurs, but whose terms
approach more and m.ore nearly the value pnlq^ that is, a?i*.
§ 16.] We are now in a position to solve the general problem
of § 12 1. Suppose, for example, that we are required to find the
fraction, whose denominator does not exceed D, which shall
approximate most closely by defect to the quantity Xi. What we
have to do is to convert Xi into a simple continued fraction, form
the series (A) of last section, and select that fraction from it whose
denominator is either D, or, failing that, less than hut nearest
to D, say PjQ. For, if there were any fraction nearer to Xi than
PjQ, it would lie to the right oiPjQ in the series; that is to say,
would fall between PjQ and the next fraction P'jQ of the series,
or between two fractions still more complex. Hence the denom-
inator of the supposed fraction will be greater than Q', and hence
greater than D.
Similarly, the fraction which most nearly approximates to Xi
by excess, and whose denominator does not exceed D, is obtained
* This may also be seen from the fact that the continued fraction
a, -4 ... — may also be written a, H . . . ; that is to
Bay, we may consider the last quotient to bo oo , and the last convergent
t The first general solution of this problem was given by Wallis (see
his Algebra (1685), chap, x.) ; Huyghens also was led to discuss it when
designing the toothed wheels of his Planetarium (see his Descriptio Automati
Planetarii, 1682). One of the earlier appearances of continued fractions in
mathematics was the value of 4/7r given by Lord Brouncker (about 1655).
While discussing Brounckcr's Fraction in his Arithmetica Injinitorum (1656),
Wallis gives a good many of the elementary properties of the convergents
to a general continued fraction, including the rule for their formation.
Saunderson, Euler, and Lambert all helped in developing the theory of
the subject. See two interesting bibliographical papers by Giinther and
Favaro, Bulletino di Bibliographia e di Storia dclle Scienze Mathematiclie e
Fisiche, t. vii. In this chapter we have mainly followed Lagrange, who gave
the first full exposition of it in his additions to the French edition of Euler's
Algebra (1795). We may here direct the attention of the reader to a series
of comprehensive articles on continued fractions by Stern, Crelle^s Jour., x.,
XI., xviii.
§§ 15, 16 EXAMPLES 44d
by taking tJiat fraction in series (B) of last section whose de-
nominator most nearly equals without exceeding D.
N.B. — If the denominator in the (A) series which most
nearly equals without exceeding D be the denominator of an
intermediate convergent, the denominator in the (B) series which
most nearly equals without exceeding D will be the denominator
of a principal convergent.
Example 1.
To find the fraction, whose denominator does not exceed GO, which
779
approximates most closely to -— ^ .
TTT T- 779 , 1 1 1 1 1
We have _:.3 + j-^ — j^- ^^ ^.
0 3 15 143
The odd convergent s are t > t » "j
38
1 4 64 779
the even convergents ^ , j- , j= , ^ .
The two series are
01237111579143922 1701 2480
1' I' I' 1' 2' 3 ' 4 ' 21' 38 ' 245' 452 ' 659 ' * ' ' ^ ''
1419344964207350493636779
0' I' 5' 9' 13' 17' 55' 93' 131' 169' 207 ^''
Hence, of the fractions whose denominators do not exceed 60, 143/38 is the
closest by defect and 207/55 the closest by excess to 779/207.
Of these two it happens that 143/38 is the closer, although its denomin-
ator is less than that of 207/55; for we have 143/38 = 3-76315 . . ., 207/55
= 3-76363 . . . , and 779/207 = 3-76328 . . . For a rule enabling us in most
cases to save calculation in deciding between the closeness of the (A) and (B)
approximations, see Exercises xxx., 10.
Example 2.
Adopting La Caille's determination of the length of the tropical year as
365'' 5'' 48' 49", so that it exceeds the civil year by 5'' 48' 49", we are required
to find the various ways of rectifying the calendar by intercalating an integral
number of days at equal intervals of an integral number of years. (Lagrange.)
20929^
The intercalation must be at the rate of „- .,^ per year ; that is to say,
ob4U0
at the rate of 20929 days in 86400 years. If, therefore, we were to intercalate
20929 days at the end of every 864 centuries we should exactly represent La
Caille's determination. Such a method of rectifying the calendar is open to
very obvious objections, and consequently we seek to obtain an approximate
rectification by intercalating a smaller number of days at shorter intervals.
c. II. 29
4 33 161
1 ' 8 ' 39 '
2865
'694
6
9
13
17 21 25 29
62
1'
2'
3 '
4 ' 5 ' 6 ' 7 '
15'
450 EXAMPLES CH. XXXII
If we turn 86400/20929 into a continued fraction and form the (A) and (B)
series of convergents, we have (omitting the earlier terms)
8434 14003 . .
' 2043' "3392"' *^- ^ >'
95 128 289 450 611
23' 31 ' 70 ' 109' US'
772 933 1094
187' 226' "265 ' *°* ^ ''
Hence, if we take approximations which err by excess, we may with increas-
ing accuracy intercalate 1 day every 4 years, 8 every 33, 39 every 161, and
so on* J and be assured that each of these gives us the greatest accuracy
obtainable by taking an integral number of days less than that indicated in
the next of the series.
The (B) series may be used in a similar mannerf.
Example 3.
An eclipse of the sun will happen if at the time of new moon the earth be
within about 13° of the line of nodes of the orbits of earth and moon. The
period between two new moons is on the average 29*5306 days, and the mean
synodic period of the earth and moon is 346-6196 days. It is required to
calculate the simpler periods for the recurring of eclipses.
Suppose that after any the same time from a new moon the moon and earth
have made respectively the multiples x and yoia, revolution, then a; x 29 '5306 =
y X 346-6196. Hence ylx = 295306/3466196 -O + j^-j-^^ipj-j-j-j-— ...
The successive convergents to this fraction are 1/11, 1/12, 3/35, 4/47, 19/223,
61/716.
Suppose we take the convergent 4/47, the error incurred thereby Vvill be
< 1/47 X 223 in excess, and we may write on the most unfavourable supposition
y_±_ 1
a; ~ 47 47 x 223 '
* The fraction 4/1 corresponds to the Julian intercalation, introduced by
Julius Caesar (45 e.g.). 33/8 gives the so-called Persian intercalation, said to
be due to the mathematician Omar Alkhayami (1079 a.d.). The method in
present use among most European nations is the Gregorian, which corrects the
Julian intercalation by omitting 3 days every 4 centuries. This corresponds
to the fraction 400/97, which is not one in the above series; in fact, 70 days
every 289 years would be more accurate. The Gregorian method has, how-
ever, the advantage of proceeding by multiples of a century. The Greeks and
Ilussians still use the Julian intercalation, and in consequence there is a
difference of 12 days between their calendar and ours. See art. " Calendar,"
Encyclopcedia Britannica, 9th ed.
t See Lagrange's additions to the French edition of Euler's Algebra (Paris,
1807), 1. 11., p. 312.
§ 16 EXERCISES XXX 461
Hence, if a; = 47, y = 4- 1/223. But 3607223 = 1° -61. Hence, 47 lunations
after total eclipse, new moon will happen when the earth is less than 1°-61
from the line of nodes, 47 lunations after that again when the earth is less
than 3° '2 from the line of nodes, and so on. Hence, since 47 lunations = 1388
days, eclipses will recur after a total eclipse for a considerable number of
periods of 1388 days.
If we take the next convergent we find for the period of recurrence 223
lunations, which amounts to 18 years and 10 or 11 days, according as five or
four leap years occur in the interval. The displacement from the node in this
case is certainly less than 360°/716, that is, less than half a degree, so that
this is a far more certain cycle than the last; in fact, it is the famous
" saros " of antiquity which was known to the Chaldean astronomers.
Still more accurate results may of course be obtained by taking higher
convergents.
Exercises XXX.
(1.) Find the first eight convergents to l + ^r — -„— -. — :, — • . • , and find
^ ' ^ ^ 2+ 3+ 4+ 1+ '
the fraction nearest to it whose denominator does not exceed 600.
(2.) Work out the problem of Exercise xxix., 4, using intermediate as
well as principal convergents.
(3.) Work out all the convergents to 27r whose denominators do not
exceed 1000.
(4.) Solve the same problem for the base of the Napierian system of
logarithms e = 2-71828183 ....
(5.) Two scales, such that 1873 parts of the one is equal to 1860 parts of
the other, are superposed so that the zeros coincide : find where approximate
coincidences occur and estimate the divergence in each case.
(6.) Two pendulums are hung up, one in front of the other. The first
beats seconds exactly ; the second loses 5 min. 37 sec. in 24 hours. They
pass the vertical together at 12 o'clock noon. Find the times during the day
at which the first passes the vertical, and the second does so approximately
at the same time.
(7.) Along the side AB and diagonal ^C of a square field round posts are
erected at equal intervals, the interval in the two cases being the same. A
person looking from a distance in a direction perpendicular io AB sees in the
perspective of the two rows of posts places where the posts seem very close
together ("ghosts"), and places where the intervals are clear owing to
approximate coincidences. Calculate the distances of the centres of the
ghosts from A, and show that they grow broader and sparser as they recede
from A.
(8.) Show that between two given fractions pjq and p'jq', such that
pq' -p'q = l, an infinite number of fractions in order of magnitude can be
inserted such that between any consecutive two of the series no fraction can
be found less complex than either of them.
29—2
452 EXERCISES XXX CH. XXXII
(9.) In the series of fractions whose denominators are 1, 2, 3, . . . , «
there is at least one whose denominator is v, say, such that it differs from a
given irrational quantity x by less than I/hi/. (For a proof of this theorem,
due to Dirichlet, not depending on the theory of continued fractions, see
Serret, Alg. Sup., 4™« ed., t. i., p. 27.)
(10.) If the nearest rational approximation in excess or defect (see § 16)
be an intermediate convergent PjQ, where <3 = Xg-n-i + (?„_2 , show that the
approximation in defect or excess will be nearer unless Q > ^q^ + qn-J^^n+i ■
(11.) If zero partial quotients be (contrary to the usual understanding)
admitted, show that every continued fraction may be written in the form
« 1 1 1
V-\ ; ; . • . , where a,, a,. «3, • • • are each either 0 or 1. Show
the bearing of this on the theory of the so-called intermediate convergents.
(12.) 0-0 = 0, ra-i =r 1, ■u:r = an+r W^-i + ^,-2; show that Pa+rl<ln+r~Pnl9n =
"^rllnln+r 5 ^1 " Pnlin = ( ''^r+fn+r'^r-l)l(In{'ln+r +fn+rQn+r-l) . where /„ = X„ - a^.
(Hargreaves, Mess. Math., Feb. 1898.)
CHAPTER XXXIII.
On Recurring Continued Fractions.
EVERY SIMPLE QUADRATIC SURD NUMBER IS EQUAL
TO A RECURRING CONTINUED FRACTION.
§ 1.] We have already seen in two particular instances
(chap. XXXII., § 5) that a simple surd number can be expressed
as a recurring continued fraction. We proceed in the present
chapter to discuss this matter more closely*.
Let us consider the simple surd number (P^ + sjB)IQi. We
suppose that its value is positive ; and we arrange, as we always
may, that Pi, Qi, B shall be integers, and that sfB shall have
the positive sign as indicated. It will of course always be
positive ; but Pi and Qi may be either positive or negative. It
is further supposed that R - P^ is exactly divisible by Q^. This
is allowable, for, ii B — Pi were, say, prime to Qi, then we might
write (Pi + JB)iq, = (PiQi + JqmiQi' = {P^ + ^R')IQx,
where B' - P/^ { = Q,^ (B - P,^) = {B- P,') Q,'} is exactly divisible
by Qx'.
For example, to pnt t ( ^ ~ \/ 9 ) ^°*° *^^ standard form contemplated,
we must write
so that in this case P^= -16, Qi= -32, iJ=96; J?-Pi'*=96-256= -160,
which is exactly divisible by Qi= -32.
* The following theory is due in the main to Lagrange. For the details
of its exposition we are considerably indebted to Serret, Alg. Sup., chap. n.
454 EECURRENCE-FORMULA FOR P„ AND Qn CH. XXXIII
§ 2.] If we adopt the process and notation of chap, xxxii.,
§§ 3 and 5, the calculation of the partial and complete quotients
of the continued fraction which represents (Pi + \/B)/Qi proceeds
as follows : —
Pi+JE 1
A'a - Q - ^2 +
_Pn+jjR_ ^J_
(1),
where it will be remembered that ai, a2> • • • are the greatest
integers which do not exceed x^, x^, . . . respectively ; and
Xz, Xs, . . . are each positive, and not less than unity.
It should be noticed, however, that since we keep the radical
iJB unaltered in our arrangement of the complete quotients, it
by no means follows that P^, Q2, Ps, Qs, &c., are integers, much
less that they are positive integers.
The connection between any two consecutive pairs, say Pn,
Qn and P„+i, Qn+i, follows from the equation
Qn
= «„.+
{Pn^, + sllt)IQ,
or
{{Pn - an Qn) Pn+l " Qn Qn+Z + i^} + {Pn " «« Qn + Pn+l} s/E = 0
(3).
It follows from (3), by chap, xi., § 8, that
(Pn - «« Q,>) Pn+i - Qn Qn+i + li=0,
Pn ~ (^n Qn + Pn+l — " >
whence .
Pn+l ~ ^n ^n ~ J^n
Pn+l + Qn Qn+l = B
If we write w - 1 for w in (5), we have
Pn'-^Qn-lQn-B
(4),
(5).
(6).
I
§§ 2, 3 EXPBESSIONS FOR P„ AND Qn 455
From (5), by means of (4) and (6), we have
= Pn'+Qn-,Qn-(anQn-PnT,
SO that Qn+i = Qn-i + ''iCln Pn " C^n Qn ,
= Qn-i + an{Pn-Pn+i) (7).
The formulse (4) and (7) give a convenient means of cal-
culating Pa, Pa, Qsy P4, Qi, &c., and hence the successive
complete quotients ^2, ^3, • - •
Qa is given by the equation
P^' + Q^Qi^jR,
namely, ^2 = —q ,
= ^^^2a,P,-a,'q,.
From this last equation it follows, since by hypothesis
{R-P^)IQi is an integer, that ^2 is an integer. Hence, since
Pi, Qi are integers, it follows, by (4) and (7), that Pa, P3, . . .,
Pn,Qs, • • •> Qn are also all integers.
§ 3.] We shall now investigate formulae connecting P„ and
Qn with the numerators and denominators of the convergents
to the continued fraction which represents {Pi+ JE)/Qi.
We have (chap, xxxii., § 9)
Pl+ JB ^ Pn-lO^n+Pn-i /^x
^1 ^n-l^n + Qn-2
^ Pn-i Pn + Pn-1 Qn + Pn-1 N^
qn-, Pn + qn-2 Qn + ^n-l '^E
Hence
(Pi + JE) (g„_, Pn + qn-2 Qn + Qn-l ^ E)
= Ql (Pn-l Pn +Pn-2 Qn+ Pn-1 V^) (l).
From (1) we derive
qn-lPn + qn-2 Qn = Ql Pn-1 -PlQn-l (2);
E-P"
Pn-1 Pn + Pn-2 Qn = PlPii-l + —Q-^ 3'»-l (3)-
456 Pn < \/R, Qn < 2 \/R, a„ < 2 ^/R ch. xxxiii
From (2) and (3) we obtain, since pn-i qn-i —Pn-2 ^n-i
=(-ir-
( - l)**-' Pn = Pi (Pn-l qn-1 ^ Pn-i qn--)
R-P"^
+ — ^- qn-\ qn-2- Qi Pn-l Pn-l (4) ;
( - l)*^-^ Q. = - 1pn-l qn-l Pi - ^^^ qn-l' + QlPn-l' (5).
The formulae (4) and (5) give us the required expressions,
and furnish another proof that P2, P3, . . ., Pn, Q2, Qa, • • •, Qn
are all integral.
§ 4.] If in equation (2) of last paragraph we replace Pi by-
its value Qi{pn-i^n+Pn-2)l(qn-ii«n + qn-2)- 'JR, derived from
equation (A), we have
qn-l Pn + qn-. Qn = j " ^T^^' + ^^^^^ (l)"
qn-l'^n ■•" qn-2
Also, since cc^ = (Pn + JR)/Qn, we have
Pn-a^nQn = -'JR (2).
From equations (1) and (2) we derive, by direct calculation,
the following four : —
Pn =
i^ii^^ii^-^^^-^-^)^^^^^^
Qn =
L {q^_^ (q^_^a!n + qn-.) 2^R + {- if-' Qi} (4) ;
\qn-l^n + 9'»-2/
JR-Pn =
J- — ^TTT-v^^^^A'^'^-^'-^) ^-(-^T-'Q] (5);
(qn-ia^n + qn-.) I V ^n / J
2jR-Qn =
-^,{(^;^qn-i+qn-2){qn-iXn+qn-2)2jR-(-lT-'Qi}{G).
(qn-li^n+qn-.T
The coefficients of JR and 2\/S in these four formulae are
positive, and increase without limit when n is increased without
limit. Hence, since Qi is a fixed quantity, it follows that for
§§ 3, 4 CYCLE OF (P, + ^R)IQ, 457
some value of n, say n = v, and for all greater values, P„, Q^,
JlR-Pn, 2jK-Qn will all be positive. In other words, on
and after a certain value of n, n = v say, P„ and Qn will be
positive; and Pn<jB, and Qn<2jB.
Cor. 1. Since Pn and Q„ are integers, it follows that
after n = v P^ cannot have more than JR different values, and
Qn cannot have more than 2 JB different values; so that Xn
= {Pn + jR)IQn cannot have more than JR x 2 JR - 2R different
values. In other words, after the vtli complete quotient, the
complete quotients must recur within 2R steps at most.
Hence the continued fraction which represents (Pi + J R)IQi
must recur in a cycle of 2R steps at most.
Since ever after n=v Pn and Qn remain positive, it is clear
that in the cycle of complete quotients there cannot occur any one
in which Pn and Qn are not both positive.
It should be noticed that it is merely the fact that P„ and
Qn ultimately become positive that causes the recurrence.
If we knew that, on and after n = v, Pn remains positive, then
it would follow, from § 2 (4), that Qv and all following remain
positive ; and it would follow, from § 2 (5), that P^+i and all
following are each < JR ; and hence, from (4), that Qv+x and all
following are each < 2jR ; and we should thus establish the
recurrence of the continued fraction by a somewhat different
process of reasoning.
Cor. 2. Since a„ is the greatest integer in {Pn + jR)IQn,
and since, \{ n>v, Pn and Qn are both positive, and Pn<jR,
and Qn>lj it follows that, if n>v, an<2jR.
It follows, therefore, that none of the partial quotients in the
cycle can exceed the greatest integer in 2 JR.
Cor. 3. By means of (3) and (4), we can show that ultimately
Pn+Qn>jR (7).
Cor. 4. From § 2 (5), we can also show that ultimately
Pn+Qn-^>JR (8).
458 PURE RECURRING C.F. CH. XXXIII
Cor. 5. Since JJi>Pjn, it follows from Cor. 3 and Cor. 4
that ultimately
■t^m, ~ Ln"^ vent
<Qn-l (9).
EVERY RECURRING CONTINUED FRACTION IS EQUAL TO A
SIMPLE QUADRATIC SURD NUMBER.
§ 5.] We shall next prove the converse of the main pro-
position which has just been established, namely, we shall show
that every recurring continued fraction, pure or mixed, is
equal to a simple quadratic surd number.
First, let us consider the pure recurring continued fraction
^i = «i + — 7 ... — 7 . . . (I).
*
Let the two last convergents to
1 1
tti + — . . -
be p'/q' and p/q.
From (1) we have
1 1 1
^1 = ai + . . . - — - — ,
02+ Or + iTj
_ jt?a?i +p' ^
~ qxi + q"
whence
g^i' + {q -p)xi-p' = 0 (2).
The quadratic equation (2) has two real roots ; but one of
them is negative and therefore not in question, hence the other
must be the value of a^i required.
We have, therefore,
^1 - — 2^— (3),
L + JW
= — ]g— >say;
which proves the proposition in the present case.
§§ 4, 5 MIXED RECURRING C.F. 459
It should be noticed that, since ai4=0, p/q>l', so that
p>q>g'' Hence p — q' cannot vanish, and a pure recurring
fraction can never represent a surd number of the form JNJM.
Next, consider the general case of a mixed recurring con-
tinued fraction.
Let
1 111 1
Xl = al^ . • . . . . — . . ,
(4).
* \ /
Also let
1 1
2/i = «i + —— . . . . . .
* "2 + a, +
(5).
«
Then, by (3),
L + J]V
y^^ M •
From (4) we have
1 1 1
Xi = ai+ . . . ,
a. + cir + yi
whence, if P'jQ' and PJQ be the two last convergents to
1 1
«! + . . . -,
a<i+ ar
Pyi + P'
_PL + P'M+PjN
/'fi^
QL + Q'M+QjN
Hence, rationalising the denominator, we deduce
_ 1/+ VJN
Example 1.
Evaluate a;, = 1 + — - . . . .
* 2+1+ 1 +
«
The two last convergents to 1 + — - are 3/2 and 4/3 ; hence
2+1
_4xi + 3
^^~3xi + 2*
We therefore have
3a;i2-2xi-3 = 0,
the positive root of which is
1 + x/iO
4(50 C.F. FOR \I{GID) CH. XXXIII
Example 2.
t:. 1 . ,11111
Evaluate ?/, = 3 + -. — :; — -pr— -i — ^i — ....
•'1 4+1+2+1+1 +
* «
The two last convergents to 3 + j are 3/1 and 13/4; and, by Example 1
above,
1_J_ _i + Vio
* 2+l+***~ 8 •
1 1
We have, therefore,
2/1 = 3 +
4+ (l^VIO)/3'
^13(l + >yi0)/3 + 3
4(l + s/l0)/3 + l '
_22 + 13VlO
~ 7 + 4^10 '
_366-3ViO
~ 111
_ 122-^10
~ 37 •
ON THE CONTINUED FRACTION WHICH REPRESENTS \/{CJD).
§ 6.] The square root of every positive rational number, say
J{C/D), where C and I) are positive integers, and C/I) is not
the square of a commensurable number, can be put into the form
JN/M, where N^CD and M=D. Since N/M = C is an
integer, we know from what precedes that JNJM can be
developed, and that in one way only, as a continued fraction of
the form
1 111 1
Xi = ai+ . . . . . . . . . (1).
^2 + a,. + tti + oo + a, + ^ '
* " *
We have, in fact, merely to put Pi = 0, R = N, Qi = M m our
previous formulae.
We suppose that J NjM is greater than unity, so that Oj + 0.
If J NjM were less than unity, then we have only to consider
MIJN = JM^NIN, which is greater than unity.
The acyclic part a^ + — — ... — must consist of one term at
ffj + U/f
§§ 5, 6 ACYCLIC PART OF '^NfM 461
least, for we saw, in § 5, that a pure recurring continued fraction
cannot represent a surd number of the form JJV/M. Let us
suppose that there are at least two terms in this part of the
fraction ; and let P'/Q', P/Q be the two last convergents to
tti + . . . — ; and p'/q', p/q the two last convergents to
(X'2 "T Ctff
1 111 1 ^, .„
a, + . . . . . . — . Then, it
aQ + ar+ 0-1 + <h+ ««
1 1
we have *
1 11
02+ «r + ^1
1 1 1 ]_ 1 1
02 + * * * <*r + "l + «2 + * * * "s + 3/1 '
Hence
^ ^Py, + P' ^py,+p'
' %i + Q' qyi + q ^ ''
Eliminating ^i from the equations (2), we have
m - Q'q) X,' - W - Q'P + Pq - Pq) ^1 + (Pp - P'p) = 0 (3).
Now, if Xi = JNJM, we must have
M^x^^-N=^ (4).
In order that the equations (3) and (4) may agree, we must
have
qp'-Q'p + P<]l-P'q = 0 (5);
and
Pp'-Pp_ N
Qq'-Q'q M' ^^^'
It is easy to show that equation (6) cannot be satisfied. We
have, in fact,
Pp' - P'p _ P'p PIP -pIp
Qq'-Q'q Q'q QIQ'-qlq
But, by chap, xxxii., § 7,
P_£^ 1 j^_^ 1_ I
P p' " ttr-i + ' ' ' tti * a^_i +' ' ' Ui
where/ is a proper fraction.
(7).
462 CYCLE OF QUOTIENTS FOR ^JNjM CH. XXXIII
Similarly
Q q _ \ 1 1 1
^ q a,.-! + «o a^_i + a,
= «r - «s ±/',
where/' is a proper fraction.
Now ttr-ois cannot be zero, for, if that were so, we should
have ttr^oLs, that is to say, the cycle of partial quotients would
begin one place sooner, and would be ag, oj, Og, . . . , ag_i, and not
tti, 02, . . . , ag, as was supposed. It follows then that a^- «« is
a positive or negative integral number. Hence the signs of
F/P'-p/p and Q/Q'-q/q' are either both positive or both
negative, and the sign of the quotient of the two is positive.
Hence the left-hand side of (6) is positive, and the right-hand
side negative.
There cannot, therefore, be more than one partial quotient in
the acyclic part of {\).
Let us, then, write
11 11
Xi = a + . . . ... (8),
* ' *
11 11
= a +
ttj + ao + ' ' ag + l/(^j - a) '
Hence
p]{x^-a)+p'
x.^'
' ql{x,-a) + q"
which gives
q'xi^ - (p + q'a -q)xi-{p- ap) = 0 (9).
From (9) we obtain
^ __P' + q'<^-q ^ J(P + g'a -qY + 4.{p- ap') 7
^ 2g' 2q' ^ ''
In order that (10) may agree with Xi = JjV/M, we must have
p' + q'a-q=^0 (11);
and
q"N/M'=={p-ap')q' (12).
Cor. 1. By equation (11) we have
p'/q' + a = q/q'.
^ 6, 7 CYCLE OF QUOTIENTS FOR ^/NjM 463
Hence, by chap, xxxii., § 7, Cor. 2,
oil 1^1 1
2a + — . . . = a« + . . . — .
Oi + Qa + ag_i a5_i + «,
It follows, therefore, by chap, xxxii., § 3, that
In other words, the last partial quotient of the cyclical part of
the continued fraction which represents J NIM is double the
unique partial quotient which forms the acyclical part; and the
rest of the cycle is reciprocal, that is to say, the partial quotients
equidistant from the two extremes are equal.
In short, we may write
jN^ll 1111 ..„.
^^^ = a + . . . -z . . . (13).
* «
Cor. 2. If we use the value of q'a given by (11), we may
throw (12) into the form
q^NlW =pq' -p' (q -p) ;
wlience
q'^'NIM'' -p'^ =pq' -p'q,
-±l (14),
the upper sign being taken if pjq be an even convergent, the lower
if it be an odd convergent.
§ 7.] All the results already established for {Pi + jR)IQi
apply to JN(M. For convenience, we modify the notation as
follows : — _
a, =a, x, = {P, + jK)iq, = {0 + jN)IM;
as =a,_i, Xs={Ps + jR)IQs^{Ls-i + jN)IMs-i;
as+^ = 2a,
0/8+2— °-l, .......
From § 2 (4), we then have
Ln = «n-l Mn-i - Ln-\ ( 1 ) ;
and, in particular, when n=\,
L, = aM (1').
464 CYCLES OF DIVIDENDS AND DIVISORS CH. XXXIII
From § 2 (5), we have
Lr,' + M^^,Mr, = N (2);
and, in particular,
L^^MM^ = N (2').
From § 3 (4) and (5), we have
( - ifL^ = {NIM) qnqn-i - MpnPn-l (3) ;
{-fM,= Mp,'-{NIM)q^' (4).
These formulse are often useful in particular applications.
It will be a good exercise for the student to establish them
directly.
§ 8.] Let us call L^, L^, &c., the Bational Dividends and M,
Ml , M2, &c., the Divisors belonging to the development of JN/M.
Then, from the results of § 4, we see that
None of the rational dividends can exceed JW; none of the
partial quotients and none of the divisors can exceed 2jN.
All the rational dividends, and all the divisors, are positive.
It is, of course, obvious that the rational dividends and the
divisors form cycles collateral with the cycle of the partial and
total quotients; namely, just as we have
so we have
Ls+\ = Li, Ls+'i = L.2, (1),
and
itf,+i = ifi, Ms^^ = M^, (2).
We can also show that the cycles of the rational dividends
and of the divisors have a reciprocal property like the cycle of
the partial quotients ; namely, we have
Ls-i = L.„ Ms-i = Mi;
Ls-2 = Ls, Ma-2 = M-i ;
(3).
For, by § 7 (2),
£,+,' + M,^JI, = L,' + MJI;
but Ls+i = Li and Ms+i = Mi, hence
31, = 31 (4).
§§ 7, 8 THE COLLATERAL CYCLES 465
Again, by § 7 (1),
Ls+i = o.gMg-Lg;
but Z,+i =Li,a,= 2a, Ma = M, hence we have
A = 2aM- Ls.
Now, by § 7 (1'), Li = aM, hence
A = 2A-A,
therefore Lg = A (5).
Again, by § 7 (2),
whence, bearing in mind what we have already proved, we have
Mg., = M, (6).
Once more, by § 7 (1),
Ls = ««-] Mg-i — Lg-i,
i/2=a,iV/i-Xi.
Now Mg-i = Ml and a,_i = a^, hence
Xg — ^2 = Z-i — Xg-i.
But Lg-Li, hence
X/g_i = X/2.
Proceeding step by step, in this way, we establish all the
equations (3).
It appears, then, that we may write the cycles of the rational
dividends and of the divisors thus —
L\, L^t Ls, . . ., Lz, Z-j, Z-i;
M„ M„ M,, . . ., Ms, M„ M„ M.
Since J/ precedes Mi, we may make the cycle of the divisors
commence one step earlier, and we thus have for partial quotients,
rational dividends, and divisors the following cycles : —
«i5 "2> «3, • . •» «3> «2, «i, 2a; oi.
Z-i, L.2, Lz, • . -, Ls, Za, i/i ; X].
M, Mu M„ M„ . . ., M^, M, ; M, M,.
That is to say, the cycle of the rational dividends is collateral
with the cycle ojthe partial quotients, and is completely reciprocal;
c. II. 30
466 TESTS FOR MIDDLE OF CYCLE CH. XXXIII
the cycle of the divisors begins one step earlier* (that is, from the
very beginning), and is reciprocal after the first term.
§ 9.] The following theorem forms, in a certain sense, a
converse to the propositions just established regarding the cycles
of the continued fraction which represents JNJM.
If Lm =Ln+i, Mm =Mn, O-m - «»)
then Xm-i = i/n+2, Mm-i=Mn+i, a.m-l = 0-n+l (l)-
We have, by § 7 (2),
whence, remembering our data, we deduce
Mm-^ = Mn+, (2).
Again, by § 7 (1),
whence, since Lm = Ln+i by data,
= («m-l - «n+l) Mn+i (3).
If Lm~i>Ln+^, we may write (3)
(Im-l — Ln+^IMn+\ = Ctm-i - a„+i (4) ;
ii Lm-i<In+i, we may write
(-^n+2 - Ln--)IM,n-i = O-n+l " ^m-1 (5).
But, by § 4 (9), the left-hand sides of (4) and (5) (if they
differ from 0) are each <1, while the right-hand sides are each
positive integers (if they differ from 0).
It follows, then, that each side of equation (3) must vanish,
so that
J^m-l — ■L'n+i (6),
«m-l = O-n+l (7),
which completes the proof.
* The fact that the cycle of the divisors begins one step earlier than the
cycles of the partial quotients and rational dividends is true for the general
recurring continued fraction. Several other propositions proved for the
special case now under consideration have a more general application. The
circumstances are left for the reader himself to discover.
§§ 8, 9 TESTS FOR MIDDLE OF CYCLE 467
Cor. 1. Starting with the equations in the second line o/{l)
as data, we could in like manner prove that
and so on, forwards and backwards.
Cor. 2. If we put m = n, the conditions in (1) become
in other words, the conditions reduce to
and the conclusion becomes
Jjn-\— -1-^71+2} -^n-l — -^Jn+lf ''■n-l — ^n+l'
Hence, if two consecutive rational dividends he equal, they are
the middle terms of the cycle of rational dividends, which must there-
fore be an even cycle ; and the partial quotient and divisor cor-
responding to the first of the two rational dividends will be the middle
terms of their respective cycles, which must therefore be odd cycles.
Cor. 3. If we put m = n + l, the conditions in (1) reduce to
■^n+i — -^Im "ti+i = ct» j
and the conclusion gives
Using this conclusion as data in (1), we have as conclusion
Ln-i = L/n+S, i)y„_i = ilz„+2j "■ji-i = "■n+2 )
and so on.
Hence, if two consecutive divisors (Mn, Mn+i) be equal, and also
the two corresponding partial quotients (a„, a„+i) be equal, these two
pairs are the middle terms of their respective cycles, which are both
even ; and the rational dividend (Ln+i) corresponding to the second
member of either pair is the middle term of its cycle, which is odd.
These theorems enable us to save about half the labour of
calculating the constituents of the continued fraction which
represents slNjM. In certain cases they are useful also in
reducing surds of the more general form {L + 'JN)/M to con-
tinued fractions.
Example 1.
Express ,^8463/39 as a simple continued fraction ; and exhibit the cycles
of the rational dividends and of the divisors.
30—2
468 EXAMPLES CH. XXXIII
We have
J8m^^ I -'78+V8463_g^ 1
39 39 (78+v'8463)/61
78+^/8463_„ -44+^8463^^ ^ ^
61 61 (44 + ^8463)/107'
44+js/8463_ -63 + ^8463^^ 1
107 ~ "'' 107 ~ (63 + V8463)/42
63+ V8463_g ^ -63+^/8463^g ^ 1
42 42 (63 + ^463)/107
63+^^^8463^^^^^
Since we have now two successive rational dividends each equal to 63, we
know that the cycle of partial quotients has culminated in 3. Hence the
cycles of partial quotients, rational dividends, and divisors are —
Partial quotients . . 2, 1, 3, 1, 2, 4 ;
Rational dividends . 78, 44, 63, 63, 44, 78;
Divisors ... 39, 61, 107, 42, 107, 61;
and we have
Jsm 1 1 _L_L J^ J^
39 2+ 1+ 3+ 1+ 2+ 4+ ■ ■ ■ ■
« *
Example 2.
If c denote the number of partial quotients in the cycle of the continued
fraction which represents JnjM, prove the following formulae : —
Ifc=2t,
Pc_Pt+j'lt+Pt9t-i
if c = 2« + l,
if 771 be any positive integer
9c Qti<lt+i + 9t-i)
Pc^Pt+i(It+i+PtQt
1c It+i^ + Qt^
'p2r«._P,rJ+(NIM^)q„
(I.);
(II.);
(III.).
Qzmc ^Pmc imc
For brevity we shall prove (III.) alone. The reader will find that (I.)
and (II.) may be proved in a similar manner. For a different kind of demon-
stration, see chap, xxxiv., § 6.
We have
^»?«-a+— - . . . ^ — ... — (2m cycles),
gam* «i+ «i+ 2a+ a^ ^
1 11 1 1 , , ,
= a H . . . ,T — . . . i — (m cycles),
ai+ ai+ 2a+ <^i+ a+Pmckrw
_ (a +Pmclimc)Pmc +Pmc-\
(a +Pmcllmc) Imc + 9mc-\ '
_ (fflj^mt+Prnf-l) linc+Pm^
3tnc{«?mc + 9'»nc-l+l>»nc)
(o).
§ 9 EXERCISES XXXI 469
Now the equations (2) and (3) of § 3 give us
Qmc ^mc+l + 9)nc-l Vmc+1 — ^Pme
Pmc Pmc+l +P,nc-1 Qmc+1 = (^1^) q„
In the present case,
Qmc+l = Qc+l = ^^c = M-
The equations (/3) therefore give
^9inc + 1me-l —Pmc \
aPmc +Pmc-1 = i^lM^) qmc\
From (a) and (7) (III.) follows at once.
The formulae (I.), (II.), (III.) enable us, after a certain number of con-
vergents to Jn/M have been calculated, to calculate high convergents
without finding all the intermediate ones.
Consider, for example,
x/8463_ 111111
03).
(7)-
39 2+ 1+ 3+ 1+ 2+ 4+ '
* *
Here c=6, t = 3, and we have for the first four convergents 2/1, 5/2, 7/3,
26/11; hence
Pe ^P-i<}3+P3<l2
Qe ^3(34 + 32)'
_26x3 + 7x2_92
3(11 + 2) ~39'
Also P_.^Pl±iflMW^
_ 92^ + (8463/39'^). 39^ _ 16927 ^
"■ 2 X 92 X 39 ~ 7170 '
Pit ^Pjl+iNimiq^
924 ^Pu'il2
_ 16927^ X 39^+8463 x 7176^
~ 2x392x16927x7176 *
The rapidity and elegance of this method of forming rational approximations
cannot fail to strike the reader.
Exercises XXXI.
Express the following surd numbers as simple continued fractions, and
exhibit the cycles of the partial quotients, rational dividends, and divisors : —
(1.) V(lOl). (2.) iV(63)- (3-) n/(H)-
(7.) Express the positive root of a;^ - a; - 4=0 as a continued fraction, and
find the 6th convergent to it.
(8.) Express both roots of 2x--6x-l = 0 as continued fractions, and
point out the relations between the various cycles in the two fractions.
470 EXERCISES XXXT CH. XXXIII
(9.) Show that
J{a^ + l) = a + ^,..;
(10.) Express ij{a^ + l) as a simple continued fraction, and find an
expression for the 7tth convergent.
Evaluate the following recurring continued fractions, and find, where you
can, closed expressions for their nth convergents; also obtain recurring
forniula3 for simplifying the calculation of high convergents: —
1
(11.)
a + — -. .
a +
*
(12.)
1
a-
*
(13.)
1 1
a+ b +
# ♦
Show, in this case, that
* *
where the cycle consists of n units followed by 2.
(15.) Show that
\x+ 4lX+ J \2x+ J
« ♦ •
is independent of x.
(16.) Show that
(-■*-2^---y-("-2Tr---j^-^-
« «
(17.) If x=a + r^ — . . ., y^h + —~...,
« *
11 ,1
* a+ 6+ * a + b + c +
*
show that
2{x + y + z)-(a + b + c) _ 1 1 1
2u-{a + b + c)-abc ~bc + l ca + 1 ab + l'
(18.) Show that
\b+ ' ' ') ~2a + b-- • ' ' '
§ 9 EXERCISES XXXI 471
(19.) If p be the numerator of any convergent to ,^2, then 2p2±i ^iH
also be the numerator of a convergent, the upper or lower sign being taken
according aspjq is an odd or an even convergent; also, if q, q' be two con-
secutive denominators, q'^ + q'^ will be a denominator.
(20.) Evaluate
Jl J_ 1
1+ 1+ ' * 'm+ ■ ■ ' '
« *
where the cycle consists of n - 1 units followed by ii.
(21.) In the case of :j — t — . . . , prove that
* *
P,n = (l,n+l= {(v/2 + l)2»+l + (V2- 1)2»+1}/2V2,
i'2«-i = i32n = {(V2 + l)^"- (v/2- 1)^»}/V2.
(22.) Convert the positive root of ax^ + abx-b=:0 into a simple con-
tinued fraction; and show that p^ and q.^ are the coefficients of a;" in
(x+bx'-x^)l(l- ab + 2.x'^ + x*) and (ax + ab + l.x^ + x*)l(l - ab + 2.x^ + x^)
respectively.
Hence, or otherwise, show that if a, /3 be the roots of 1- {ab + 2)z + z^=0,
then
pH /an
ap^,, = bq2„-^ = ab
P-2n+l — l2n ~ '
(a"+i-j3"+i)-(a"- ^)
(23.) If the number of quotients in the cycle of
V^V 1 1 111
-^rr =aH . . . ~ ... be c,
M a^+ a., + 00+ a,+ 2a +
* " *
show that
a-\ . . . ^ . . . (m cycles) =--—~^.
«!+ ai+2a+ai+ Oj+a'' ■' ' M^p^^
(24.)* If c be the number of quotients in the cycle of ^JNjM, show that
if c = 2< + l,
gVr-l + ffVr ^"
r = 0, 1, . . ., t-1;
and if c = 2^
Pt-r-iPt~r-\ +Pt+r-^Pt+r _ ^
qt-r-^qt-r-l + it+r-lit+r -3^'*
(25. )t lisJZ = a-\ ; ...- — ~ .. ., and if the convergent
«!+ Oo+ a„+ ai+ 2rt+ ^
« ' *
* For solutions of Exercises 24 and 26-29 see Muir's valuable little tract
on The Expression of a Quadratic Surd as a Continued Fraction, Glasgow
(Maclehose), 1874.
t In connection with Exercises 25 and 30-32 see Serret's Cours
d'Algebre Superieure, 3™" ed., t. i., chaps, i. and ii.
472 EXERCISES XXXI CH. XXXIII .
obtained by taking 1, 2 i periods, ending in each case with a^, be
Zi, Z„ . . ., Zi, and if Z^^PJQ^, . . ., Zi = PilQi, P^ and Q^ being
integers prime to each other as usual, then
Pi - Qis/^={Pi-i - Qi-is/^) {Pi - Qxsiz),
= (P,-Q,JZ)i;
Zi + ^Z^fZ. + ^ZV
Zi-^Z \Zi-^Zj '
(26.) If N be an integer, and if a cyclical partial quotient occur in the
development of »JN equal to the acyclic partial quotient a, that quotient
will be the middle term of the reciprocal part of the cycle ; and no cyclical
partial quotient can occur lying between a and 2a.
(27.) When 2^ is a prime integer, the cycle of partial quotients is even,
and the middle term of the reciprocal part of the cycle is a or a - 1, according
as a is odd or even.
