r£) '^^
ALGEBRA
AN
ELEMENTARY TEXT-BOOK
By Uie same Author
INTRODUCTION TO
ALGEBRA
FOR THE USE OF SECONDARY SCHOOLS
AND TECHNICAL COLLEGES
Fourth Edition. Crown 8uo. C'fo/A
FRtCE 7s. 6d,
EXCERPT /ro«l ihc PREFACE
I have kept the fundanicntjkl principles of the subject well
to the front from the very bcvjinnin;;. At the same time
I have not forgotten, what every mathcniaticAl (and other)
teacher should have perpetiwlly in mind, that a general
pro|>osition is a projicrty of no value to one that has not
Wfustercd the piirticiiliir.s. The utmost rigour of iiocumto
logic'il dedmtion hivs therefore hi-on icvt my aim than a
gradual development of algebraic idcji-s. In arranging the
exercises I have acted on a .similar principle of keeping out
as far as possible questions that have no theorcticjd or
practical interest
A. & C. BLACK. LTD. 1, 5 & 6, SOOO SQUARE, LONDO.N. W. 1.
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AVHTIutAHU . Till. IHfonI I'nIrerKltr I'tom
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I'AXAKA . . . Tl ■• , ,,„,H.. Llil
: >(n-.:t. Tombto 3
I.MjIA . . M
utu
.'.'■1 .M I.. .1 II I....III M.iiir;«
ALGEBEA
AN ELEMENTARY TEXT-BOOK
FOR THE
HIGHER CLASSES OF SECONDARY SCHOOLS
AND FOR COLLEGES
J!Y
G. CHRYSTAL, M.A., LL.D.
LATE HONORARY FELLOW OF CORrDS CHRISTI COLLEOF,, CAMBniDOE ;
AUD PROFESSOK OF MATUE1IATIC3 IN THE UNIVERSIII OF EDINBOUfiH
PART II
SECOND EDITION
#
A. AND C. BLACK, LTD.
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1926
Kr«* Kitilim puhluhrd \nrrmlirT ISgg
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iOvnnUd in 190«. 1916, 1919, ijjj and last
/•
pL.
- . '^
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Printwl In (Ireat nritain
PREFACE TO THE SECOND EDITION
OF PART II.
The present edition of this volume has been earefiiliy
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.
Kdinbcroh, 3rd March, 1900,
PIIEFACE TO FITIST EDITION.
The delay in tlie appearance nf this volume finds an apology
partly in circumstances of a private character, pixrtly in
public engagements that could not be declined, but most of
all in the growth of the work itself as it pmgressed 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 iu finitely 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 Difierential and Integral Calculus before they have
grasj>ed 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 matheinalics threatened
fur a time lu become its grave. Mathematicians hud fallen
PREFACE vii
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 d/da;
and fdx 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 modem 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 te.xt-books.
Chapters xxix. and xxx. may be regarded as an
elementary illustration of the application of the modem
Theur} of Functions. They are intended to pave the way
VIII rriEFACE
for tho study of the recent works of continental mntlie-
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 it^self : 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 ol 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 n<jt been defined. I have concluditi chapter XXX. with
a careful discussion of the Reversion of Scries and of the
E.\pansion 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 adniinible collection of examples of their use.
The other innovations call for little explanation, as they
aim merely at gixater coiiipleleuesa on the old lines, la
PREFACE
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
fiicilitate 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
• ''Alltz eu avuut, el iu ioi vous viendia."
X PltEFACE
Analyse AlgMtrique, Scrret'H Algdbre Supirienre, and RchlS-
niilch'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 criticifira, 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. Fraseu, and to my present or former pupils —
Messrs. B. B. P. Brandford, J. W. Butters, J. Cbockett,
J. GOODWILME, C. TWEEDIE.
In taking leave of this w^ork, 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,
remembeiiog 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
Edimbubob, lit Hovember 1889.
CONTENTS.
The principal technical terms a/re printed in italics in the
following table.
CHAPTER XXIII.
PERMUTATIONS AND COMBINATIONS.