(28.) If N be an integer, and the cycle of sJN be odd, then A is the sum
of the squares of two integers which are prime to each other.
Exhibit 365, as the sum of two squares.
(29.) The general expression for every integer whose square root has a
cycle of c terms, the reciprocal part of which has the terms a^ , Og , . . . , ao , aj ,
is
(^pm-{-lYp'q')^+p'm-[-iyq'\
where m is any positive integer, and p'jq', pjq are the two last convergents to
111
^ 02+ a2+ Oi
Find an expression for all the integers that have 1, 2, 1 for the reciprocal
part of the cycle of their square root.
(30.) If two positive irrational quantities, x and x', can be developed
in continued fractions which are identical on and after a certain constituent,
show that
x'= {ax + h)j{a'x + h'),
where a, h, a\ h', are integers such that alf - a'b = ± 1 ; and that this con-
dition is sufficient.
(31.) The equation of the 2nd degree with rational coefficients which is
satisfied by a given recurring continued fraction has its roots of opposite
signs if the fraction is purely recurring, and of the same sign if it is mixed
and has more than one acyclic partial quotient.
(32.) Investigate the relation between the cycles of the partial and
complete quotients of the two continued fractions which represent the
numerical values of the two roots of an equation of the 2nd degree with
rational coefficients.
Illustrate with 27 x- - 97x + 77 = 0.
^ 10, 11 DIOPHANTINE PROBLEMS 473
APPLICATIONS TO THE SOLUTION OF DIOPHANTINE PROBLEMS.
§ 10.] When an equation or a system of equations is in-
determinate, we may limit the solution by certain extraneous
conditions, and then the indeterminateness may become less in
degree or may cease, or it may even happen that there is no
solution at all of the kind demanded.
Thus, for example, we may require (I.) that the solution be
in rational numbers ; (II.) that it be in integral numbers ; or,
still more particularly, (III.) that it be in positive integral num-
bers. Problems of this kind are called Diophantine Problems,
in honour of the Alexandrine mathematician Diophantos, who,
so far as we know, was the first to systematically discuss such
problems, and who showed extraordinary skill in solving them*.
We shall confine ourselves here mainly to the third class of
Diophantine problems, where positive integral solutions are
required, and shall consider the first and second classes merely
as stepping-stones toward the solution of the third. We shall
also treat the subject merely in so far as it illustrates the use of
continued fractions : its complete development belongs to the
higher arithmetic, on which it is beyond the purpose of the
present work to enter t.
Eqitations of the 1st Degree in Two Variables.
§ 11.] Since we are ultimately concerned only with positive
integral solutions, we need only consider equations of the form
ax±hy- c, where a, b, c are positive integers. We shall suppose
that any factor common to the three coefficients has been
• See Heath's Diophantos of Alexandria (Camb. 1885).
t The reader who wishes to pursue the study of the higher arithmetic
should first read Theory of Numbers, Part I. (1892) by G. B. Mathews,
M.A. ; then the late Henry Smith's series of Reports on the Theory of
Numbers, published in the Annual Reports of the British Association (1859-
60-61-62) ; then Legendre, TMorie des Nombres ; Dirichlet's Vorlesungen
iiber Zahlentheorie, ed. by Dedekind; and finally Gauss's Disquisitiones
Arithmeticee. He will then be in a position to master the various special
memoirs in which Jacobi, Hermite, Kummer, Henry Smith, and others have
developed this great branch of pure mathematics.
474 ax — hy = c CH. xxxiii
removed. We may obviously confine ourselves to the cases
where a is prime to h ; for, if x and y be integers, any factor
common to a and b must be a factor in c. In other words, if a
be not prime to h, the equation ax±hy^c has no integral solution.
§ 12.] To find all the integral solutions of a^-hy = c; and to
separate the positive integral solutions.
We can always find a particular integral solution of
ax-hy-c (1).
For, since a is prime to h, if we convert ajb into a continued
fraction, its last convergent will be ajb. Let the penultimate
convergent be pjq, then, by chap, xxxii., § 8,
aq-ph=±i (2).
Therefore
a{±cq)-b{±cp)^c (3).
Hence
x' -± cq, y =±cp (4)
is a particular integral solution of (1).
Next, let {x, y) be any integral solution of (1) whatever.
Then from (1) and (3) by subtraction we derive
a{x-{±cq)]-b{y-{±cp)] = 0.
Therefore
{^ - ( ± cq)]l{y -{±cp)] = b/a (5).
Since a is prime to b, it follows from (5), by chap, iii., Exercises
IV., 1, that
x-{±cq) = bt, y-(±cp) = at,
where f is zero or some integer positive or negative. Hence
every integral solution of (1) is included in
x = ±cq + bt, y = ±cp + at (6),
where the upper or lower sign must be taken according as the
upper or lower sign is to be taken in (2).
Finally, let us discuss the number of possible integral solu-
tions, and separate those which are positive.
1°. If a/b>plq, then the upper sign must be taken in (2),
and we have
x-cq + bt, y^cp + at (6).
§§11-13 ax + hy=^c 475
There are obviously an infinity of integral solutions. To get
positive values for x and y we must (since cpla<cqlb) give to
t values such that - cpja 1^t'if>+ go . There are, therefore, an
infinite number of positive integral solutions.
2°. If a/b<p/q, so that cp/a>cqlb, we must write
x = — cq + bt, y = - cp + at (6").
All our conclusions remain as before, except that for positive
solutions we must have cp/a1f>t':}(> + oo .
We see, therefore, that ax — by = c Jms in all cases an infinite
number of positive integral solutions.
§ 13.] To find all the integral solutions of
ax + by = c (7),
and to separate the positive integral solutions.
We can always find an integral solution of (7) ; for, if p and
q have the same meaning as in last paragraph, we have
{±cq)a + {^cp)b = c (8),
that is, x' = ±cq, y =+cp\& & particular integral solution of (7).
By exactly the same reasoning as before, we show that all
the integral solutions of (7) are given by
x = ±cq-bt, y= + cp + at (9);
so that there are in this case also an infinity of integral
solutions.
To get the positive integral solutions : —
1°. Let us suppose that a/6 >j3/g', so that cp/a<cg'/6. Then
the general solution is
x = cq-bt, y = - cp + at (9').
Hence for positive integral solutions we must have cpjali^t
>cq/b.
2°. Let us suppose that alb<p/q, so that cpla>cq/b, then
x = -cq-bt, y = cp + at (9").
Hence for positive integral solutions we must have —cpja:lf>t
>-cq/b.
476 EXAMPLES CH. XXXIII
In both these cases the number of positive integral solutions
is limited. In fact, the number of such solutions cannot exceed
1 + I cqjh - cp/a \ ; that is, since | aq -pb | = 1, the number of
positive integral solutions of the equation ax + by = c cannot
exceed 1 + clab.
Example 1, To find all the integral and all the positive integral solutions
of 8a; + 13?/ = 159.
We have
8 1 1 1 11
r3~i+ 1+ 1+ 1+ 2'
The penultimate convergent is 3/5 ; and we have
8x6-13x3 = 1,
8 (795) + 13 (-477) = 159.
Hence a particular solution of the given equation is a;' = 795, y'= - 477; and
the general solution is
.'c = 795-13<, 2/=-477 + 8t.
For positive integral solutions we must have 795/13 <t:««t 477/8, that is,
61j'V't*'*59|. The only admissible values of t are therefore 60 and 61;
these give x = 15, y = 3, and x = 2,y = ll, which are the only positive integral
solutions.
Example 2. Find all th« positive integral solutions of 3a; + 2?/ + 3z = 8.
We may write this equation in the form
3x + 22/ = 8-3«,
from which it appears that those solutions alone are admissible for which
2 = 0, 1, or 2.
The general integral solution of the given equation is obviously
x = 8-3z-2^ 7/=-8 + 32 + 3t.
In order to obtain positive values for x and y, we must give to t integral
values lying between + 4 - f 2 and + 2§ - z. The admissible values of t are
3 and 4, when 2=0; 2, when 2 = 1; and 1, when 2 = 2. Hence the only
positive integral solutions are
x = 2, 0, 1, 0;
2/ = l, 4, 1, 1;
2 = 0, 0, 1, 2.
In a similar way we may treat any single equation involving more than
two variables.
§ 14.] Any system of equations in which the number of
variables exceeds the number of equations may be treated by
methods which depend ultimately on what has been already
done.
^ 13, 14 SYSTEM OF TWO EQUATIONS 477
Consider, for example, the system
ax + hy + cz = d (1),
a'x + b'y + c'z = d' (2),
where a, b, c, d, a', &c. denote any integers positive or negative.
This system is equivalent to the following : —
- {ca) X + {he') y = (dc) (3),
ax + by + cz = d (4),
where (ca') stands for ca - ca, &c.
Let 8 be the G.C.M. of the integers {ac\ {he). Then, if 8
be not a factor in {dc), (3) has no integral solution, and conse-
quently the system (1) and (2) has no integral solution.
If, however, 8 be a factor in {dc'), then (3) will have integral
solutions the general form of which is
x^x" -v {be') t/B, y = y" + {ca) t/8 (5),
where {x", y") is any particular integral solution of (3), and t is
any integer whatever.
If we use (5) in (4), we reduce (4) to
cz-c {ah') t/8 = d- ax" - by' (6),
where c {ab')l8 is obviously integral.
In order that the system (1), (2) may be soluble in integers,
(6) must have an integral solution. Let any particular solution
of (6) hez^z,t = t'. Then
z-z' _ {ab')
t-t' ~ 8 '
Hence, if e be the G.C.M. of {ab') and 8, that is, the G.C.M.
of {he), {ca), {ab'), then
z = z' + {ab') u/e, t^t' + Sii/e (7),
where u is any integer.
From (5) and (7) we now have
x = x' + {be') u/e, y = y' + {ca') w/c, z = z' + {ab') w/e (8),
where x == x" + {be) t'/S, y = y" + {cd) t'jh.
If in (8) we put w = 0, we get x = x, y = y', z = z' ; therefore
{x', y', z) is a particular integral solution of the system (1), (2).
A little consideration will show that we might replace {x, y, z')
by any particular integral solution whatever. Hence (8) gives all
478 fermat's problem ch. xxxiii
the integral solutions of (1), (2), {x\ y, z) being any particular
integral solution, e the G.C.M. of {he), (ca), {ah'), and u any
integer whatever.
The positive integral solutions can be found by properly
limiting u.
Example.
3a; + 4?/ + 27z = 34, 3a: + 5y + 21z = 29.
Here {&c')= -51, (ca') = 18, (a6') = 3. Hence e = 3; a particular integral
solution is (1, 1, 1) ; and we have for the general integral solution
a;=l-17w, j/ = l + 6u, « = l + w.
The only positive integral solution is a;=l, y = l, z = l.
Equations of tJie 2nd Degree in Two Variables.
§ 15.] It follows from § 7 (4) that, if pnjqn be the wth con-
vergent and Mn the {n + l)th rational divisor belonging to the
development of ^{C/'D) as a simple periodic continued fraction,
then
Dpn'-Cqn^={-TMn (1).
Hence the equation Da? — Cy^ = + H, where G, D, H are positive
integers, and CjD is not a perfect square, admits of an infinite
number of integral solutions provided its right-hand side occurs
among the quantities {-YMn belonging to tJie simple continued
fraction which represents J{CID) ; a7id the same is true of the
equation Da? - Gy^ = -H.
The most important case of this proposition arises when we
suppose J9 = 1. We thus get the following result : —
Tlie equation a? — Cy^ = ±H, where G and H are positive
integers, and G is not a perfect square, adfnits of an infinite
number of integral solutions provided its right-hand side occurs
among the quantities ( - )'^Mn belonging to the development of JG
as a simple continued fraction.
Cor. 1. The equation a?- Gy'^= 1, where C is positive and not
a p&rfect square, always admits of an infinite number of solutions*.
* By what seems to be a historical misnomer, this equation is commonly
spoken of as the Pellian Equation. It was originally proposed by Fermat
as a challenge to the English mathematicians. Solutions were obtained by
§§ 14-16 Lagrange's theorem Regarding x^- Cy^= ±H 4tld
For, if the number of quotients in the period of JC be
even, =25 say, then {-f^3Ls will be + 1 (since here 3I=+l).
Therefore we have
where t is any positive integer ; that is to say, we have the
system of solutions
^=i?2t„ y = q2ts (A),
for the equation or - Cy"^ = 1.
If the number of quotients in the period be odd, = 2s - 1 say,
then ( - T-'M,s-i will be - 1, but ( - )^-^iJf4.-2, ( - T'^Mss-,, . . .
will each be + 1. Hence we shall have the system of solutions
^ =P4ts~2t , y = Qits-a (B),
for the equation x"^ — Cy'^ = 1.
Cor. 2. The equation a^ — Cy^ = - 1 admits of an infinite
number of integral solutions provided there he an odd number of
quotients in the period of ^C.
§ 16.] In dealing with the equation
a?-Gf^±H (1)
we may always confine ourselves to what are called primitive
solutions, that is, those for which x is prime to y. For, if x and y
have a common factor 6, then 6" must be a factor in H, and we
could reduce (1) to x"^- Cy"^ = ±HI6'^. In this way, we could
make the complete solution of (1) depend on the primitive
solutions of as many equations like x'^- Cy"^ = ±H\(P' as ^has
square divisors.
We shall therefore, in all that follows, suppose that x is
prime to y, from which it results that x and y are prime to H.
With this understanding, we can prove the following im-
portant theorem : —
If H<JG, all the solutions of (1) are furnished by the
conver gents to JC according to th^ method of % 16.
This amounts to proving that, \{ x = p, y^qhe any primitive
integral solution of (1), ihexx pjq is a convergent to JG.
Brouncker and Wallis. The complete theory, of which the solution of this
equation is merely a part, was given by Lagrange in a series of memoirs which
form a landmark in the theory of numbers. See especially (Eiivres, t, ii.,
p. 377.
480 GENERAL SOLUTION OF a?—C\f= ±1, OR iiT CH. XXXIII
Now we have, if the upper sign be taken,
Hence Pl9l- JG = Hjq (p + J Cq),
<>jC/q{p+JCq),
<W(p/qJC-^l) (2).
Now p/q - JC is positive, therefore pjq JC> 1. Hence
plq-JC<l/2f (3).
It follows, therefore, by chap, xxxii., § 9, Cor. 4, that p/q is
one of the convergents to JC.
If the lower sign be taken, we have
q'-{l/C)f = H/C,
where JI/C<J{llC). We can therefore prove, as before, that
q/p is one of the convergents to «y(l/C), from which it follows
th&tp/q is one of the convergents to JC.
Cor. 1. All the solutions of
x''-Ctf=-l (4)
are furnished hy the penultimate convergents in the successive
or alternate periods of JC.
Cor. 2. If the number of quotients in the period of J C he
even, the equation
a^'-Cf = -l (5)
has no integral solution. If the number of quotients in the
period be odd, all the integral solutions are furnished by the
penultimate convergents in the alternate periods of JC.
§ 17.] We have seen that all the integral solutions of the
equation (4) are derivable from the convergents to JC; it is
easy to give a general expression for all the solutions in terms
of the first one, say {p, q). If we put
x + yJC={p + qJCr\ ,..
^-^JC-{p-qJCr\ ^ ^'
we have
x'-Cy^ = (p^-Cq')''=l.
Hence (6) gives a solution of (4).
In like manner, if n be any integer, aud (p, q) the first
solution of (5), a more general solution is given by
X^yJC = {p + qJCT"-\ (7)
x-yJC=(p-qjCr-'\
16, 17
EXAMPLES
481
(8)
Finally, if {p, q) be the first solution of (1), we may express
all the solutions derivable therefrom* by means of the general
solution (6) of the equation (4). For, if (r, s) be any solution
whatever of (4), we have
(pr ± Cqsf -C{ps± qry = ± //.
Therefore
w = pr±Cqs
y=ps±qr
is a solution of (1).
The formulae (6), (7), (8) may be established by means of the
relations which connect the convergents of ^C (see Exercises
XXXI., 25, and Serret, Alg. Sup., § 27 et seq.). This method of
demonstration, although more tedious, is much more satisfactory,
because, taken in conjunction with what we have established
in § 16, it shows that (6), (7), and (8) contain all the solutions
in question.
Example 1. Find the integral solutions of x^- 13j/^=l.
If we refer to chap, xxxii., § 5, we find the following tahle of values
for .yi3 :—
n
««
Pn
In
^n
1
2
3
3
4
1
1
4
3
3
7
2
3
4
11
3
4
5
18
5
1
6
7
119
137
33
38
4
3
8
256
71
3
9
393
109
4
10
G49
180
1
11
6
4287
1189
4
Hence the smallest solution of x^ - l^y^ = 1 is x = 649, y = 180. "We have,
in fact,
6492 - 13 . 1802=421201 - 421200=1.
* It must not be forgotten that there may be more than one solution in
the first period. For every such primary solution there will be a general
group like (8).
c. II. 31
482 x^ -Cif = ± H, WHEN H > \IC CH. xxxiii
From (6) above, we see that the general solution is given by
X = i {(649 + 180^13)" + (649 - 180^13)"},
y = ^{ (649 + 180^13)" -(649- 180 Vl3)"}/v/13,
where n is any positive integer.
In particular, taking n=2, we get the solution
a; = 6492 + 13 . 1802 = 842401,
y= 2.649.180=233640.
Example 2. Find the integral solutions of x'^-l%y^= - 1.
The primary solution is given by the 5th convergent to ,^13, as may be
seen by the table given in last example.
The general solution is, by (7),
x=^{(18 + 5Vl3)2»-i + (18-5V13p-'},
2/ = 2^{(18 + 5Vl3)2»-i-(18-5^13)2"-i}.
where n is any positive integer.
Example 3. Find all the integral solutions of x^ - 13j/2= 3.
The primary solution \s x = ^, y = l, as may be seen from the table above.
The general solution is therefore, by (8),
a;=4r±13s, ?/ = 4sir,
where (r, s) is any solution whatever of x^- 13?/2=1.
In particular, taking ?-=649 and s = 180, we get the two solutions, a;=256,
i/ = 71, and x=4936, 2/ = 1369.
§ 18.] Let US next consider the equation
a^-Cf = ±H (9),
where C is positive and not a perfect square, and II is positive
but>va
We propose to show that the solution of (9) can always be
made to depend on the solution of an equation of the same form
in which II<JC; that is, upon the case already completely
solved in §§ 15-17.
Let (x, y) be any primitive solution of (9), so that x is prime
to y. Then we can always determine {xi, y^ so that
xy^-yxi = ±l (10)*.
In fact, if pjq be the penultimate convergent to xjy when
converted into a simple continued fraction, we have, by § 12,
w^ = tx±p, yi = ty±q (11).
* There is no connection between the double signs here and in (9).
§ 18 LAGRANGE'S CHAIN OF REDUCTIONS 483
If we multiply both sides of (9) by Xx - Cyi, and rearrange
the left-hand side, we get
{xxx-Cyy,y- C{xy,-yx,f = ±H{x,^-Cy,%
This gives, by (10),
{XX, - Cyy.y -C=^±H {x^ - Cy^) (12).
Now
XX, - Cyy, =t{x'- Cf) ± {xp - Cyq) (13).
But we may put xp - Cyq = 8II± K,, where Kil^^II. Hence
xx,-Cyy, = (t±S)H±(±K,) (14).
Now t and the double sign in (13) are both at our disposal ;
and we may obviously so choose them that
xxi-Cyyi = Ki (15),
where
K,:^hff- (16).
"We therefore have, from (12),
K,'-C=±II{x-'-Cy,') (17).
Now, by hypothesis, JC<ff, therefore C<IP and K^'^G
<H\
Since {x,, y,) are integers, it follows from (17) that, if (9)
have an integral solution, then it must be possible to find an
integer K{i(>^H such that
{K,^-C)IH=H, (18),
where H, is some integer which is less than H'^IH, that is, <H.
If no value of Kx<\H can be found to make iK^-G)lH
integral (and, be it observed, we have only a limited number of
possible values to try, since K^lif-^H), then the equation (9) has
no integral solution.
Let us suppose that one or more such values of K,, say K,,
Ki , Ki', . . ., can be found, and let the corresponding values of
Hi be Hi, Hi, H", . . . Then it follows from our analysis that
for every integral solution of (9) we must be able to find an
integral solution of one of the limited group of equations
x^-Cyi^ = ±Hi ^
Xi^-Gyi'=^±H;
Xi'-Gyi' = ±Hi'
(19),
where Hi , Hi, Hi', ... are all less than H.
31—2
484 PRACTICAL METHOD OF SOLUTION CH. XXXIII
If it also happens that in all the equations (19) the numerical
value of the right-hand side is < JC, then these equations can
all be completely solved, as already explained.
If {xi, i/i) be a solution of any one of them, we see, by (10)
and (15), that
X = (K,x, + Cy^)IH„ y = {K,y, +_x,)IH, (20)*
or a; = [k^x^ hF Cy^lHU y = {K^'y, + Xi)/Hi,
If in any of the equations (19), say, for instance, in the first,
the condition Hi<>JG is not yet fulfilled, we can repeat the
above transformation, and deduce from it a new system,
a;i-Cyi = ±H, )
x^^-Cyi = ±H^\ (21),
where H2 and H^ are each less than Hi ; and we have
Xi = {K2X2+Cy^lH2, yi = {K2y2 + x^lH2 \
xi = {K^x., + Cy^lH;, yi = {K^y^ + x-^lH^ \ (22).
Since the //'s are all integers, the chain of successive operations
thus indicated must finally come to an end in every branch.
Thus we see that any integral solution of{d) must be deducible
from the solution of one or other of a finite group of equations of
the type
a^-Cf^IU"'^ (23),
where Hr^''^<^G.
The practical method of solution thus suggested is as
follows : —
Find all the integral values oiKx<\EiQx which {K^-G)IH
is an integer. Take any one of these, say K^ ; and let H^ be
the corresponding value of {K{^-C)/ff. Then, if Hi<JC, solve
the equation aji'^ - C?/i^ = ± ZTj generally ; take the formula (20);
and find which of the solutions {x^, y^), if any, make {x, y) integral.
We thus get a group of solutions of (9). If Hx>JC, then we
find all the values of K^K^ff^ for which (K^^ - 0)1 H^ is integral,
* Since the signs of x and y are indifferent in the solutions oi s^-Cy^=
A JZ, it is unnecessary to take account of the double signs of Ifj, H-^', &c.
For the same reason, the ambiguities of sign in (20) and (22) are independent.
§ 18 EXAMPLE 485
= 7/2 say, and, if H2<JC, solve the equation x.2—Cyi = ±H^\
then pass back to x through the two transformations (20)
and (22) ; and, finally, select the integral values of x and y thus
resulting, if there be any.
By proceeding in this way until each branch and twig, as it
were, of the solution is traced to its end, we shall get all the
possible integral solutions of (9), or else satisfy ourselves that
there are none.
The straightforward application of these principles is illus-
trated in the following example. Into the various devices for
shortening the labour of calculation we cannot enter here.
Example. Find the integral solutions of
a;2-15?/2=61 (9').
Let {/ri2-15)/61=J7i (18'),
where K^ > 30.
Then K-^=l^^-<o\TI^.
Since K^ j> 900, we have merely to select the perfect squares among the
numbers 15, 76, 137, 198, 259, 320, 381, 442, 503, 564, 625, 686, 747, 808, 869.
The only one is 625, corresponding to which we have K^ = 25 and H^ = 10.
Since Hi>»J15, we must repeat the process, and put
(K^^~15)I10 = H^ (18"),
where K^ > 5, and therefore K^^ > 25.
Since K2^=15 + 10H2, the only values of K.^^ to be examined here are 5,
15, 25. Of these the last only is suitable, corresponding to which we have
^2=5, Ifa = l.
We have now arrived at the equation
x^'-15y,^=l (21'),
the first solution of which is easily seen to be (4, 1). Hence the general
solution of (21') is
^2=|{(4+x/15)"+(4-Vl5)n ]
The general solution of (9') is connected with this by the relations
a?! = (5x2 =F 15j/2)/l, Vi = (5?/2 T x^)!! (22') ;
x = (25xi=f152/i)/10, y={25y,=fx,)ll0 (20').
Hence a;=14x2=r45i/2, y==f^3x^ + Uy^) .^g.
a; = llx2=F30jf2. 2/= =1=23:2 + 11^2 1
where x^ and j/j are given by (24). The question regarding the integrality of
X and y does not arise in this case.
As a verification put ar2=4, 1/2=1, and we get the solutions (11, 2),
(101, 20), (14, 3) and (74, 19) for (9'), which are correct.
486 REMAINING CASES OF BINOMIAL EQUATION CH. XXXIII
§ 19.] There remain two cases of the binomial equation
a^ - Cy^ ^±11 which are not covered by the above analysis —
x'-Cf = ±H (26),
where C is a perfect square, say G = W \ and
a?+Cf=-vH (27).
The equation (26) may be written
{x - By) {x + By) = ± //.
Hence we must have
X - By = u
. 7? 1 (28),
X + By = V j
where u and v are any pair of complementary factors of ±11.
We have therefore simply to solve every such pair as (28), and
select the integral solutions. The number of such solutions is
clearly limited, and there may be none.
In the case of equation (27) also the number of solutions is
obviously limited, since each of the two terms on the left is
positive, and their sum cannot exceed If. The simplest method
of solution is to give y all integral values '^J(ff/C), and
examine which of these, if any, render JI- Cy^ a perfect square.
§ 20.] In conclusion, we shall briefly indicate how the
solution of the general equation of the 2nd degree,
ax"^ + 2hxy + hy'' + 2gx + 2fy + c = Q (29),
where a, b, c, f, g, h are integers, can be made to depend on the
solution of a binomial equation.
By a slight modification of the analysis of chap, vii., § 13,
the reader will easily verify that, provided a and b be not both
zero, and c be not zero, (29) may be thrown into one or other
of the forms
{Cy + Ff-C{ax + hy + gf = -a^ (30);
or {Cx+Gf-C{kx + by+ff=^-bA (31),
where A = abc + 2/gh - ap - bg^ - c/i'', C=h^- ab, F= gh - af,
G - hf- bg ; say into the form (30). If, then, we put
ax-ifhy + g = r})
§§19, 20 GENERAL EQUATION OF 2ND DEGREE 487
(30) reduces to
e-Crf=-a^ (33),
which is a binomial form, and may be treated by the methods
already explained.
If /r>a&, then Cis positive, and, provided Cbe not a perfect
square, we fall upon cases (1) or (9).
If C be a positive and a perfect square, we have case (26).
It should be noticed that, if either « = 0 or b = 0, or both
a = 0 and & = 0, we get the leading peculiarity of this case, which
is that the left-hand side of the equation breaks up into rational
factors (see Example 2 below).
\i h^<ab, then Cis negative, and we have case (27).
lih^ = ah, then C=0, and the equation (29) may be written
{ax + hyf + 2agx + 2afy + ac = 0 (34),
which can in general by an obvious transformation be made to
depend upon the equation
rf-Qi (35),
which can easily be solved.
Example 1. Find all the positive integral solutions of
3x2 _ Qxy + 7j^a -4_x + 2y = 109.
This equation may be written
(3x - 42/ - 2)2 + 5 (?/ - 1)2 = 336,
say ^2+5^2=336.
Here we have merely to try all values of ?? from 0 to 8, and find which of
them makes 336 - 5rp a perfect square. We thus find
^=±16, ,;=±4;
|=±4, i;=±8.
Hence
3a;-4y-2=±16, j/-l=±4 (1);
3a; -4?/ -2= ±4, y-l==fc8 (2).
It is at once obvious that in order to get positive values of y the upper
sign must be taken in the second equation in each case. Hence y = 5 or
?/ = 9, To get corresponding positive integral values of x, we must take the
lower sign in the first of (1), and the upper sign in the first of ("i). Hence
the only positive integral solutions are
x—2, y — o, and x=14, 2/ = 9.
488 EXAMPLES CH. XXXIIl
Example 2. Find the positive integral solutions of
This is a case where the terms of the 2nd degree break up into two rational
factors. We may put the equation into the form
(9a; + 6?/-l)(3j/-4) = 112.
Since 3y -i is obviously less than 9x + Gy-l when both x and y are
positive, 3)/ - 4 must be equal to a minor factor of 112, that is, to 1, 2, 4, 7,
or 8; the second and the last of these alone give integral values for y, namely,
y = 2 and j/ = 4. To get the corresponding values of x, we have 9x + 6y-l
= 56 and 9a; + 6?/-l = 14, that is to say, 9x = 45 and 9x= -9. Hence the
only positive integral solution is x = 5, y = 2.
Example 3. Find all the integral solutions of
9x2 _l2xy + 42/2 + 3x + 2y = 12.
Here the terms of the 2nd degree form a complete square, and we may
write the equation thus —
(3x - 2(/)2 + (3x - 2y) + 4?/ = 12,
or 4(3a;-22/)2 + 4(3x-22/) + l + 16?/ = 49;
that is, (6.T -iy + 1)2 = 49 - 16?/.
Hence, if
u=Qx-iy + l (1),
so that u is certainly integral, we must have
?/ = (49-w2)/16 (2).
Now we may put M=16/i±s, where s is a positive integer J»8.
It then appears that y will not be integral unless (49 - s2)/16 be integral.
The only value of s for which this happens is s=l. Therefore
u=16a4±1 (3).
Hence, by (1), (2), and (3), we must have
x = 2 + 4/x(l-8/i)/3, y = 3~2fM-16fji^ (4),
or
x = 4/* + (5-32/^2)/3, j/ = 3 + 2/t-16M2 (5).
It remains to determine /j. so that x shall be integral.
Taking (4), we see that fi{l- 8/i)/3 will be integral when and only when
fj. — Si> or /j. = 3p -1.
Using these forms for /x, we get
a; = 2 + 4;'-96v2, y = 3-&p-lUv'' (6);
a;= -10 + 68«'-96;'2, t/= - ll + 90;'-144»'2 (7).
Taking (5), we find that (5-32/i2)/3 is integral when and only when
fi=3v + l or /A=8;'-l.
Using these forms, we get from (5)
x=-5-52p-Q6p\ i/=- 11-90*/ -144^2 ^g);
a; = - 13 + 76;/ - 96^2, y=-U + l02v- lUv^ (9) .
The formulas (6), (7), (8), (9), wherein v may have any integral value,
positive or negative, contain all the integral solutions of the given equation.
§20
EXERCISES XXXII 489
Exercises XXXII.
Find all the integral and also all the positive integral solutions of the
following equations: —
(1.) 5x + 7y = 2d. (2.) 16a; - 17t/ = 27,
(3.) 11a; + 72/ = 1103. (4.) 13G7x - 1013?/ = 16246.
(5.) If £x. ys. be double £y. xs., find x and y.
(6.) Find the greatest integer which can be formed in nine different
ways and no more, by adding together a positive integral multiple of 5 and a
positive integral multiple of 7.
(7.) In how many ways can £2 : 15 : 6 be paid in half-crowns and florins?
(8.) A has 200 shilling-coins, and B 200 franc-coins. In how many ways
can A pay to B a debt of 4s. ?
(9.) 4 apples cost the same as 5 plums, 3 pears the same as 7 apples, 8
apricots the same as 15 pears, and 5 apples cost twopence. How can I buy
the same number of each fruit so as to spend an exact number of pence and
spend the least possible sum ?
(10.) A woman has more than 5 dozen and less than 6 dozen of eggs in
her basket. If she counts them by fours there is one over, if by fives there
are four over. How many eggs has she ?
(11.) A woman counted her eggs by threes and found that there were two
over ; and again by sixes and found there were three over. Show that she
made a mistake.
(12.) Find the least number which has 3 for remainder when divided by
8, and 5 for remainder when divided by 7.
(13.) Find the least number which, when divided by 28, 19, 15 re-
spectively, gives the remainders 15, 12, 10 respectively.
(14.) In how many ways can £2 be paid in half-crowns, shillings, and
sixpences ?
(15.) A bookcase which will hold 250 volumes is to be filled with 3-volumed
novels, 5-volumed poems, 12-volumed histories. In how many ways can this
be done? If novels cost 10s. 6d. per volume, poems 7s. Gd., and histories 5s.,
show that the cheapest way of doing it will cost £129. 15s.
Solve the following systems, and find the positive integral solutions : —
(16.) x + 2y + 3z = 120.
(17.) x+y + z + u= 4,) (18.) 2x + 5y+ 3^ = 324,)
5?/ + 62 -f 9m = 18.) 6a; -4?/ -1-142 = 190.]"
(19.) 5a;-62/ + 72 = 173,'l (20.) 17x-i-19(/ -1-212 =400.
17a; - 4?/ -f 32 = 510.)
(21.) x+ y+ z+ « = 26,'j
3x + 2y + iz+ M=63, L
23; + 32/-l-22 + 4it = 74.J
(22.) Show how to express the general integral solution of the system
a-^T^Xi + a^.^x.^+ . . . -j-ai„a;„=di,
a2ia;i + G„2.T2-f . . .-ha2„x„=d2.
'^n-l. 1*1 + "«-!. 2^2+ • • •+''«-], n^n — "n-l
by means of determinants, when a particular solution is known.
490 EXERCISES XXXII CH. XXXIfl
Find the values of x which make the values of the following functions
integral squares : —
(23.) 2x2 + 2x. (24.) (x^-x)l5. (25.) a; + 11 and a; + 20, simultaneously.
(26.) 7x + 6 and 4x + 3, simultaneously. (27.) x^ + x + 8.
Solve the following equations, giving in each case the least integral
solution, and indicating how all the other integral solutions may be found : —
(28.) x^-Uy^=-8. (29.) x^-Uy'^=+5.
(30.) a;2-44?/2=-7. (31.) a;2-44^2^+4
(32.) x2 + 31/2 =628. (33.) x2-69?/2= - 11.
(34.) x2-472/2=+l. (35.) x^-i7y''=-l.
(36.) x2- 261/2= -1105. (37.) x^-7y^=186.
(38.) x2-(a2 + 1)2/2 = 1. (39.) x^-(a^-l)y^ = l.
(40.) x2-(a2+a)2/2=l. (41.) x^-(a^-a)y'^=l.
(42.) x2 + 5x2/-2x + 3z/ = 853. (43.) xy-2x-dy = 15.
(44.) x2-2/2 + 4x-5?/ = 27. (45.) 3x2 + 2x2/ + 02/2 =390.
(46.) x2 + 4x2/ - 111/ + 2x-86j/- 140 = 0.
(47.) !b2 - xy - 722/2 + 2x-440t/- 659 = 0.
(48.) x2 + 2x2/ -17^2 + 721/ -75=0.
(49.) 61x2 + 28x3/ + 2512/2 + 264x4-5262/ + 260=0.
(50.) Show that all the primitive solutions of Dx^- (7y'= =tH are
furnished by the convergents to ^(C/D), provided H<^{CD). Show also
how to reduce the equation Dx2- C2/2= ifl", when if>^(CD).
(51 .) Find all the solutions of
4x2-72/2= -3,
and of 4x2-72/2 = 53.
(52.) If D, E, F, II be integers, and H<^{E^-DF) (real), show that all
the solutions of
Dx2 - 2Exy + Fy"^ = ± H
are furnished by the convergents to one of the roots of
Dz^-2Ez + F=0.
(See Serret, Alg. Sup., § 35.)
(53.) If 0„=P„-a;g„, where a; is a periodic fraction having a cycle of
c quotients, and p„ and g„ have their usual meanings, then
C/„H-^=(a-/3x^+i)»J7r,
1 1
where a;,+i - «r+i + ;i — T • • • ,-; — T • • •»
* a,.+2+ «r+c +
all
and s=«H-i+ T • • • 7~ •
In particular, if x=sJ(CID), then
DPnc+r - sKGD) q^^= {aM, - /3L^ - /3V(CD)}» (Bp^ - ^(CD) q,)IM*.
Point out the bearing of this result on the solution of Dx^-Cy"= ±11.
CHAPTER XXXIV.
General Continued Fractions.
FUNDAMENTAL FORMULA.
§ 1.] The theory of the general continued fraction
^i = «i + -^-^. . . (A),
where a^, a^, a^, . . ., h.2,h, . . . are any quantities whatever,
is inferior in importance to the theory of the simple continued
fraction, and it is also much less complete. There are, how-
ever, a number of theorems regarding such fractions so closely
analogous to those already established for simple continued
fractions that we give them here, leaving the demonstrations,
where they are like those of chap, xxxii., as exercises for the
reader. There are also some analytical theories closely allied to
the general theory of continued fractions which will find an
appropriate place in the present chapter.
In dealing with the general continued fraction, where the
numerators are not all positive units, and the denominators
not necessarily positive, it must be borne in mind that the chain
of operations indicated in the primary definition of the right-
hand side of (A) may fail to have any definite meaning even
when the number of the operations is finite. Thus in forming
tlie third convergent of 1 + - — ^rz_^^' • • ^® ^^^ ^^^ ^o
1 + 1/(1 - 1) ; and in forming the fourth to 1 + 1/{1 - 1/(1 - 1)}.
It is obvious that we could not suppose the convergents of this
fraction formed by the direct process of chap, xxxii., § 6 (a), (^),
492 C.FF. OP FIRST AND SECOND CLASS CH. XXXIV
(y). It must also be remembered that no piece of reasoning
that involves the use of the value of a non-terminating continued
fraction is legitimate till we have shown that the value in
question is finite and definite.
In cases where any difficulty regarding the meaning or conver-
gency of the continued fraction taken in its primary sense arises,
we regard the form on the right of (A) merely as representing the
assemblage of convergents pijq-i, p^jq^, • . -fPnlqn whose denomi-
nators are constructed by means of the recurrence-formulae (2) and
(3) below.
That is to say, when the primary definition fails, we make
the formulae (2) and (3) the definition of the continued fraction.
In what follows we shall be most concerned with two varieties
of continued fraction, namely,
ai + — . . . (15 ),
and a, + -^-^... (C),
where ai, a.2, a^, . . ., &2, ^3, • • • are all real and positive. We
shall speak of (B) and (C) as continued fractions of the first and
second class respectively.
§ 2,] Upi/qitPi/gi) &c. be the successive convergents to
a<i+ a3 +
then
Pn = CtnPn-l + &n^n-3 (2) J
qn = Clnqn-l + bnqn-2 (3),
with the initial conditions pQ = l,pi = ai; g'l = 1, g'a = «2.
Cor. 1. In a continued fraction of the first class pn and qn
are both positive ; and, provided a„<|; 1, each of them continually
increases with n*.
In a continued fraction of the second class, subject to the
restriction a„<{:l + 6„, p^ and g„ are positive, and each of them
continually increases with n*.
* It does not necessarily follow that Z,p„=QO and Zg'„=oo , for the suc-
cessive increments here are not positive integral numbers, as in the case of
simple continued fracUons.
§§ 1-3 PROPERTIES OP CONVERGENTS 493
These conclusions follow very readily by induction from such
formulae as
Pn -Pn-l = («n - 1 )i?n-l + Kpn-i (•*)•
Cor. 2.
-^=^^4._^ AzZ_. . .^ (5);
, Pn-1 ttn-l + a»-2 + «!
_5iL = «^+_A_ A-_l_. . . h (6).
§ 3.] From (2) and (3) we deduce
Pnqn-l -Pn-ign = ( " )"^2^3 • • • &» (!)•
Cor. 1. The convergents, as calculated by the recurrence-rule,
are not necessarily at their lowest te7"ms.
Cor. 2.
Cor. 3.
Phl _ Pj^ = (-Y ^-^^ • ' -^n /gx
2n S'n-l QnQn-l
Cor. 4.
Pnqn~2-Pn-2qn = {-T"^anb2l>3 . • • ^u-i (4);
/^ra Pn-2 _ / \n-l ^nf^if^'i • ' • ^n-1 /r\
Qn qn-2 qnqn-2
Cor. 5.
/Pn _ Pn-\\ l(Pn-l _ Pn~-i\ ^ _ hnqn-2
\qn qn-J/\qn-i qn-2) qn '
hnqn-'i
(6).
(^nqn-i + bnqn-2
Cor. 6. In a continued fraction of the first class, the odd
convergents form an increasing series, and the even convergents a
decreasing series ; and every odd convergent is less than, and every
even convergent greater than, following convergents.
In a continued fraction of the second class, subject to the
restriction a„<t;l + &„, all the convergents aVB positive, and form
an increasing series.
494 CONTINUANT DEFINED CH. XXXIV
These conclusions follow at once from (2) and (5), if we
remember that, for a fraction of the second class, we have to
replace h^, . . ., bn by -h, . . ., -bn.
CONTINUANTS.
§ 4.] The functions j9„, q^ of ^i, ofa, • • ., ««; h, h, . . ., bn
which constitute the numerators and denominators of the con-
tinued fraction
, &2 h bn
(Zi H ...
a2+ CC3+ an
belong to a common class of rational integral functions*.
In fact, pn is determined by the set of equations
Pfi = a.iPi + hpo, Pz = aiPi + biPi, . . ., Pn = anPn-1 + bnPn-2
(1),
together with the initial conditions j9o = l, pi = ai; while q^ is
determined by the system
qz^a^qz + hqi, g'4 = «4g'3 + ^4(72, • • . 9'n = a»g''^-i + ^ng«-2
(2),
together with the initial conditions g'l = 1, §'2 = «2-
It is obvious, therefore, that qn is the same function ofa^, as,. . .,
««; bs, bi, . . .,bn aspn is of a^, a.2, ...,«»; b^, b^, . . ., bn-
We denote the function pn by
Pn = K( ^^'•••'M (3),
and speak of it as a continuant of tits nth order whose denomin-
ators are a^, a^, . . ., a», and whose numerators are bi, . . ., 6«.
We have then
\a^2» (^if • • • J ^n/
* This was first pointed out by Euler in his memoir entitled " Specimen
Algorithmi Singularis," Nov. Comm. Petrop. (1764). Elegant demonstrations
of Euler's results were given by Mobius, Crelle's Jour. (1830). The theory
has been treated qI l&te in connection with determinants by Sylvester and
Muir.
;}-5
FUNCTIONAL NATURE OF CONTINUANT
495
When the numerators of the continuant are all unity, it is
usual to omit them altogether, and write simply K{ax, a^, . . ., a„).
A continuant of this kind is called a simple continuant.
When it is not necessary to express the numerators and
denominators it is convenient to abbreviate both
^L !"' * " !") and ^(a„ a„ . ..,«„)
\»i , ih} • • • ) "71/
into K(l, n). In this notation we should have, if r<5,
\fl5,., (Ir+i, '
\(ls , ds—l > •
., as)
(5);
(6).
In particular, K{r, r) means simply a,., so that j^i = K{1, 1) = ai-
To make the notation complete, we shall denote po and q^ by
K{ ), which therefore stands for unity ; and, in general, when
the statement of any rule requires us to form a continuant for
which the system of numerators and denominators under con-
sideration furnishes no constituents, we shall denote that con-
tinuant by ^( ) and understand its value to be unity. It will
be found that this convention introduces great simplicity into
the enunciation of theorems regarding continuants.
§ 5.] A continuant of the nth order is an integral function of
the nth degree of its constituents.
This follows at once from the definition of the function, for
we have, by § 4 (1),
K{l,n) = anK{l,n-l) + hnK{l,n-2), \
K{1, n-l)^ an-iK(l, n-2) + K-J<:{1, n - 3),
K{1, /+ 1) = ai+xK{l, I) + hi+,K{ ),
K{l,l) = ai, K{ ) = 1.
(7).
The following rule of Hindenburg's gives a convenient
process for writing down the terms of a series of continuants,
gay Z(l, 1), K{\, 2), K{1, 3), . . . :-
496 EULER's construction for continuant CH. XXXIV
«!
as
«4
«1
^2
^4
^5
K
h.
as
^5
«!
fta
«3
K
«3
^>5
»!
^'3
h.
1st. Write down «], and enclose it in the rectangle 1, 1. The
terrain 1, 1 is^(l, 1).
2nd. Write a^ to the right of all the rows in 1, 1 ; and write
&2 underneath. Enclose all the rows thus constructed in the
rectangle 2, 2. Then the rows in 2, 2 give the products in
/r(l, 2), namely, a^az + h.
3rd. Write as at the ends of all the rows of 2, 2 ; repeat
under 2, 2 all the rows in 1, 1, and write bs at the end- of each of
them. Enclose all the rows thus constructed in 3, 3. Then
the rows in 3, 3 give the products in K{1, 3), namely,
aia-itti + hiCh + a^hz.
The law for continuing the process will now be obvious. The
scheme is, in fact, merely a graphic representation of the con-
tinual application of the recurrence-formula
K{1, n) = anK{l, n-\) + hnK{\, n-2) (8).
By considering Hindenburg's scheme we are led to the
following rule of Euler's* for writing down all the terms of a
continuant of the wth order.
Write down a^a^a^ . . . an-ittn. This is the first term. To
get the rest, omit from this product in every possible way one or
more pairs of consecutive a's, always replacing the second a of
the pair by a b of the same order.
* Euler {I.e.) gave the rule for the simple continuaiit merely. Cayley
{Phil. Mag., 1853) gave the more general form.
^ 5, 6 PROPERTIES OF CONTINUANTS 497
For example, to get tlie terms of K{1, 4). The first is a^af^a^a^. By
omitting from this, first aia2, then a2^3' then aga^, and replacing by b^, 63, 64
respectively, we get three more terms, b^a^tti, a^b^a^, a^a^b^. Then,' omitting
two pairs, we get b^^b^. We thus get all the terms of iC (1, 4).
It is easy to verify this rule up to K{\, 5); and a glance at
the recurrence-formula (8) shows that, if it holds for any two
consecutive orders of continuants, it will hold for all orders.
From Euler's rule we deduce at once the following : —
Cor. 1. The value of a continuant is not altered by reversing
the order of its constituents, that is to say,
j^r h, . . ., K\^r hn, . . ., h\ ^g^^
\ai, a^, • . •, a^ \an, flf»-i) • • •> <^i/
We could obviously form the continuant K (1, n) by starting
with anttn-i . . . a^ai instead of ai^a . . . a»-ia„, and replacing each
consecutive pair of a's in every possible way by a 6 of the same
order as the first a of the pair. In this way we should get pre-
cisely the same terms as before. Hence the theorem. We may
express it in the form
K{l,m) = K{m,l) (10).
Cor. 2. We have the following recurrence formula : —
K{1, m) = aiK{l +l,m) + bi+^K{l + 2, m) (11).
For, by Cor. 1,
K{l,m) = K{m, I),
= aiK{m, l+l) + bi+iK(m, 1+2), by (7),
= ai K{1 +\,m) + bi+i K{l + 2, m), by Cor. 1.
§ 6.] The theorems (1) and (4) of § 3 may be written in
continuant notation as follows : —
J:(1, n) Z (2, w - 1) - K{\, n- 1)^(2, n)
= {-fbA...b,,K{ )K{ ) (12),
K{\, n)K{2, n-2)-K{l,n-2)K{2, n)
= ( - T-' b,b, . . . bn-, K{ ) K{n, n) (13).
These are particular cases of the following general theorem,
originally due to Euler*: —
* Euler stated it, however, only for simple continuants. It has been
stated in the above general form and proved by Stern, Muir, and others.
C. II. 32
498 euler's continuant-theoeem en. xxxiv
K{1, n) K{1, m) - K{\, m) K{1, n)
= {-r-''%bu^ . . . ^>«+i^(l, l-2)K{m+2, n) (14),
where l<l<m<n.
This theorem is easily remembered by means of the following elegant
memoria technica, given by its discoverer : —
1, 2, . . ., 1-2, l-l, \l, . . ., m, |m + l, m + 2, . . ., n.
Draw two vertical lines enclosing the indices belonging to K{1, m); then two
horizontal lines as above ; and put dots over the indices immediately outside
the two vertical lines. The indices for the first continuant on the left of (14)
are the whole row ; those of the second are inside the vertical lines ; those of
the third and fourth under the upper and over the lower horizontal lines ;
those of the two continuants on the right outside the two vertical lines, the
dotted indices being omitted. The 6's are the 6's ot K{1, m) with one more at
the end ; and the index of the minus sign is the number of constituents in
K{l,m).
The proof of the theorem is very simple. We can show, by
means of the recurrence-formulae (7) and (11), that, if the formula
hold for /, ?» + 2, and for /, m + 1, or for 1-2, m, and for l-l, m,
it will hold for /, m. Now (12) asserts the truth of the theorem
for 1^2, m=n~l; and it is easy to deduce from (12), by
means of (7) and (11), that the theorem holds ior 1 = 3, m = n-lf
and also for 1=2, m-n- 2. The general case is therefore
established by a double mathematical induction based on the
particular case (12).
The theorem (14) might be made the basis of the whole
theory of continued fractions ; and it leads at once to a variety
of important particular results, some of which have already been
given in the two preceding chapters. Among these we shall
merely mention the following regarding what may be called
reciprocal simple continuants : —
K{a^, a^, . . ., tti, tti, . . ., a2, a^
= K((h,a2, . . ., aiY + K(ai, a,, . . ., a^-i)* (A);
K{ai, a<i, . . ., ftf-i, ai,ai-i, . . ., a2, a^
==K{ai, a., . . ., tti--,) {K{a„ a^, . . ., a^) + K{ai, a^, . . ., a<-a)}
(B).
§§ 6, 7 smith's proof of a theorem of fermat's 499
Example. Show that every prime y of the form 4\ + 1 can be exhibited as
the sum of two integral squares*.
Let Atj , A4 1 • • ., Ms be all the integers prime to ■p and < ^p ; and let simple
continued fractions be formed for j>/yUj, 2'/m2» • • •> Vli'-si each terminating so
that the last partial quotient > 1. Then each of these continued fractions has
for its last convergent the value K(a-^, a^, . . . , a,J/7i'(a2, a^, . . . , a„), where
the two continuants are of course prime to each other, and ai>l, a„>l.
From this it appears that there are as many ways, and no more, of
representing ^^ by a simple continuant (whose constituents are positive
integers the first and the last of which are each greater than unity) as there
are integers prime to p and < ^p.
Now, since K{ai, a^, . ■ ■, a^^ = K{a^, . . . , a^, Oj), and a„>l, it is
obvious that -K'(a„, • . . , ag, flj) must arise from one of the other fractions pj/i.
Hence, given any fraction pjfi, it is possible to find another also belonging to
the series which shall have the same partial quotients in the reverse order.
Let p be a prime of the form 4\ + l, then the greatest integer in ^p is 2\,
which is even. Since, therefore, the number of continuants which are equal
to p must be even, and since K (p) is one of them, there must, among the
remaining odd number, be one at least which gives rise to no new fraction
when we reverse its constituents, that is to say, which is reciprocal. Now
the reciprocal continuant in question cannot be of the form K{ai, 02, . . .,
«i-i. flji «i-i» • • •> ^2> '''i)' ^^^ ^* follows from (B) that such a continuant
cannot represent a prime, unless i = l, or else i = 2, and 0^ = 1, all of which are
obviously excluded.
We must therefore have an equation of the form
p = K{ai, aj, . . ., a<, o,-, . . ., a^, a^),
K(ai, aj, . . ., ai)^ + K(a^, a^, . . ., ai_i)2,
by (A), which proves the theorem in question.
As an example, take 13 = 3x4 + 1.
_ , 13 ,„ 13 ^ 1 13 , 1 13 ^ 1 13 „ 111
Wehavey=13; ^=6 + ^; —4+^; -^=3 + ^; -^=2 + ^^^;
H=2 + L So that 13=£:(13)=ir(6, 2)=K(4, 3)=K(B, 4:)=K(2, 1, 1, 2)
6 D
=K(2, 6); and, in particular, 13=jB:(2, 1, 1, 2) = K(2, l)2+j: (2)2 =32 + 22.
§ 7.] By considering the system of equations (1) of § 4, it is
easy to see that, if we multiply ar, K, h+i by c^, the result is
the same as if we multiplied the continuant K{1, n) {n>r) by
Cr. Hence we have
jy I C2O2J CipJ^Zi C3C4O4, • • ., Cn-\CrPn
O/i, C<^-2} CsQis, 64(14, • • .J C.
\ai, «2, . . ., «;
■nfitn/
^ (15).
* The following elegant proof of this well-known theorem of Fermat's was
given by the late Professor Henry Smith of Oxford (Grelle's Jour., 1855).
32—2
500 EEDUCTION TO SIMPLE CONTINUANT CH. XXXIV
We may so determine c^, Cs, . . ■ , c^ that all the numerators
of the continuant become equal. In fact, if we put
Cj)2^\ 026^)3 = \ . . ., Cn-iCnbn=\
we get
Ca = '^bJ)ilbJ)J)2, ... ,
Hence
/ \, X, X, . . A
= {ll^ybJ>n-A-.- . ■''-^[a„Xa,lb„a.A/b3,XaA/bA, • ■ ■)
(16),
where p is the number of even integers (excluding 0) which do
not exceed n.
Cor. Every continuant can be reduced to a simple continuant,
or to a cmtinuant each, of whose numerators is - 1.
Thus, if we put X = + 1 and X = - 1, we have
j^/ &2, . . •, bn\
= Z>A-2 . • .xX(ai, ^2/^2, ttsb^bs, aA/bA, - - -,
anbn-ibn-3 • • •/M»-2 • • •) (17),
= {-ybnbn-, . . .xK(^^^ _^^i^^^ aAlbJ,-aA/bA, . • •',
{-f-^ dubn-lbn-S ' . -Ibnbn-^ • • 7
§ 8.] The connection between a continuant and a continued
fraction follows readily from (11). For we have, provided
K(2, n), K{3, n), K{^, n), . . . are all different from zero,
K{l,n)_^ b,
KA^) ""^'^ K (2, n)/K (3, n)'
K(2^ ^ ^ bs
K{S,n)~'^ K{3,n)IK{4:,ny -
Hence
-^(1>^)_^ , ^a ba ^r /,qn
Kj^) ~'''^a,+ aa+ - ' ' K{r, n)IK{r + 1, n) ^'''^-
§§ 7-10 C.F. IN TERMS OF CONTINUANTS 501
If in this last equation we put r = n, and remember that
here K{n+l, n) = K{ ) = 1, we get
^Ji4=^+AA...*» (20),
a result which was obvious from the considerations of § 4.
§ 9.] When the continuant equation
K{1, n) = anK{l, n-l) + bnK(l, n-2),
or Pn = anPn-i + hPn-2,
which may be regarded as a finite difference equation of the
second order, can be solved, we can at once derive from (20) an
expression for
a I ^^ ^^ ^
^ aa + as +'"'«» *
When ttn and 6„ are constants, the problem is simply that of
finding the general term of a recurring series, already solved in
chap. XXXL, § 7.
Example. To find an expression for the nth convergent to
_1 1^ J^
^-■^ + 1+ 1+ • • • 1+ • • • •
Here we have to solve the equation Pn=Pn-i+Pn-2f "^'^^^ the initial con-
ditions 2>o = 1 ) i^i = !• The result is
K{1, n) =i5n= {(1 + v/5)"+i - (1 - V5)"+n/2"+V5.
Hghcg
P^^Kjl, w)_ {(l + V5)"+i-(l-V5)"+i}/2'*+V5
g„ K{2,n) {(l + V5)''-(l-^/5)»}/2V5 '
_ (l + ^5)"+^-(l-^5)"+i
* (l + ^5)™-(l-V5)" *
From the expression for K{l,n) (all the terms in which reduce in this case
to + 1) we see incidentally that the number of different terms in a continuant
of the nth order is
2n+l /5 — 2^ l«+l'^l + ^n+l<^3 + ^ n+1^5+ • • •/•
§ 10.] When two continued fractions i^ and F' are so related
that every convergent of F is equal to the convergent of F' of
the same order, they are said to be equivalent*.
* We may also have an {m, ra)-equivalence, that is, Prmllrm^Prnllrn-
See Exercises xxxiii., 2, 17, &o.
502
BEDUCTION TO SIMPLE C.F.
CH. XXXIV
It follows at once from §§ 7 and 8 (and is, indeed, otherwise
obvious, provided the continued fraction has a definite meaning
according to its primary definition) that we may multiply a^, br,
and br+i by any quantity m{=¥0) without disturbing the equi-
valence of the fraction. Hence we may reduce every continued
fraction to an equivalent one which has all its numerators equal
to + 1 or to - 1. Thus we have
a, +
= ai +
tta + «3 + »4 + *
1 1
1
02/^2+ a3V^3+ «A/^A+ ' " * anK-iK-
lbnbn-2
(21).
11.] If we treat the equations (1) as a linear system to
determine K{1, 1), K{1, 2),
minant notation, we get
, K{1, n), and use the deter-
a,
h.
0
0
0 . .
. 0
0
0
1
^2
^'3
0
0 . .
. 0
0
0
0-
-1
«3
h
0 . .
. 0
0
0
0
0-
1
a^
h,. .
. 0
0
0
0
0
0
0
0 . .
.-1
««-
1 b,,
0
0
0
0
0 .
. 0-
-1
an
which gives an expression for a continuant as a determinant.
The theory of continuants has been considered from this point
of view by Sylvester and Muir* ; and many of the theorems
regarding them can thus be proved in a very simple and natural
manner.
Exercises XXXIII.
(1.) Assuming that both the fractions
a b c _ a h c
^~a+&Tc+*""' ^~b+c+'d+'''
are convergent, show that
x{a + l-\-y) = a-\-y.
See Muir's Theory of Determinants, chap. iii.
§§ 10, 1 1 EXERCISES XXXIII 503
(2.) If piq and p'lq' be the ultimate and penultimate convergents to
a + ; — . . . T , show that
b+ k
1 1 ^ . , 1 r 1 1 1-1
a + , — . . . - — ... to n periods=- P^—, -, ...-;,
* h+ k+ ^ qlf 5 +p=F g +p=F qj
*
where the quotient q' +p is repeated n-1 times, and the upper or the lower
sign is to be taken according as pjq is an even or an odd convergent.
(3.) Evaluate a -J . . . to n quotients, a being any real quantity
positive or negative. Show from your result that the continued fraction in
question always converges to the numerically greatest root of .t^- ax -1 = 0*.
(4.) Deduce from the results of (2) and (3) that a recurring continued
fraction whose numerators and denominators are real quantities in general
converges to a finite limit ; and indicate the nature of the exceptional cases.
(5.) Evaluate 2 - - — - — - — ... to n terms.
14 2 2 2
(6.) Show that the nth convergent to - — - — - — - — - — . . . , every sub-
o— o— o — o — o —
2
sequent component being - , is (2"- 1)/(2" + 1).
X s* x"^* — X
(7.) Show that ; — . . . to n terms = — rn — t.
^ ' x + l-x + 1- a;"+i - 1
(8.) :; z ^ — -. . . (;t + l components)
^ ' 1- a + 1- a + 2- "• ^ '
= l + a + a(a + l)+. • .+a(a + l) . . . (a + n-1).
(9.) If 0 (n) = . . . n quotients, then
<p{m + n) = {<f> (m) + i>(n)- a<l> {m) <p (n)}/{l + 0 (m) 0 (n)}.
(Clausen.)
(10.) Show that
K{0, 02, as, . . ., a„) = K{a3, . . ., aj ;
K(. . . a,b,c,0,e,f,g, . . .)=K{. . . a,b,c + e,f,g, . . .);
K(. . . a, b, c,0, 0,0, e,f,g, . . .) = K(. . . a,b,c + e,f,g, . . .);
K(. . . a,b, c, 0, 0, e, f, . . .) = k{. . . a, b, c, e, f, . . .).
(Muir, Determinants, p. 159.)
(11.) Show that the number of terms in a continuant of the nth order is
, / ,\ (re-2)(n-8) (n-3)(re-4)(n-5)
l + (n-l) + ^^ ^1 i + ^ 3! ^+- • • •
(Sylvester.)
(12.) If2>„=Jir( ^' 3' ■ • ■' "), show that there exists a relation of
the form
^Pn^ + Spn-i" + Cp^-^ + Dp^-s' = 0,
where A, B, C, D are integral functions of a„, 6„, a„_i, 6„_i.
* This is a particular case of the theorem (due to Euler?) that the
numerically greatest root of x^-px + q = 0 is p — - — — • . . •
504.
EXERCISES XXXIII
CH. XXXIV
(13.) Show that
and deduce the theorem of § 19. (Muir, I.e.)
Taking {a, b, c, . . ., k) to denote the continued fraction — ■ t —
. . . -T, and [a, h, c, . . ., /c], or, when no confusion 18 likely, [a, k], to
denote k( ~ ' ~ ' * * *' ) prove the following theorems*: —
\a, 6, c, . . ., «/
(14.) If a; = (a, fe, c, . . ., c, y), then y = (e, . . .,c, h, a, x) ;
xy-{e, . . ., a)x-(a, . . ., e)y + {e a) (a d)=0;
(a, . . ., c) (e, . . ., b) = {c, . . ., a) {a, . . ., d);
{^-(« c)}{y-{e,. . ., a)}
= {e,. . .,a)2(d,...,a)2(c,. . .,a)2. . . (a)^.
(15.) (a,. . .,c)-{a,. . .,d) = ie,. . .,a){d,. . .,af[c,. . .,a)K..(a)\
(16.) [a, b, c, d, e] = ll(a, b, c, d, c) [b, c, d, e) (c, d, e) (d, c) (e).
(17.) Prove the following equivalence theorem : —
(a, . . .,e,f, a',. . ., e', f, a", . . ., e". f", a'", . . ., e'", /'")
-J_ir7 1 , K e'] [a, e][a", e"] [a', e'][a"', e'"] [a", e"] ]
- [a, e] r' ^J "^ [a, e'] - [a', e"] - [a", e'"] - [a"\ e"']f"' - [«'". d"']i *
(18.) (a,f, a',f', a",f", a"',f"', . . .)
_1 j a' aa^ a'a'"
~ a \ afa' -a-a' - a' J 'a" -a'- a" - a"/" a'" - a" - a"' -
Jl_JLJ L J_
^ '■' Tn+ b+ vi+ c+ 171+ ' ' '
If, 1 1
= — -{am + l-- — =;
m { 2 + bm- 2 + cm-
(20.) v/2 = l + 2^2T---=H^-*'itl^l^'"}-
■■}■
■■]■
(21.) (a, . . ., e, f, a, . . ., e, /', a, . . ., e, f", . . . ad oo)
- {e, . . . , a, f, e, . . ., a, f\ e, . . .,a, /", . . . ad oo )
= (a, . . .,e)-{c, . . .,a).
(22.) Show that the successive constituents a, j3, 7, . . .,X, n,v may be
omitted from the continued fraction (. . . o, 6, a, jS, 7, . . . , X, /jl, v, c, d, . . .)
without altering its value, provided [j3, . ., ;u]=±l, a=^[y, . . ., mL
and »/=±[|8, . . ., X]; and construct examples.
* *
(23.) If a; = (a, . . ., e, /, . . .), the other root of the quadi-atic equation
to which this leads is x = (/, c, . . ., a, . . .).
1
(24.) If 6 + ^. ^ '
bm+ a+ ' ' ' a™ +
. . be one root of a quadratic
* The notation and the order of ideas used in (14) to (23), as well as
some of the special results, are due ^ Mobius {Crelle's Jour., 1830).
§ 12 CONVERGENCE OF A C.F. 505
equation, the other is
1 111 111
b +
» »
(Stern, Crelle's Jour., 1827.)*
(25.) I{q>p, show that
^^q -p pq {q -p) pq jq -pf
- a^-ry^- /j2_«a_ • •
{q-p)q = q^-p
q- q'-p^- q -p
*
.. „a i>g(g-y)'
q^-pi-
CONVERGENCE OF INFINITE CONTINUED FRACTIONS.
§ 12.] By the value or limit of an infinite continued fraction
is meant the limit, if any such exist, towards which the con-
vergent Pn/qn approaches when n is made infinitely great. It
may happen that this limit is finite and definite ; the fraction is
then said to be convergent. It may happen that L Pn/qn fluctuates
n=<»
between a certain number of finite values according to the
integral character of n ; the fraction is then said to oscillate.
Finally, it may happen that L pnfqn tends constantly towards
n=oo
± 00 ; in this case the fraction is said to be divergent.
We have already seen that all simple continued fractions are convergent.
The fraction 1 — - — - — - — ... is an obvious example of oscillation, its
value being 1, 0, or - oo according as n=3m + l, 3m + 2, or 3m + 3,
The fraction 1 — , ,_ - — :j— :; — . . . diverges to - oo , for - — - — - —
-h + iJ5- 1+1+1+ ^ 1+1+1 +
. . . converges to -^ + is/5, as may be easily seen from the expression for
its nth convergent given in § 9.
The last example brings into view a fact which it is important
to notice, namely, that the divergence of an infinite continued
fraction is something quite difi"erent from the divergence of an
infinite series. The divergence of the fraction is, in fact, an
accidental phenomenon, and will in general disappear if we
modify the fraction by omitting a constituent. It is therefore
* (23) and (21) are generalisations of an older theorem of Galois', See
Gergonne Ann. d. Math.^ t. xix.
506 PARTIAL CRITERION FOR C.F. OF FIRST CLASS CH. XXXIV
not safe in general to argue that a continued fraction does not
diverge because the continued fraction formed by taking all its
constituents after a certain order converges.
With the exception of simple continued fractions and recur-
ring continued fractions (whether simple or not), the only cases
where rules of any generality have been found for testing con-
vergency are continued fractions of the " first " and " second
class." To these we shall confine ourselves in what follows*.
§ 13.] A continued fraction of the first class cannot he
divergent ; and it will he convergent or oscillating if any one of
tlie residual fractions x<i, Xz, . . .,Xn,. . . converge or oscillate.
The latter part of this proposition is at once obvious from the
equation
62 ^3 hn
Xi = ai-\ ... — .
a2 "T a^ "T *c/ji
Again, since (§ 3, Cor. 6) the odd convergents continually
increase and the even convergents continually decrease, while any
even convergent is greater than any following odd convergent, it
follows that Lpsn/^in = A and Lp2n-i/(l2n-i = -S, where A and B are
two finite quantities, and A <^B. U A=B, the fraction is con-
vergent ; if A>B, it oscillates ; and no other case can arise.
§ 14.] A continued fraction of the first class is convergent if
the series 5a„_ia„/6» be divergent.
We have, since all the quantities involved are positive,
Qn ~ Cf'n^n-l + hngn-2 j
9'n-2 = Cin-2^n-3 + t'n-22'»-4> Qn-2^ (^n-2^n-3 >
Qi = atqs + 642-2 , g'4 > ^4 5-3 ;
^3 = 053^2 + &33'x, q3>aiq2\
* Our knowledge of the convergence of continued fi-actions is chiefly due
to Schlomilch, Handb. d. Algebraischen Analysis (1845) ; Arndt, Disquisitiones
Nommllce de Fractionihus Continuis, Sundi® (1845) ; Seidel, Untersuchungen
uher die Convergenz und Divergenz der Kcttenhriiche (Habilitationsschrift
Miinchen, 1846) ; also Ahluindlungen d. Math. Classe d. K. Baxjerischen Akad,
d, Wiss., Bd. VH. (1855) ; and Stern, Crelle's Joiir.^ xxxvii. (1848).
^12-15 COMPLETE CRITERION FOR C.F. OF FlllST CLASS 507
Hence
Therefore
qnqn-l>qiq2 (h + a^a,-^ {h + «3a4) . . . (&n + aJTi-i^Ti),
and, since qx = l, g'2 = «2>
_3^
'r^>l(-x)(^^T)---('^""e)''>-
Now, since ^an-iajbn is divergent, n (1 + a„_ia„/J„) diverges
to + Qo (chap. XXVI., § 23), therefore Lqnqn-ilhhz . . . 6„= + go.
Hence
^q^n q^in-J q-2!n,q2n-\
\q2n q^n-V
-0,
that is, the continued fraction is convergent.
Cor. 1. The fraction in question is convergent i/Lan-i dn/K > 0.
Cor. 2. Also i/Lan/bn>0, and 2a„ be divergent.
Cor. 3. Also if Zan+ibn/an-ibn+i > 1.
The ahove criterion is simple in practice ; but it is not
complete, inasmuch as it is not proved that oscillation follows
if ^aji-ittn/bn be convergent. The theorem of next paragraph
supplies this defect.
§ 15.] If a continued fraction of the first class be reduced to
the form
^ ^ _1_ _1_ J_ J_ , .
^ d2+ d3 + d4+' ' ' dn+ ' ' '
so that
J _ J _^^ /7 — *^3^2 n _ ^4^3
^^^a^A-A-3. . . ^5j^
UnOn-2 • • •
then it is convergent if at least one of the series
ds^ ds^ dr+ . . . (6)
C?2 + C?4 + <^6 + . . . (7)
be divergent, oscillating if both these scries be convergent.
508 COMPLETE CEITERION FOR C.F. OF FIRST CLASS CH. XXXIV
This proposition depends on the following inequalities be-
tween the q's and d's of the fraction (4) : —
0<g'„<(l + d.2) (l + d,) . . . (1 + dn) (8) ;
q2n>d2 + di+ . . . +d.:ai (9);
g2n-a>l (10).
These follow at once from Euler's law for the formation of
the terms in qn, which, in the present case, runs as follows : —
Write down dzd^ . . . dn and all the terms that can be formed
therefrom by omitting any number of pairs of consecutive (f s.
We thus see that qn contains fewer terms than the product
(1 + ^2) (1 + c^s) • • • (1 + ^») ; and, since the terms are all positive,
(8) follows. Again, in forming the terms of the 1st degree
in ganj we can only have letters that stand in odd places in the
succession d^d^di . . . c?27» ; hence (9) ; and (10) is obvious from a
similar consideration.
To apply this to our present purpose, we observe that, since
the numerators are all equal to 1, we have
P-M P^n-l _
(11).
q^n qin-l q-2nqin-l
If we suppose d^ + 0, neither q^n nor q^n-i can vanish. Hence,
if both Lq^ and Lq^^-i be finite, the fraction will oscillate, and
if one of them be infinite it will converge.
Now, if both the series (6) and (7) converge, the series
d2 + ds + di+ . . . + dn will converge ; and the product on the
right of (8) will be finite when n= co. In this case, therefore,
both q2n and q^n-i will be finite ; and the fraction (4) will
oscillate.
If the series c?2 + «^4 + ^o + • • • diverge, then by (9) Lq^n = <» ,
and the fraction (4) will converge.
By the same reasoning, if the series c?3 + c?5 + c?? + . . . diverge,
then the fraction
^^^J l_ J_ _ '
ds+ di+ ' ' dn +
will converge ; and consequently the fraction (4) will converge.
§§ 15, 16 EXAMPLES 509
Remark. — We might deduce the criterion of last paragraph
from the above. For we have
did2 = aia2lbg,, did3 = a^ajb-i, . . ., dn-\dn = an-\anjbn-
Now, if the series %dn converge, the series formed by adding
together the products of every possible pair of its terms must,
by chap, xxx., § 2, converge : a fortiori, the series 2c?„_iC?„, that
is, ^an-ian/bn, must converge. Hence, if this last series diverge,
2c?n cannot converge. 5c?„ must therefore diverge, since it cannot
oscillate, all its terms being positive. Therefore either (6) or (7)
must diverge, that is to say, the fraction (4) must converge.
Example 1. Consider the fraction
^ + 2+2+2+ • • • •
_2(2«-l)2(2ra-3)'^. . ■ 3'.P
Here d^^i+i - (2n)2 (2/1 - 2)2 . . .4-'.22 *
It may be shown, by the third criterion of chap, xxvi., § 6, Cor. 5, that
the series 2i2„+i is divergent. Or we may use Stirling's Theorem, Thus,
when n is very great, we have very nearly
=2 [{J{2w2n) (2k/c)2»}/{22'» (2irn) (n/e)2»}]2,
= 2/7rn.
The convergence of Sdan+i is therefore comparable with that of 21/n, which
is divergent.
Hence the continued fraction in question converges.
Example 2.
X x^ afi
a+ a+ a +
oscillates or converges according as a;>l or > 1.
Example 3.
2+ 3+4+ • • • *
Here i«n-i ""nl^n =L{?i-l)nl{n + l) = co,
therefore the fraction is convergent.
§ 16.] There is no comprehensive criterion for the con-
vergence of fractions of the second class ; but the following
theorem embraces a large number of important cases : —
If an infinite continued fraction of the second class of the form
F = (V\
Oa - tta - ' ' " «rt - ' ' '
510 CRITERION FOR C.F. OF SECOND CLASS Cfl. XXXIV
be stick that
an^bn + 1 (2)
for all values of n, it converges to a finite limit F not greater than
unity.
If the sign > occur at least once among the conditions (2), then
F<1.
If the sign = alone occur, then F=l — l/S, where
S=l + b2 + h^hi + hj)ibi + . . . + ^2^3 • • . ^» + . • . ad cx) (A),
so that F =■ or <1 according as the series in (A) is divergent or
convergent.
These results follow from the following characteristic pro-
perties of the restricted fraction (1) : —
Pn -Pn-l ^hh. . .bn (3)
Pn^b2 + bibs + b^b-ibi + . . . + bibs . . . 6„ (4)
qn - qn-1 ~ hbj . . .bn (5)
qn = l + b2 + bibs + . . . + bibs . . .b^ (6)
qn -Pn = qn-X - Pn-1 ^. • • ^q^-Pi^l (7).
To prove (3) we observe that
Pn -Pn-l = («n " l)i?n-l " bnPn~2-
Hence, since pn, qn are positive and increase with n (§ 2,
Cor. 1),
Pn - Pn-1 = bn {Pn-1 - Pn-2),
Pn-l -Pn-2 = bn-i {pn-2 -Pn-3),
acc. BUS an^bn + 1;
Ps-Pi^ bsbi. acc. as ag > ig + 1.
Therefore j3„—j3„-i ^bibs . . . bn, where the upper sign must
be taken if it occur anywhere among the conditions to the right
of the vertical line.
To prove (4), we have merely to put in (3) n — 1, w-2,
. . ., 3 in place of n, adjoin the equation Pi = bi, and add all
the resulting equations.
(5) and (6) are established in precisely the same way.
It follows, of course, that pn and qn both remain finite or
both become infinite when w = co , according as the series in (6)
is convergent or divergent.
§ 16 CRITERION FOR C.F. OF SECOND CLASS 511
To prove (7), we have
= fe-1 -^n-l) + K {(<7n-l -i?n-l) " {qn-^- Pn-^\
according as a» = Z>„+ 1, provided qn-\—Pn-\ is positive.
This shows that, if any one of the relations in (7) hold, the
next in order follows. Now q<^—p^ = a<i-h2.'^\, according as
tta = ^2 + 1 ; and qz- pz = a^a^ — h- has ^ (a^ - h) {bs + 1) - &3
^ (oa - ^2) + ^3 («2 - ^2 - 1), according as aa^bs+l; hence the
theorem. It is important to observe that the first > that occurs
among the relations a2 = b2+l, ^3 = 63+1, . . . determines the
first > that occurs among the relations (7) : all the signs to the
right of this one will be = , all those to the left > .
The convergency theorems for the restricted fraction of the
second class follow at once. In the first place, as we have
already seen in § 3, the convergents to (1) form an increasing
series of positive quantities, so that there can be no oscillation.
Also, since g„-j9„ = 1, it follows that
Pn/qn ^ 1 - 1/qn (8). '
Therefore, since qn>l, it follows that i^ converges to a finite
limit ^1.
If the sign > occur at least once among the relations (2),
the sign < must be taken in (8); that is, i^<l.
If the sign = occur throughout, we have
Lpn/qn = 1 - i^l/gn = 1 - l/S,
where >S^ is the sum to infinity of the series (6). Hence, if (6)
converge, F< 1; if it diverge, F= 1.
If we dismiss from our minds the question of convergency,
and therefore remove the restriction that b^, 63, . . ., 6„ be
positive, but still put a„ = 6„ + 1, a„_i = bn-i + 1 , . . ., aa-bs + l,
052 = ^2 + 1> we get by the above reasoning
Pn/qn^l-1/qn (8');
qn=l+b.2 + b.A+ ' . . +bA. . 'h (6').
512 INCOMMENSURABLE C.FF. CH. XXXIV
Now (8') gives us qn=l/(l-pn/Qn)- Hence tlie following
remarkable transformation theorem : —
Cor. Ifh<i,. . ., bnbe any quantities whatsoever, then
1 + ^2 + l^it'i + . . . + ^2^3 . . .bn
l-b^+l-h+l-' ' ■&„+! ^^^'
from which, putting ^1 = 62, u^^b^b-i, . . ., m« = ^2^3. • • 6«+i,
we readily derive
1 + Wi + Mo + • • ' +Un
1 Ui Wg 111 W3 U2Ui
1 — 1 + Ml - Ml + Ma — «<2 + M3 — «*3 + W4 - *
«„-2 + M„_i - M„-i + Z*„
an important theorem of Euler's to which we shall return
presently.
INCOMMENSURABILITY OF CERTAIN CONTINUED FRACTIONS.
§ 17.] If CTa, «3, . . ., an, b^, h, . . ., hn he all positive
integers, then
I. The infinite continued fraction
bz h bn ' /jv
a2+ a3+ ' ' ' an+ ' ' ' '^
converges to an incommensurable limit provided that after some
finite value of n the condition ofnH^^re be always satisfied.
II. The infinite continued fraction
h ba bn /g)
ftj ~~ ttj — an —
converges to an incommensurable limit provided that after some
finite value of n the condition a„ = 6„ + 1 6^ always satisfied, wJiere
the sign > need not always occur but must occur infinitely often*.
To prove II., let us first suppose that the condition
«„^6„ + l holds from the first. Then (2) converges, by § 16,
• These theorems are due to Legendre, JEltments de G6om4trie, note nr.
§§ 16, 17 INCOMMENSURABLE C.FF. 513
to a positive value < 1. Let us assume that it converges to a
commensurable limit, say K/K, where Aj, Xj are positive integers,
and Xi>X2.
Let now
ba hi
Pz = . • • .
ih - ^4 -
Since the sign > must occur among the conditions ^3 ^ ^3 + 1,
0^4 = ^4+1, • • ., Ps must be a positive quantity <1. Now, by
our hypothesis,
KIK = hl{a2 - Pz),
therefore Pa = {a^ K - K K)lK ,
= A3/A2, say,
where X3 = a.2X2- ^2^1 is an integer, which must be positive and
<X2, since Pi is positive and < 1.
Next, put
hi h
Pi = • ... .
tti-Us-
Then, exactly as before, we can show that P4 = X4/X3, where X4 is a
positive integer <X3.
Since the sign > occurs infinitely often among the conditions
an = bn+ 1, this process can be repeated as often 'as we please.
The hypothesis that the fraction (2) is commensurable therefore
requires the existence of an infinite number of positive integers
Xi, X2, X3, X4, . . . such that Xi>X2>X3>X4> . . . ; but this is
impossible, since X^ is finite. Hence (2) is incommensurable.
Next suppose the condition a„ ^ 6„ + 1 to hold after n = m.
Then, by what has been shown,
y = ...
is incommensurable.
Now we have
K bi bm
F=
a^-az- am-y
„ 4-1,, TP \(^m~y)Pm-l t^mPm-2
consequently F= /^z^— TTV^ '
_Pm-yPvi-\
qm-yqm-\
II. 33
(3),
514 EULER's transformation CH. XXXIV
where pjqm, Pm-i/Qm-i are the ultimate and penultimate con-
vergents of
It results from (3) that
1/ (Fqm-i -Pm-l) = Fq„^ -pra (4).
JNow Fqm-\-pm-i and Fq^-pm cannot both be zero, for
that would involve the equality Pm/qm=Pm-ilqm-i, which is
inconsistent with the equation (2) of § 3. Hence, if F were
commensurable, (4) would give a commensurable value for the
incommensurable ^. F must therefore be incommensurable.
The proof of I. is exactly similar, for the condition a„<t&»
secures that each of the residual fractions of (1) shall be positive
and less than unity.
These two theorems do not by any means include all cases of
incommensurability in convergent infinite continued fractions.
1^ 32 ^2
Brouncker's fraction, for example, 1 + - — - — - — . . . ,
converges to the incommensurable value if-n-, and yet violates the
condition of Proposition I.
CONVERSION OF SERIES AND CONTINUED PRODUCTS INTO
CONTINUED FRACTIONS.
§ 18.] To convert the series
U1 + U2+ ' ' ' +Un+ . . .
into an "equivalent" continued fraction 0/ the form
«!- Oa- ' * ' «»-'
A continued fraction is said to be "equivalent" to a series
when the nth. convergent of the former is equal to the sum of n
terms of the latter for all values of n.
Since the convergents merely are given, we may leave the
denominators qi, 5-2, . . . , qn arbitrary (we take g'o = 1, as
usual).
§§ 17, 18 euler's transformation 515
For the fraction (1) we have
Pn/qn-pn-i/qn-i = bib2. . . Vg'»-i2'» (2);
g'i = «i, q2 = a'iqi-K, . . ., qn = anqn-i-hnqn-2 (3);
Pi/qi = hlqi (4).
Since
Pjqn ^Ui + ll.2+ . . . + tin (5),
we get from (2) and (5)
Un = bib.2. . . bn/qn-iqn,
Un-1 — Oibz • . . bfi-i/qn-2qn-ly
(6).
ti2 = bib2/qiq2,
Ui = bifqi.
From (6), by using successive pairs of the equations, we get
bi = qi1h, ^2 = ^2^2^, b3 = q3U3/qiU2, . . ., bn-=qnUnlqn-2'Un-l
(7).
Combining (3) with (7), we also find
(ii = qi, a^ = q2('Ui + U2)/qiUi, a3 = g'3(«2 + M3)/?2«2, • • •,
Cln = qn{Un-l + ttn}/qn-lttn-l (8).
Hence
S„ = Ui + U2+ • ■ ■ +tln,
^qiUi q2ih/ui qaUs/qiU^
qi- q2{Ui + U2)/qiUi- q3{u2 + Uz)lq^u^-' '
qnlln/ qn-2Un-l /n\
qn{Un-l + Un)/qn-lUn-l
It will be observed that the q's may be cleared out of the
fraction. Thus, for example, we get rid of qi by multiplying
the first and second numerators and the first denominator by
1/qi, and the second and third numerators and the second
denominator by qi ; and so on. We thus get foy /8« the
equivalent fraction
^ ^ «*i_ IhfUi U3/U2 uJUn-i ,jQX
which may be thrown into the form
S =— ^2 U1U3 Un-2Un /j.x
"1- Mi + «a- t^ + Ws-* ■ ' Un-i+Un
33—2
516 EXAMPLES — BROUNCKER's FRACTION CH. XXXIV
This formula is practically the same as the one obtained
incidentally in § 16 ; it was first given, along with many applica-
tions, by Euler in his memoir, "De Transformatione Serierum
in Fractiones Continuas," Opuscula Analytica, t. ii. (1785).
It is important to remark that, since the continued fraction
(10) or (11) is equivalent to the series, it must converge if the
series converges, and that to the same limit.