PAGE
Definition of r-permutation and r-combination .... 1
Methods of Demonstration 2
Permutations ^~°
Number of r-permutations of n letters 2
Kramp's Notation for Fi\ctorial-n (n!) 4
Linear and Circular Permutations 4
Number of r-permutations with repetition 4
Permutations of letters having groups alike .... 5
Examples ......■•••• "
Combinations " ^•'
Combinations from Sets "
Number of r-combinations of n letters 6
Various properties of ^C^— Vandermonde'a Theorem . . 8-9
Combinations when certain letters are alike .... 10
Combinations with repetition 1"
Properties of „Hr— Number of r-ary Products .... 12
Exercises I
Binomial and Multinomial Theorems 14-18
Ifi
Examples
Exercises II
Examples of the application of the Law of Distribution . . 21-22
Distributions and Derangements 22-2o
Distribution Problem 22
Derangement Problem 24
Subfactorial n (n;) defined 25
Theory of Substitutions 25-32
Notation for Substitutions 26
Order and Group "'
Cj/ciic SuOstitutiuiis and Transpositions 27
b-2
xii coNTE>rrs
PIOK
Cycles of a SnbBtitntion 27
Decompoaition into Trauspositioiis 28
Odd and n-en Suh$tituliont 29
Eierci6c8 IIL 82
Exercises IV 83
CHAPTER XXIV.
GENERAL TIIEOBY OP INEQUALITIES.
Definition of Algebraic Inequality 35
Elementary Theorems 86
Examples 38
Derived Theorems 41-50
A Mean-Theorem for Fractions 41
(x»>-l)/p><(x«-l)/j 42
mx"-i(x-I)^(*"-l)*m(x-l) 43
ma"'-'((i-t)^o"'-b'»*jni'^'(a-i) 45
Inequality of Arithmetic and Geometric Means ... 4G
2pa"'/2p><(Zpa/Sp)'» 48
Exercises V 60
Applications to Maxima- and Minima-Theorems .... 5'2-GI
Fnndamental Theorem 52
Reciprocity Theorem 63
Ten Theorems deduced 53-69
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 .... 06
Enumeration of Elementary Indeterminate Forms ... 09
Extension of Fundamental Operations to Limiting Values . . 09
Limit of a Sum 70
Limit of a Product 70
Limit of a Quotient 71
Limit of a Function of Limits 71
T.imiting fomu for Rational Fuuctions 72
Forms 0/0 and od/oo 72
Fundamental Alk'cbraic Limit 7, (x"- I)/(i- 1) when r = l . 74
Exnmplib — Ll'{i + l}ll'i, LVx'ifx, when x = x, dtc . . 70
CONTENTS XIU
PAGK
Exponential Limits 77-81
i(l + l/x)^ when a:=<», Napierian Base 77
i (l + x)V^, L {1 + ylxf, L {l + xij)y\ L {a='-l)lx . . 79
35=0 x=« X=0 Z=0 «
Exponential and Logarithmic Inequalities .... 80-81
Euler's Constant 81
General Limit Theorems 82
L{/(x)}*(^>={L/(x)}^W 82
L{f{x + i)-f{x)] = tf{x)lxvihcnx=7> 83
r/(x+l)//(a;) = L{/(x)}'/== when x=oo 84
Exponential Limits Resumed 85
L a'jx, L logax/x, L xlogaX 85
X=ao X^OD X=0
Examples— Z. a:"/Hl, L m(m~l) , . . {m~n + l)ln\ . . 86
L i»'=l 87
»-+o
General Theorem regarding the form 0" 88
Cases where 0» + l 88
Forms ao» and 1" 89
Trigonometrical Limits 89
Fundamental Inequalities 90
Xsinx/x, Ltanx/x, when x = 0 91
^(^-i/sT- ^(=°=-xT' i(tan|/^y,whenx = =o . 91
Limit of the Sum of an Infinite Number of Infinitely Small Terms 92
i(l'- + 2'+. . . + n'-)/ft'-+i 92
Dirichlet's Theorem 94
Geometrical Applications 95
Notion of a Limit in General, Abstract Theory of Irrational Numbers 97-109
The Rational Onefold 99
Dedekind's Theory of Sections 99
Systematic Bepresentation 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 11*
Necessary and Suflicient Conditions for Convergeney ... 115
Residue and Partial Residue 117
xiv CONTENTS
man
Four Elementary Coraparison Theorems IIB
Ilalio of Conrergencf 120
Abtolutely Comergrnt and Semi-Convfrgent Scries ... 120
Special Tests of Convorgeuoy for Scries of Positive Terms . . 120-132
/,ii,""<>l 121
Lu„^J"n<^1- • ■ ''^1
Examples — Integro-Gcomotric, Logarithmic, Exponential, Bi-
nomial Series 122-123
Cauchy's Condenmlion Test 123
Logarithmic Criteria, first form 125
Logarithmic Scale of Convorgency 128
Logarithmic Criteria, second form 129
Examples— Hypergcometric and Binomial Scries . .130-132