By giving to Wi, Wg, . . ., w„ various values, and modifying
the fraction by introducing multipliers as above, we can deduce
a variety of results, among which the following are specially
useful : —
(12);
ViX + V^X- + . . . + VnX
ViX V^X V1V3X
V«-2«H^
~ 1— «! + V2X — V2 + V3X — '
■ Vn-1 + VnX
X x^ «"
-+ - + . . . + —
'Ox «2 'On
X ViX V2X
lOn-xX
Vx- VxX + V^- V2X + V3-
' Vn-iX + Vn
bx 0x02 0x02 . , . On
ttxX bxttzX bzCtsX
bn-iUnX
h- b2 + a2X- ba + a^x—' ' ' bn + anX
Example 1. If -iir<x<^ir, then
X 12x2 32a;2 52^2
(13);
(14).
~H-3-a;2+ 5-3a;2+ 7-5x2+' • •'
and, in particular,
7r_J_ ii 3^ 52^
4~1+ 2+ 2+ 2+ ■ * ■'
•which is Brouncker's formula for the quadrature of the circle.
Example 2. If x < 1,
(1.^\m-i,^ 1 {m-l)x 2(m-2)x 3(m-3)x
^ ■*" ' ~ "^1- 2+(m-l)x- 3 + (m-2)x- 4 + (m-3)x-
§§ 18-20 REDUCTION OF INFINITE PRODUCT TO C.F. 517
Also, if m> -1,
2m = 1 . JL IKli) 2(»t-2) 3(m-3)
1- m + 1- m + 1- m + 1- ' ' ''
and, if m > 0,
m l(m-l) 2(m-2) 3(wt-3)
0 = 1-
1+ 3-m+ 5-m+ 7-m +
§ 19.] 77^6 analysis of last paragraph enables us to construct
a continued fraction, say of the form (1), whose first n conver gents
shall he any given quantities fi,f 2, . . .,/„ respectively.
All we have to do is to replace Mi, lu, . . ., u^ in (10) or (11)
by /i , /2 -/i , . . . , fn. -fn-x respectively.
The required fraction is, therefore,
/i f.-fxf{fz-A) (A-A)(A-f) ^
1 — /2 — fi ~fl ~ Ji ~/2 ~
C/n-2 ~/n-s) \Jn, ~Jn-l)
Jn Jn—2
Cor. Hence we can express any continued product, say
a\(xi . . . w^
61^2 . . . «}i
as a continued fraction.
We have merely to i^\xt fi = d^/ci, f2 = didz/eie^, . . ., effect
some obvious reductions, and we find
p _ di ei{d2-e^d^i{d3-e^diei{d2-e^{di-e^diei{d^-e^{d!i-e6)
"~ Ci- d^- d^d^-e^i- d^di-e^i- dSh-e^e^-
• • • d d P P ^^^''
§ 20.] Instead of requiring that the continued fraction be
equivalent to the series, or to the function f(n, x), which it is to
represent, we may require that the sum to infinity of the series
(or/(co , x))he reduced to a fraction of a given form, say
1- i_ i_---i_--- u;>
where /?o, A, . . . , )8« are all independent of x.
There is a process, originally given in Lambert's Beytrdge
* A similar formula, given by Stern, Crelle's Jour., x., p. 267 (1833), may
be obtained by a slight modification of the above process.
518 Lambert's transformation ch. xxxiv
(th. II., p. 75), for reducing to the form (1) the quotient of two
convergent series, say F{1, a;)/F(0, a;).
We suppose that the absobite terms of i^(l, a;) and F{0, x)
do not vanish, and, for simplicity, we take each of these terms to
be 1. Then we can establish an equation of the form
F{1, x) - F{0, x) = /3^xF{2, x) (2j),
where F{2, x) is a convergent series whose absolute term we
suppose again not to vanish, and /3i is the coefficient of x in
jP(1, x)-F{0, x), which also is supposed not to vanish*.
In like manner we establish the series of equations
Fi'i, x) - F{\, x) - li,xF{^, x) (2a),
F{3,x)-Fi2,x)^(3sxF(4,,x) (2^),
F(n + l,x)- F{n, x) = (3n+ixF{n + 2, x) (2„+i).
Let us, in the meantime, suppose that none of the functions
F becomes 0 for the value of x in question. We may then put
G (n, x) = F{n + 1, x)/F(n, x) (3),
where G (n, x) is a definite function of n and x which becomes
neither 0 nor oo for the value of x in question.
The equation (2n+i) may now be written
G{n, x)-l= Pn-hixG (n+l, x)G (n, x),
that is, G {n, x) = 1/{1 - Pn^-,xG {n + 1, x)} (4).
If in (4) we put successively n = 6, n=l, . . ., we derive
the following : —
^(0, ^) = 1- r: • • • i-(i-ilG{n,x)) (^>5
^~'G{n,x)~ 1- ' ' ' l-{l-llG{n + m,x)) ^^''
* The vanishing of one or more of these coeflScients would lead to a more
general form than (1), namely,
1- i- • ■ • *
General expressions have been found for /3„ , j8, , . . • by Heilermann, Crellc's
Jour. (1846), and by Muir, Proc. L.M.S. (1876).
§ 20 LAMBERT'S TRANSFORMATION 519
In order that we may be able to assert tlie equality
G^(0,^) = j^f^. . .f^. . .ado) (7),
it is necessary, and it is sufficient, that it be possible by making
m sufficiently great to cause 1 - IjG (n, x) to differ from the mih.
convergent of the residual fraction
1- 1- • • • 1- • • •
by as little as we please.
Let us denote the convergents of (8) by Pi/qi, ihl^i,
Pml^m^ Then, from (6), we see that
{l-llG{n, X)]-Pralqm
^ Pm-Pm-l{l-i/G(n + m, X)} _ Prn
qm-qm-i{l-'i-JG{n + m,x)} q^'
^ {1 - 1/G (n + m, X)} (Pm/qm-Pm-l/qm-l)
qm/qm-1 - {1 - l/G {n + m, x)}
(8)
(9),
_{l-l/G(n + m, X)} fin+t Pn+2 ■ . • ^n+m^"* /j^x
qm[qm-qm-i{^-'^IG(n + m,x)]]
The necessary and sufficient condition for the subsistence of (7)
is, therefore, that the right-hand side of (9), or of (10), shall
vanish when m = co .
Concerning these conditions it should be remarked that while
either of them secures the convergence of the infinite continued
fraction in (7), the convergence of the fraction is not necessarily
by itself a sufficient condition for the subsistence of the equation
(7).
In what precedes we have supposed that none of the functions
F{7i, x) vanish. This restriction may be partly removed. It is
obvious that no two consecutive F's can vanish, for then (by
the equations (2)) all the preceding F'b would vanish, and
G(0, x) would not be determinate. Suppose, however, that
F{r + 1, x) = 0, so that G (r, x') = 0; then (5) furnishes for
G (0, x) the closed continued fraction
520 EXAMPLE CH. XXXIV
In order that this may be identical with the \alue given by
(7), it is necessary and sufficient that G(r + 1, w), as given by
(6), should become co , that is, it is necessary and sufficient that
the residual fraction
1-
. . ad
should converge to 1 ; but this condition will in general be
satisfied if the relation (4) subsist for all values of n, and the
condition (9) be also satisfied when w<i:r + 2,
§ 21.] As an example of the process of last paragrapli, let
Fin, x) = l + -pT V + wri—, — w~^ rT\ + . . . (11).
^ l!(y + w) 2! (y + «) (y + w+ 1)
Then
F{n ^ 1, .) - F{n, .) = - ^^,^^^%^,,^ F{n . 2, .) (2') ;
and
G{n,x)^\l[l+-. ry^ -G{n + l,x)\ (4'),
where G (n, x) = F{n + 1, x)/F(n, x).
Hence
r^a ^\- 1 ^/y(y+l)^/(y+l)(y + 2) xl{y^n-l){y + n)
f^l"»'^;-l+ 1+ 1+ ~ ' ' • \-{l-llG{n,x)}
(5');
and
1 _ a?/(y + w) (y + ?2- + 1)
G{n,x)~ 1 +
x/{y + n + m-l)(y + n + m) . ,.
l-{l-l/G{n + m,x)} ^ ^'
The series (11) will be convergent for all finite values of x,
and for all positive integral values of n, including 0, provided y
be not 0 or a negative integer. Hence we have obviously, for
all finite values of x, LG {n + m, x) = l when m=cc.
Let us suppose that x is positive. Then the residual con-
tinued fraction
§§ 20, 21 EXAMPLE 521
xl{y + n) {y + n + 1) a;/(y + w + 1) (y + y?, + 2)
1+ 1+ • • •
xl{y + n + 'm-l){y + n + m)
1 +
(8')
is (by the criterion of § 14) evidently convergent. Hence the
factor Pmlqm-Pm-\lqm-\ in the expression (9) vanishes when
m= 00.
Also, since the q'?, continually increase, Lqmlqm-\^ 1.
Therefore we may continue the fraction to infinity when x is
positive.
Next suppose x negative, =-y say ; we then have
(^(0^ ^)^ 1 y/y(y + i) y/(y+i)(y + 2)^ •
y/{y + n~l)(y + n) - „. ,
l-{l-llG{n,-y)} ^^^'
and
_ 1 ^i/l(y + n)(y + n + l)
G{n,-y) 1-
yl{y + 7i + m-l)(y + n + m) , „.
~ l-{l-l/G{n + m,y)} ^^ ^
The fraction (8) in this case is "equivalent" to
1 f ?/ y y \ (8")
y + wly + w + l — y + w + 2-*''7 + w + m— '"'J ''
which is obviously convergent (by § 16), if y have any finite
value whatever. Hence the i&ctoT pm/qm-Pm-i/qm-i belonging
to the equivalent fraction (8) must vanish.
Again, by § 2 (6),
qm
5'm-i
_ yl(y + n + m- l)(y + n + m) yj{y + n -^^ m-2) {y + n + m—\)
= 1 Y^z iz
y/{y + n)(y + n + l)
1 - -^— I ^^ -^ o • . --^1 (12).
y + w + w(.y + w + m-l-y + w + ?»-2- y + 7U^
1
r
522 C.FF. FOR TAN SC AND TANH 00 CH. XXXIV
If only n be taken large enough, the fraction inside the
brackets satisfies the condition of § 16 throughout : its value is
therefore < 1, however great m may be ; and it follows from (12)
that Lqmlqm-i = 1 when m= co.
Since LG (n + m, —y) = l when m=co, it follows that all the
requisite conditions are fulfilled in the present case also.
We have thus shown that
F{l,a;)^ l_ xly{y+l) ^/(y+l)(y + 2)
F{Q,w) 1+ 1+ 1+ ■ ■ •
xl{y + n-\){y + n)
r+ ""' '
whence, by an obvious reduction,
F{\, OR) y X X a
ad oo (13),
(14),
F(0, x) 7 + 7 + 1 + 7 + 2 + ' *'y + 71+'*
a result which holds for all finite real values of x, except such
as render jP(0, x) zero*, and for all values of y, except zero
and negative integers.
If we put +a^l4t: in place of x in the functions F{0, x) and
F{\, x), and at the same time put 7 = ^, we get
F{Q, - a? I A) = cos X, F(l,- x'jA) = sin x/x ;
F{0, 0^/4:) = cosh X, F{1, ^^4) = sinh x/x.
Cor. 1. Hence, from (14), we get at once
. tAj tAj Uj w *■- _■!
1-3-5- 2n+l- ^ ' '
tanh X = - — r — ^— . . . . . . (16).
1+3 + 5+ 2n + l+ ^ ^
Cor. 2. T/ie numerical constants -n- and ir^ are incommensurable.
For, if TT were commensurable, 7r/4 would be commensurable,
say = Xf/JL. Hence we should have, by (15),
* In a sense it will hold even then, for the fraction
1 J X X \
y ('^■'■7 + 1+ 7 + 2+ * • • )
which represents ^(0, x)IF(l, x) will converge to 0. Of course, two consecu-
tive functions F(n, x), F(n + 1, x) cannot vanish for the same value of x^
otherwise we should have i*' (oo , a;) = 0, which is impossible, since F{cd ,x) — l.
21, 22 INCOMMENSURABILITY OF TT AND e 523
1- 3- 5- •
X7/.2
* 2?J + 1-' '
/tA — 3/A — 5/A — '
■ (2/^+1) /A-
(17).
Now, since X and fj. are fixed finite integers, if we take n large
enough we shall have {2ti + l)/i>X^ + 1. Hence, by § 17, the
fraction in (17) converges to an incommensurable limit, which
is impossible since 1 is commensurable.
That TT^ is also incommensurable follows in like manner very
readily from (15).
By using (16) in a similar way we can easily show that
Cor. 3. Any commensurable power of e is incommensurable*.
§ 22.] The development of last paragraph is in reality a
particular case of the following general theorem regarding the
hypergeometric series, given by Gauss in his classical memoir
on that subject (1812) t:—
If
J^{a,IJ,y,a;)-l + ^^^W+ i2.y(y+l) ^'-•••'
and
G (a, A y, x) = F(a, /? + 1, y + 1 , a:)/F{a, /8, y, x),
then
r< /„ o ., ™\ _ _i_ /^i^ P^ Pin^
^\<^,P,y,X)-^_ i_ i_. . •l/G(a + w,/3 + 7i, y + 2«)
(18),
where
a. «(y-/^) (^+l)(y-H-a)
^'~y{y^iy ^^" (y + i)(r + 2) '
^ (a+l)(y+l-^) _(;8 + 2)(y+2-a)
P' (y+2)(y+3) ' ''' (y + 3)(y + 4) '
^ ^(a + W-l)(y + y^-l-^) _ (^ + w)(y + W-a)
^^'^-^ {y + 2n-2){y + 2n-\) ' ^'" "(y + 2;^ - 1) (y + 2«) '
* The results of this paragraph were first given by Lambert in a memoir
which is very important in the history of continued fractions (Hist. d. I'Ac.
Roy. d. Berlin, 1761). The arrangement of the analysis is taken from Legendre
(I.e.), the general idea of the discussion of the convergence of the fraction
from Schlomilch. t Weike, Bd. in., p. 134.
524 gauss's C.F. FOR HYPERGEOMETRIC SERIES CH. XXXTV
After what has been done, the proof of this theorem should
present no difficulty.
The discussion of the question of convergence is also com-
paratively simple when x is positive ; but presents some difficulty
in the case where x is negative. In fact, we are not aware that
any complete elementary discussion of this latter point has been
given. * -
Cor. If in (18) we put (3 = 0, and write y- 1 in place of y,
we get the transformation
a a(a + l) , a(a+l)(a + 2) ,
y y(r + i) y(y + i)(y+2)
_ 1 )8i^^ , .
~ 1 - 1 - 1 - • • • ^^^^'
where
^-y' . ^^"y(y+i)'
n_. (^+l)y 2(y + l-a)
'^^-(y+l)(y + 2)' ^' (y + 2)(y + 3)'
^ (a + n-l){y + n-2) ^ n(y + n-l-a)
/^2»-i -(y + 2n-3){y + 2n - 2) ' ^'" ~ (y + 2w - 2) (y + 2n - 1) '
Gauss's Theorem is a very general one ; for the hypergeometric
series includes nearly all the ordinary elementary series.
Thus, for example, we have, as the reader may easily verify,
{l+xr = F{-m,(3,p,-x);
\og{l+x)--xF{l,l,2,-x);
sinh x = a; L L F{h, h\ f , x'l^kk') ;
fc=oo fe'=oo
8mx = x L L F(k, k', |, - arjikk') ;
fc=ao Jt'=oo
sm-^x = xF(^, i,|,^^);
= xJ{l-x')F{l,l,^,x')',
Un-'x=^xF(l,l,^,-x'').
§ 22 EXERCISES XXXIV 525
Exercises XXXIV.
Examine the convergence of tbe following : —
. J_ 1 1^
^ ' "*"l2+23+ 32+ • • • •
,„ , , 12 12 . 22 22 . 3-
/rx I 12 3
<^-) i+irnriT--- •
1" 2" 3*
(7.) X+ ; ... .
^ ' x+ x+ x +
(9.)
2 13.3 2».4 33.5
1+ 1+ 1+ 1+
(2.)
12 2= 32
+ 3+5+ 7+ * • •
(4.)
, 1 1.2 2.3
' + 1+1+ 1+'-
(6.)
m2 {Hi + n)2 {m + 2T>]
n+ n+ n+
(8.)
, 1.3 3.5 5.7
'+1+ 1+ !+••
2 22 23 2*
(10.)
1+ rTi+ 1+ * * '
(11.) Show that the fraction of the second class, a, ~ . ■ -, con-
verges to a positive limit if, for all values of n,
a^bib^ + aslb^bsi-. . . +a„+i/6„6„4.i>l.
(Stern, Gdtt. Nach., 1845.)
(12.) Show that — ? ^- . . . , where o„ > 0, converges if a„+i :t> a„ + 1.
Oj — a2 — O3 —
(13.) Show that the series of fractions (Fn -i'n-i)/(5'n - 3n-i) forms a
descending series of convergents to the infinite continued fraction of the
second class, provided a^^h^ + 1, and the sign > occurs at least once among
these conditions.
(14.) Show that
XXX . •
x + 1- x + 1- a; + l- ' * *'
where x > 0, is equal to x or 1 according as a; < or <i: 1.
(16.) Evaluate 2^ 3Z 43 • • ' '
, TO ra + 1 TO + 2
and
TO+1- TO +'2- TO + 3- * * ''
where m is any integer.
Show that
nfi\ 1 ,« , «(« + !) , _, a (a + l)6 (a + 2)(6 + l)
^^°-' '^b'^ b{b + iy' ' '~ '^b- a+b + 2- a + & + 4-
(17.) sma;-^^ 2.3-x2+ 4.5-x2+ 6.7-x2+ • ' ' '
/1Q^ 1 /I L ^ ^ 1'^ 22x 32x
(18.) log(l + x)= - ^-^ 3-^-^ ^-^-^ ....
* Exercises (5) to (10) are taken from Stern's memoir, Crelle's Jour., xsxvii.
626, EXERCISES XXXIV CH. XXXIV
„„, , 1« 22 32
(19-) 1 = 3353 7-- •• •
(20.) log ^ -j^ 2X-1+ 3X-2+ 4X-3+ ' ' ' '
(2n-l)2(y2-i)3
4 (n2/2 + y + 7i) - ' * *
/99\ aj-J:. _^ ^.^ 2z 3x :
(22.) € _^_ j^^^_ 2^^_ 3 + a._ 4^^_ ■ • • •
Evaluate the following : —
,„, , , 1 1 2 3 4 ,_, , 1 2 3 4
(23.) 1 + 1-3-43 5363- •• ' (24.) ____... .
,„, , 1 12 22 32 ,--,12 3 4
(25-) iTiTiTiT • • • • (''•) 2T3T4T5T • • • -
(27.) Show that tanx and tanhx are incommensurable if x be commen-
surable.
Establish the following transformations : —
(28.) e _j— j-^ 23 3:; 23 5+ 23 7> ' ' ■ '
, , , „ , a; 12x Px 22x 22x 32a; 32x
(29.) log(l + x) = — ______... .
,„« V . , X 12x2 22x2 32x2
(30.) tan-x = ^-3^ 5T 7 + ' ' ' *
, , X 12x2 22x2 32x2
tanh-ix = ^ ____,^. . . .
n tan x (n2 - 1^) tan2 x (n2 - 22) tan2 x
(31.) tan nx= ^_ ^ g^; ^ ^ ....
(Euler, Mem. Acad. Pet., 1813.)
sin (n + 1) X „ 1 1
(32.) -. i- = 2cosx-j7 ji . . .,
^ ' smnx 2cosx- 2cosx-
where there are n partial quotients.
(33.) If
0 (o, /3, 7, x)
(3 - 1) (2^ - 1) (? - 1) (2" - 1) (9^ - 1) (2''''' - 1)
then
^(tt, /3 + 1. 7 + 1. x) _ 1 j3iX j3aX
0(0, /3, 7, X) ~1- 1- 1- * • ■'
§ 22 EXERCISES XXXIV 527
where
_(/+^-l)(gV+>-'-l) +,-,
^(g°+*--l)(,>+--^-l) B-H-
(Heine, Crelle^s Jour., xxxii.)
(34.) Show that
" ~ r "'"2{a-l)+ 2(a-l)+ 2(a-l)+ * * "j
^ t""*" 2(a + l)+ 2(a + l)+ 2(a + l)+ ' 'j"
Wallis (see Muir, PhiL Mag., 1877).
CHAPTER XXXV.
General Properties of Integral Numbers.
NUMBERS WHICH ARE CONGRUENT WITH RESPECT TO
A GIVEN MODULUS.
§ 1.] Ifmhe any positive integer whatever, which we call the
modulus, two integers, M and N, which leave the same remainder
when divided by m are' said to he congruent with respect to the
modulus m,*.
In other words, if M=pm + r, and N- qm + r, M and N are
said to be congruent with respect to the modulus m. Gauss,
who made the notion of congruence the fundamental idea in his
famous Disquisitiones Arithmetics, uses for this relation between
M and N the symbolism
M=N {modim);
or simply M=N,
if there is no doubt about the modulus, and no danger of con-
fusion with the use of s to denote algebraical identity.
Cor. 1. If two numbers M and N be congruent with respect
to modulus m, then they differ by a multiple of m; so that we
have, say, M=N+pm.
Cor. 2. If either M or N have any factor in common with m,
then the other must also have that factor; and if either he prime
to m, the other must be prime to m also.
In the present chapter we shall use only the most elementary
consequences of the theory of congruent numbers.
* To save repetition, let it be understood, when nothing else is indicated,
that throughout this chapter every letter stands for a positive or negative
integer.
§§ 1-3 PERIODICITY OF INTEGERS 529
Our object here is simply to give the reader a conspectus
of the more elementary methods of demonstration which are
employed in establishing properties of integral numbers ; and to
illustrate these methods by proving some of the elementary
theorems which he is likely to meet with in an ordinary course
of mathematical study. Further developments must be sought
for in special treatises on the theory of numbers.
§ 2.] If we select any "modulus" w, then it follows, from
chap, ni., § 11, that all integral numbers can he arranged into
successive groups of m, such that each of the integers in one of these
groups is congruent with one and with one only of the set
0, 1, 2, . . ., {m-2\ {m-l) (A),
or, if we choose, of the set
0, 1, 2, . . ., -2, -1 (B),
where there are m integers.
Another way of expressing the above is to say that, if we
take any m consecutive integers whatever, and divide them hy m,
their remainders taken in order will be a cyclical permutation of
the integers (A).
Example. It we take m=5, the set (A) is 0, 1, 2, 3, 4. Now if we take
the 5 consecutive integers 63, 64, 65, 66, 67 and divide them by 5, the
remainders are 3, 4, 0, 1, 2, which is a cyclical permutation of 0, 1, 2, 3, 4.
§ 3.] A large number of curious properties of integral
numbers can be directly deduced from the simple principle of
classification just explained.
Example 1. Every integer which is a perfect cube is of the form Ip, or
Tpil. Bearing in mind that every integer N has one or other of the forms
7m, 7m ±1, 7wi±2, 7m ±3,
also that {Im ± rf = (Imf =t 3 (7m)2 r + 3 (Im) r^ ± r^,
= (72/713 dk 21wi2r + 3»!r2) 7 ± r^,
we see that in the four possible cases we have
2^=(7m)3=(7%3)7;
2V3=(7/H±l)3=i»/7±1;
^3= (7m ±2)3,
= ilf7±8 = (J/±l)7±l;
JV8=(7m±3)3=(M±4)7=rl.
c. n. 34
530 EXAMPLES CH. XXXV
In every case, therefore, the cube has one or other of the forms 7p or
7iJ±l.
Example 2. Prove that 32»+i + 2'»+2 is divisible by 7 (Wolstenholme).
We have 32"+i + 2»+2= (7 - 4)2"+i + 2"+2.
Now (see above. Example 1, or below, § 4)
(7 _ 4)2»i+i = Ml - 42JI+1.
Hence 32"+i + 2»+2=ilf7-42»+i + 2»+2,
=Jf7-2«+2(23»-l).
But 2^™- 1 is divisible by 2^ - 1 (see chap, v., § 17), that is, by 7. Hence
2n+2 (231-1) =2^7.
Finally, therefore, 32»+i + 2"+2 =(M-N)7,
which proves the theorem.
Example 3. The product of 3 successive integers is always divisible by
1,2.3.
Let the product in question be m (m + 1) (m + 2) . Then , since m must have
one or other of the three forms, 3m, 37)i + l, 3m -1, we have the following
cases to consider: —
3m(3m + l)(3OT + 2) (1);
(3TO + l)(3m + 2)(3m + 3) (2);
(3m-l)3m(3m + l) (3).
In (1) the proposition is at once evident ; for 3m is divisible by 3, and
(3m + 1) (3m + 2) by 2. The same is true in (2) .
In case (3) we have to show that (3m - 1) m (37« + 1) is divisible by 2.
Now this must be so ; because, if m is even, m is divisible by 2 ; and if m be
odd, both 3m -1 and 3m + 1 are even; that is, both 3m -1 and 3m + 1 are
divisible by 2.
In all cases, therefore, the theorem holds.
Example 4. To show that the product of p successive integers is always
divisible by 1 . 2 . 3 . . .p.
Let us suppose that it has been shown, 1st, that the product of any p-1
successive integers whatever is divisible by 1.2. 3. . .p-1; 2nd, that the
product of jp successive integers beginning with any integer up to x is divisible
by 1.2.3 . . .p-1. p.
Consider the product of p successive integers beginning with a; + l. We
have
(x + l)(x + 2) ...{x+p-l){x+p)
=p(x + l){x + 2) . . . {x+p-l) + x{x + l){x+2) . . . (x+p-1) . . . (1).
Now, by our first supposition, (x + 1) (a; + 2) . . . {x+p-1) is divisible by
1.2. . . p-1 ; and, by our second, x{x + l){x + 2) . . . (x +p - 1) is divisible
by 1 . 2 . 3 . . .p.
Hence each member on the right of (1) is divisible by 1 . 2 . 3 . . .p.
It follows, therefore, that, if our two suppositions be right, then the pro-
duct oip successive integers beginning with x + 1 is divisible by 1 . 2 . 3 . . .p.
But we have shown in Example 3 that the product of 3 consecutive integers
is always divisible by 1 . 2 .3; and it is self-evident that the product of 4 con-
§3
PYTHAGOREAN PROBLEM 631
secntive integers beginning with 1 is divisible by 1 . 2 . 3 . 4. It follows, there-
fore, that the product of 4 consecutive integers beginning with 2 is divisible
by 1 . 2 . 3 . 4. Using Example 3 again, and the result just established, we
prove that 4 consecutive integers beginning with 3 is divisible by 1.2.3.4;
and thus we finally establish that the product of any 4 consecutive integers
whatever is divisible by 1 . 2 . 3 . 4.
Proceeding in exactly the same way, we next show that our theorem holds
when p=5; and so on. Hence it holds generally.
This demonstration is a good example of " mathematical induction."
Example 5. If a, h, c be three integers such that a^+b'^=c^, then they are
represented in the most general way possible by the forms
a = \(m2-n2), b = 2\mn, c=\{m?+n^).
First of all, it is obvious, on account of the relation a^ + b-^c^, that, if
any two of the numbers have a common factor X, then that factor must occur
in the other also ; so that we may write a = \a', b = \b', c = Xc', where a', 6', c'
are prime to each other, and we have
o'2 + 6'2=c'» (1).
No two of the three, a', b', c', therefore, can be even ; also both a' and b'
cannot be odd, for then a'^ + b''^ would be of the form 4n + 2, which is an
impossible form for the number c'^.
It appears, then, that one of the two, a', b', say b' (=2|3), must be even, and
that a' and c' must be odd. Hence (c' + a')/2 and (c' - a')/2 must be integers ;
and these integers must be prime to each other ; for, if they had a common
factor, it must divide their sum which is c' and their difference which is a' ;
but c' and a' have by hypothesis no common factor.
Now we have from (1)
c'2-a'2=6':!=:4p2,
whence
m<-^>^ <^)-
Therefore, since (c' + a')/2 is prime to (c' - a')/2, each of these must be a
perfect square ; so that we must have
^'="" (3),
^=»' (4),
/3=mra (5),
where m is prime to n.
From (3) and (4), we have, by subtraction and addition,
a'=n?-v?, c'=m2 + n2;
and, from (5), b'=2^ = 2mn.
Returning, therefore, to our original case, we must have generally ,
a=X(7n2_n=), 6 = 2Xmn, c=X(m2 + n2),
This is the complete analytical solution of the famous Pythagorean
problem— to find a right-angled triangle whose sides shall be commensurable.
34—2
532 PROPERTY OF AN INTEGRAL FUNCTION CH. XXXV
§ 4.] The following theorem may be deduced very readily
from the principles of § 2. Let /(a:) stand for jOo+i^i^+/'2i^^ +
. . . + pn^'^, where po, Pi, . . ., Pn are positive or negative
integers, and x any positive integer; then, if x he congruent
with r with respect to the modulus m, f{x) will be congruent with
f{r) with respect to modulus m.
By the binomial expansion, we have
{qm + r)" = {qmY + „(7i {qmY'^r + . . . + nCn-i {qm) r"-^ + r*,
= (g«;^'i-i + JJ^q^-^m''-''r + . . . + nCn-iqr''-^) m + r",
= Mnm + r" ;
where Mn is some integer, since all the numbers nG\, n^2, • • .,
nCn-i are, by § 3, Example 4, or by their law of formation (see
chap. IV., § 14) necessarily integers.
Similarly
{qm + r)"-^ = Jf„_i w + r""^,
Hence, \i x = qm-^r,
f{x)=^p^+p,r+p2r'' + . . .+pnr'' + (piMi+p23l2 + . . .+pn3I„)m,
=f{r) + Mm.
Hence /(x) is congruent with /(r) with respect to modulus m.
Cor. 1. Since all integers are congruent (with respect to
modulus m) with one or other of the series
0, 1, 2, . . ., m-1,
it follows that to test the divisibility/ of f{x) by m for all integral
values of x, we need only test the divisibility by m off{0), /(I),
/(2), . . .,/(m-l).
Example 1. Let/(a;) = x(x + l) (2x + 1) ; and let it be required to find when
f{x) is divisible by 6. We have/(0) = 0, /(I) = 6, /(2) = 30, /(3) = 84, /(4) = 180,
/(5) = 330. Each of these is divisible by 6 ; and every integer is congruent
(mod 6) with one of the six numbers 0, 1, 2, 3, 4, 5 ; hence x{x+l) (2x + l)
is always divisible by 6.
Cor. 2. f{qf{r) + r} is always divisible by f{r); for
f{qf{r) + r} =Mf(r) +f{r) = {M+ l)f{r).
Hence an infinite number of values of x can always be found
which will make f(x) a composite number.
^ 4, 6 DIFFERENCE TEST OF DIVISIBILITY 633
This result is sometimes stated by saying that no integral
function of x can furnish prime numbers only.
Example 2. Show that x* - 1 is divisible by 5 if a; be prime to 5, but not
otherwise.
With modulus 5 all integral values of x are congruent with 0, ±1, ±2.
If/(x) = x*-l,/(0)=-l,/(--fcl) = 0,/(±2) = 15. Now 0 and 15 are each
divisible by 5 ; but - 1 is not divisible by 5. Hence a;^ - 1 is divisible by 5
when X is prime to 6, but not otherwise.
Example 3. To show that x'^+x + VJ is not divisible by any number less
than 17, and that it is divisible by 17 when and only when x is of the form
17morl77n-l.
Here
/(0) = 17, /{ + 1) = 19, /( + 2) = 23, /( + 3) = 29, /( + 4) = 37, /( + 5) = 47,
/( + 6)=:59, /( + 7) = 73, /( + 8) = 89, /{-1) = 17, /(-2) = 19, /(-3) = 23,
/(-4) = 29, /(-5)-37, /(-6) = 47, /(-7) = 59, /(-8) = 73.
These numbers are all primes, hence no number less than 17 will divide
x^+x + n, whatever the value of x may be; and 17 will do so only when
a;=jul7 or a;=ml7-l.
§ 5.] Method of Differences. — There is another method for
testing the divisibility of integral functions, which may be given
here, although it belongs, strictly speaking, to an order of ideas
somewhat different from that which we are now following.
Let fn {cc) denote an integral function of the wth degree.
/„ {x + 1) -/„ {x) =Pq +pi{x+l) + . . . +pn-i (x + 1)"-^ +pn (x + If
-po-piX-. . .-pn-iX''-''-pnX'' (1).
Now on the right-hand side the highest power of x, namely
x^, disappears ; and the whole becomes an integral function of
the n - 1th degree, fn~i (x), say. Thus, if m be the divisor,
we have
/„ (X + 1) -/, (X) ^fn-i (x) ,
m - m ^ ''
It may happen that the question of divisibility can be at
once settled for the simpler function fn-i{x). Suppose, for
example, that it turns out that /„_i {x) is always divisible by m,
whatever x may be ; then/„ {x+\) —fn (x) is always divisible by
m, whatever x may be. Suppose, further, that /„ (0) is divisible
by m ; then, since /„ (1) — /„ (0), as we have just seen, is divisible
by m, it follows that/,i(l) is divisible by m. Similarly, it may
be shown that /„ (2) is divisible by m ; and so on.
534 EXERCISES XXXV CH. XXXV
If the divisibility or non-divisibility of /„_i(a;) be not at once
evident, we may proceed with /„_i {x) as we did before with
/„ {x), and make the question depend on a function of still lower
degree ; and so on.
Example, /g (x) = x^-xis always divisible by 5.
f^{x + \)-f^{x) = {x + lf-{x + l)-x^ + x,
= 5a;*+10a;3 + 10a;2 + 5x,
=M6.
Now /6(1)=0,
therefore f^{2)-f^[l) = M^5,
and J\{2)=M,b.
Similarly, /sCS) -/s (2) = ilfi5,
therefore /s (3) = (M^ + ill j) 5 ;
aud so on.
Thus we prove that/j (1), /j (2), /^ (3), &c., are all divisible by 5 ; in other
words, that x^ - x is always divisible by 5.
Exercises XXXV.
(1.) The snm of two odd squares cannot be a square.
(2.) Every prime greater than 3 is of the form 6?irfc 1.
(3.) Every prime, except 2, has one or other of the forms 47t±l.
(4.) Every integer of the form 4n-l which is not prime has an odd
number of factors of the form 47i - 1.
(5.) Every prime greater than 5 has the form 30m + n, where n = 1, 7, 11,
13, 17, 19, 23, or 29.
(C.) The square of every prime greater than 3 is of the form 24m + 1 ; and
the square of every integer which is not divisible by 2 or 3 is of the same
form.
(7.) If two odd primes differ by a power of 2, their sum is a multiple of 3.
(8.) The difference of the squares of any two odd primes is divisible by 24.
(9.) None of the forms (Sm + 2) n^ + 3, Amn - m - 1, Amn -m-n can repre-
sent a square integer. (Goldbach and Euler.)
(10.) The nth power of an odd number greater than unity can be presented
as the difference of two square numbers in n different ways.
(11.) If N differ from the two successive squares between which it lies by
X and y respectively, prove that N -xy is a, square.
(12.) The cube of every rational number is the difference of the squares of
two rational numbers.
(13.) Any uneven cube, n^ is the sum of n consecutive uneven numbers,
of which r? is the middle one.
(14.) There can always be found n consecutive integers, each of which is
not a prime, however great n may be.
§5
EXERCISES XXXV 535
(15.) In the scale of 7 every square integer must have 0, 1, 2, or 4 for its
unit digit.
(16.) The scale in which 34 denotes a square integer has a radix of the
form ?i(3n + 4) or (n + 2) (3n + 2).
(17.) There cannot in any scale be found three different digits such that
the three integers formed by placing each digit differently in each integer
shall be in Arithmetical Progression, unless the radix of the scale be of the
form 3p + 1. If this condition be satisfied, there are 2{p-l} such sets of
digits ; and the common difference of the A.P. is the same in all cases.
(18.) If X > 2, x* - ix^ + 5x^ - 2x is divisible by 12.
(19.) a;»/5 + a;*/2 + x3/3-a;/30, and a*/6 + a;«/2 + 5a;'*/12 - a;2/12 are both in-
tegral for all integral values of x.
(20.) If X, y, z be three consecutive integers, {Zx)^-d'Zx^ is divisible
by 108.
(21.) i3_ a; jg divisible by 6.
(22.) Find the form of x in order that x^ + 1 may be divisible by 17.
(23.) Examine how far the forms x- + x + 4:l, 2a;'^ + 29 represent prime
numbers.
(24.) Find the least value of x for which 2* - 1 is divisible by 47.
(25.) Find the least value of x for which 2*- 1 is divisible by 23.
(26.) Find the values of x and y for which 7^ - y is divisible by 22.
(27.) Show that the remainder of 22'''*'^ + 1 with respect to 22^+ 1 is 2.
(28.) 32^ ~ 2^" is divisible by 5, if x ~ y = 2.
(29.) Show that 22^t1 + 1 jg always divisible by 3.
(30.) 433:±i + 2s^±i + 1 is divisible by 7.
(31.) x^"'' + x"^'" + 1 never represents a prime unless a; =0 or cb = 1.
(32.) If P be prime and =a^ + b^, show that P" can be resolved into the
sum of two squares in ^n ways or ^ (n + 1) ways, according as n is even or odd,
and give one of these resolutions.
(33.) li x'^ + y-=z'^,x, y, z being integers, then xyz = 0 (mod 60) ; and if x
be prime and >3, j/sO (mod 12). Show also that one of the three numbers
= 0 (mod 5).
(34.) The solution in integers of x^ + tj^=2z^ can be deduced from that of
x^ + y"^ — z'. Hence, or otherwise, find the two lowest solutions in integers of
the first of these equations.
(35.) If the equation x^ + ?/=*= ^^ had an integral solution, show that one of
the three x, y, z must be of the form Im, and one of the form 3?re.
(36.) The area of a right-angled trisingle with commensurable sides cannot
be a square number.
(37.) The sum of two integral fourth powers cannot be an integral square.
(38.) Show that (3 + ^5)* + (3-^5)* is divisible by 2^.
(39.) If X be any odd integer, not divisible by 3, prove that the integral
part of 4* - (2 + v/2)« is a multiple of 112. '
(40.) If n be odd, show that l + »C4 + „C8+„Cm+ ... is divisible by
536 LIMIT AND SCHEME FOR DIVISORS OF N CH. XXXV
ON THE DIVISORS OF A GIVEN INTEOER.
§ 6.] We have already seen (chap, iii., § 7) that every
composite integer iV can be represented in the form a'^h^cy . . .,
where a, b, c, . . . are primes. If iVbe a perfect square, all the
indices must be even, and we have N - a^"' b^^' c'y' . . . ; so that
jN^a^'b^'cy' ....
In this case N is divisible by J N.
If N be not a perfect square, then one at least of the indices
must be odd ; and we have, say,
N^-a'"^'+^b'''^'c-y' . . .=a'^'b^'cy' . . . a^'+^b^' cy' . . .,
so that N is divisible by a'^b^'cy' , . . , which is obviously less
than J N. Hence
Every composite number has a factor which is not greater than
its square root.
This proposition is useful as a guide in finding the least
factors of large numbers. This has been done, once for all, in a
systematic, but more or less tentative, manner, and the results
published for the first nine million integers in the Factor Tables
of Burckhard, Dase, and the British Association*
§ 7.] The divisors of any given number N=a'^b^cy . . . are
all of the form a'^'b^'cy . . . , where a, fi\ y', . . . may have any
values from 0 up to a, from 0 up to /?, from 0 up to y, . . .
respectively. Hence, if we include 1 and iV itself among the
divisors, the divisors of N-a°-b^cy . . . are the various terms
obtained by distributing the product
(1 + a + a^ + . . . + a")
X (1 + ^> + ^)2 + . . . +b^)
X (1 + c + c^ + . . . + cy)
X (1).
* For an interesting account of the construction and use of these tables,
see J. W. L. Glaisher's Report, Rep. Brit. Assoc. (1877).
6, 7 SUM AND NUMBER OF FACTORS 537
Cor. 1.
Since
l + a + a^+. . .+«» = ;— ,
a- 1
1 +b + b^+ . . . +b^= , _ ,
and so on,
It follows that the sum of the divisors of N-a'^l^c'/ . . . is
(g'+^-lX&P+^-l). . .
(a-l)(6-l). . . •
If in (1) we put a = 1, 6 = 1, c = 1, . . . , each divisor, that is,
each term of the distributed product, becomes unity ; and the
sum of the whole is simply the number of the different divisors.
Hence, since there are a + 1 terms in the first bracket, ^ + 1 in
the second, and so on, it follows that
Cor. 2. The number of the divisors of N=a'^b^cy . . . is
(a+l){/3 + l)(y+l). . . .
Cor. 3. The number of ways in which* N=a'^b^cy . . . can
be resolved into two factors is |{1 + (« + 1) (;8 + 1) (y + 1) . . .}, or
|(a + l)()8+l)(y + l). . ., according as N is or is not a square
number.
For every factor has a complementary factor, that is to
say, every factorisation corresponds to two divisors ; unless N be
a square number, and then one factor, namely JN, has itself
for complementary factor, and therefore the factorisation
N=jNy.jN corresponds to only one divisor.
Cor. 4. The number of ways in which N=a'^b^c''' , . . can be
resolved into two factors that are prime to each other is 2"~^,
n being the number of prime factors a", b^, cy, . . . .
For, in this kind of resolution, no single prime factor, a" for
example, can be split between the two factors. The number
of different divisors is therefore the same as if a, ^, y, . . .
* This result is given by Wallis in his Discourse of Combinations, Alterna-
tions, and Aliquot Parts (1685), chap, iii., § 12. In the same work are given
most of the results of §§ 6 and 7 above.
538 EXAMPLES CH. XXXV
were each equal to unity. Hence the number of ways is
^(1 + 1) (1 + 1) (1 + 1). . . (to factors) = 1.2'^= 2"-\
Example 1. Find the different divisors of 360, their sum, and their
number.
We have 360 = 233=5.
The divisors are therefore the terms in the distributed product
(1 + 2 + 22 + 23) (1 + 3 + 32) (1 + 5); that is to say,
1, 2, 4, 8, 3, 6, 12, 24, 9, 18, 36, 72, 5, 10, 20, 40, 15, 30, 60, 120,
45, 90, 180, 360.
Their sum is (2^ - 1) (.S^ - 1) (52 - l)/(2 - 1) (3 - 1) (5 - 1) = 1170.
Their number is (1 + 3) (1 + 2) (1 + 1) = 24.
Example 2. Find the least number which has 30 divisors. Let the
number be N=a°-h^cy. There cannot be more than three prime factors ; for
30=2x3x5, which has at most three factors, must =(a + l) (^3 + 1) (7 + I).
There might of course be only two, and then we must have 30 = (a + 1) (|3 + 1) ;
or there might be only one, and then 30 = a + 1.
In the" first case a = l, ^ = 2, 7 = 4. Taking the three least primes,
2, 3, 5, and putting the larger indices to the smaller primes, we have
2^=2'».32.5 = 720.
In the second case we should get 2i4 . 3, 2^ . 3^ or 29 . 32.
In the last case, 229.
It will be found that the least of all these is 2* . 32 . 5 ; so that 720 is the
required number.
Example 3. Show that, if 2» - 1 be a prime number, then 2"-i (2" - 1) is
equal to the sum of its divisors (itself excluded)*.
Since 2" - 1 is supposed to be prime, the prime factors of the given number
are 2""^ and 2»-l. Hence the sum of its divisors, excluding itself, is, by
Cor. 1 above,
= (2»-l){2»-2"-i},
= 2"-! (2"- 1) {2-1},
= 2»-i(2"-l);
as was to be shown.
ON THE NUMBER OF INTEGERS LESS THAN A GIVEN
INTEGER AND PRIME TO IT.
§ 8.] If we consider all the integers less than a given one, N,
a certain number of these have factors in common with N, and
the rest have none. The number of the latter is usually denoted
* In the language of the ancients such a number was called a Perfect
Number. 6, 28, 496, 8128 are perfect numbers.
§§ 7, 8 euler's theorems regarding ^ (N) 539
by <f> (N). Thus <^ (N) is taken to denote the number of integers
(including 1) which are less than N and prime to N.
We have the following important theorem, first given by
Euler : —
If N=a^^^a^^'ai''i . . . a„"", then
The proof of this theorem which we shall give is that which
follows most naturally from the principles of § 7.
Proof. — Let us find the number of all the integers, not
greater than Ny which have some factor in common with N.
That factor must be a product of powers of one or more of the
primes a^, a^, a^, . . ., a„.
Now all the multiples of «! which do not exceed N are
Ifti, 2ai, 3ai, . . ., {Nja^ai, iVV^i in number (3);
all the multiples of a^ which do not exceed N are
l^a, 2a2, 3052, . . •, {Nja^^a^, iV7a2 in number (4);
and so on.
All the multiples of «ia2 which do not exceed N are
laitta, 2aia2, 3aaa2> • • •, {Njaia^a^a^,
Nja^a^ in number (5) ;
and so on.
Similarly, for a^a^a^
we have
laiagfts, 2aia2«3, 3ai
a^tts, . . ., (N/aia^a^) a^a^as,
N/aia^aa in number
(6),
Let us now consider the number
N
N N
+ — +— +. . .
«2 as '
N
N N
ttitti
a^as Uitti
N
+
aia^a^
jsr N
+ + + . . .
a^a^Ui, a^a^ai
N
aittiaiat
(7).
540 euler's theorems regarding <f> (N) ch, xxxv
The number of terms in the first line is „<7i. The number
in the second line is nC^, since every possible group of 2 out of
the n letters ai«2 • • • «n occurs among the denominators. The
number in the third line is nOa for a similar reason. And so on.
Now consider every multiple of the r letters aia^a^ . . . ar
which does not exceed N; in other words, every number, not
exceeding N, that has in common with it a factor of the form
a^'^a-t'^ . . • «/'• This multiple will be enumerated in the first
line, once as a multiple of ai, once as a multiple of a^, and so
on ; that is, once for every letter in it, that is, rOi times.
In the second line the same multiple will be enumerated once
as a multiple of aia^, once as a multiple of ai«3, and so on ; that
is, once for every group of two that can be formed out of the r
letters aia^ . . . «r, that is, rO^ times. And so on. Hence,
paying attention to the signs, the multiple in question will in
the whole expression (7) be enumerated
times ; that is, just once. This proof holds, of course, whatever
the r letters in the group may be, and whether there be 1, 2, 3,
or any number up to w in the group.
It follows, therefore, that (7) enumerates, without repetition
or omission, every integer which has a factor in common with N.
But, from formula (1), chap, iv., § 10, we see that (7) is simply
\ aj\ ttg/ ■ " \ a J
To obtain the number of integers less than N which are
prime to i\r, we have merely to subtract (8) from AT. We thus
obtain
.W=^(x-i)(i-i)...(.-l),
which establishes Euler's formula.
Example. 77=100=22.52; 0(lOO) = 22.52(l-i) (1-|)=40.
§ 9.] 1/ M= PQ, where P and Q are pi-ime to each other, then
i>{M) = <l>{P)^(Q) (1).
§§8,9 <l>(PQR...) = <l>{P)<f>{Q)<}>iR)... 541
For, since P and Q are prime to each other, we must have
Q = bM^^- . .,
where none of the prime factors are common ; and therefore
M=a^^^a^^^ . . . h^P^h^^ . . .,
where ai, «2, . . ., h, bz, . . . are all primes.
But, by § 8, we then have
=<.«,"... .(i-l)(i-i)...V.V... . (i-i)(i-i;)---.
Cor. I/PQES . . . be prime to each other, then
<t>(PQES, . .) = <l>{P)^{Q)^{B)<f.{S) . . . (2).
For, since P is prime to Q, E, S, . . ., it follows that P is
prime to the product QES . . . Hence, by the above proposition,
<I>{PQES. . .) = <I>{P)<I>{QES. . .).
Repeating the same reasoning, we have
<f>{QES. . .) = cf>{Q)cf>{ES. . .);
and so on.
Hence, finally,
<ly{PQES. . .) = <f>(P)<f>{Q)<l.(E)<}>{S). . . .
Eemark. — There is no difficulty in establishing the theorem
^ {PQ) = <f> (P) ^ (Q) « priori. This may be done, for example, by
means of § 13 below (see Gross' Algebra, § 230). The theorem
of § 8 above can then be deduced from <f>{PQE . . .) =
0 (P) <f>{Q)f}> (E) . . . The course followed above, though not
so neat, is, we think, more instructive for the learner.
Example. 56 = 7x8,
^(56) = 24,
^(7) = 6,
0(8)=4;
0(56) = 0(7)x^(8).
542 gauss's theorem regarding divisors of N ch. xxxv
§ 10.] 1/ di, di, dg, . . ., <&c., denote all the divisors of tJw
integer N, then*
^{d,) + cf>{d,) + <l>(ds) + . . . = N. . . (1).
(Gauss, Bisq. Arith., § 39.)
For the divisors, (?i, d^, ds, . . ., are the terms in the
distribution of the product
(l+ai + ai^+. . . +«i"')(l + «2 + «/ + . . .+«2"0. . . .
If we take any one of these terms, say dr = ai^a.^i . . .,
then, by § 9, Cor.,
since a-^, a^, . . . are primes.
It follows that the left-hand side of (1) is the same as
{l + «^(«i) + <^K)+. • .+^«')}
x{l4-«^(a2) + «^(a2^) + . . .^<^{a^^)\
. . ..... (2).
But <^ {a{) = a{ (l - ^) = ^i*" - «/"'•
Hence
l + «/»(ai) + «^(«i') + . • • + «^«')
= 1 + On.- 1 + tti^- Oj + . . . + «!"■- al»l-^
= ai"' ;
and so on.
It appears, therefore, that (2) is equal to a^^a^^ . . ., that
is, equal to N.
Example. iV=315 = 32.5.7.
The divisors are 1, 3, 5, 7, 9, 15, 21, 35, 45, 63, 105, 315, and we have
^(l) + ^(3) + ^(5)+ . . . +0(315)
= 1 + 2 + 4+6+6 + 8 + 12 + 24 + 24 + 36 + 48 + 144 = 815.
* Here and in what follows 1 is included among the divisors, and, for con- •
venience, 0 (1) is taken to stand for 1. Strictly speaking, ^ (1) has no meaning
at all.
^ 10, 11 PRIME DIVISORS OF w! 643
PROPERTIES OF m!
§ 11.] The following theorem enables us to prove some
important properties of m\: —
The highest power of the prime p which divides ml exactly is
■(3MiMf^
where J( — ), ^("s)* • • • denote the integral parts of m/p,
m/p% . . . ; and the series is continued until the greatest power of
p is reached which does not exceed m.
To prove this, we remark that the numbers in the series
l,2,...,m
which are divisible by p are evidently
\p, 2p, 3^, . . ., Jcp,
where kp is the greatest multiple of pl^m. In other words,
i: = I{m/p). Hence I (m/p) is the number of the factors in ml
which are divisible by 2?-
If to this we add the number of those that are divisible by
p^, namely /(m/p^), and again the number of those that are
divisible by p^, namely I{m/p^), and so on, the sum will be the
power in which p occurs in ml.
Hence, since p is a prime, the highest power of p that will
divide ml exactly is
It is convenient for practical purposes to remark that
For, if
then
mlp'-^ = i + klp'-'{k<p'-')
(IX
mlp' = ilp + klp'
(2),
=j+llp + klp'-{l<p)
(3).
544
EXAMPLES
cn. XXXV
Now
lip + Wl^ip - i)/p + (p'-' - i)Ip%
>(p^^'-pW^\
<1.
Hence, by (3),
But, since i/p =j + l/p,
We may therefore proceed as follows : — Divide m by p; take
the integral quotient and divide again hy p; and so on; until the
integral quotient becomes zero ; then add all the integral quotients,
and the result is the highest power of p which will divide m\ exactly.
Example 1. To find the highest power of 7 which divides 1000! exactly.
In dividing successively by 7 the integral quotients are 142, 20, 2 ; the
sum of these is 164. Hence 7^^^ is the power of 7 required.
Example 2. To decompose 25! into its prime factors.
Write down all the primes less than 25 ; write under each the successive
quotients ; and then add. We thus obtain
2
3
5
7
11
13
17
19
23
12
6
3
1
8
2
5
1
3
2
1
1
1
1
22
10
G
3
2
1
1
1
1
Hence 25! = 2«2 . 3i» .5^.1^. IP . 13 . 17 . 19 . 23.
Example 3. Express 39!/25! in its simplest form as a product of prime
factors.
Eesult, 213 , 38 . 52 . 72 . 11 . 132 . 17 . 19 . 29 , 31 . 37.
Example 4. Find the highest power of 5 that will divide 27 . 28 . 29 . . . 100
exactly.
Eesult, 518.
Example 5. If m be expressed in the scale oip, in the form
the highest power of 2> that will divide ml exactly is the
m-Po-Pi-Pi-
-Pt
th.
p-1
Example 6. If m = 2* + 2^ + 2'>'+ . . . {k terms), where o</3<7<. . .,
the greatest power of 2 that will divide ml is the (m - k)th.
§§ 11, 12 PROPERTIES OF ml/flglh] . . 545
§ 12.] 1/ /+ g + h+ . . .:^w, then ml/flglh] ... is an
integer*.
To prove this, it will be sufficient to show that, if any prime
factor, p say, appear in /\g\h\ . . . , it will appear in at least
as high a power iu ml In other words (§ 11), we have to
show that
+ . . . . (1).
Now, if d be any integer whatever, we have
f/d^/'+r/d (/'>d-i),
g/d^g' + g"/d (g":!^d~l),
h/d^k' + k"/d (r>c^-l),
• • • • t ,
and we obtain by addition
/+g + h+. . . ., , „ f" + g" + h"+. . .
Hence, if/" + g" + //'+. . .<d,
^4MtX^
If, on the other hand, /" + g" + h" + . . . >d,i then
^\ d r-^ +g +h+. . .,
* This theorem might, of course, be inferred from the fact that
m\lf\g\h\ . . . represents the number of permutations of m things / of
which are alike, g alike, h alike, &c.
t If n be the number of the letters f, g, h, . . ., the utmost value of
f"+g" + h" + . . . is n (d - 1). Hence the utmost difference between the two
sidesof {2)i8l{n(d-l)/d}.
c. II. 35
546 EXERCISES XXXVI CII. XXXV
It appears, tlierefore, that, even if m =/+ g + h + . . .,
(jM^^K^^--- <^'-
A fortiori is this so if m >/+ g + h+ . . . ,
If now we give d the successive values p, p^, . • . , r.nd com-
bine by addition the inequalities thus obtained from (3), the
truth of (1) is at once established.
Cor. 1. If f-^g + h-^ . . .I^m, and none of tlw numbers
f,g,h,. . . is equal to m, the integer ml/flglhl . . . is divisible
hymifm be a prime.
Cor. 2. The product of r successive integers is exactly
divisible by r\.
The proofs of these, so far as they require proof, we leave to
the reader. Cor. 2 has already been established by a totally
different kind of reasoning in § 3, Example 6.
Exercises XXXVI.
(1.) What is the least multiplier that will convert 945 into a complete
square ?
(2.) Find the number of the divisors of 2160, and their sum.
(3.) Find the integral solutions of
a;z/ = 100x + 102/ + l (o) ;
2/3= 108a! (7).
(4.) No number of the form x* + 4 except 5 is prime.
(5.) No number of the form 2"^+^^ ^ except 5 is prime.
(6.) To find a number of the form 2" . 3 . a (a being prime) which shall be
equal to half the sum of its divisors (itself excluded).
(7.) To find a number N of the form 2'^ahc ... (a, 6, c being unequal
primes) such that N is one-third the sum of its divisors.
(8.) Show how to obtain two "amicable" numbers of the forms 2"pg,2*r,
•where j3, q, r are primes. (Two numbers are amicable when each is the sum
of the divisors of the other, the number itself not being reckoned as a
divisor.)
(9.) To find a cube the sum of whose divisors shall be a square.
(One of Fermat's challenges to Wallis and the English mathematicians.
Var. Op. Math., pp. 188, 190.)
(10.) If N be any integer, n the number of its divisors, and P the product
of them all, then iV»=:P2,
§§12,13 EXERCISES XXXVI 547
(11.) The sum and the sum of the squares of all the numbers less than
N and prime to it are ^N (a - 1) (& - 1) (c - 1) . . . and ^N^ (1 - 1/a) (1 - 1/6)
. . . + ^iV^ (1 - o) (1 - 6) . . . respectively. ( Wolstenholme. )
(12,) If p, q,r, . . . be prime to each other, and d (N) denote the sum of
the divisors of N, show that
d(pqr . . .) = d(p)d{q)d{r) . . . .
(13.) If N=abc, where a, b, c are prime to each other, then the product of
all the numbers less than N and prime to N is
{abc - 1)! n {(a - l)l/(6c - 1)! a(6-i)(<'-i)}.
(Gouv. and Caius Coll., 1882.)
(14.) The number of integers less than (j-^ + l)" which are divisible by r
but not by r* is (r - 1) {(r^ + 1)" - Ij/r^.
(16.) Prove that
(16.) In a given set of N consecutive integers beginning with A, find the
number of terms not divisible by any one of a given set of relatively prime
integers. (Cayley. )
(17.) If m - 1 be prime to n + 1, show that ^C'„ is divisible by n + 1.
(18.) (a + l){a + 2). . .2axb{b + l) . . . 2bl{a + b)l is an integer.
(19.) The product of any r consecutive terms of the series x-1, x^-1,
x^ -1, , . .is exactly divisible by the product of the first r terms.
(20.) If p be prime, the highest power of p which divides n! is the
greatest integer in {n-S (n)}l{p - l)™, where S (n) is the sum of the digits of
n when expressed in the scale of p.
If S (m) have the above meaning, prove that S {7n-n)<tS (m) - S (n) for any
radix. Hence show that (n + l) (n + 2) . . . (n + m) is divisible by ml.
{Camb. Math. Jour. (1839), vol. i., p. 226.)
(21.) If /(n) denote the sum of the uneven, and F (71) the sum of the even,
divisors of n, and 1, 8, 6, 10, , , . be the "triangular numbers," then
/(n)+/(n-l)+/(n-3)+/(n-6)+. . .
= F{n)+F{n-l)+F{n-3) + F(n-Q)+. . .,
it being understood that /(n -n) = 0, F{n-n)=n.
ON THE RESIDUES OF A SERIES OF INTEGERS IN
ARITHMETICAL PROGRESSION.
§ 13.] The least positive remainders of the series of numbers
k, k + a, k + 2a, . . . , k + (m- l)a
with respect to m, where m is prime to a, are a permutation of the
numbers of the series
0,1,2,. . .,(m-l).
35—2
548 PROPERTIES OF AN INTEGRAL A.P. CH. XXXV
All the remainders must be different ; for, if any two
different numbers of the Series had the same remainders, then
we should have
k + ra = fjt-m + p, and k-\- sa = ixm + p,
whence
{r - s) a = (fx - fx') m, and {r - s) a/m = p.- p!.
Now this is impossible, since a is prime to m, and r and s aro
each < m, and therefore r-s<m. Hence, since there are only
m possible remainders, namely, 0, 1, 2, . . ., (w- 1), the proposi-
tion follows.
Cor. 1. If the remainders of h and a with respect to m he
k' and a\ the remainders will occur as follows: —
Jc, Jc' + a, Jc + 2a', . . . , k' + ra,
until we reach a number that equals or surpasses m ; this we must
diminish hy m, and then proceed to add a' at each step as before.
Thus, if 7: = 11, a = 25, m = 7, the series is
11, 36, 61, 86, 111, 136, 161.
We have /i;' = 4 and a' = 4, hence the remainders are
4, 4 + 4-7 = 1, 5, 5 + 4-7 = 2, &c. ;
in fact,
4, 1, 5, 2, 6, 3, 0.
Cor. 2. If the progression of numbers be continued beyond
m terms, the remainders will repeat in the same order as before ;
and in this periodic series the number of remainders intervening
between two that differ by unity is always the same.
Cor. 3. There are as many terms in the series
k, k-\-a, k+2a, . . ., k + {m-l)a
which are prime to m, as there are in the series
0,1.2,. . .(m-l).
Tlmt is, the .number of terms in the series in question which are
prime to m is <^ (m). See § 8.
This follows from the fact that two numbers which are
congment with respect to m are either both prime or both non-
prime to m.
Cor. 4. If out of the series of numbers
0, 1,2,. . .,(m-l)
§§ 13, 14 PROPERTIES OF AN INTEGRAL A.P. 549
we select those that are less than m and prime to it, say
Tu ra, . . ., r„
(the number n being ^ (w) ), then the numbers
k + r^a, k + r^a, . . ., ^ + r„a,
V)here k = 0 or a multiple of m, and a prims to m as before, are
all prime to m ; and their remainders with respect to m are a
permutation of
n, ra, . . ., r„.
For, as we have seen already, all tlie n remainders are unlike,
and every remainder must be prime to m ; for, if we had
k + rta = fim + p, where p is not prime to m, then rta = pm+p-k
would have a factor in common with m, which is impossible,
since r^ and a are both prime to m.
Hence the remainders must be the numbers r^, rg, . . ., r,j
in some order or other.
§ 14.] Ifm be not prime to a, but have with it the G.C.M. g,
so that a = ga, m - gm', the remainders of the series
k, k + a, k+2a, . . ., k+(m-l)a
with respect to m will recur in a shorter cycle of m'.
Consider any two terms of the series out of the first m', say
k^-ra, k+sa. These two must have different remainders, otherwise
(r-s)a would be exactly divisible by m : that is, {r - s) ga'/gm'
would be an integer ; that is, (r - s) a'/m' would be an integer ;
which is impossible, since a' is prime to m' and r — s<m'.
Again, consider any term beyond the m'th, say the {m' + r)th.,
then, since
{k + (m' + r)a} — {k + ra] — m'a,
= gm'a',
= ma',
it follows that the {m' + r)th term has the same remainder with
respect to w as the rth.
In other words, the first m' remainders are all different, and
after that they recur periodically, the increment being ga",
where a" is the remainder of a with respect to m', subject to
diminution by m as in last article.
Example. If k = ll, a=25, wi = 15, we have the series
11, 36, 61, 86, 111, 136, 161, 186, 211, 236, 261, . . . ;
550 fermat's theorem ch. xxxv
and here g = 5; a'=5; m'-3; a"=2; fc'=ll; gra" = 10. Hence the re-
mainders are
11, 6, 1, 11, 6, 1, 11, 6, 1, 11, 6, . . . .
Cor. J[ftke G.G.M., g,ofa and m divide k exactly, and, in
particular, ifk=^0, the remainders of the series
k, k + a, k + 2a, ...
are the numbers
Og, lg,2g,3g,. . .,(m'-l)g
continually repeated in a certain order.
For, in this case, since k = gK, we have {k + ra)lm = (k + ra)lm',
hence the remainders are those of the series
K, K + a, K + 2a', . . .
with respect to m' which is prime to a, each multiplied by g.
Hence the result follows by § 13.
Example. Let fc:=10, a = 25, m = 15 ; then the series of numbers is
10, 35, 60, 85, 110, 135, 160, 185, ....
We have (7 = 5; a' — 5; m' = 3; k=2; and the remainders are
10, 5, 0, 10, 5, 0, 10, 5, ... ;
that is to say,
2x5, 1x5, 0x5, ... .
§ 15.] From § 13 we can at once deduce Fermat's Theorem*,
which is one of the corner-stones of the theory of numbers.
If m be a prime number, and a be prime to m, a"'~^ - 1 is
divisible by m.
If a be prime to m, then we have
la = /Mi7w + pi,
'ia — [h^m + P2)
(m - 1) a = i^m-\m + p,„_i,
where the numbers pi, P2, • • •, 9m-\ are the numbers
1, 2, . . ., (w-1) written in a certain order.
* Great historical interest attaches to this theorem. It was, with several
other striking results in the theory of numbers, published without demonstra-
tion among Fermat's notes to an edition of Bachet de Meziriac's Diophantus
(1670). For many years no demonstration was found. Finally, Euler {Com-
ment. Acad. Petrop., viii., 1741, and Comment. Nov. Acad. Petrop., vn., 1761)
gave two proofs. Another, due to Lagrange [Nouv. Mem. de V Ac. de Berlin,
1771), is reproduced in § 18. The proof given above is akin to Euler'e second
and to that given by Gauss, Disq. Arith., § 49.
^14-17 EULEr's GENERALISATION OF FERMAT'S THEOREM 551
Hence
1.2. . . (m - 1) a"*-' = {^L^m + pi) {fx^m + p.^ . . . {i^m-im + Pm~i),
= Mm + pipn . . . p,n-i,
= Mm + 1.2 . . . {m-l).
We therefore have
1.2 . . .{m-l) (a'"-^ - l)^Mm.
Now, m being a prime number, all the factors of 1 . 2 . . . {m-\)
are prime to it. Hence m must divide a'"~^ - 1.
It is very easy, by the method of differences, explained in § 5,
to establish the following theorem : —
If m he a prime, a^-a is exactly divisible hy m*.
Since a"* - a = a (a™"^ - 1), if a be prime to m, this is simply
Fermat's Theorem in another form.
§ 16.] By using Cor. 4 of § 13 we arrive at the following
generalisation of Fermat's Theorem, due to Euler : —
If m be any integer, and a be prime to m, then a*^*") — 1 is
exactly divisible by m.
Here <^ (m) denotes, as usual, the number of integers which
are less than m and prime to it.
For, if ri, 7-2, • • • . ^« be the integers less than m and prime
to it, we have, by the corollary in question,
ria = fi^m + pi,
r2a = fj^m + p2,
rnCt = H-nm + pn,
where the numbers pi, p^, . . ., p„ are simply Ti, r^, . . ,, Tn
written in a certain order.
We have therefore, just as in last paragraph,
n^2 . • • ^'»(a"- l) = Mm,
whence, since Ti, r^, . . ., r„ are all prime to m, it follows that
a"- 1, that is, a'''^'"' - 1, is divisible by m.
§ 17.] The famous theorem of Wilson can also be established
by means of the principles of § 13.
* For another proof of this theorem see § 18 below.
552 WlLSON^S THEOREM — GAUSSES PROOF CH. XXXV
Auy two integers whose product has the remainder + 1 with
respect to a given modulus m may be called, after Euler, Allied
Numbers.
Consider all the integers,
1,2,3,. . .,(m-l),
less than any prime number m (the number of them is of course
even). We shall prove that, if we except the first and last, they
can be exhaustively arranged in allied pairs.
For, take any one of them, say r, then, since r is prime to m,
the remainders of
r.l, r.2, . . ., r{m-l)
are the numbers
1, 2, . . .,(m-l)
in some order. Hence, some one of the series, say rr', must have
the remainder 1 ; then rr will be allies.
The same number r cannot have two different allies, since all
the remainders are different.
Nor can the two, r and r', be equal, unless r=l or = m-l;
for, if we have
r'- fim + 1,
then r^-l=fim; that is, (r+1) (r-1) must be divisible by m.
But, since m is prime, this involves that either r+1 or r-1 be
divisible by m, and, since r cannot be greater than m, this involves
in the one case that r = m - 1, in the other that r = 1.
Excluding, then, 1 and m-1, we can arrange the series
2, 3,. . .,{m-2)
in allied pairs. Now every product of two allies is of the form
fim+1; hence the product 2.3. . . {m-2) is of the form
(fiiin + l) {fi^m + 1) . . ., which reduces to the form 3Im + 1.
Hence
2.3 . . . (m-2) = Mm + l;
and, multiplying by w» - 1, we get
1.2.3. . .(m-2) {m-l) = Mmim-l) + m-l.
Whence
1.2.3 . . . (m- 1) + 1 = Am,
^ 17, 18 THEOREM OF LAGRANGE 553
That is, if m he a prime, {m-l)\ + 1 is divisible by m, which is
Wilson's Theorem*.
It should be observed that, if m be not a prime, (w- 1)! + I
is not divisible by m.
For, if m be not a prime, its factors occur among the numbers
2, 3, . . ., {m-\), each of which divides {m-\)\, and, there-
fore, none of which divide {m - 1)! + 1.
§ 18.] The following Theorem of Lagrange embraces both
Fermat's Theorem and Wilson's Theorem as particular cases : —
lf{x + l) {x + '2). . . {x+p-\)
= x^-'^ + A-,x'P-'^+. . . + Ap-^x + Ap-1,
andp be prime, then Ai, A^, . . ., Ap-2 are all divisible by p.
We have
{x-^-p){ap-'^-\-AiX^-^ + . . .+Ap-2X+ Ap--i]
= {x + l) {{x + If-' + .Ix {x + l)^-'^ + . . . + J[p-2 (^ + 1) + Ap-,].
Hence
pxP~^-^pAixP~^+pA2xP~^ + . . . + pAp-2X + pAp-i
={(x+iy-af}+A,{(x+iy-'-xp-'}+A,{(x+iy-''-xp-^}+. . .
Therefore
pAi=pC2 + p-iCiAi,
pAs = pCj +P.1C2A1 + p^-iC^Az,
pA^ = pCi + p-iCaA^ + P-0C2A2 + P-3O1A3.
Hence, since p-iCi, p-^Ci, psCu ... are not divisible hy p
if j9 be prime, we get, by successive steps, the proof that Ai, A^,
A3, . . . are all divisible hy p.
* This theorem was first published by Waring in his Meditationes Alge-
braica (1770). He there attributes it to Sir John Wilson, but gives no proof.
The first demonstration was given by Lagrange (Nouv. Mem. de VAc. de
Berlin, 1771) ; this is reproduced in § 18. A second proof was given by Euler
in his Opuscula Anahjtica (1783), vol. 1., p. 329, depending on the theory of
the residues of powers.
The proof above is that given by Gauss {Bisq. Arith., §§ 77, 78), who
generalises the theorem as follows : — " The product of aU the numbers less
than m and prime to it is congruent with - 1, if m =p'^ or = 2p'^, where p
is any prime but 2, or, again, if m = 4; but is congruent with +1 in every
other case." This extension depends on the theory of t^uadratic residues.
554 EXERCISES XXXVII CH. XXXV
Cor. 1. Put a; = l, and we get
2.3. . .p = l + (Ai + A2 + . . . + J.p_2) + Ap-i,
Therefore Ap-i + 1, that is, (p-l)\ + 1, is divisible by p.
Cor. 2. Multiplying by x and transposing, we get
w'^ — x = x {x + '[) . . . (x +p — 1)
-{l+Ap-i)x-(AixP-^ + A^iif-^ + . . .+Ap-^a^).
But x{x+\) . . . {x+p-1), being the product of ^ con-
secutive integers, must be divisible by p. Also, if p be prime,
1 + Ap-1 is divisible by p.
Therefore, x^ — x is divisible by p if p be prime. From which
Fermat's Theorem follows at once if x be prime to p.
Exercises XXXVII.
(1.) x^^-x is divisible by 2730.
(2.) If a; be a prime greater than 13, x^^ - 1 is divisible by 21840,
(3.) If the nth power of every number end with the same digit as the
number itself, then n = 4^ + l.
Give a rule for determining by inspection the cube root of every perfect
cube less than a million.
(4.) If the radix, r, of the scale of notation be prime, show that the rth
power of every integer has the same final digit as the integer itself, and that
the (r - l)th power of every integer has for its final digit 1.
(5.) If n be prime, and x prime to n, then either a;("~i)/'2 - 1 or x'""!'/^ + 1
is divisible by n.
(6.) If n be prime, and x prime to n, then either a;"("-')/2 - 1 or a;»i"-^)/2 + 1
is divisible by n^.
(7.) If m and n be primes, then
m^-^ + ra^'-i^l (mod. rmi).
(8.) If o, j3, 7, . . . be primes, and N=a^y . . ., then
S(A^/a)"~^=l (mod. a^y . . .).
(9.) If 71 be an odd prime, show that
(a + l)"-(a" + l) = 0 (mod. 2ra).
Hence show that, if n be an odd prime and p an integer, then any integer
expressed in the scale of 2n will end in the same digit as its {pn -p + l)th
power. Deduce Fermat's Theorem. (Math. Trip., 1879.)
(10.) If n be prime and >x, then
.T"-2 + a;n-3^ . . . +x + l = 0 (mod. n).
(11.) If n be an odd prime, then
l + 2(n + l) + 22(n + l)2+. . . +2"-«(n + l)»-2=0 (mod.n).
(12.) If nbeodd, l'» + 2»+. . . +(n-l)"=0 (mod. n).
§§ 18, 19 NOTATION FOR NUMBER OF PARTITIONS 555
(13.) If n be prime, and p<n,
(p - 1)1 (n-p)l-{- 1)P= 0 (mod, n),
and, in particular,
[{i(ji-l)}!p + (-l)(»-i)/2 = 0(mod. n).
(Waring.)
(14.) Find in what cases one of the two {^ (n - 1)}I ± 1 is divisible by n.
What determines which of them is so ?
(15.) If p be prime, and n not divisible by ^ - 1, tlicn
l»4-2"+. . .+(p-l)" = 0(mod.23).
(16.) If J) be any odd prime, m any number > 1 which is not divisible by
p-1, then
12'» + 22'"+. . .+ fLlP\^=0{mod.p).
(17.) If neither a nor b be divisible by a prime of the form 4n - 1, then
0^411-2 _ J4H-2 yf[ii Qot be exactly divisible by a prime of that form.
Hence show that a*'^-^ + 6<»-2 is not divisible by any integer (prime or not)
of the form 4n - 1.
Also that a^ + b^ is not divisible by any integer of the form 4n - 1 which
does not divide both a and b. Also, that any divisor of the sum of two
integral squares, which does not divide each of them, is of the form 4n + l.
(Euler.)
(18.) Show, by means of (17), that no square integer can have the form
imn — m — n'^, where m, n, a are positive integers. (Euler.)
PARTITION OF NUMBERS.
Elder's Theory of the Enumeration of Partitions.
§ 19.] By the partition of a given integer n is meant the
division of the integer into a number of others of which it is the
sum. Thus 1 + 2 + 2 + 3 + 3, 1+3 + 7, are partitions of 11.
There are two main classes of partitions, namely, (I.) those' in
which the parts may be equal or unequal ; (II.) those in which
the parts are all unequal. When the word " Partition " is used
without qualification, class (I.) is understood.
We shall use a quadripartite symbol to denote the number
of partitions of a given species. Thus P ( | ] ) and Pu ( | | ) are
used to denote partitions of the classes (I.) and (11.) respectively.
In the first blank inside the bracket is inserted the number to
be partitioned ; in the second, an indication of the number of the
parts ; in the third, an indication of the magnitude or nature of
556 EXPANSIONS AND PARTITIONS CII. XXXV
the parts. It is always implied, unless the contrary is stated,
that the least part admissible is 1 ; so that ^m means any
integer of the series 1, 2, . . ., m. An asterisk is used to mean
any integer of the series 1, 2, . . . , oo , or that no restriction is
to be put on the number of the parts other than what arises
from the nature of the partition otherwise.
Thus P {n\p\q) means the number of partitions of n into p
parts the greatest of which is q; P{n\p\^q) the number of
partitions of n into p parts no one of which exceeds q ;
P {n\*\^q) the number of partitions of n into any number of
parts no one of which is to exceed q ; Pu {n \ l^p \ * ) the
number of partitions of n into p or any less number of unequal
parts unrestricted in magnitude; Pw(w|jt?|odd) the number of
partitions of n into p unequal parts each of which is an odd
integer; P{n\*\\, 2, 2^ 2^*, . . .) the number of partitions of
n into any number of parts, each part being a number in the
series 1, 2, 2^ 2^ . . . ; and so on.
The theory of partitions has risen into great importance of
late in connection with the researches of Sylvester and his
followers on the theory of invariants. It is also closely con-
nected with the theory of series, as will be seen from Euler's
enumeration of certain species of partitions, which we shall
now briefly explain.
§20.] If we develop the product {\ -^ zx) {I + zap) . . .
(1 + zofl), it is obvious that we get the term z^x^ in as many
different ways as we can produce n by adding together p of the
integers 1, 2, . . ., q, each to be taken only once. Hence we
have the following equation : —
(1 + zx) (1 + zx") . . .{l+zafl) = l + '^Pu {n \p \:l^q) z^x"* (1).
Again, if to the product on the left of (1) we adjoin the
factor l + z + z'' + 2^+. . . adco (that is, 1/(1 - z) ), we shall
evidently get z^x"' as often as we can produce n by adding
together any number not exceeding^ of the integers 1, 2, . . ., q.
Therefore
(1 + zx) (1 + zx') . . . (1 + zafl)/(l - z)
= 1 + ^Pu (n \:!f>p i:^g) s^af (2).
§§ 10-21 EXPANSIONS AND PARTITIONS 567
.In like manner, we have
{l + a;)(l+ar'). . . (1 +^) = 1 + 2Ptt(w | * |>g)a;" (3);
(1 + zw) (1 + za^) . . . ad GO = 1 + ^Pu (n \p\^) z^x'' (4) ;
{l+x){l+a^) . . .a.dico = \ + -S,Pu{n.\*\*)x'' (5),
Also, as will be easily seen, we have
1/(1 -zx){l-za^)...{l-zafi)^l + ^P{n\p\ :)>^)cV (6)
\l{l-z){l-zx). . .{l-zofi) = l^'ZP{n\1^p\-^q)z^x'' (7)
\l{\-x){l-a?). . .{l-xfi) = \+^P{n\*\-^q)x'' (8)
\l{l-zx){l-zx'). . .a,^co = \ + ^pln\p\*)zPx'' (9)
1/(1 - z){\-zx){\ -zap) . . .ad 00 = 1 + 2P (7i |>>j9 1 *) z^^" (10)
ll(l-x){l-x^). . .adoo = l + 2P(w|*|*)^" (11)
and so on.
By means of these equations, coupled with the theorems
given in chap, xxx., § 2, and Exercises xxi., a considerable
number of theorems regarding the enumeration of partitions
can be deduced at once.
§ 21.] To find a recurrence-formula for enumerating the
partitions of n into any number of parts none of which exceeds
q; and thus to calculate a table for P{n\* \1^q).
By (8), we have
ll{\-x){\-x''). . .{I- xf^)=l + %P {n\^\-^q) x^.
Hence, multiplying on both sides by 1-af^, and replacing
1/(1 -x){l-x^) . . . (1 - xfl~^) by its equivalent, we derive
l + %P{n\*\-i(>q-\)af'
= l+%{P{n\*\:^q)-P{n-q\*\1^q)]x^ (12),
where we understand P(0, | * |:f>g') to be 1.
Hence, if «<j:g',
P{n\*\1^q) = P{n\*\-^q-l) + P{n-q\^\1^q) (13);
and, if w<5',
P{n\*\1^q) = P{n\*\1^q-l) (14).
By means of (13) and (14) we can readily calculate a table of
double entry for P{n\*\1f>q), as follows : —
558
EULER's table for P {n \ ■» \ :1^ q) CH. XXXV
A
1 2 3
4
5
66 7
c 8
9
10
11 12 13 14
15
16
17 18
19
20
1
1 1 1
1
1
1
1
1
1
1
1111
1
1
1 1
1
1 D
2
. 2 2
3
3
4
4
5
5
6
6 7 7 8
8
9
9 10
10
11
3
. . 3
4
5
7
8
10
12
14
16 19 21 24
27
30
33 37
40
44
4
5
6
9
11
15
18
23
27 34 39 47
54
64
72 84
94
103
5
7
10
13
18
23
30
37 47 57 70
84
101
119 141
164
192
6
11
14
20
26
35
44 i 58 71 90
110
136
163 199
235
282
9
7
15
21
28
38
49 65 I 82 105
131
164
201 248
300
364
8
,
22
29
40
52 70 89 1 116
146
186
230 288
352
434
9
.
30
41
54 73 94 123
157
201
252 318
393
488
10
,
42
55 75 97 128
164
212
267 340
423
530
11
E
•
•
56 76 99 131
F
169
219
278 355
1
445
560
B
a
d
Take a rectangle of squared paper BA G\ and enter the values
of n at the heads of the vertical columns, and the values of q
at the ends of the horizontal lines. We remark, first of all, that
it follows from (14) that all the values in the part of any vertical
column below the diagonal AF are the same. We therefore
leave all the corresponding spaces blank, the last entry in the
column being understood to be repeated indefinitely.
Next, write the values of P{\\*\^\\ P(2l*|>l), . . .,
that is, 1, 1, . . ., in the row headed 1.
To fill the other rows, construct a piece of paper of the form
ahcd. Its use will be understood from the following rule, which
is simply a translation of (13) : —
To fill the blank immediately after the end of any step, add
to the entry above that blank the number which is found at the
left-hand end of the step.
Thus, to get the number 23, which stands at the end of the
step lying on the fourth horizontal line, we add to 14 the number
9, which lies to the immediate left of ah in the same line as
the blank. Again, in the ninth line 157 = 146+11; and
so on.
By sliding ahcd backwards and forwards, so that he always
lies on AB, we can fill in the table rapidly with little chance of
error. We shall speak of the table thus constructed as Euler's
§§ 21-23 ENUMERATIONS EEDUCIBLE TO EULER's TABLE 559
Table. It will be found in a considerably extended form in his
Introductio, Lib. I., chap, xvi.
A variety of problems in the enumeration of partitions can
be solved b)^ means of Euler's Table, as we shall now show.
§ 22.] To find by means of Eulers Table the number of
partitions of n into p parts of unrestricted magnitude.
Let us first consider P{n\p\*). By (9) above, we have
l + %P{n\p\*)x^z^=ll{l-zx){l-zx'). . . ad ao,
= 1 + tx^z^lll -x){l-aP). . .{\- x^),
by Exercises xxi. (18).
Hence
^P{n\p\^)x'' = '^aFl{l-x){l-x'). . .(I-xp),
= SP(7J I *!>/?) a;"+^ by (8).
Therefore
F(n\p\^.) = F{n-p\*\:^p) (15).
Again,
1 + SPw {n\p\*) x'^zP = (1 + zx) (1 + zx') . . . ad oo ,
= 1 + Sari^U'+i) ^p/^ -x)(l-x''). . . (1 - ^),
by chap, xxx., § 2, Example 2.
Hence
^Pu (n\p\i^)x'' = xlP <P+^)/( l-x)(l-x") . . . (l-x^),
= SP (w I * I >j9) ^"+ii'(^+'), by (8).
Therefore
Pu{n\p\*) = P{n-y{p+l)\*\:lf>p) (16).
Examplel. P(20 | 5 | «) = P(15 | « |>5)=84,
Example 2. Pm(20 | 5 | •) = P(5| « |>5)=7.
§ 23.] If we take any partition of n into p parts in which
the largest part is q, and remove that part, we shaU leave a parti-
tion o( n-q into p-1 parts no one of which exceeds q. Hence
we have the identity
P{n\p\q) = P(n-q\p-l\::^q) (17);
and, if we make p infinite, as a particular case, we have
P{n\*\q) = P{n-q\*\:^q) (18).
It will be observed that (18) makes the solution of a certain
class of problems depend on Euler's Table.
5 GO THEOREMS OF CONJUGACY CH. XXXV
By comparing (15) and (18), we have the theorem
P{n\*\q) = P{n\q\*l
which, however, is only a particular case of a theorem regarding
conjugacy, to be proved presently.
§ 24.] Tliem^ems regarding conjugacy.
(I.) P{n\-^p\>q) = P{n\>q\>p) (19).
(II.) P{n-p\q-l\1^p)^P{n-q\p-l\1^q) (20).
(III.) P{n\p\q) = P{n\q\p) (21).
To prove (I.) we observe that, by (7), we have
l + ^P{n\-^p\1^q)z^x^==ll{l-z){l-zx). ..{l-zofi),
„(l-;r'^^')(l-^+^). ..(1-^^P)
-i + ^z (i-x){l-a^)...{l-a^) '
Hgiic6
^T,/ 1^ ,< X ™ (1-3^+0(1-^+'). . .(1-^+^)
^P{n\>p\1^q)x^=' (l-j)(l-.-)...(l-.^) '
{l-x){\-x'). . .{l-afl+P)
~{l-x){\-x')...{l-afl){l-x){l-x')...(\-xP)'
Since the function last written is symmetrical as regards p
and q, it must also be the equivalent of %P{n\'^q\'^p)af^.
Hence Theorem (I.).
Theorem (II.) follows from (6) in the same way.
Since, by (17), we have
P{n\p\q) = P{n-q\p-l\-^q),
P{n\q\p) = P{n-p\q-l\-i^p);
therefore, by (II.),
P{n\p\q) = P{n\q\p),
which establishes Theorem (III.)-
The following particular cases are obtained by making p or
a infinite : —
P{n\-^p\^) = P{n\*\>p) (22);
P{:n\p\*) = P{n\*\p) (23).
§§ 23-26 FURTHER REDUCTIONS TO EULEr's TABLE 561
§ 25.] Tlie following theorems enable us to solve a number
of additional problems by means of Euler's Table : —
P{n\p\-if>q) = P{n-p\*\l!f>p)-^P{n-iH-p\*\>p)
-2P(?^-/^-jo|*|:}>^)
(24).
Here the summations are with respect to fii, /.t2, . . . ; and
fix is any one of the numbers q, q + l, . . . , q+p - 1, ju.2 the sum
of any two of them, /i.3 the sum of any three, and so on. The
series of sums is to be continued so long as n — iXr—p^O. If
P{n\p\:lf>q) come out 0 or negative, this indicates that the
partition in question is impossible.
P{n\:if>p\:!f>q)=P{n\*\::f>p)-:^P(n-v,\^\:^p)
+ %P{n-v^\*\:!^p)
-2P(w-v3|#|:t>j9)
. . . . (25).
Here vi, Vg, . . . have the same meanings with regard to
q + 1, q + 2, . . ., q +p a,8 formerly fii, /j^, . . . with regard to
q, q+1, . . ., q+p-1.
P(n\*\*)
= P{n-l\*\1f>l) + P(n-2\*\:!f>2) + . . . +P(0 | * 1:^^^) (26).
The demonstrations will present no difficulty after what has
already been given above.
CONSTRUCTIVE THEORY OF PARTITIONS.
§ 26.] Instead of making the theory of partitions depend on
series, we might contemplate the various partitions directly, and
develop their properties from their inherent character. Sylvester
has recently considered the subject from this point of view, and
has given what he calls a Constructive Theory of Partitions, which
throws a new light on many parts of the subject, and greatly
simplifies some of the fundamental demonstrations*. Into this
• Amer. Jour. Math. (1882).
c. II. 36
562 GRAPH OF A PARTITION CH. XXXV
theory we cannot within our present limits enter ; but we desire,
before leaving the subject, to call the attention of our readers to
the graphic method of dealing with partitions, which is one of
the chief weapons of the new theory.
By the graph of a partition is meant a series of rows of
asterisks, each row containing as many asterisks as there are
units in a corresponding part of the partition. Thus
* * *
# * * * #
I * * *
is the graph of the partition 3 + 5 + 3 of the number 11.
For many purposes it is convenient to arrange the graph so
that the parts come in order of magnitude, and all the initial
asterisks are in one column. Thus the above may be written —
The graph is then said to be regular.
The direct contemplation of the graph at once
gives us intuitive demonstrations of some of the
foregoing theorems.
For example, if we turn the columns of the graph last
written into rows, we have
where there are as many asterisks as before. The new
graph, therefore, represents a new partition of 11, which
may be said to be conjugate to the former partition-
Thus to every partition of n into p parts the greatest of
which is q, there is a conjugate partition into q parts the
greatest of which is p. Hence
P{n\p\q) = P{n\q\p\
an old result.
Again, to every partition of n into p parts no one of which
exceeds q, there will he a conjugate partition into q or fewer parts
the greatest of which is p. Hence
P{n\p\>q) = P{n\1^q\p) (27),
a new result ; and so on*.
* According to Sylvester (I.e.), this way of proving the theorems of
conjugacy originated with Ferrers.
§^ 2G, 27 EXTENSION AND CONTRACTION OF GRAPHS 563
§ 27.J The following proof, given by Franklin*, of Euler'o
famous theorem that
(l-x)(l-ai'){l-af). . .ada,=l(- )P;»«'^p'^p) (28)t,
is an excellent illustration of the peculiar power of the graphic
method.
The coeflficient of of* in the expansion in question is obviously
Pu {n I even | * ) - Fa {n ] odd | * ) (29).
Let us arrange the graphs of the partitions (into unequal
parts) regularly in descending order. Then the right-hand edge
of the graph will form a series of terraces all having slopes of
the same angle (this slope may, however, consist of a single
asterisk), thus —
A B
* *
if * *
* * *
* * * *
* * * # ■::•
* * * >'f * #
'^ if- ^ % i^ *^
* * * i( * » #
"We can transform the graph A by removing the top row and
placing it along the slope of the last terrace, thus —
, We then have a regular graph A'
representing a partition into unequal parts.
This process may be called contraction.
We cannot transform B in this way;
but we may extend B by removing the
slope of its last terrace, and placing it
above the top row, thus —
j^, We then have a regular graph B repre-
senting a partition into unequal parts.
Every graph can be transformed by con-
traction or by extension, except when the top
row meets the slope of the last terrace ; and in
this case also, provided it does not happen that
the number of asterisks in the top row is equal
• Comptes Rendus (1880).
+ Euler originally discovered this theorem by induction from particular
cases, and was for long unable to prove it. For other demonstrations, sea
Legeudre, TMorie des Noinhrcs, t. n., § 15, and Sylvester {l.c.).
36—2
564 franklin's proof of EULER's expansion CH. XXXV
to the number in the last slope or exceeds it only by one,
as, for example, in
;,; * * * * * *
* * >¥ * ■»**■:!■ ■»
* ♦ # * * ******
Contraction or extension in the first of these would produce
an irregular graph ; contraction in the second would produce an
irregular graph ; and extension would produce a graph which
corresponds to a partition having two parts equal. These two
cases may be spoken of as unconjugate ; they can only arise when
the p parts of the partition are
p, p+1, p + 2, . . ., 2;?-l,
and the number
n=p + {p+l) + . . . +(2i?-l) = ^(:V-jp);
or when the p parts are
^+1, p + 2, p + 3, . . ., 2p,
and
n = {p + l) + (p + 2)+ . . . +2p = ^{3p'' +p).
Since contraction or extension always converts a partition
having an even or an odd number of parts into one having
an odd or an even number of parts respectivel}'^, we see
that, unless n be a number of the form i{Sp^±p),
Pu (n I even | * ) = Pu (n \ odd [ * ).
When n has one or other of the forms | {^p^±p), there will
be one unconjugate partition which will be even or odd
according as p is even or odd ; all the others will occur in pairs
which are conjugate in Franklin's transformation. Hence
Pu (i (3j3^ ±p) I even \*)-Pu(^ (Sjo^ ±i>) 1 odd | * ) = ( - 1)" (30).
Euler's Theorem follows at once.
Exercises XXXVIII.
(1.) Show how to evaluate Fu(n\ >p\*) by means of Euler'e Table.
Evaluate
(2.) P(13|5|>3). (3.) P(13|>6|>3).
(4.) P(10|.|»). (5.) P(20|9|l>3).
§ 27 EXERCISES XXXVIII 565
Establish the following : — ■
(6.) Pu (n I « I • ) = P (M - i? (g + 1) I * 1 1> q), where lq{q + l) just > n.
(7.) Pu{n\v\*)=P(n-\p{Tp-\)\p\*).
(8.) P(n|p|») = Pw(n + ip(i)-l)|i)|*).
(9.) Pu{n\p\>q) = P{n-iiP(p-l)\p\>q-p + l).
(10.) Is the theorem P{n-p\q~\\*):=P{n-q\p-l\*) universally
true?
(11.) Show how to form a table for the values of P (n | * | 2, 3, . . ., q).
(See Proc. Edinb. Math. Sec, 188.3-4.)
(12.) Show how to form a table for the number of partitions of n into an
indefinite number of odd parts.
Establish the following : —
(13.) P(ra|«|l, 2, 22,23,. . .) = 1.
(14.) Pu(n|p|l, 3, . . .,23-l)=P(7i-2)2+p|p|l, 3, . ..,2<7-l).
(15.) P(n|j>|2, 4, . . ., Q.q) = P{n-p\p\l, 3, . . ., 2q-l).
(16.) P(H|*|odd)=Pu(H|*|«).
(17.) P(n\>p\2,4:, . . .,2g)=P(M|}>gl2, 4, . . .,2p).
(18.) P(w+p1p11, 3, . . .,2g + l)=P(ji + g|(z|l, 3 2^-fl).
(19.) Pu(n + p^\p\l, 3, . . ., 2q + l)=Pii.{n + q-\q\l, 3, . . .,2p + l).
(20.) P(n^2p\p\2, 4, . . ., 2g' + 2) = P(u + 23 | (^ | 2, 4, . . ., 2i> + 2).
(21.) Show that P {n\p\*) = P{n-l\p-l\*) + P(n~p\p\*)\ and
hence construct a table for P (;t |2> | «). (See Whitworth, Choice and Chance,
chap, in.)
CHAPTEK XXXVI.
Probability, or the Theory of Averages.
§ 1.] An elementary account of the Theory of Probability,
or, as we should prefer to call it, the Theory of Averages, has
usually found a place in English text-books on algebra. This
custom is justified by several considerations. The theory in
question affords an excellent illustration of the application of the
theory of permutations and combinations which is the funda-
mental part of the algebra of discrete quantity ; it forms in its
elementary parts an excellent logical exercise in the accurate use
of terms and in the nice discrimination of shades of meaning ;
and, above all, it enters, as we shall see, into the regulation of
some of the most important practical concerns of modern life.
The student is probably aware that there are certain occur-
rences, or classes of events, of such a nature that, although we
cannot with the smallest degTce of certainty assert a particular
proposition regarding any one of them taken singly, yet we can
assert the same proposition regarding a large number iV of them
with a degree of certainty which increases (with or without limit,
as the case may be) as the number N increases.
For example, if we take any particular man of 20 years of age,
nothing could be more uncertain than the statement that he will
live to be 25 ; but, if we consider 1000 such men, we may assert
with considerable confidence that 96 per cent, of them will live to
be 25 ; and, if we take a million, we might with much greater con-
fidence assign the proportion with even closer accuracy. In so
doing, however, it would be necessary to state the limits both of
habitat and epoch within which the men are to be taken ; and,
even with a million cases, we must not expect to be able to assign
§ 1 DEFINITION OF PROBABILITY 567
the proportion of those who survive for 5 years with absolute
accuracy, but be prepared, when we take one million with
another, to find occasional small fluctuations about the indicated
percentage.
We may, for illustration, indicate the limits just spoken of
by saying that "man of 20" is to mean a healthy man or
woman living in England in the 18 th century. The " event,"
as it is technically called, here in question is the living for 5
years more of a man of 20 ; the alternative to this event is not
living for 5 years more. The whole, made up of an event and
its alternative or alternatives, we call its universe. The alternative
or alternatives to an event taken collectively we often call the
Complementary Event. The living or not living of all the men
of 20 in England during the 18th century we may, following
Mr Venn*, call the series of the event. It will be observed
that on every occasion embraced by the series the event we are
considering is in question ; and we express the above result of
observation by saying that the probability that a man of 20
living under the assigned conditions reached the age of 25 is '96.
We are thus led to the following abstract definition of the
Probability or Chance of an Event : —
I/on taking any very large number N out of a series of cases
in which an event A is in question, A happens on pN occasions,
the probability of the event A is said to be p.
In the framing of this definition we have, as is often done in
mathematical theories, substituted an ideal for the actual state
of matters usually observed in nature. In practice the number
p, which for the purposes of calculation we suppose a definite
quantity, would fluctuate to an extent depending on the nature
of the series of cases considered and on the number N of specimen
cases selected!. Moreover, the mathematical definition contains
no indication of the extent or character of the series of cases.
* Logic of Chance.
t We might take more explicit notice of this point by wording the
definition thus: — "If, on the average, inN out of a series of cases, dkc."
But, from the point of view of the ideal or »iathematical theory, nothing
would thus be gained.
568 REMARKS ON THE DEFINITION CH. XXXVI
How far the possible fluctuations of p, the extent of the series,
and the magnitude of N will affect the bearing of any con-
clusion on practice must be judged by the light of circumstances.
It is obvious, for instance, that it would be unwise to apply to
the 14th century the probability of the duration of human life
deduced from statistics taken in the 18th. This leads us also to
remark that the application of the theory of probability is not
merely historical, as the definition might suggest. Into most of
the important practical applications there enters an element of
induction*'. Thus we do in fact apply in the 19th century a
table of mortality statistics deduced from observations in the
18th century. The warranty for this extension of the series of
cases by induction must be sought in experience, and cannot in
most cases be obtained a priori.
There are, however, some cases where the circumstances are
so simple that the probability of the event can be deduced,
without elaborate collecting and sifting of observations, merely
from our definition of the circumstances under which the event
is to take place. The best examples of such cases are games of
hazard played with cards, dice, &c. If, for example, we assert
regarding the tossing of a halfpenny that out of a large number
of trials heads will come up nearly as often as tails — in other
words, that the probability of heads is ^, what we mean thereby
is that all the causes which tend to bring up heads are to
neutralise the causes that tend to bring up tails. In every
series of cases in question, the assumption, well or ill justified,
is made that this counterbalancing of causes takes place. That
this is really the right point of view will be best brought home
to us if we reflect that undoubtedly a machine could be con-
structed which would infallibly toss a halfpenny so as always
to land it head-up on a thickly sanded floor, provided the coin
were always placed the same way into the machine ; also, that the
coin might have two heads or two tails ; and so on.
In cases where the statement of probability rests on grounds
so simple as this, the difficulty regarding the extension of the
series by induction is less prominent. The ideal theory in such
• In the proper, logical sense of the word.
^ 1, 2 COROLLARIES ON THE DEFINITION 669
cases approximates more closely than usual to the actual circum-
stances. It is for this reason that the illustrations of the
elementary rules of probability are usually drawn from games of
hazard. The reader must not on that account suppose that the
main importance of the theory lies in its application to such
cases ; nor must he forget that its other applications, however
important, are subject to restrictions and limitations which are
not apparent in such physically simple cases as the theory of
cards and dice.
Before closing this discussion of the definition of probability
as a mathematical quantity, it will be well to warn the learner
that probability is not an attribute of any particular event
happening on any particular occasion. It can only be predicated
of an event happening or conceived to happen on a very large
number of " occasions," or, in popular language, of an event " on
the average" or in the "long run." Unless an event can happen,
or be conceived to happen, a great many times, there is no sense
in speaking of its probability, or at least no sense that appears to
us to be admissible in the following theory. The idea conveyed
by the definition here adopted would be better expressed by
substituting the word frequency for the word probability ; but,
after the above caution, we shall adhere to the accepted term.
§ 2.] The following corollaries and extensions may be added
to the definition.
Cor. 1. If the 'probability of an event be p, then out of N
cases in which it is in question it will happen pN times, N being
any very large number*.
This is merely a transposition of the words of the definition.
As an example, let it be required to find the number out of 5000 men of
20 years of age who will on the average live to be 25. The probability of a
man of 20 living to be 25 may be taken to be '96; hence the number
required is -96x5000 = 4800.
Cor. 2. If the probability of an event be jt?, the probability of
its failing is 1-p.
For out of a large number iV of cases the event will happen
on pN occasions ; hence it will fail to happen on N-pN
* It is essential that pN also be a very large number. See Simmons,
Pj-oc. L. M. S., XXVI., p. 307 (1895).
570 COROLLARIES ON THE DEFINITION CH. XXXVI
= (1 -p) N occasions. Hence, by the definition, the probability
of the failing of the event is 1-p.
Cor. 3. If the universe of an event he made up ofn alternatives,
or, in other words, if an event must happen and that in one out of
n ways, and if the respective probabilities of its happening in these
ways be pi, Pi, . . .,pn, thenpi+p, + . . .+pn=l.
For on every one of N occasions the event will happen ; and
it will happen in the first way on p^N occasions, in the second on
p^N' occasions, and so on. Hence N^piN^-p^N^ . . .+pnN;
that is, 1 =pi +P2 + . . . +Pn'
Cor. 4. Ifati event is certain to happen, its probability is 1 ;
if it is certain not to happen, its probability is 0.
For in the former case the event happens on 1 . N cases out
of N cases ; in the latter on 0 . iV cases out of N.
The probability of every event is thus a positive number
lying between 0 and 1.
Cor. 5. If an event must happen in one out of n ways all
equally probable, or if one out ofn events must happen and all are
equally probable, then the probability of each way of happening in
the first case, or of each event happening in the second, is Ijn.
This follows at once from Cor. 3 by making p^ ^p^ = . . . = j9„.
As a particular case, it follows that, if an event be equally
likely to happen or to fail, its probability is \.
Definition. — The ratio of the probability of the happening of
an event to the probability of its failing to happen is called ths
odds in favour of the event, and the reciprocal of this ratio is called
the odds against it.
Thus, if the probability of an event be p, the odds in favour
are p:\-p', the odds against 1-p -.p. Also, if the odds in
favour be m : w, the probability of the event is ml{m + n). If the
probability of the event be \, that is, if it be equally likely to
happen or to fail, the odds in favour are 1:1, and are said to
be even.
Cor. 6. If the universe of an event can be analysed into m + n
cases each of which in the long run will occur equally often*, and
* Tliis is usually expressed by saying that all the cases are equally likely.
§§ 2, 3 DIRECT CALCULATION OF PROBABILITIES 571
if ill m of these cases the event will happen and in the remaining
nfail to happen, the probability of the event is m/{m + n).
After what has been said this will be obvious.
DIRECT CALCULATION OF PROBABILITIES. .
§ 3.] The following examples of the calculation of proba-
bilities require no special knowledge beyond the definition of
probability and the principles of chap, xxiii.
Example 1, There are 5 men ia a company of 20 soldiers who have
made up their minds to desert to the enemy whenever they are put on
outpost duty. If 3 men be taken from the company and sent on outpost
duty, what is the probability that all of them desert ?
The 3 men may be chosen from among the 20 in j^Cj ways, all of which
are equally likely. Three deserters may be chosen from among the 5 in 5C3
ways, all equally likely. The probability of the event in question is therefore
8^s/2oW-i 2.3/ 1.2.3 -^'^^*-
Example 2. If n people seat themselves at a round table, what is the
chance that two named individuals be neighbours ?
There are (see chap, xxiii., § 4) (n-l)l different ways, all equally likely,
in which the people may seat themselves. Among these we may have A and B
or B and 4 together along with the (n-2)! different arrangements of the
rest ; that is, we have 2 (n - 2)! cases favourable to the event and all equally
likely. The required chance is therefore 2 (n- 2)!/(?i- l)!=2/(ji-l).
When n=3, this gives chance =1, as it ought to do. The odds against
the event are in general n - 3 to 2 ; the odds will therefore be even when the
number of people is 5.
Example 3. If a be a prime integer, and n=a'", and if any integer Iii>n
be taken at random, find the chance that I contains a as a factor s times
and no more.
The integer I must be of the form Xa», where X is any integer less than
a*"-* and prime to a*"-'. Now, by chap, xxxv., § 8, the number of integers
less than a^~* and prime to it is a'"~*(l - 1/a). Also the number of integers
> n is oT. Hence the required chance is a*"-* (1 - Ha)la^=a~* (1 - 1/a) = 1/a*
- Ha^\
Example 4. Find the probability that two men A and B oim and n years
of age respectively both survive for p years.
The mortality tables (see § 15 below) give us the numbers out of 100,000
individuals of 10 years of age who complete their mth, Hth, m + pth, n+pth
years. Let these numbers be Z^, Z„, Im+p, i„+p. The probabilities that A
and B live to be m+p and n+p years of age respectively are Im+pl^my 'nWn
respectively. Consider now two large groups of men numbering M and N
respectively. We suppose A to be always selected from the first and B always
572 DIRECT CALCULATION OF PROBABILITIES CH. XXXVI
from the second. In this way we could select altogether MN pairs of men
who may be alive or dead after p years have elapsed. The number out of
the M men living after p years is Ml^+pjl^, by § 2, Cor. 1. Similarly the
number living out of the N men is Nln+pjl^. Out of these we could form
MNl^^pln+pjlm^n pairs. This last number will be the number of pairs
of survivors out of the MN pairs with which we started. Hence the
probability required is im+pWp/^m'»=('m+p/^m) (Wp/^n); in other words, it
is the product of the probabilities that the two men singly each survive for
p years. The student should study this example carefully, as it furnishes a
direct proof of a result which would usually be deduced from the law for
the multiplication of probabilities. See below, § 6.
Example 5. A number of balls is to be drawn from an urn, 1, 2, . . ., n
being all equally likely. What is the probability that the number drawn
be even?
We can draw 1, 2, . . ,, « respectively in ^Cj, ^Cj, . . ., „(7„ ways
respectively. Hence we may consider the universe of the event as consisting
of „Ci + ^Ca + . . . + nC„ =(1 + 1)™-! = 2" -1 equally likely cases. The number
of these in which the drawing is even is „C2 + „C4+ . . .=^{(1 + 1)"
+ (1 - 1)" - 2} = 1(2" - 2) = 2™-i - 1. The number of ways in which an odd
drawing can be made is ^(^1 + ^03+. . . =i {(1 + 1)"- (1-I)''} = i2"=2"-^
Hence the chance that the drawing be even is (2"~i- l)/(2"-l), that it be
odd 2'*-'/(2"- 1)- The sum of these is unity, as it ought to be; since, if
the drawing is not odd, it must be even. In general, an odd drawing is more
likely than an even drawing, the odds in its favour being 2"~* : 2""! - 1 ; but
the odds become more nearly even as n increases.
Example 6. A white rook and two black pawns are placed at random on
a chess-board in any of the positions which they might occupy in an actual
game. Find the ratio of the chance that the rook can take one or both of
the pawns to the chance that either or both of the pawns can take the rook.
Let us look at the board from the side of white ; and calculate in the first
place the whole number of possible arrangements of the pieces. No black
pawn can lie on any of the front squares ; hence we may have the rook on
any of these 8 and the two pawns on any two of the remaining 56 ; in all,
8 X 2 jigC2= 8 X 56 X 55 arrangements. Again, we may have the rook on any one
of the 56 squares and the two pawns on any two of the remaining 55 squares ;
in all, 56 X 55 X 64 arrangements. The universe may therefore be supposed
to contain 62 x 56 x 55 equally likely cases.
Instead of calculating the chance that the rook can take either or both of
the pawns, it is simpler, as often happens, to calculate the chance of the
complementary event, namely, that the rook can take neither of the pawns.
If the rook lie on one of the front row of squares, neither of the pawns can
lie on the corresponding column, that is, the pawns may occupy any two out
of 49 squares ; this gives 8 x 49 x 48 arrangements. If the rook lies in any
one of the remaining 56 squares, neither of the pawns must lie in the row or
column belonging to that square; hence there are for the two pawns 42 x 41
positions. We thus have 56 x 42 x 41 arrangements. Altogether we have
§§ 3, 4 DIRECT CALCULATION OF PROBABILITIES 573
8 X 49 X 48 + 56 X 42 X 41 = 56 X 49 X 42 arrangements in which the rook can
take neither pawn. Hence the chance that the rook can take neither pawn
is 56 X 49 X 42/62 x 56 x 55 = 1029/1705, The chance that the rook can take
one or both of the pawns is therefore 1 - 1029/1705 = 676/1705.
Consider now the attack on the rook. If he is on a side square, he can
only be attacked by either of the two pawns from one square. For the side
squares we have therefore only 24 x 54 arrangements in which the rook can
be taken. There remain 36 squares on each of which the rook can be taken
from two squares, that is, in 6 ways. For the 36 squares we therefore have
36 X 2 + 36 X 4 X 53 arrangements in which the rook can be taken by one or by
both the pawns. Altogether there are 9000 arrangements in which the rook
may be taken. Hence the chance that he be in danger is 9000/62 x 56 x 55 =
225/4774, The ratio of the two chances is 9464 : 1125.
§ 4.] A considerable number of interesting examples can be
solved by the method of chap, xxiii., § 15. Let there be r bags,
the first of which contains cfi, 6i, Cj, . , ,, h^ counters, marked
with the numbers «!, /3i, yi, . . ., /Ci; the second, a^,h^, Cj, . . .k^,
marked o^, P%,y^, . . ., K-i', and so on. If a counter be drawn
from each bag, what is the chance that the sum of the numbers
drawn is w ?
By chap, xxiii., § 15, the number of ways in which the sum
of the drawings can amount to n is the coefficient, An say, of af^
in the distribution of the product
(ai^;"' + haf' + , , . + ^i.^"')
X {a^af^ + b^a^^ + , . . + k^of')
X {UrX'^ + brO^' + . . . + hrX"'').
Again, the whole number of drawings possible is the sum of
all the coefficients ; that is to say,
(«! + 6i + . . . + ^i)
X (aa + 62 + . . . + ita)
X (ar + ^r + • • • + ^v) = A say.
Hence the required chance is AJD.
Example 1. A throw has been made with three dice. The sum is known
to be 12 ; required the probability that the throw was 4, 4, 4.
The number of ways in which 12 can be thrown with three dice is the
coefficient of x^^ in
(x^-^x^ + x^ + x^ + x^-^x^f,
574 DIRECT CALCULATION OF PROBABILITIES CH. XXXVl
tliat ia to say, of x^ in
{I + X + x'^+ x^ + X* + x^)K
Now the coefficients in (l + x+ . • . +x^y^ up to the term in x^ are (see
chap. IV., § 15) 1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2. Hence the coefficient of x*
in the cube of the multinomial is 5 + 6 + 5 + 4 + 3 + 2=25.* The required
probability is therefore 1/25.
Example 2. One die has 3 faces marked 1, 2 marked 2, and 1 marked 3;
another has 1 face marked 1, 2 marked 2, and 3 marked 3. What is the
most probable throw with the two dice, and what the chance of that throw?
The numbers of ways in which the sums 2, 3, 4, 5, 6 can be made are the
coefficients of x'^, a;*, x^, x^, x^ in the expansion of (3a; + 2a;^ + x") {x + 2x^ + 3x'').
Now this product is equal to
3x2 + 8x3 ^ 14^4 + sx" + 3x6.
The sum that will occur oftenest in the long run is therefore 4. The
whole number of different ways in which the different throws may turn out
is (3 + 2 + 1) (1 + 2 + 3) = 36. Hence the probability of the sum 4 is 14/36
= 7/18.
Example 3. An urn contains m counters marked with the numbers
1, 2, . . ., m. A counter is drawn and replaced r times; what is the
chance that the sum of the numbers drawn is n?t
The whole number of possible different drawings is ni'*.
The number of those which give the sum n is the coefficient of a;" in
(x + x2+. . . + x"»)^ that is to say, of x"-*' in (l + x+. . . + x'^-y. Now
1 + X + . . . + x™"^ = (1 - x'^)l(l - x). We have therefore to find the coefficient
of x"-*" in
(l-x'»)'-(l-.T)-'-={l-^CiX'" + rC'2X2«-rC3x3"*+. . .}
'^r+i''+ 1.2 "^ + 1.2.3 '^+- • -f*
The coefficient in question is
_r(r + l). . .(«-!) r(r + l). . .(w-m-l)r
^ n-r - (jT^) ! (n -r-vi)[ 11
r()+l). . .(7i-27?t-l)r(r-l)
"*" (7i-j--2m)!21 -• . . .
The required probability is ^„_r/m'".
Example 4. If m odd and n even integers (n<i.m- 1) be written down at
random, show that the chance that no two odd integers are adjacent is
n! {n + l)\l(m + n)\{n-m + l)l.
In order to find in how many different ways we can write down the
integers so that no two odd ones come together, we may suppose the m odd
integers written down in any one of the ml possible ways, and consider the
VI -1 spaces between them together with the two spaces to the right and left
of the row. The problem now is to find in how many ways we can fill the
* We might also have found the coefficient of x* by expanding
(1 - x')3 (1 - x)~8, as in Example 4 below.
t This is generally called Demoivre's Problem. For an interesting aocoant
of its history see Todhunter, Hist. Frob., pp. 59, 85.
§§ 4, 5 ADDITION RULE 575
n even integers into the spaces so that there shall always be one at least in
every one of the m-1 spaces. A little consideration will show that the
number of ways, irrespective of order, is the coefficient of x" in
{l + x + x^ + . . . ad Qo)2(x + x2 + . . .adoo)"*-!;
that is, of a;»-"»+i in {l+x + x^+. . .)^(l + x + x^+. . O"*"!;
that is, of a;»-"»+i in (1 - a;)-("»+>).
This coefficient is
(m + l)(m + 2). . .(n + 1)^ (« + !)!
{n-m + l)l ~m!(n-m + l)I*
If we remember that every distribution of the n integers among the m + 1
spaces can be permutated in n\ ways, we now see that the number of ways
in which the m + n integers can be arranged as required is
n\m\ (n+l)!/m! (7i-m + l)! = n!(n + l)!/(n-m + l)I.
The whole number of ways in which the m + n integers can be arranged is
{m + 7i)\, hence the probability required is n!(/i + l)!/(ri- wi + l)!(m + n)!.
ADDITION AND MULTIPLICATION OF PROBABILITIES.
§ 5.] In many cases we have to consider the probabilities of
a set of events which are of such a nature that the happening of
any one of them upon any occasion excludes the happening of
any other upon that particular occasion. A set of events so
related are said to be mutually exclusive. The set of events
considered may be merely different ways of happening of the
same event, provided these ways of happening are mutually
exclusive.
In such cases the following rule, which we may call the
Addition Ruh, applies : —
If the probabilities of n mutually exclusive events be p^, p^,
. . ., Pn, the chance that one out of these n events happens on any
particulwr occasion on which all of them a/re in question ispi+p2 +
. . . +Pn'
To prove this rule, consider any large number N of occasions
where all the events are in question. Out of these N occasions
the n events will happen on p^N, p-^N, . . ., Pn^ occasions re-
spectively. There is no cross classification here, since no more
than one of the events can happen on any one occasion. Out of
iV^ occasions, therefore, one or other of the n events will happen
on piN + P2N + . . . + Pn^= (pi +P2 + • • • + Pn) ^ occasions.
Hence the probability that one out of the n events happens on
any one occasion is pi +p2 + . . . +Pn-
576 MULTIPLICATION RULE CH. XXXVI
It should be observed that the reasoning would lose all force
if the events were not mutually exclusive, for then it might be
that on the j9iiV occasions on which the first event happens one
or more of the others happen. We shall give the proper formula
in this case presently.
As an illustration of the application of this rule, let us suppose that a
throw is made with two ordinary dice, and calculate the probability that the
throw does not exceed 8. There are 7 ways in which the event in question
may happen, namely, the throw may be 2, 3, 4, 5, 6, 7, or 8 ; and these ways
are of course mutually exclusive. Now (see § 4, Example 1) the probabilities
of these 7 throws are 1/36, 2/36, 3/36, 4/36, 5/36, 6/36, 5/36 respectively.
Hence the probability that a throw with two dice does not exceed 8 is
(l + 2 + 3 + 4 + 5 + 6 + 5)/36=26/36 = 13/18.
§ 6.] When a set of events is such that the happening of
any one of them in no way affects the happening of any other,
we say that the events are mutually independent. For such a set
of events we have the following Multiplication Rule : —
Xf the respective probabilities of n independent events he pi,
Ps, ■ • -yPn, the probability that they all happen on any occasion
in which all of them are in question is piPi . . . pn-
In proof of this rule we may reason as follows : — Out of
any large number iV of cases where all the events are in question,
the first event will happen on piN' occasions. Out of these j9iiV
occasions the second event will also happen on p^iPiN^ =PiP2^
occasions ; so that out of N there are p^p^N occasions on
which both the first and second events happen. Continuing
in this way, we show that out of N occasions there are
jt7ijt?2 . . . PnN occasions on which all the n events happen.
The probability that all the n events happen on any occasion
is therefore Pi.p-i . . . pn-
It should be noticed that the above reasoning would stand
if the events were not independent, provided p^ denote the
probability that event 2 happen after event 1 has happened, p^.
the probability that 3 happen after 1 and 2 have happened, and
so on.
It must be observed, however, that the probability calculated
is then that the events happen in the order 1, 2, 3, . . ., w.
Hence the following conclusion : —
§§5-7 EXAMPLES OF ADDITION AND MULTIPLICATION 577
Cor. If the events 1, 2, . . ., n be interdependent and pi
denote the probability of \,p2 the probability that 2 happen after
1 has happened, ps the probability that 3 happen after 1 and 2
have happened, and so on, then tlie probability that the events
1, 2, . . .,n happen in the order indicated isp^p-i . . . Pn-
As an illustration of the multiplication rule, let us suppose that a die is
thrown twice, and calculate the probability that the result is such that the
first throw does not exceed 3 and the second does not exceed 5.
The probability that the first throw does not exceed 3 is, by the addition
rule, 3/6 ; the probability that the second does not exceed 5 is 5/6. The result
of the first throw in no way affects the result of the second ; hence the
probability that the result of the two throws is as indicated is, by the
multiplication rule, (3/6) x (5/6) = 5/12.
As an example of the effect of a slight alteration in the wording of the
question, consider the following : — A die has been thrown twice : what is the
probability that one of the throws does not exceed 3 and the other does not
exceed 5 ?
Since the particular throws are now not specified, the event in question
happens — 1st, if the first throw does not exceed 3 and the second does not
exceed 5 ; 2ud, if the first throw is 4 or 5 and the second does not exceed 3.
These cases are mutually exclusive, and the respective probabilities are 5/12
and 1/6. Hence, by the addition rule, the probability of the event in question
is 7/12.
§ 7.] The following examples will illustrate the application
of the addition and multiplication of probabilities.
Example 1. One urn. A, contains m balls, p7i being white, (l-^)m black;
another, B, contains n balls, qn white, {l-q)n black. A person selects one of
the two urns at random, and draws a ball ; calculate the chance that it be
white ; and compare with the chance of drawing a white ball when all the
VI + n balls are in one urn.
There are two ways, mutually exclusive, in which a white ball may be
drawn, namely, from A or from B.
The chance that the drawer selects the urn A is 1/2, and if he selects that
urn the chance of a white ball is p. Hence the chance that a white ball is
drawn from A is (§ 6, Cor.) ^p. Similarly the chance that a white ball
is drawn from B is \q. The whole chance of drawing a white ball is there-
fore (iJ + g)/2.
If all the balls be in one urn, the chance is {pm + qn)l{m-\-n).
Now (pm-{-qn)l{m-\-n)> = <[p + q)l2,
according as 2{pjn-\-qn)> = <{p + q) [m+n),
according as (m-n){p-q)> = <0.
Hence the chance of drawing a white ball will be unaltered by mixing if
either the numbers of balls in A and B be equal, or the proportion of white
balls in each be the same.
c. II. 37
578 EXAMPLES OF MULTIPLICATION AND ADDITION CH. XXXVI
If the number of balls be unequal, and the proportions of white be un-
equal, then the mixing of the balls will increase the chance of drawing a
white if the urn which contains most balls have also the larger proportion of
white; and will diminish the chance of drawing a white if the urn which
contains most balls have the smaller proportion of white.
De Morgan* has used a particular case of this example to point out the
danger of a fallacious use of the addition rule. Let us suppose the two urns
to be as follows: A (3 wh., 4 bl.) ; B (4 wh., 3 bl.). We might then with
some plausibility reason thus ; — The drawer must select either A or B. If he
select A., the chance of white is 3/7 ; if he select B, the chance of white is
4/7. Hence, by the addition rule, the whole chance of white is 3/7 + 4/7 = 1.
In other words, white is certain to be drawn, which is absurd. The mistake
consists in not taking account of the fact that the drawer has a choice of urns
and that the chance of his selecting A must therefore be multiplied into his
chance of drawing white after he has selected A. The chance should there-
fore be 3/14+4/14=1/2.
The necessity for introducing the factor 1/2 will be best seen by reasoning
directly from the fundamental definition. Let us suppose the drawer to make
the experiment any large number ^ of times. In the long run the one urn
will be selected as often as the other. Hence out of iV times A will be selected
JY/2 times. Out of these 2^/2 times white will be drawn from A (3/7) (^/2)
= JV (3/14) times. Similarly, we see that white will be drawn from B 2^(4/14)
times. Hence, on the whole, out of 2^ trials white will be drawn
(3/14 + 4/14)2^ times. The chance is therefore 3/14 + 4/14.
Example 2. Four cards are drawn from an ordinary pack of 52 ; what is
the chance that they be all of different suits ?
We may treat this as an example of § 6, Cor. The chance that the
first card drawn be of one of the 4 suits is, of course, 1. The chance, after one
suit is thus represented, that the next card drawn be of a different suit is,
since there are now only 3 suits allowable and only 51 cards to choose
from, 3.13/51. After two cards of different suits are drawn, the chance that
the next is of a different suit is 2.13/50. Finally, the chance that the last
card is of a different suit from the first three is 13/49. By the principle just
mentioned the whole chance is therefore 3.13.2.13.13/51.50.49=133/17.25.49
= 1/10 roughly.
Example 3. How many times must a man be allowed to toss a penny in
order that the odds may be 100 to 1 that he gets at least one head ?
Let X be the number of tosses. The complementary event to " one head
at least " is " all tails." Since the chance of a tail each time is 1/2, and the
result of each toss is independent of the result of every other, the chance of
«' all tails " in x tosses is (1/2)*. The chance of one head at least is therefore
1 - (1/2)='. By the conditions of the question, we must therefore have
1- (1/2)* =100/101;
* Art. "Theory of Probability," £;jcy. Metrop. Eepublished £7icy. Pwra
Math. (1847), p. 399.
§ 7 EXAMPLES OF MULTIPLICATION AND ADDITION 579
hence 2* =101,
a;=logl01/log2,
=2-0043/-3010,
=6-6 ....
It appears, therefore, that in 6 tosses the odds are less than 100 to 1, and in
7 tosses more.
Example 4. A man tosses 10 pennies, removes all that fall head up;
tosses the remainder, and again removes all that fall head up ; and so on.
How many times ought he to be allowed to repeat this operation in order
that there maj' be an even chance that before he is done all the pennies have
been removed?
Let X be the number of times, then it is clearly necessary and sufficient
for his success that each of the 10 pennies shall have turned up head at least
once. The chance that each penny come up head at least once in x trials is
1 - (1/2)*. Hence the chance that each of the 10 has turned up heads at least
once is {l-(l/2)"^}^<*. By the conditions of the problem we must therefore
have
{l-(l/2)*}io = l/2;
■ (1/2)*= l-(l/2)iAo = . 06697;
a;=-log •06697/log2,
= 3-9 very nearly.
Hence he must have 4 trials to secure an even chance.
Example 5. A man is to gain a shilling on the following conditions. He
draws twice (replacing each time) out of an urn containing one white and one
black ball. If he draws white twice he wins. If he fails a black ball is added,
he tries twice again, and wins if he draws white twice. If he fails another
black ball is added; and so on, ad infinitum. What is his chance of gaining
the shilling? (Laurent, Calcul des ProhahiliUs (1873), p. 69.)
The chances of drawing white in the various trials are 1/2^, 1/3^, . . .
1/n^, . . . The chances of failing in the various trials are 1 - 1/2^,
1 - 1/32, . . . , 1 - l/ra", . . . Hence the chance of failing in all the trials
is (1 - 1/22) (1 _ iy32) . . . (1 _ i/,i2) ... ad 00 .
Now
.L^4)('4^)-(-.^)
_ ^ {1.3}{2.4} . . . {(n-3)(n-l)}{(«-2)n}{(n-l)(n + l)}
12.22...n2 '
= L
_ n{n + l)
The chance of failing to gain the shilling is therefore 1/2. Hence the chance
of gaining the shilling is 1/2.
We might have calculated the chance of gaining the shilling directly, by
37—2
580 EXAMPLES OF MULTIPLICATION AND ADDITION CH. XXXVI
observing that it is the sum of the chances of the following events : 1°,
gaining in the first trial; 2°, failing in 1st and gaining in 2nd; 3°, failing
in 1st and 2nd and gaining in the 3rd ; and so on. In this way the chance
presents itself as the following infinite series:—
The sum of this series to infinity must therefore be 1/2. That this is so may
be easily verified. The present is one example among many in which the
theory of probability suggests interesting algebraical identities.
Example 6. A and B cast alternately with a pair of ordinary dice. A
wins if he throws 6 before B throws 7, and B if he throws 7 before A throws
6. If A begin, show that his chance of winning : B's=SO : 31. (Huyghens,
De Ratiociniis in Ludo Alece, 1657.)
Let p and q be the chances of throwing and of failing to throw 6 at a
single cast with two dice ; r and s the corresponding chances for 7.
A may win in the following ways : 1°, A succeed at 1st throw ; 2°, A fail
at 1st, B fail at 2nd, A succeed at 3rd ; and so on. His chance is therefore
represented by the following infinite series : —
p + qsp + qsqsp + . . .=p {l + {qs) + {qs)^ + . . .},
=PI(1 - qs),
B may win in the following ways : — 1°, A fail at 1st, B succeed at 2nd ;
2°, A fail at 1st, B fail at 2nd, A fail at 3rd, B succeed at 4th ; and so on.
His chance is therefore
qr + qsqr + qsqsqr + . . . = qr{l + (qs) + (qsY + . . .},
= qrl{l-qs).
Hence A's chance : B^s=p : qr.
Now (see § 4, Example 1) iJ = 5/36, g = 31/36, r=6/36 ; hence
^'s chance : B's=5/36 : 6 . 31/362,
= 30:31.
For Huyghens' own solution see Todhunter, Hist. Prob., p. 21.
Example 7. A coin is tossed m + n times {m>n). Prove that the chance
of at least m consecutive heads appearing is {n + 2)/2"*"*'i.
The event in question happens if there appear — 1st, exactly m; 2nd,
exactly m + 1; . . .; (n + l)th, exactly m + n consecutive heads.
Now a run of exactly m consecutive heads may commence with the 1st,
2nd, 3rd, n-lth, nth, n + lth throw. Since m>n, there cannot be more
than one run of m or more consecutive heads, so that the complication due
to repetition of runs does not occur in the present problem. The chances
of the first and last of these cases are each 1/2"*+^, the chances of the others
1^2"»+2^ Hence the chance of a run of exactly m consecutive heads is
2/2"»+i + (n - 1)/2'»+2 = (n + 3)/2»^+2.
In like manner, we see that the chance of a run of m + 1 consecutive
heads is (n + 2)/2"'+^; and so on, up to m + 7i-2. Also the chances of a run
of exactly m + n - 1 and of exactly vi + n consecutive heads are l/2'"+'»-i and
1/2W+" respectively.
§§ 7, 8 PROBABILITY OF COMPOUND EVENTS 581
Hence the chance ^ of a run of at least m heads is given by
_n + 3 n + 2 5 4_ 1
P ~ 2»»+2 ''' 2'»»+3 + • • • + 2m-t-» "'" 2»n+n+l 2'"+" '
The summation of the series on the left-hand side is effected (sec
chap. XX., § 13) by multiplying by (1 - 1/2)2= 1/4. We thus find
4 _n + 3 n + 2 ra + l 4
2(n+3) _ 2(n + 2) _ _ 2.5 _ 2.4
~ 2"*+* 2"*+* ' * * 2"*"'^+^ 2"'+"+2
n+3 ,6.5.4.1
2»»+4 ' • • ' 2V^+n+i ^ 2"*+"+^ ' 2'''+''+3 2"*+"+^ '
_n+3_n+4 3 2 1
iP ~ 2>n+a 2"'+* 2"*+"+* 2"'+''+^ 2"'+"+2 '
_n + 2
Hence i) = (n + 2)/2^+i.
GENERAL THEOREMS REGARDING THE PROBABILITY OF
COMPOUND EVENTS.
§ 8.] The probability that an event, wJiose probability is p,
happen on exactly r out of n occasions in which it is in question is
nCrP^q^'^ where q=l —p is the probability that the event fail.
The probability that the event happen on r specified occasions
and fail on the remaining n — r is by the multiplication rule
2)pqpqq . . . where there are r p's and n-r qs, that is, j^''^'""'".
Now the occasions are not specified ; in other words, the happen-
ing, and failing, may occur in any order. There are as many
ways of arranging the r happenings and n — r failings as there
are permutations of n things r of which are alike and n-r alike,
that is to say, w!/r! (w — r)! =„6y. There are therefore nOr
mutually exclusive ways in which the event with which we are
concerned may happen ; and the probability of each of these is
pTgn-r^ Hence, by the addition rule, the probability in question
is nOrP^'q'"'''-
It will be observed that the probabilities that the event
happen exactly n, n-1, . . ., 2, 1, 0 times respectively, are the
1st, 2nd, 3rd, . . ., {n+ l)th terms of the expansion of {p + q)\
Since, if we make n trials, the event must happen either 0,
582 PROBABILITY OF COMPOUND EVENTS CH. XXXVI
or 1, or 2, . . ,, or w times, the sum of all these probabilities
ought to be unity. This is so ; for, since p + q = l, (p + qY = 1.
It will be seen without further demonstration that the pro-
position just established is merely a particular case of the
following general theorem : —
If there he m events A, B, G, . . . one hut not more of which
must happen on every occasion, and if their prohahilities he p, q, r,
. . , respectively, the prohahility that on n occasions A happen
exactly a times, B exactly ft times, C exactly y times, . . . is
n\p'^q^n . . .Ia\(3\y\. . .,
where a + (3 +y + . . .-n.
It should be observed that the expression just written is
the general term in the expansion of the multinomial
(^ + g' + r + . . .y.
Example 1. The faces of a cubical die are marked 1, 2, 2, 4, 4, 6;
required the probability that in 8 throws 1, 2, 4 turn up exactly 3, 2, 3 times
respectively.
By the general theorem just stated the probability is
8! / ly /I Y /ly _ 7.5.2
312I3lV6y V^y U/ ~ 38 '
= qZ approximately.
Example 2. Out of n occasions in which an event of probability 2? is in
question, on what number of occasions is it most likely to happen ?
We have here to determine r so that ^^rP^q^'^ may be a maximum.
Now „(?rJP''3"^/nCr-l2''^'2"~'^^ = (w - r + 1) pjrq.
Hence the probability will increase as r increases, so long as
{ji-r+l)p>rq,
that is, (n + l)jj>r(p + g),
that is r<{n+l)p
If ()i+ l)jj be an integer, =« say, then the event will be equally likely to
happen on s - 1 or on s occasions, and more likely to happen s - 1 or s times
than any other number of times.
If {n+l)p be not an integer, and 8 be the greatest integer in (ra + 1)^, then
the event is most likely to happen on « occasions*.
* When n is very large, {n-\-l)p differs inappreciably from np. Hence
out of a very large number n of occasions an event is most likely to happen
on pn occasions. This, of course, is simply the fundamental principle of § 2,
Cor. 1, arrived at by a circuitous route starting from itself in the first
instance.
|§ 8, 9 pascal's problem 583
As a numerical instance, suppose an ordinary die is thrown 20 times,
what is the number of aces most likely to appear?
Here n = 20; i> = l/6; {n + l)p = 3^.
The most likely number of aces is therefore 3.
§ 9.] The probability that an event happen on at least r out
of n occasions where it is in question is
nCrPY-'' + nCr+^p'+Y-'-' + • • • + nCn-iP'^-'q + p\ . . (1).
For an event happens at least r times if it happen either
exactly r ; or exactly r+1 ; . . . ; or exactly n times. Hence
the probability that it happens at least r times is the sum of
the probabilities that it happens exactly r, exactly r+1, . . .,
exactly w times ; and this, by § 8, gives the expression (1).
Another expression for the probability just found may be
deduced as follows : — Suppose we watch the sequence of the
happenings and failings in a series of different cases. After we
have observed the event to have happened just r times, we may
withdraw our attention and proceed to consider another case;
and so on. Looking at the matter in this way, we see that the
r happenings may be just made up on the rth, or on the r+ 1th,
. . ., or on the nth. occasion.
If the r happenings have been made up in just s occasions,
then the event must have happened on the sth occasion and on
any r - 1 of the preceding s - 1 occasions. The probability of
this contingency is
p X ,.^Cr-^p'-Y~' = ,.^Cs-rP''(t"''
Hence the probability that the event happen at least r times in
n trials is
p' + rC^p'-q + r-^,C^p'q'+. . . + n-iGn-rP'Y-"
=i?'-{l + rC^q + r+iC.q' + . . . + „_:(7«_,^''-'-} (2).
As the two expressions (1) and (2) are outwardly very different, it may be
•well to show that they are really identical. To do this, we have to prove that
=i--'{i+.c.(i)..c.(i)V....„<,„(i)"-],
584 GENJERAL FORMULA FOR COMPOUND EVENT CH. XXXVt
The expression last written is, up to the (n - r)th power of q, identical with
(1 - g)"-'-{l + g/(l - <7)1» = (1 - g)"-^/(l - 3)™= (1 - <?)-'•.
Now, as may be readily verified,
(l-q)-'=l+rG^q + r+jC^q^+ • • • +n-l'^«-r3"-'"+ • • ■ •
The required identity is therefore established.
Example. A and B play a game which must be either lost or won ; the
probability that A gains any game is p, that B gains it l-p = q; what is the
chance that A gains m games before B gains ra? (Pascal's Problem.)*
The issue in question must be decided in m + n-1 games at the utmost.
The chance required is in fact the chance that A gains m games at least out
of m + n-1, that is, by (1) above,
P^+^-^ + m+n-lG,p^+^-^q+ . . . +,n+„-iC^p'"3"-l (1').
We might adopt the second way of looking at the question given above,
and thus arrive at the expression
P^{l+mGiq + m+lG2l'+ ' ' ■ +,n+™-2C™-i9"-'} (2'),
for the required chance.
§ 10.] The results just arrived at may be considerably-
generalised. Let us consider n independent events Ai, A2,
. . ., An, whose respective probabilities are pi, 2hi • • •» Pn-
In the first place, in contrast to ^ 8, 9, let us calculate the
chance that one at least of the n events happen.
The complementary event is that none of the n events happen.
The probability of this is (1 —p^ (1 -p-^ ... (1 -p^^. Hence the
probability that one at least happen is
= ^Px-%PiPi-^^PiP-2Pz- • ' -{-T'^PiPi- ' -Pn (1).
Next let us find the probabiliti/ that one and no more of the n
events happen.
The probability that any particular event, say A^, and none
of the others happen is, pi{l-p^ {1 -ps) ... (1 -pn). Hence
the required probability is
^Pi(l-P^)(l-P3). . -i^-Pn)
=:%-2(7i2/?,Jt?2 + 3C,2/?i^2i?3-. . ■(-T~\On-iPiPz. . .pn (2).
* Famous in the history of mathematics. It was first solved for the
particular case p = g by Pascal (1654). The more general result (1') above
was given by John Bernoulli (1710). The other formula (2') seems to be due
to Montmort (1714). See Todhunter, Hist. Frob., p. 98.
I 10 GEKERALlSAl'ION OF PASCAL's PROBLEM 585
For the products two and two arise from - 2jt?i (P2+P3+ • . .
+Pn), and each pair will come in once for every letter in it. Again,
the products three and three arise from Spi {p-zPs +PiPi + • • •) ;
hence each triad will come in once for every pair of letters that
can be selected from it ; and so on.
By precisely similar reasoning, we can show that the probability
that r and no more of the n events happen is
%i?2 ' • • ^r (1 -Pr+^) (1 -Pr+2) ... (1 -Pn)
= ^PiP2- • 'Pr-r+iCi^PiP2 ' . .Pr+1
+ r+^Gi^PlPl • • . Pr+i
( - Yr+sCs'^PiPi . . . Pr+s
{-T-\Cn-rPyP2. • . Pn (3).
We can now calculate the probability that r at least out of the
n events happen.
To do so we have merely to sum all the values of (3) obtained
by giving r the values r, r + 1, r + 2, . . ., n successively.
In this summation the coefficient of ^pip-2 . . . pr+a is
( ~ )* {r+s^s ~ r+st/«_i + r+sCg_2 — . . . ( — )*~V+«^l + ( — 1)*}.
Now the expression within the brackets is the coefficient of
af in (1 + ivy+'' X (1 + w)-\ that is to say, in (l+a;Y+''-\ Tliis
coefficient is r+s-iOg. Hence the coefficient of ^pip.2 . . . pr+g is
( ~ ) r+8-i^«.
The probability that r at least out of the n events happen is
therefore
^PlP2. ■ ■ Pr-rOi^PiP2. . .Pr+l
+ r^xC^PlPl • . . i?r+2
( - )\-^»-xG^PlP2 ■ . . Pr+S
( - )'^-\-iCn-rPlP2 ■ • -Pn (4).
Since the happening of the same event on n different occasions
may be regarded as the happening of n different events whose
586 THIRD SOLUTION OF PASCAL'S PROBLEM CH. XXXVI
probabilities are all equal, the formulae (3) and (4) above ought,
when pi -p2 = . . . =pn each = p, to reduce to nCrP^q^~^ and
the expression (1) or (2) of § 9 respectively.
If the reader observe that, when p^ ^p^ = . . . =j3n =P,
%PiP2 . . • Pr = nOrP'^, &c. , ho will havB no difficulty in showing
that (3) is actually identical with JJrP''(t~^ in the particular
case in question.
The particular result derived from (4) is more interesting.
We find, for the probability that an event of probability p will
happen r times at least out of n occasions, the expression
(-)"-VlC„-.i?» (5).
Here we have yet another expression equivalent to (1) and
(2) of § 9. It is not very difficult to transform either of the two
expressions of § 9 into the one now found ; the details may be
left to the reader.
Example. The probabilities of three independent events are ]>, q, r;
required the probability of happening —
1st. Of one of the events but not more ;
2nd. Of two but not more ;
3rd. Of one at least ;
4th, Of two at least ;
5th. Of one at most ;
6th. Of two at most.
The results are as follows : —
1st. p + q + r-2{pq+pr + qr) + ^pqr;
2nd. fq + pr + qr - ipqr ;
3rd. p + q + r-{'pq+'pr + qr)+pqr ;
Ath.. pq+pr + qr-2pqr \
6th. l-(pq-\-pr-\-qr) + 2pqr',
6th. 1-pqr.
The first four are particular cases of preceding formula ; 5 is comple-
mentary to 4 ; and 6 is complementary to " of all three."
§ 1 1.] The Recurrence or Finite Difference Method for solving
problems in the theory of probability possesses great historical and
practical interest, on account of the use that has been made
of it in the solution of some of the most difficult questions in
the subject. The spirit of the method may be explained thus.
§§10, 11 RECURRENCE METHOD 587
Suppose, for simplicity, that the required probability is a function
of one variable x ; and let us denote it by u^. Reasoning from
the data of the problem, we deduce a relation connecting the
values of «» for a number of successive values of x\ say the
relation
/(Mx+2, ««+:, «x) = 0 (A).
We then discuss the analytical problem of finding a function
Ux which will satisfy the equation (A).
It is not by any means necessary to solve the equation (A)
completely. Since we know that our problem is definite, all
that we require is a form for m-b which will satisfy (A) and at the
same time agree with the conditions of the problem in certain
particular cases. The following examples will sufficiently illus-
trate the method from an elementary point of view.
Example 1. A and B play a game in which the probabilities that A and
B win are a and ^ respectively, and the probability that the game be drawn
is 7. To start with, A has m and B has n counters. Each time the game
is won the winner takes a counter from the loser. If A and B agree to play
until one of them loses all his counters, find their respective chances of
winning in the end*.
Let Mj. and Uj. denote the chances that A and B win in the end when each
has X counters. If we put 111+%=^, the respective chances at any stage of
the game are u^ and Vp_^.
Consider A'& chance when he has x + l counters. The next round he
may, 1st, win ; 2nd, lose ; 3rd, draw the game. The chances of his
ultimately winning on these hypotheses are au^^^ '■> /^"a i 7"a;+i respectively.
Hence, by the addition rule,
"x+l = «"xf 2 + ^"x + 7Wa:+l •
If we notice that 0 + ^ + 7=! (for the game must be either won, lost, or
drawn), we deduce from the equation just written
aWx+2 - (a + ^) «;^i + ^"a = 0 (!)•
It is obvious that u^=AK'', where A and X are constants, will be a
solution of (1), provided
a\2-(a + /3)X + /3 = 0 (2),
that is, provided X=l or X = /3/a. Hence u^=A and Uji.=B {^la)" are both
solutions of (1) ; and it is further obvious that u^=A + B{piaY is a solution
of (1).
We have now the means of solving our problem, for it is clear from (1)
that, if we knew two particular values of u^ , say Mq and u-y , then all other
• First proposed by Huyghens in a particular case; and solved by
James Bernoulli. See Todhunter, Hist. Prob., p. Gl.
588 PROBLEM REGARDING DURATION OF PLAY CH. XXXVI
values could be calculated by the recurrence formula (1) itself. The solution
Mj.=4+jB (/3/a)*, containing two undetermined constants A and J5, is
therefore sufficiently general for our purpose *. We may in fact determine
A and B most simply by remarking that when A has none of the counters his
chance is 0, and when he has all the counters his chance is 1. We thus have
^+i? = 0, ^ + jB(|3/a)P=l,
whence ^ = aP/(aP-/3P), B--aPl{aP- ^p).
We therefore have
u^ = aP-'= (a* - /3^)/(a»' - i3») ;
and, in like manner,
The chances at the beginning of the game are given by
M^=a«(a'»-^)/(aP-|8P),
t7„=/3™ (a» - j8™)/(aP - ^»^).
Cor. 1. Ifa=p, then (see chap, xxv., § 12)
The odds on A in this particular case are m to n.
It might be supposed that when the skill of the players is unequal this
could be compensated by a disparity of counters. There is, however, a
limit, as the following proposition will show : —
Cor. 2. The utmost disparity of counters cannot reduce the odds in A's
favour to less than a-j8 to )3.
For, if we give A 1 counter, and B n counters, the odds in ^'s favour are
a"(a-/3)//3(a™-/3") : 1; that is, (a-;8)/i3 {1- (^/a)"} : 1. Now, if o>|8, this
can be diminished by increasing n; but, since L (^/a)" = 0, it cannot become
less than (a - )3)/(3 : 1, that is, a - j3 : /3.
Hence we see that, if A be twice as skilful as B{a=2^), we cannot by
any disparity of counters (so long as we give him any at all) make the odds
in his favour less than even.
Example 2. A pack of n different cards is laid face downwards. A
person names a card ; and that card and all above it are removed and shown
to him. He then names another ; and so on, until none are left. Required
the chance that during the operation he names the top card once at least t.
Let M„ be the chance of succeeding when there are n cards ; so that u,j_i
is the chance of succeeding when there are n-1; and so on. At the first
trial the player may name the 1st, 2nd, 3rd, . . . , or the nth card, the
chance of each of these events being 1/n. Now his chances of ultimately
succeeding in the n cases just mentioned are 1, m„_2, «„_3, . . . , u^, 0
respectively. Hence
M„=l/n + M„_2/n + w„_3/n+ . . . +ujn + ujn.
We have therefore
?lM„=l + ttl+W2+ . . . +M„-2 (!)•
* This piece of reasoning may be replaced by the considerations of
chap. XXXI., § 8.
+ Reprint of Problems from the Ed. Times, vol. xlii., p. 69.
EVALUATION OF PROBABILITIES INVOLVING FACTORIALS 589
From (1) we deduce
(n-l)«»-i = l + Mi + W2+- • • + w»-3 (2).
From (1) and (2)
««»-(»-l)"n-l = W»-2.
that IS,
n («« - ««-i) = - («„-i - w„-o) (3).
Hence
(n - 1) (m„_i - M„_2) = - (m„_2 - M„_3),
{n - 2) («„_2 - W„_3) = - (tt„_3 - «„_,),
3(»3-M2)=-("2-Wl)-
Hence, multiplying together the last n-2 equations, we deduce
inl (u„ - w„_i) = ( - !)»-» (m3 - «,).
Since «! = !, Mj= J, this gives
«»-«»-i=(-l)"-V«I (4).
Hence, again,
«.-i-"»-2=(-l)»-7(«-l)!.
t/2-Mi = (-1)1/2!,
Ml -0=1.
From the last n equations we derive, by addition,
«„= 1 - 1/2! + 1/3! -... + (- 1)»-Vk1 (5).
Introducing the sub-factorial notation of chap, xxiii., § 18, we may write
the result obtained in (5) in the form w„=l-ni/n!.
From Whitworth's Table* we see that the chance when n = 8 is -632119.
When n=co the chance is 1 - l/e= -632121 ; so that the chance does not
diminish greatly after the number of cards reaches 8.
EVALUATION OF PROBABILITIES WHERE FACTORIALS OF
LARGE NUMBERS ARE INVOLVED.
§ 12.] In many cases, as has been seen, the calculation of
probabilities depends on the evaluation of factorial functions.
When the numbers involved are large, this evaluation, if pursued
directly, would lead to calculations of enormous length t, and the
greater part of this labour would be utterly wasted, since all
that is required is usually the first few significant figures of the
probability. The difficulty which thus arises is evaded by the
use of Stirling's Theorem regarding the approximate value of x^
• Choice and Chance, chap. rv.
t In some cases the process of chap, xxxv., % 11, Examples 2 and 3 is
useful.
590 EXERCISES XXXIX CH. XXXVI
when X is large. In its modern form this theorem may be
stated thus —
(see chap, xxx., § 17).
From this it appears that, if iz; be a large number, x\ may-
be replaced by ^{^t^cd) afe''', the error thereby committed being
of the order 1/1 2^h of the value of x\.
As an example of the use of Stirling's Theorem, let us consider the follow-
ing problem : — A pack of 4n cards consists of 4 suits, each consisting of n
cards. The pack is shuffled and dealt out to four players; required the
chance that the whole of a particular suit falls to one particular player. The
chance in question is easily found to be given by
p = (3n)Inl/(4w)I.
Hence, by Stirling's Theorem, we have
_V(27r3n) (3n)8"e-»»J(2im)n»e-"
^~ V(27r4w)(4n)4"e-4« '
the error being comparable with 1/llwth oip. Hence, approximately,
2) = V(3tW2)(27/25G)".
Example. Let 4ra = 52, w=13, then
23 = ,y(3 X 3-1416 X 13/2) (27/256)i3.
This can be readily evaluated by means of a table of logarithms. We
find
^ = 156/101*.
The event in question is therefore not one that would occur often in the
experience of one individual.
Exercises XXXIX.
(1.) A starts at half-past one to walk up Princes Street; what is the
probability that he meet B, who may have started to walk down any time
between one and two o'clock ? Given that it takes A 12 minutes to walk up,
and B 10 minutes to walk down.
(2.) A bag contains 3 white, 4 red, and 5 black balls. Three balls are
drawn ; required the probability — 1st, that all three colours ; 2nd, that only
two colours ; 3rd, that only one colour, may be represented.
(3.) A bag contains m white and n black balls. One is drawn and then a
second ; what is the chance of drawing at least one white — 1st, when the first
ball is replaced; 2nd, when it is not replaced?
(4.) If n persons meet by chance, what is the probability that they all
have the same birthday, supposing every fourth year to be a leap year ?
(5.) If a queen and a knight be placed at random on a chess-board, what
is the chance that one of the two may be able to take the other ?
§12 EXERCISES XXXIX 591
(6.) Three dice are thrown ; show that the cast is most likely to be 10 or
11, the probability of each being ^.
(7.) There are three bags, the first of which contains 1, 2, 1 counters,
marked 1, 2, 3 respectively ; the second 1, 4, 6, 4, 1, marked 1, 2, 3, 4, 6 re-
spectively ; the third 1, 6, 15, 20, marked 1, 2, 3, 4 respectively. A counter
is drawn from each bag ; what is the probability of drawing 6 exactly, and of
drawing some number not exceeding 6 ?
(8.) Six men are bracketed in an examination, the extreme difference of
their marks being 6. Find the chance that their marks are all different.
(9.) From 2n tickets marked 0, 1, 2, . . ., (2;i- 1), 2 are drawn; find the
probability that the sum of the numbers is 2n.
(10. ) A pack of 4 suits of 13 cards each is dealt to 4 players. Find the
chance — 1st, that a particular player has no card of a named suit ; 2nd, that
there is one suit of which he has no card. Show that the odds against the
dealer having all the 13 trumps is 158,753,389,899 to 1,
(11.) If I set down any r-permutation of n letters, what is the chance that
two assigned letters be adjacent?
(12.) There are 3 tickets in a bag, marked 1, 2, 3. A ticket is drawn
and replaced four times in succession ; show that it is 41 to 40 that the sum
of the numbers drawn is even.
(13.) What is the most likely throw with n dice, when n > 6 ?
(14.) Out of a pack of n cards a card is drawn and replaced^ The opera-
tion is repeated until a card has been drawn twice. On an average how many
drawings will there be ?
(15.) Ten different numbers, each >100, are selected at random and
multiplied together; find the chance that the product is divisible by 2, 3,
4, 5, 6, 7, 8, 9, 10 respectively.
(16.) A undertakes to throw at least one six in a single throw with six
dice; B in the same way to throw at least two sixes with twelve dice; and C
to throw at least three sixes with eighteen dice. Which has the best chance
of succeeding? (Solved by Newton; see Pepys' Diarij and Correspondence,
ed. by Mynors Bright, vol. vi., p. 179.)
(17.) A pitcher is to be taken to the well every day for 4 years. If the
odds be 1000 : 1 against its being broken on any particular day, show that the
chance of its ultimately surviving is rather less than J.
(18.) Five men toss a coin in order till one wins by tossing head ; calculate
their respective chances of winning.
(19. ) A and B, of equal skill, agree to play till one is 5 games ahead.
Calculate their respective chances of winning at any stage, supposing that
the game cannot be drawn. (Pascal and Fermat.)
(20.) What are the odds against throwing 7 twice at least in 3 throws
with 2 dice?
(21.) Show that the chance of throwing doublets with 2 dice, 1 of which
is loaded and the other true, is the same as if both were true.
592 EXERCISES XXXIX CH. XXXVI
(22.) A and B throw for a stake; A's die is marked 10, 13, 16, 20, 21, 25,
and B'3 5, 10, 15, 20, 25, 30. The highest throw is to win and equal throws
to go for nothing ; show that A'b chance of winning is 17/33.
(23.) A pack of 2n cards, n red, n black, is divided at random into 2 equal
parts and a card is drawn from each ; find the chance that the 2 drawn are
of the same colour, and compare with the chance of drawing 2 of the same
colour from the undivided pack.
(24.) 4m cards, numbered in 4 sets of m, are distributed into m stacks of
4 each, face up ; find the chance that in no stack is a higher one of any set
above one with a lower number in the same set.
(25.) Out of m men in a ring 3 are selected at random ; show that the
chance that no 2 of them are neighbours is
(m-4)(m-5)/(m-l)(m-2).
(26.) If m things be given to a men and b women, prove that the chance
that the number received by the group of men is odd is
{4(6 + a)'"-|(6-o)™}/(6 + a)"'.
(Math. Trip., 1881.)
(27.) A and B each take 12 counters and play with 3 dice on this condi-
tion, that if 11 is thrown A gives a counter to B, and if 14 is thrown B gives
a counter to A ; and he wins the game who first obtains all the counters.
Show that ^'s chance is to B'a as
244,140,625 : 282,429,536,481.
(Huyghens. See Todh., Hist. Prob., p. 25.)
(28.) A and B play with 2 dice ; if 7 is thrown A wins, if 10 B wins,
if any other number the game is drawn. Show that A's chance of winning
is to B's as 13 : 11. (Huyghens. See Todh., Hist. Prob., p. 23.)
(29.) In a game of mingled chance and skill, which cannot be drawn, the
odds are 3 to 1 that any game is decided by skill and not by luck. If A
beats B 2 games out of 3, show that the odds are 3 to 1 that he is the better
player. If B beats C 2 games out of 3, show that the chance of .4's winning
3 games running from C is 103/352.
(30.) There are m posts in a straight line at equal distances of a yard
apart. A man starts from any one and walks to any other; prove that the
average distance which he will travel after doing this at random a great
many times is ^{m + 1) yards.
(31.) The chance of throwing / named faces in n casts with a, p + 1- faced
die is
j(p + l)n_Zpn+/(/_j±)(p_l)n_ . . .| ^(p + l)n.
(Demoivre, Doctrine of Chances. )
(32.) If n cards be thrown into a bag and drawn out successively, the
chance that one card at least is drawn in the order that its number indicates
is
1-1/21 + 1/3!- . . . (-l)»-V/i!.
(This is known as the Treize Problem. It was originally solved by
Montmort and Bernoulli.)
§ 13 VALUE OF AN EXPECTATION 593
(33.) A and B play a game in which their respective chances of winning
are a and /3. They start with a given number of counters p divided between
them ; each gives up one to the other when he loses ; and they play till one
is mined. Show that inequality of counters can be made to compensate for
ineriuality of skill, provided a/|3 is less than the positive root of the equation
xP - 2xP-^ + 1 = 0. It phe large, show that, to a second approximation, this
root is 2 - 2^- 1^.
MATHEMATICAL MEASURE OF THE VALUE OF AN EXPECTATION.
§ 13.] If a man were asked what he would pay for the
privilege of tossing a halfpenny once and no more, with the
understanding that he is to receive £50 if the coin turn up head,
and nothing if it turn up tail, he might give various estimates,
according as his nature were more or less sanguine, of what is
sometimes called the value of his expectation of the sum of £50.
It is obvious, however, that in the case where only one trial
is to be allowed the expectation has in reality no definite value
whatever — the player may get £50 or he may get nothing ;
and no more can be said.
If, however, the player be allowed to repeat the game a large
number of times on condition of paying the same sum each time
for his privilege, then it will be seen that £25 is an equitable
payment to request from the player; for it is assumed that
the game is to be so conducted that, in the long run, the coin
will turn up heads and tails equally often ; that is to say, that
in a very large number of games the player will win about as
often as he loses. With the above understanding, we may speak
of £25 as the value of the player's expectation of £50 ; and it
will be observed that the value of the expectation is the sum
expected multiplied by the probability of getting it.
This idea of the value of an expectation may be more fully
illustrated by the case of a lottery. Let us suppose that there
are prizes of the value of £a, £b, £c, . . . , the respective prob-
abilities of obtaining which by means of a single ticket are
p, q, r, . . . If the lottery were held a large number iV^ of
times, the holder of a single ticket would get £a on pN
c. II. 38
594 ADDITION OF EXPECTATIONS CH. XXXVI
occasions, £h on qN occasions, £c on rN occasions, . . . Hence
the holder of a single ticket in each of the N lotteries would get
£,{pNa + qNb + rNc + ...). If, therefore, he is to pay the same
price £t for his ticket each time, we ought to have, for equity,
Nt =pNa + qNb + rNc + . . . ,
that is,
t = pa + qh + rc+ . . . .
Hence the price of his ticket is made up of parts corresponding
to the various prizes, namely, pa, qh, re, . . . These parts are
called the values of the expectations of the respective prizes ; and
we have the rule that the value of the expectation of a sum of
money is that sum multiplied hy the chance of getting it.
The student must, however, remember the understanding
upon which this definition has been based. It would have no
meaning if the lottery were to be held once for all.
Example. A player throws a six-faced die, and is to receive 20s. if he
throws ace the first throw ; half that sum if he throws ace the second throw;
quarter that sum if he throws ace the third throw ; and so on. Eequired the
value of his expectation.
The player may get 20, 20/2, 20/2^, 20/2^, . . . shillings. His chances of
getting these sums are 1/6, 5/6^, 5^/6^, 5^/6-*, . . . Hence the respective
values of the corresponding parts of his expectation are 20/6, 20 . 5/6^ . 2,
20 . 52/6^. 22, 20 . 5'/6*. 2"S . . . shillings. The whole value of his expectation
is therefore
¥{'4HHyKAy- • --} 47(-A)=f —
that is, 5s. S^d.
§ 14.] It is important to notice that the rule which directs
us to add the component parts of an expectation applies whether
the separate contingencies be mutually exclusive or not. Thus,
if pi, p^, ps, . . . be the whole probabilities of obtaining the
separate sums ai, a^, «3, . . ., then the value of the expectation
is piai + p-jjiz + Pad's + • • •> ^ven if the expectant may get more
than one of the sums in question. Observe, however, that pi must
be the whole probability of getting a^, that is, the probability of
getting the sum ai irrespective of getting or failing to get the
other sums.
If the expectant may get any number of the sums ai, a^,
^ 13-15 ADDITION OF EXPECTATIONS 695
. . ., an,VTe might calculate his expectation by dividing it into
the following mutually exclusive contingencies: — ai, a^, . . . , Un',
Oi + aa, tti + tfs, &c. ; cti + a^ + ch, &c. ; . . . ; «! + aa + • • . + ««•
Hence the value of his expectation is
2a,pi (1 -^a) (1 -ps) ... (1 -pn)
+ 2 (ai + a2)piP2 (1 -ps) ... (1 -pn)
+ 2 («! + aa + o.^PiV^V^ (1 -^4) ... (1 -i?n)
+ («! + tta + . • . + «n)i'lP2i?3 . • . i?».
By the general principle above enunciated the value in
question is also "Za^px. The comparison of the values gives a
curious algebraic identity, which the student may verify either
in general or in particular cases.
Example. A man may get one or other or both of the sums a and 6.
The chance of getting a is y, and of getting h is q. Eequired the value of
his expectation.
He may get a alone, or 6 alone, or a + & ; and the respective chances are
p (1 - g), q (1 -p), pg. Hence the value of his expectation is op (1 - q)
+ 6g(l-2)) + (a + 6)pg, which reduces to ap + bq, as it ought to do by the
general principle.
N.B. — If the man were to get one or other, but not both of the sums a
and 6, and his respective chances were p and q, the value of his expectation
would still be ap + bq; hutp and g would no longer have the same meanings
as in last case.
LIFE CONTINGENCIES.
§ 15.] The best example of the mathematical theory of the
value of expectations is to be found in the valuation of benefits
which are contingent upon the duration or termination of one or
more human lives. The data required for such calculations are
mainly of two kinds — 1st, knowledge, or forecast as accurate as
may be, of the interest likely to be yielded by investment of
capital on good and easily convertible security ; 2nd, statistics
regarding the average duration of human life, usually embodied
in what are called Mortality Tables.
The table printed below illustrates the arrangement of
mortality statistics most commonly used in the calculation of
life contingencies : —
38—2
596
MORTALITY TABLE
CH. XXXVI
The IP' Table of the Institute of Jctuarics.
Age.
Number
Decre-
Age.
Number
Decre-
Age.
Number
Decre-
Living.
ment.
Living.
ment,
Living.
ment.
X
h
dx
X
Ix
d^
X
h
dx
10
100,000
490
40
82,284
848
70
38,124
2371
11
99,510
397
41
81,436
854
71
35,753
2433
12
99,113
329
42
80,582
865
72
33,320
2497
13
98,784
288
43
79,717
887
73
30,823
2554
14
98,496
272
44
78,830
911
74
28,269
2578
15
98,224
282
45
77,919
950
75
25,691
2527
16
97,942
318
46
76,969
996
76
23,164
2464
17
97,624
379
47
75,973
1041
77
20,700
2374
18
97,245
466
48
74,932
1082
78
18,326
2258
19
96,779
556
49
73,850
1124
79
16,068
2138
20
96,223
609
50
72,726
1160
80
13,930
2015
21
95,614
643
51
71,566
1193
81
11,915
1883
22
94,971
650
52
70,373
1235
82
10,032
1719
23
94,321
638
53
69,138
1286
83
8,313
1545
24
93,683
622
54
67,852
1339
84
6,768
1346
25
93,061
617
55
66,513
1399
85
5,422
1138
26
92,444
618
56
65,114
1462
86
4,284
941
27
91,826
634
57
63,652
1527
87
3,343
773
28
91,192
654
58
02,125
1592
88
2,570
615
29
90,538
673
59
60,633
1667
89
1,955
495
30
89,865
694
60
58,866
1747
90
1,460
408
31
89,171
706
61
57,119
1830
91
1,052
329
32
88,465
717
62
55,289
1915
92
723
254
33
87,748
727
63
53,374
2001
93
469
195
34
87,021
740
64
51,373
2076
94
274
139
35
86,281
757
65
49,297
2141
95
135
86
36
85,524
779
66
47,156
2196
96
49
40
37
84,745
802
67
44,960
2243
97
9
9
38
83,943
821
68
42,717
2274
98
0
39
83,122
838
69
40,443
2319
In the first column are entered the ages 10, 11, 12, . . .
Opposite 10 is entered an arbitrary number 100,000 of children
that reach their tenth birthday; opposite 11 the number of these
that reach their eleventh birthday ; opposite 12 the number that
reach their twelfth birthday; and so on. We shall denote these
numbers by ko, hi, liz, • • - In a t^^ird column are entered the
differences, or "decrements," of the numbers in the second
column ; these we shall denote by c?io, dn, c?i2, • • • It is obvious
that d^ gives the number out of the 100,000 that die between
their wth. and x + 1th birthdays. It is impossible here to discuss
the methods employed in constructing a table of mortality, or
§§15, 16 USES OF MORTALITY TABLE 597
to indicate the limits of its use ; we merely remark that in
applying it in any calculation the assumption made is that the
lives dealt with will fall according to the law indicated by the
numbers in the table. This law, which we may call the Law of
Mortality, is of course only imperfectly indicated by the table
itself ; for although we are told that dj. die between the ages of
X and ;» + 1, we are not told how these deaths are distributed
throughout the intervening year. For rough purposes it is
sufficient to assume that the distribution of deaths throughout
each year is uniform ; although the variation of the decrements
from one part of the table to another shows that uniform
decrease* is by no means the general law of mortality.
§ 16.] By means of a Mortality Table a great many interesting
problems regarding the duration of life may be solved which do
not involve the consideration of money. The following are
examples.
Example 1. By the probable duration n of the life of a man of m years
of age is meant the number of years which he has an even chance of adding
to his life. To find this number.
By hypothesis we have imW»n=l/2- Hence i^^^=i^/2. IJ2 will in
general lie between two numbers in the table, say Ip and Zp+, . Hence tn-^n
must lie between jp and ■p-\-\. We can get a closer approximation by the
rule of proportional parts (see chap. xxi. , § 13).
Example 2. To find the "mean duration" or " expectancy of life " for a
man of m years of age.
By this is meant the average J'T (arithmetical mean) of the number of
additional years of life enjoyed by all men of m years of age.
Let us take as specimen lives the /^ men of the table who pass their mth
birthday ; suppose them all living at a particular epoch ; and trace their
lives till they all die.
In the first year l^- l^^i die. If wo suppose these deaths to be equally
distributed through the year, as many of the l^ - Z^+j will live any assigned
amount over half a year as will live by the same amount under half a year.
Hence the l^ - l^+i lives that have failed will contribute ^ (Z^ - Z^+j) years to
the united life of the l^ specimen lives. Again, each of the Z„^i who live
through the year will contribute one year to the united life. Hence the
whole contribution to the united life during the first year is ^(Zto-Z^^j)
+ Wi = i('n» + Wi)- Similarly, the contribution during the second year is
h ( Wi + ^m+i) ; *°^ so o°- Hence the united life is
H^m+Wl)+MWl + W2)+ • • • =i^m + Wl+'m+s+ • • • (1),
* Demoivre's hypothesis.
598 EXAMPLES CH. XXXVI
the series continuing so long as the numbers in the table have any significant
value.
If we now divide the united life by the number of original lives, we find
for the mean duration
^=H(Wl + W2+ • • ■)llm (2).
Owing to our assumption regarding the uniform distribution of deaths over
the intervals between the tabular epochs, this expression is of course merely
an approximation.
Example 3. A and B, whose ages are a and b respectively, are both
living at a particular epoch ; find the chance that A survive B.
The compound event whose chance is required may be divided into
mutually exclusive contingencies as follows: —
1st. B may die in the first year, and A survive ;
2nd. „ second ,, ;
and so on.
The 1st contingency may be again divided into two : —
(a) A and B may both die within the year, B dying first ;
(j3) B may die within the year, and A live beyond the year.
The chance that A and B both die within the first year is (Z„ - Z^^j)
ih ~ ^b+i) I ^ah- Since the deaths are equally distributed through the year, if
A and B both die during the year, one is as likely to survive as the other ;
hence the chance of A surviving B on the present hypothesis is J. The
chance of the contingency (a) is therefore (la~^a+i){h~^b+i)l^^ah' ^^^
chance of (/3) is obviously la+i{h~h+i)l^ah'
Hence the whole chance of the 1st contingency, being the sum of the
chances of (a) and (/3), is (la + la+iWb-^}H-i)l^hh-
In like manner, we can show that the chance of the 2nd contingency is
(h+l + ^af 2) (^6+1 - h+i)l^lah •
Hence the whole chance that A survive B is given by
Sa,h={{h+h+l){h-lh+l) + {^<^l + la+2Wh+l-lh^^}+- ' •}mah (!)•
The reader will have no difficulty in seeing that (1) may be written in the
following form, which is more convenient for arithmetical computation : —
r=oo
^a,h = \+{ 2 W (^6+r-] - 'fc+r+i) - iJb+i}l^lJb (2),
r=l
where 00 stands for the greatest age in the table for which a significant value
of l^ is given.
If we denote by S^^^ the chance that B survive A, we have, of course,
If a=b, it will be found that (2) gives fif„^j,= l/2 ; as it ought to do.
§ 17.] Let US now consider the following money problem in
life contingencies : — What should an Insurance Office ask for
undertaking to pay an annuity of SA to a man of m years of age.
§ 17 ANNUITY PROBLEMS — AVERAGE ACCOUNTING 599
the first payment to he made w + 1 years hence*, the second w + 2
years hence ; and so on, for t years, if the annuitant live so long.
We suppose that the office makes no charges for the use of
the shareholders' capital, for management, and for "margin" to
cover the uncertainty of the data of even the best tables of
mortality. Allowances on this head are not matters of pure
calculation, and differ in different offices, as is well known. We
suppose also that the rate of interest on the invested funds of
tlie office is £i per £1, so that the present value, v, of £1 due
one year hence is £1/(1 + i). The solution of the problem is then
a mere matter of average accounting.
Let ni««m denote the present value of the annuity; and let
us suppose that the office sells an annuity of the kind in
question t to every one of l^ men of m years of age supposed to
be all living at the present date.
The office receives at once n|««mC pounds. On the other
hand, it will be called upon to pay
n+ \, w + 2, . . . , n + t
years hence respectively. Reducing all these sums to present
value, and balancing outgoings and incomings on account of the
Im lives, we have, by chap, xxii., § 3,
n\t(^m''m— '^ tni+n+l + "^ ^ni+n+2 + . • . + "W t„(+„+{.
Hence
n\tO'm — \V^^ 'm+n+1 + '*^" 4»+n+2 + • • • + 'W""^ lm+n+t)/lm>
- -y" 2 ;„+„+,. 'yV/,^ (1).
The same result might be arrived at by using the theory of
expectation.
• This is what is meant by saying that the annuity begins to run n years
hence.
t The annuity need not necessarily be sold to the person ("nominee")
on whose life it is to depend. The life of the nominee merely concerns the
definition of the "status" of the annuity, that is, the conditions under
which it is to last.
600 PEOBLEMS SOLVABLE BY ANNUITY TABLE CH. XXXVI
The annuity whose value we have just calculated would be
technically described as a deferred temporary annuity.
If the annuity be an immediate temporary annuity, that is,
if it commence to run at once, and continue for t years provided
the nominee live so long, we must put n-^. Then, using the
actuarial notation, we have
r=l
dm — 2 ^m+r^ Ihn (2).
If the annuity be complete, that is, if it is to run during the
whole life of the nominee, the summation must be continued as
long as the terms of the series have any significant value ; this
we may indicate by putting t= <x>. Then, according as the
annuity is or is not deferred, we have
„ I a,„ = -y" 2 In+n+r 'O^jlm (3).
r=l
r=oo
0'm= 2 Im+rV^jlm (4;).
r=l
§ 18.] The function am, which gives the value of an im-
mediate complete annuity on a life of m years, is of fundamental
importance in the calculation of contingencies which depend on
a single life. Its values have been deduced from various tables
of mortality, and tabulated. By means of such tables we can
readily solve a variety of problems. Thus, for example, nia^,
\tdm, nit^m cau all bc found from the annuity tables; for we
have
n|<^m~'P '■vi+ndm+n/'m (5)
It^m — Oim ~ '^ Li+t Cl'vi+t/l'm. (6)
n\tdm — V^ im+n dm+n ~ '^ l'm+n+tdm+n+t)l'm \J)
as the reader may easily verify by means of formulas (1) to (4).
These results may also be readily established a priori by
means of the theory of expectation.
§ 19.] Let us next find ak,m ths present value of an im-
mediate complete annuity of £1 on tlie joint lives of two nominees
ofk and m years of age respectively.
Jhe understanding here is that the annuity is to be paid sp
§§ 17-19 SEVERAL NOMINEES — METHOD OF EXPECTATIONS 601
long as both nominees are living and to cease when either of
them dies.
The present values of the expectations of the 1st, 2nd, 3rd,
. . . instalments are
Hence we have
0'k,m= (vlk+llm+l + "^ 4+2 im+2 + • • -j/'kimi
r=oo
= 2 V^lk+rL+r/Um (l).
r=l
Just as in § 18, we obviously have
n\(^k,m~'^ (fk+n , m+n 'k+n 'm+n/'k I'm >
\t(^k,m — (^k,m~'^ <^fc+«,m+t 4+t 'm+trklm 5
n\t^k,m — {'^ ^k+n , m+n 'k+n 'm+n
— V dk+n+t , m+n+t ''k+n+t ^m+n+t)/ik Im j
and it will now be obvious that all these formulae can be easily-
extended to the case of an annuity on the joint lives of any
number of nominees.
Tables for ajs,,^ have been calculated; and, by combining
them with tables for «,„, a large number of problems can be solved.
Example 1. To find the present value of an immediate annuity on the
last survivor of two lives m and n, usually denoted by a,-^.
Let Pr, q^ be the probabUities that the nominees are living r years after
the present date ; then the probability that one at least is living r years
hereafter is Pr + 'lr~Pr1r-
Hence
a;;:;;, = Sy'' ( p^ + ?r - Pr^r)y
1
= -Z.v'-pr + SVg^ - 2 «'>,.(?, ,
This is also obvious from the consideration that, if we paid an annuity
on each of the lives, we should pay £1 too much for every year that both
lives were in existence.
Example 2. Find the present value a^, „, „ of an annuity to be paid so
long as any one of three nominees shall be alive, the respective ages being
fe, m, n.
J£ Pg, Qg, Tg be the chances that the respective nominees be alive after s
years, then
at:^„ = 2j)H1 - (1 -P,) (1 -?,) (1 -'•,)} .
= 2i;» (i)g + g, + r, - (Z/g - rg pg - i),^, + jj,g,r,),
= ak + '^m + '^n- <^m,n- <^n,k - <^k,m + ^k.tmn-
The numerical solution of this problem would require a table of annuities
pn three joint lives, or some other means of calculating a^^ ,„^ ^^ .
602 LIFE INSURANCE PREMIUM CH. XXXVI
§ 20.] A contract of life insurance is of the following
nature : — A man A agrees to make certain payments to an
insurance office, on condition that the office pay at some stated
time after his death a certain sum to his heirs. As regards A,
he enters into the contract knowing that he may pay less or
more than the value of what his heirs ultimately receive accord-
ing as he lives less or more than the average of human life ; his
advantage is that he makes the provision for his heirs a certainty,
so far as his life is concerned, instead of a contingency. As
regards the office, it is their business to see that the charge made
for J.'s insurance is such that they shall not ultimately lose if
they enter into a large number of contracts of the kind made
with A ; but, on the contrary, earn a certain percentage to cover
expenses of management, interest on shareholders' capital, &c.
The usual form of problem is as follows : —
What annual premium Pm must a man of m years of age pay
{in advance) during all the years of his life, on condition that the
office shall pay the sum of £1 to his heirs at the end of the year in
which he dies?
Pm is to be the "net premium," that is, we suppose no
allowance made for profit, &c., to the office. Suppose that the
office insures L lives of m years, and let us trace the incomings
and outgoings on account of these lives alone. The office
receives in premiums XP^L, £Pmlm+i, ... at the beginning
of the 1st, 2nd, . . . years respectively. It pays out on lives
failed £{L-lm+i), £(^+1-^2), ... at the end of the 1st,
2nd, . . . years respectively. Hence, to balance the account,
we must have, when all these sums are reduced to present
value,
Pm{lm + lm+if^ + L+2V^+ • • •)
= {L - L+i) V + {Im+i - L+2) V" + {L+2 - lm+3) V* + . . . (1),
the summation- to be continued as long as the table gives signi-
ficant values of 4-
Since dm = lm- L+i , we deduce from (1)
"* ~ L+ Im+lV + L+2V^ + . . .
^ 20, 21 RECURRENCE METHOD FOR ANNUITIES 603
Dividing by i«, we deduce from (1)
= V + v{lm+iV + lrn+2V'+ . . .)/lm
Hence
Fm = v-aJ{l+am) (3).
The last equation shows that the premium for a given life
can be deduced from the present value of an immediate com-
plete annuity on the same life. In other words, life insurance
premiums can be calculated by means of a table of life annuities.
§ 21.] It is not necessary to enter further here into the
details of actuarial calculations ; but the mathematical student
will find it useful to take a glance at two methods which are in
use for calculating annuities and life insurances. They are good
specimens of methods for dealing with a mass of statistical
information.
Recurrence MetJiod for Calculating Life Annuities.
The reader will have no difficulty in showing, by means of
the formulae of § 17, that
am = v{l+ a;„+i) lm+\llm. (l).
From this it follows that we can calculate the present value
of an annuity on a life of m years from the present value on a life
of w + 1 years. We might therefore begin at the bottom of the
table of mortality, calculate backwards step by step, and thus
gradually construct a life annuity table, without using the com-
plicated formula (4) of § 17 for each step.
A similar process could be employed to calculate a table for
two joint lives differing by a given amount.
Columnar or Commutation Method.
Let us construct a table as follows : —
In the 1st column tabulate 4 ;
„ 2nd „ d^;
3rd „ v"4 = D^, say ;
„ 4th „ '^^^dx=C^,sa.Y.
604 COMMUTATION METHOD CH. XXXVI
Next form the 5th column by adding the numbers in the
3rd column from the bottom upwards. In other words, tabulate
in the 5th column the values of
Nx = Dx+i + Dx+1 + Dx+z + . . . .
In like manner, in the 6 th column tabulate
Mx = Cx+ Cx+i + Cx+2 + . . .
All this can be done systematically, the main part of the
labour being the multiplications in calculating Bx and C4,.
From a table of this kind we can calculate annuities and
life premiums with great ease. Referring to the formulae above,
the reader will see that we have
a,„ - J^JDm (2) ;
\tam = {Nrr, - Nm+t)/Dm (4) ;
Frn^MJNm-, (6).
§ 22.] In the foregoing chapter the object has been to
illustrate as many as possible of the elementary mathematical
methods that have been used in the Calculus of Probabilities ;
and at the same time to indicate practical applications of the theory.
All matter of debatable character or of doubtful utility has
been excluded. Under this head fall, in our opinion, the
theory of a priori or inverse probability, and the applications to
the theory of evidence. The very meaning of some of the pro-
positions usually stated in parts of these theories seems to us to
be doubtful. Notwithstanding the weighty support of Laplace,
Poisson, De Morgan, and others, we think that many of the
criticisms of Mr Venn on this part of the doctrine of chances
are unanswerable. The mildest judgment we could pronounce
would be the following words of De Morgan himself, who seems,
after all, to have "doubted": — "My own impression, derived
from this [a point in the theory of errors] and many other cir-
cumstances connected with the analysis of probabilities, is, that
mathematical results have outrun their interpretation*."
* "An Essay on Probabilities and on their Application to Life Contin-
gencies and Insurance Ollices " (De Morgan), Cabinet Cyclopcsdia, App.,
p. xxvi.
§§ 21, 22 GENERAL REMARKS— REFERENCES 605
The reader who wishes for further information should consult
the elementary works of De Morgan (just quoted) and of Whit-
worth {Choice and Chance) ; also the following, of a more advanced
character : — Laurent, Traite du Calcul des Prohabilites (Paris,
1873) ; Meyer, Vorlesungen iiber Wahrscheinlichkeitsrechnung
(Leipzig, 1879); Articles, "Annuities," "Insurance," "Proha-
bilities," Encyclopaedia Britannica, 9th edition.
The classical works on the subject are Montmort's Essai
d' Analyse sur les Jeux de Hazards, 1708, 1714 ; James Bernoulli's
Ai's Conjectandi, 1713 ; Demoivre's Doctrine of Chances, 1718,
1738, 1756 ; Laplace's Theorie Analytique des Prohabilites, 1812,
1820 ; and Todhunter's History of the Theory of Probability,
1865. The work last mentioned is a mine of information on all
parts of the subject ; a perusal of the preface alone will give the
reader a better idea of the historical development of the subject
than any note that could be inserted here. Suffice it to say that
few branches of mathematics have engaged the attention of so
many distinguished cultivators, and few have been so fruitful of
novel analytical processes, as the theory of probability.
Exercises XL.
(1.) A bag contains 4 shillings and 4 sovereigns. Three coins are
drawn ; find the value of the expectation.
(2.) A bag contains 3 sovereigns and 9 shillings. A man has the option,
1st, of drawing 2 coins at once, or, 2nd, of drawing first one coin and after-
wards another, provided the first be a shilling. Which had he better do?
(3.) One bag contains 10 sovereigns, another 10 shillings. One is taken
out of each and placed in the other. This is done twice ; find the probable
value of the contents of each bag thereafter.
(4.) A player throws n coins and takes all that turn up head ; aU that
do not turn up head he throws up again, and takes all the heads as before ;
and so on r times. Find the value of his expectation ; and the chance that
all will have turned up head in r throws at most. (St John's Coll., Camb.,
1870.)
(5.) Two men throw for a guinea, equal throws to divide the stake.
A uses an ordinary die, but B, when his turn comes, uses a die marked
2, 3, 4, 5, 6, 6 ; show that B thereby increases the value of his expectation
by 5/18ths.
(G.) The Jeu des Noyaux was played with 8 discs, black on one side and
606 EXERCISES XL CH. XXXVI
white on the other. A stake S was named. The discs were tossed up by the
player; if the number of blacks turned up was odd the player won S, if all
were blacks or all whites he won 2S, otherwise he lost S to his opponent.
Show that the expectations of the player and opponent are 1315'/256 and
125S/256 respectively. (Montmort. See Todh., Hist. Prob., p. 95.)
(7.) A promises to give B a shilling if he throws 6 at the first throw
with 2 dice, 2 shillings if he throws 6 at the second throw, and so on, until
a 6 is thrown. Calculate the value of B's expectation.
(8.) A man is allowed one throw with 2 ordinary dice and is to gain a
number of shillings equal to the greater of the two numbers thrown ; what
ought he to pay for each throw ? Generalise the result by supposing that
each die has n faces.
(9.) A bag contains a certain number of balls, some of which are white.
I am to get a shilling for every ball so long as I continue to draw white only
(the balls drawn not being replaced). But an additional ball not white
having been introduced, I claim as a compensation to be allowed to replace
every white ball I draw. Show that this is fair.
(10.) A person throws up a coin ji times ; for every sequence of m (m > n)
heads or vi tails he is to receive 2™ - 1 shillings ; prove that the value of his
expectation is n(M + 3)/4 shillings.
(11.) A manufacturer has n sewing machines, each requiring one worker,
and each yielding every day it works q times the worker's wages as net profit.
The machines are never all in working order at once ; and it is equally likely
that 1, 2, 3, . . . , or any number of them, are out of repair. The worker's
wages must be paid whether there is a machine for him or not. Prove that
the most profitable number of workers to engage permanently is the integer
next to nqliq + 1) - 4 . (Math. Trip., 1 875.)
(12.) A blackleg bets £5 to £4, £7 to £6, £9 to £5 against horses whose
chances of winning are f , ^, ^ respectively. Calculate the most and the
least that he can win, and the value of his expectation.
(13.) The odds against n horses which start for a race are o : 1 ; a + 1 : 1 ;
. . ., a + n-1 :1. Show that it is possible for a bookmaker, by properly
laying bets of different amounts, to make certain to win if n>(a + l) (e + 1),
and impossible iin<.a(e- 1), where e is the Napierian base.
(14.) If Ap denote the value of an annuity to last during the joint lives
of p persons of the same age, prove that the value of an equal annuity, to
continue so long as there is a survivor out of n persons of that age, may be
found by means of the formula
n(n-l) n(n-l)(«-2) ^
n^i- 21 2T 3f -^8 • • • — -^n'
(15.) M is a number of married couples, the husbands being m years of
age, the wives n years of age. "What is the number of living pairs, widows,
widowers, and dead pairs after t years ?
Work out the case where M=500, m=40, ra=30.
(16.) If So, 6 have the meaning of § 16, show that
2^o'6'^a,6 - ^h+lh+l^a+U b+1 = ('o + 'o+l) ih - ^b+l)'
§ 22 EXERCISES XL 607
(17.) Find the probability that a man of 80 survive one or other of two
men of 90 and 95 respectively.
(18.) If «j,„,,„, . . . denote the present value of an immediate complete
annuity of £1 on the joint lives of a set of men of I, m, n, . . . years of age
respectively, show that the present value of an immediate annuity of £1
which is to continue so long as there is a survivor out of k men whose ages
are I, m, n, . . . respectively is
(19.) What annual premium must a married couple of ages m and w
respectively pay in order that the survivor of them may enjoy an annuity of
£1 when the other dies?
(20.) Calculate the annual premium to insure a sum to be paid n years
hence, or on the death of the nominee, if he dies within that time.
(21.) Show how to calculate the annual premium for insuring a sum which
diminishes in arithmetical progression as the life of the nominee lengthens.
(22.) An annuity, payable so long as either A (m years of age) or B (n
years of age) survive C (p years of age), is to be divided equally between A
and B so long as both are alive, and is to go to the survivor when one of
them dies. Show that the present values of the interests of A and B are
and «n-i«m,n-«n,P + i«»n,«,P
respectively.
(23.) If the population increase in a geometrical progression whose ratio
is r, show that the proportion of men of n years of age in any large number
of the community taken at random is ('n/»")/2 (^n/^")-
0
RESULTS OF EXEBCISES.
I.
(1.) 504000. (2.) 1210809600. (3.) 720. (4.) 12. (5.) 6. (8.) 5040;
64864800. (9.) 1235520. (10.) 6188; 3003; 3185. (U.) 408688; 18 ways of
setting together on the front, 10 ways of setting at equal distances all round.
(12.) (19C4 inCi + 1^G^ J2C3 9C1 + 17C4 12^2 gC^ + 16C4 12^1 9C3 + ,5(74 gC^)^P^.
(13.) 10^2 2o<?5 30^10 60^20- (".) 172800. (16.) 267148. (16.) 1814400, if
clock and counter-clock order be not distinguished. (17.) 2(271^^ - 3n + 2)(2n- 2)1.
(18.) 960. (19.) 9C4 ;(73 jP, ; 9C4 ,0^ 4P4 s^s- (20.) 62!/(13!)* ; 39!/(131)3.
(21.) 321/(121)28!. (22.) 64!/(2!)6(81)232!. (23.) 26; 136. (24.) 286; 84.
(26.) (jp + q)llpl 3I ; (p + qryipl {qi jl ; a little over six years.
11.
(1.) 448266240x2. (2.) -2093. (3.) 2». 1.3 . . . (2re-l)/n!. (4.)
(-)"+^(2n)!/(n + r)!(n-r)l. (6.) 22". 1.3. . . (47i- l)/(2«)!. (6.) If n be
even, the middle term is {n!/(Jn)!}x"/^; if n be odd, the two middle terms
are {nl/i(n- l)!i(n+l)l} {2a;("-i)/2 + ia;("+i)/=}. (11.) (2^3 + 3)2"*
+ (2^3-3)2^-1; (2^3 + 3)2^+1 -(2^3-3)2™+!. (16.) ^n{n+l). (16.)
2"-i(2 + n). (27.) r + 1. (28.) 10. (29.) i(v?+lln). (32.) 190274064.
(33.) 2a7 + 72a«/; + 212a«62 + i2I,a^bc + 35-2a*P + 105I,a*b^c+ 210 ^a*bcd +
1402a363c + 2102a362c2 + 420Sa='62cd! + 630Sa262c2d. (37.) 23!/(4!)55».
III.
(1.) 944. (2.) 20. (3.) (n + l)(n + 2)(n + 3)(7i + 4)(n + 5)/5! if the
separate numbers thrown be attended to ; 5?! + 1 if the sum of the numbers
thrown be alone attended to. (4.) 231. (6.) p+iC„. (7.) 62. (8.) 15„Cj.
(11.) (2re)!/2»n!. (13.) {N+a + b + c -3)\lalb\c\. (16.) 1 or 0 according as
nisevenorodd; {(1 + ^5)"+^- (1- V5)"+4/2"+V5- (17.) ^^-iCr-m-iGr-i-
(18.) 116280.
V.
(1.) xjy must not lie between 1 and b^ja^. (2.) x must lie between
i(7-V53) and ^ (7 + ^/53). (3.) x between {dc - b^)l(ad - be) and
(d^-ab)l(ad-bc), and y between (ab-c'^)l{ad-bc) and (a'-cd)l{ad-bc),
(16.) Greater. (17.) Less. (39.) 3^/^.
c. II. 39
610 RESULTS OF EXERCISES
VL
(1.) 3ahc, (2.) abclBjS. (4.) d'^jS^-^ is a minimum value if m do
not lie between 0 and 1, otherwise a maximum. (5.) Minimum when
apxP=hqy'i=crz'^. (7.) There is a maximum or minimum when (x + l) log a
= {y + m) log b = {z + n) log c, according as log a log b log c is positive or nega-
tive. (8.) a; = (n?>/ma)'/(™+n), (9.) x=l, a: = 38/15 give maxima; a; = 2, x = 3
minima. (10.) ^abc. (11.) Minimum when x = 7)ic/(m-n), 2/ = nc/(m-n).
(15.) Minimum 2^(a&)/(a + 6).
VII.
(1.) 3,00. (2.) 9/4. (3.) log 13/7. (4.) ^n{n + l). (5.) 0. (6.)
a^-^-p-imfp. (7.) a^-'^mjn. (8.) n^, co , n" according as jj> = <gf. (9.)
{vi^ -mn + v?)l{w?+mn + n'^). (10.) l/2a. (11.) aii-PVPiqlp. (12.) o*.
(13.) 16a/9. (14.) 1. (15.) i?. (16.) -in(n-l)2"-2. (17.) a^+^-P-V* ('"-«)/
n^p{p-q). (18.) (n-l)/2a. (19.) log a. (20.) 1. (21.) 1. (22.) 1.
(23.) 1. (24.) 00. (25.) oo if a; = l + 0, 0 if a; = l -0. (26.) e* (27.)
0 if 71 be negative, if n be positive 0 or oo according as a< >1. (28.) 1.
(29.) 1. (30.) 0 or 00 according as m> <.n. (31.) oo or 0 according as
axl. (32.) 1. (33.) eK (34.) e\ (35.) ^(a&). (36.) Exp (2^/3).
(37.) 00 or 0 according as \{ar-bj.) is positive or negative. If a^=bf,
a^-i + ftr-i. the limit is e^'(*-» ~ *■- '>''*' ; &c. (38.) 1/e. (39.) 0. (40.) a/6.
(41.) 2. (42.) 1. (43.) 1. (44.) 1. (45.) ^tt. (46.) 0. (47.) COS a.
(48.) 0. (49.) -8. (50.) 1. (51.) i- (52.) 1. (53.) 1. (54.) 0. (65.) 0.
(56.) 1. (57.) logm/logn. (58.) 1. (59.) 1. (60.) 1. (61.) 1. (62.) e-i'"'''"
(63.) e-2"^'/"'. (64.) e2/\ (65.) 2/7r. (74.) See chap. xxx., § 23.
VIII.
(1.) Div. (2.) Div. (3.) Conv. if a; be positive. (4.) Conv. (5.) Div.
(6. ) Div. if mod xi>a; conv. if mod x>a. (7.) Conv. if a; < 4 ; div. if x <t 4.
(8.) Conv. (9.) Div., (a;<l). (10.) Conv. (11.) Div. (12.) Conv. if a>l;
div. ifat»l. (13.) Div. (14.) Div. (16.) Abs. conv. (16.) Div.
IX.
(1.) (-)'-23.1.1.3. . . (2r-5)/2. 4.6.8. . . 2r. (2.) 1 .3 . . . (2r-l)/
2.4 . . . 2r. (3.) 3.7.11 ... (4r-l)/4. 8.12. .. 4r. (4.) 2, 1 .4 . 7 . . .
(3r- 5) 22/3/12. 24.36.48 . . . 12r. (6.) (-)'-U.2 . . . (3r - 4) ai/s-^rl .
(6.) -1.2.5 . . . (3r-4)ai-373.6.9 . . . 3r. (7.) -(n-l)(2n-l) . . .
{nr-n-l)lr\. (8.) 1.4.7 . . . (3r/2-2)/(r/2)! if r be even; 0 if r be odd.
(9.) (-)™«(n + l) . . . {n + \{r-n)-l)l{\{r-n)}\. (10.) 1 + |(x/a) + g (x/a)"
+ |2(a;/a)3. (11.) The first. (12.) The third. (13.) The fourth and fifth.
(14.) The eighth. (15.) If n=l, the 2nd and 3rd; if n=2, the 2nd; if n<(: 3,
the Ist. (19.) If m = 0, S = a; if m=l, 5 = 6; if n>l, S = 0: if n<l(4=0)
the series is divergent. (22.) I - ^^2. (23.) If m <t 1, S = wj (nt - 1| ?"'-' ; if
m = 0, 5 = 0.
RESULTS OP EXERCISES 611
X.
(1.) 21/a'-(c-a)(a-6). (2.) 0. (3.) Sl/a'-"'-2/(c-a) (a-Z;). (4.)
2r + l + l/2''+i. (6.) »-,if r beeven; r-l,if r = 4( + l; r + l,if r=4t-l. (6.)
nffr9'"-m<?i-m^r-iP2'^^ + mC2.„ir^_22;V-'+ ••• (15-) l(n + l)(n + 2){n + 3).
(19.) 1-1.3... (2n - l)/2"ttl . (20.) 7 . 10 . . . (3m + 1)/3 . G . . . (3n - 8).
XL
(2.) 275/128. (S.) 869699/256. (4.) 48; 0. (6.) 11989305/2048. (6.)
(-)'-{(r-l) + (r + 5)/2'-+='}. (10.) 1-0001005084; 1-0004000805. (11.) 2mx.
(12.) l + 2a;(l-j»)/(l-r). (13.) l + (-)»-ix/2».
XII.
(1.) -367879. (2.) -04165. (6.) {l-x^e'. (6.) 3(e-l). (7.) e + 1.
(8.) lie. (9.) 15e.
XIII.
(4.) 917. (6.) 21og{(a;-l)/(a: + l)}+log{(a; + 2)/(a;-2)}. (6.) log(12e).
(7.) (l + l/x)Iog(H-a;)-l. (8.) i{x-x-^)log {(l + x)l(l-x)} +i. (9.)
When x = l the sum is 18-24 log2. (10.) f. (12.) S {a;3«-2/(3n - 2)
+ x3»-V(3n - 1) - 2x3»/3n}.
XXV.
(1.) ^n(n + l) + -Hr-2)n(w + l)(n-l). (2.) in(n + l) (7i + 4)(n + 5). (3.)
3/4-l/2rt-l/2(n + l). (4.) l/15-l/5(5?i + 3). (6.) 1/12- l/4(2?i + l)(2?i + 3).
(6.) l/18-l/3(n + l)(n + 2)(w + 3). (7.) a/2 + 6/4-a/(« + 2) - &/2(ra + l)(n + 2).
(8.) l/8-(4n + 3)/8(2n + l)(2n + 3). (9.) 7/36-(3rt + 7)/(n + l) (n + 2) (ra + S).
(10.) ll/180-(G?i + ll)/12(2?i + l)(2H + 3)(27H-5). (11.) 3/4 + n-(2w + 3)/
2{n + l){n + 2). (12.) M„=(n + l)3(n + 3) (n + 5)/n(« + l) . . . (n + 6); apply
§ 3, Example 4. (13.) sin 6 sec (n + 1) ^ sec 9. (14.) cot (^/2«)/2» - cot 0.
(16.) tan-^na". (16.) tan-il + tau-il/2 - tan-il/n- tan-U/jn + l). (17.)
(m + n)!/(7ft + l)(n-l)l. (18.) {l/(m- 1)1 - (n + l)!/(m + n-l)!}/(TO-2).
(19.) (-)»m-aC„. (21.) {m-l-(n)!/»ii"-ii}/(TO-2). (22.) (al"+'-7ci»i -
air+iij/(a_c + r + l). (23.) (ai"+2i/cl"+^ii -a/c"-')/(a-c-?- + l). (24.)
{(a-l)i'"-»7ci'»-ii-(a + ?i)""-^'/(c + n+l)""-^l}/(»i-l)(a-c-l). (28.)
Deduce from (24). (26.) Deduce from (24). (27.) 2m { 1 - ( - )»2« {m - 1)
(m-2) . . . (m-?i)/1.3 . . . (2?i- l)}/(2m- 1).
XXVI.
(1.) 2»+»+i(3"+i-3). (2.) l{l + (-l)"} + 6-3{i"+i + (-i)"+i}-
V-{i"-(-i)"}- (3.) ll{l-(4x)™+i}/{l-4x}-9{l-(3x)»+i}/{l-3x};
{2 + 3x)l(l-7x + 12x^'), x<l. (4.) 3 {1- (2x)"+i}/{l -2a;} + 2 {l-(3a;)»+J}/
{l-Sxj; (5-13x)/(l-5x + 6x2), x<4. (5.) ^{1 - (3x)»+i}/(l-3x) +
Hl-(5a;)"+i}/(l-5x); (l-4x)/(l-8x + 15x2), x<^. (6.) 3 {1 -(2x)«+i}/
{l-2x}-2{l-x'»+i}/{l-x}; (l + x)/(l-3x + 2x'^), x-^J.
612 RESULTS OF EXERCISES
XXVII.
. (1.) (1 + 2.x^)l{l - .tS)2. (2.) - [log {[l-x)l{l + x + x^)}- V3 tan-i {^ixj
(2 + .r)}]/3x; \ {6=^ + 26-^1^ cob Qixft)}. (4.) \[e-^ + e''l^ {cos {^1^x12) + sj^
sin(V3x/2)}]. (6.) i(2'" + 2cos.nnr/3);|3™/2cos.m7r/6. (6.) l/2-l/(n + 2)I.
(7.) {2"»+3-l-(m + 3)(m + 4)/2}/(m + l)(m + 2)(TO + 3). (8.) 1/(1 + a;) -
log(l+a;). (9.)icos^-Jcos2^. (10.) l-{2n + 3)/(/i + 2)2. (11.) 2-41og2.
(14.) sin mirjimr; coshmjr.
XXVIII.
The partial quotients are as follows : —
(1.) 0, 4, 1, 6, 2. (2.) 0, 2, 4, 8, 16. (3.) 1, 15, 1, 1, 1, 3, 1, 14, 1, 1,
5. (4.) 31, 1, 1, 1, 1, 1, 1, 1, 1, 8. (5.) 2, 1, 2, 1, 1, 4, 1, 1, 6, 3, 12, 3,
5, 1, 2. (6.) 0, 126, 1, 1, 2, 1, 1, 6. (7.) 1, 2. (8.) 2, 4. (9.) 3, 3, 6.
(10.) 3, 6. (11.) 3, 2, 6. (12.) 1, 4, 2. (13.) 2, 1, 2. (14.) 3, 1, 5,
(16.) 0, 2, 1; 0, 1. (17.) a, 2, 2a', a-1, 2, 2(rt-l).
XXIX.
(1.) The 1st, 2nd, 3rd, . . . convergents are 1, 2/3, 9/13, 20/29, 29/42,
78/113, . . .: the errors corresponding less than 1/3, 1/39, 1/377, 1/1218,
1/4746, 1/17515, . . . (2.) 972/1393. (3.) 2177/528. (4.) Transits at
the same node will occur 8, 243, . . . years after : after 8 years Venus will
be less than 1° "5 from the node. (5.) Transits at the same node will occur
13, 33, . . . years after,
XXXI.
(1.) 10,20;
(2.) 0, 1, 126, 2;
0, lb.
0, 0, 6*3, 6*3 ;
*
1.
64, 6*3, 1.
(3.) 1, 5, 3, 1, 8, 1,3, 5, 2;
0,1*2,13, 8,12,12,8,13,1*2;
1*2, 5, 7,20, 3,20,7, 5.
(4.) 0, 7, 1, 4, 3, 1, 2, 2, 1, 3, 4, 1, 1*4 ;
0,0, 7,5,7,5,4,6,4,5,7, 5, 7;
61, i, 12, 3, 4, 9, 5, 6, 9, 4, 3, 1*2.
(6.) 1, 2,10, 2, 1; (6.) 2,4;
1*0,15,25,25,1*5; 2,2;
25, 20, 5, 20, 2*5. §, 1.
RESULTS OP EXETICISES 613
« ♦ * » •
(10.) 0 + 2 — ; a + (a"-i-/3"-^)/(a'»-/3"), a and p being the roots of
*
x--2ax-l=:0. (11.) i{<^ + VK + 4)}; (a""*'^-/3'''^')/(a"-/3"). whereaand
/3 are the roots of a:2- ax -1 = 0. (12.) i{a-J{a^-i)} ; (a" - /3")/(a»+i - jS^+i),
•where a and /S are the roots of x^- ox + 1 = 0. (13.) {- ab + ^(a^b'' + 4:ab)}l2a;
if o, /3 be the roots of x^- (a6 + 2)x + l = 0, then P2n=*(a"-i3")/(«-/3),
92» = (a"+'-^"+'-«" + i3»)/(a-i3), and P^-,= {p^^-p^-^)lh, q,n-i = {q2n-
g,„_2)/6. (14.) - 1 + V [{3 (a" - /S-^) + 2 (a"-^ - /S»-i) }/(a»+i - (8»+i)], where a
and/3arethe roots of x2-x- 1 = 0. (20.) -iw+V[{(in'^ + n) (a"-i-/3»-i) +
(i ra'' + 1) (a"-2 - p»-2) }/(a» - j3")], where a and /3 are the roots of x^ - x - 1 = 0.
XXXII.
(1.) 3 + 7t,2-5t. (2.) 17< + 7, 16t + 5. (3.) 2206 - 7«, lit - 3309. (4.)
1013t - 3021756, 1367« - 4077746. (6.) 13. (6.) 280. (7.) 6. (8.) If
25fr. = 20«., 41. (9.) Buy 300 of each and spend 1021d. (10.) 69. (12.)
19. (13.) 715. (14.) 697.
XXXIV.
' (1.) Converges. (2.) Converges. (3.) Oscillates. (4.) Converges. (5.)
Converges. (6.) Converges. (7.) Converges if fc > 2, oscillates if fe > 2. (8.)
Converges. (9.) Oscillates. (10.) Oscillates. (IB.) Each of the fractions
converges to 1. (23.) e. (24.) 1/(1 -c). (25.) logg2, (26.) (3-e)/(e-2).
XXXIX.
(1.) 11/30. (2. ) 3/11, 29/44, 3/44. (3.) m (m + 2n)/(m + n)2, m (m + 2n - 1)/
{m+n){m+n-l). (4.) (365 .4»+l)/(1461)™. (6.) 4/9. (7.) 55/672, 299/2688.
(8.) 1/42. (9.) (n-l)/n(2n-l). (10.) (39!)2/26!52!, 4(39!)2/26!52!. (11.)
2(r- l)/n{7i- 1). (13.) 7n/2, or, if this be not integral, the two integers on
n+l
either side of it. (14.) S r (r- l)n(n-l) . . . (ra-r+2)/n'-. (18.) 16/31,
r=2
8/31, 4/31, 2/31, 1/31. (19.) The chances in A'a favour are 6/10, 7/10, 8/10,
9/10, when he is 1, 2, 3, 4 up respectively. (20.) 25 to 2. (23.) (1 - l/n)/2,
(l-l/n)/(2-l/n).
XL.
(1.) £1 : 11 : 6. (2.) His expectations are lis. 6d. and 10s. 4^d. respect-
ively. (3.) £8 : 5 : 9i , £2 : 4 : 2i. (4.) n(l - 1/2'-), (1 - 1/2')". (7.) 7«. 2id. ;
(n + 1) (4n - l)/6n. (12.) £6, £1, £4 : 2 : 2^ .
INDEX OF PROPER NAMES,
PARTS I. AND II.
The Roman numeral refers to the parts, the Arabic to the page.
Abel, ii. 132, 136, 142, 144, 152,
184, 287
Adams, ii. 231, 243, 251
Alkhayami, ii. 450
Allardice, i. 441
Archimedes, ii. 99, 442
Argand, i. 222, 254
Arudt, ii. 506
Babbage, ii. 180
Bernoulli, James, ii. 228, 233, 276,
403, 405, 587, 605
Bernoulli, John, ii. 275, 298, 366,
403, 584
Bertrand, ii. 125, 132, 183
Bezout, i. 358
Biermann, ii. 98
Blissard, 1. 84
Bombelli, i. 201
Bonnet, ii. 63, 132, 183
Boole (Moulton), ii. 231, 398
Bourguet, ii. 183, 253
Briggs, i. 529; ii. 241
Briot and Bouquet, ii. 396
Brouncker, ii. 351, 448, 479, 516
Burckhardt, ii. 536
Biirgi, i. 558
Burnside, ii. 32
Cantoe, ii. 98
Cardano, i. 253
Catalan, ii. 132, 183, 220, 251, 253,
416
Cauchy, i. 77, 254; ii. 42, 47, 83,
110, 115, 123, 132, 138, 142, 150,
171, 188, 226, 239, 287, 340, 344,
390
Cayley, ii. 33, 312, 325, 371, 496
Clausen, ii. 346, 503
Clerk-Maxwell, ii. 325
Cossali, i. 191
Cotes, i. 247
Cramer, ii. 396
Dase, ii. 536
Dedekind, ii. 98
De Gua, ii. 396
De Morgan, i. 254, 346; ii. 125, 132,
384, 396, 417, 421, 578, 604
Demoivre, i. 239, 247; ii. 298, 306,
401, 403, 405, 407, 411, 574, 592,
597, 605
Desboves, ii. 03
Descartes, i. 201
Diophantos, ii. 473
Dirichlet, ii. 95, 140, 473
Du Bois Reymond, ii. 133, 147, 148,
184
Dur^ge, ii. 396
Ely, ii. 231, 344
Euclid, i. 47, 272
Euler, i. 254; ii. 81, 110, 188, 231,
252, 280, 341, 342, 343, 344, 345,
348, 358, 363, 365, 366, 408, 419,
448, 494, 496, 512, 515, 516, 526,
539, 550, 551, 553, 555, 556, 563
Favaro, ii. 448
Fermat, ii. 478, 499, 546, 550, 591
Ferrers, ii. 502
Fibonacci, i. 202
Forsyth, ii. 396
Fort, ii. 77
INDEX
615
Fourier, ii. 135
Franklin, ii. 88, 504
Frost, ii. 96, 112, 396, 397
Galois, ii. 505
Gauss, i. 46, 254; ii. 81, 132, 184,
333, 345, 473, 523, 542, 550, 563
Glaisher, i. 172, 530 ; ii. 81, 240,
313, 357, 371, 397, 410, 421, 536
Goldbach, ii. 422
Grassmann, i. 254
Gray, ii. 243
Greenhill, ii. 313
Gregory, ii. 110
Gregory, James, ii. 333, 351
Grillet, ii. 59
Gronau, ii. 313
Gross, ii. 541
Gudermann, ii. 312, 313
Gunther, ii. 812, 448
Hamilton, i. 254
Hankel, i. 5, 254
Hargreaves, ii. 447, 452
Harkness and Morley, ii. 106, 148,
163, 396
Harriot, i. 201
Heath, u. 473
Heilermann, ii. 518
Heine, ii. 95, 98, 527
Heis, ii. 313
Herigone, i. 201
Hermite, ii. 473
Hero, i. 83
Hindenburg, ii. 495
Horner, i. 346
Houel, ii. 312 '
Hutton, i. 201
Huyghens, ii. 448, 580, 587, 592
Jacobi, ii. 473
Jensen, ii. 184
Jordan, i. 76; ii. 32
KoHN, ii. 125, 133
Kramp, ii. 4, 403
Kronecker, ii. 237
Kummer, ii. 133, 184, 473
La Caille, ii. 449
Lagrange, i. 57, 451; ii. 396, 448,
450, 453, 479, 550, 553
Laisant, ii. 313, 336, 358
Lambert, i. 176; ii. 312, 345, 448,
517, 523
Laplace, ii. 50, 605
Laurent, ii. 184, 579, 605
Legendre, ii. 473, 512, 523, 503
Leibnitz, ii. 333, 403
Lionnet, ii. 249, 252
Lock, ii. 271
Longchamps, ii. 110
Macdonald, i. 530
Machin, ii. 333
Malmsten, ii. 80, 132
Mascheroni, ii. 81
Mathews, ii. 473
Mayer, F. C, ii. 312
Meray, ii. 98
Mercator, ii. 312
Mertens, ii. 142
Metius, ii. 442
Meyer, ii. 605
Mobius, ii. 397, 494, 504
Montmort, ii. 405, 407, 584, 592,
605, 606
Muir, i. 358; ii. 334, 471, 494, 497,
502, 504, 518, 527
Napier, i. 171, 201, 254, 529; ii. 78
Netto, ii. 32
Newton, i. 201, 430, 472, 474, 479;
ii. 14, 280, 330, 335, 351, 373,
386, 392, 396, 401, 591
Nicolai, ii. 81
Ohm, ii. 140, 231
Osgood, ii. 146
Oughtred, i. 201, 256
Pacioli, i. 202
Pascal, i. 67; ii. 584, 591
Paucker, ii. 133
Peacock, i. 254
Pfaff, ii. 335
Pringsheim, ii. 98, 133, 156, 185
Puiseux, ii. 396
Purkiss, ii. 61
Pythagoras, ii. 531
Kaabe, ii. 132, 372
Kecorde, i. 216
Eeiff, ii. 145
Reynaud and Duhamel, ii. 49
Eiemann, i. 254; ii. 140, 205, 325
Rudolf, i. 200
Salmon, i. 440
Sang, i. 630
Saunderson, ii. 443
Scheubel, i, 201
616
INDEX
Schlomilch, ii. 45, 51, 80, 111, 184,
210, 359, 373, 506, 523
Seidel, ii. 145, 606
Serret, i. 76; ii. 32, 443, 453, 471,
481, 490
Shanks, ii. 334
Sharp, ii. 333
Simpson, ii, 417
Smith, Henry, ii. 473, 499
Sprague, i. 631; ii. 88
Stainville, ii. 335
Staudt, ii. 231
Stern, ii. 342, 448, 497, 505, 506,
517, 625
Stevin, i. 171, 201
Stifel, i. 81, 200
Stirling, ii. 368, 401, 404, 422, 589
Stokes, ii. 145
Stolz, ii. 98, 163, 181, 185, 396
Sutton, i. 531
Sylvester, i. 48, 176; ii. 342, 494,
503, 556, 561
Tab, ii. 253
Tartaglia, i. 191
Tchebichef, ii. 183
ThomsB, ii. 184, 396
Todhunter, ii. 271, 276, 574, 580,
584, 587, 592, 605
Van Cetjlen, ii. 333
Vandermonde, ii. 9
Venn, ii. 667
Viete, i. 201; ii. 276
Vlacq, i. 530
Wallace, ii. 312, 314, 315
Wallis, ii. 351, 448, 479, 527, 537
Waring, ii. 132, 417, 553, 555
Weber, ii. 98
Weierstrass, i. 230; ii. 98, 151, 160,
168, 185
Whitworth, ii. 22, 25, 33, 665, 589,
605
Wilson, ii. 651
Wolstenholme, i. 443; ii. 17, 33,
372, 547
Wronski, ii. 212
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