Historical Note 132
Semi-Convergent Scries 133-1.37
Example of Direct Discussion 134
"i-«j + "j- ^^^
Trigonometrical Scries 135
Abel's Inequality 130
Convergence of a Series of Complex Terms .... 137
Necesi<ary and Sufficient Condition for Convergcncy . . 138
Convergence of the Series of Moduli sufficient .... 138
Examples— Exponential and Logarithmic Series, &8. . . 138
Application of the Fundamental Laws of Algebra to Infinite Series 139-143
Law of Association 139
Iav! of Commutation 140
Addition of Infinite Series 141
Law of Distribution 142
Theorem of Cauchy and MiTlcns 142
Uniformity and Non-Uniforniily in the Convergence of Series whose
terms ore functions of a variable 143-148
Vni/orm and Non-Uniform Convergence 144
Continuity of the sum of a Uniformly Converging Scries . 140
Du Bois-Keymond's Theorem 149
Special Discussion of the Power Series 148-157
Condition for Absolute and Uniform Convergcncy of Tower Scries 149
Circle and liadiiu of Convergence ...... 1 19
Cauchy's Kules for the Rndius of Convergence . . . l.'iO
Behaviour of Power Scries on the Circle of Convergence . . 151
Abel's Theorems ref-ardiug Continuity at the Circle of Convergence 152
Princii>lo of Indeterminate CoeJJicientt 150
Infinite Products 157-108
Convergent, Divergent, and Oscillating Products . . . 158
Discussion by inians of 2;log(l-ni,) 158
Oriterin from li/„ 159
Independent Criteria 100
CONTENTS
XV
Convergence of Complex Products . .
General Properties of Infinite Products
Estimation of the Residue of a Series or Product
Convergence of Double Series
Four ways of Summation ....
Double Series of Positive Terms
Cauchy's Test for Absolute Convergency .
Examples of Exceptional Cases .
Imaginary Double Series ....
General Theorem regarding Double Power-Series
Exercises VIII.
PAGE
160
161
168
171-183
172
174
177
179
181
182
182
CHAPTER XXVII.
BINOMIAL AND MULTINOMIAL SERIES FOB 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
IntegroBinomial Series 194
Examples 196
Exercises IX 199
Series deduced by Expansion of Rational Functions . . . 200-210
Expression of a" + /3" and (a"+i - /3"+')/(a - ^) in terms of a^ 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 assrmied . . 221
Deduction from Binomial Theorem 222
Calculation of e 224
Cauchy's Proof 226
Addition Theorem for the Exponential Seriei .... 227
XVI CONTENTS
SentouIWi Sumbef 22H-233
Expansions of z/(l -<-'), x (<■' + <-')/(<-»- <-^, Ac. . . 22V, 232
Bernoalli'e Eiprension for l'' + 2''+ . . . + n' . . . . - '•
Summations by mv&ns of Exponential Tbvorom .... 333~'J.iO
Integro-Exponential Series 233
Examples '2M
Exercises XII. - '■
Logarithmic Series 237 '.' 1
Expansion of log(l + x) - -
Derived Expansions 2o.'
Calculation of log 2, log 3, &c 241
Factor Method for calcalating Logarithms . . . . 218
First Difference of log z 245
Summations by Logarithmic Scries 245-250
2^(n)i«/(n + a)(n + 6) 246
Examples — Certain Semi-Convergent Series, <t'C. . . . 248
Inequality and Limit Theorems 250
Exercises XUI 251
CHAPTER XXIX.
SUUMATION OF THE FUXDAUENTAL POWER-SEEIES FOB COMPLEX
VALUES OF THE VARIADLK.
Preliminary Matter 254-272
Definition and Properties of the Circular Functions . . . 254-262
Evenneu, OdJruu, Periodicity 255
Graphs of the Circular Functions 256
Addition Formula for the Circular Functiom .... 258
Inverte Circular Functiom 269
Multiple-valuednen S60
Principal and other Branchei S60
Inversion of ui = r" and u>»'=z« 262-271
Circulo-Spiral Ornpht tor ie = z* 264
Multiplicity and Continuity ot ^i/it 265
Riemann'$ Surface 265
Principal and other Branches of ^le 267
Circulo-Spiral Graphs for w'=e* 2Ca
Principal and other Values of ^/u* 270
Exercises XIV 271
Geometric and Integro-Geometric Series 272
2r*cos(o + iifl), *c 273
Formula connecl«l with Dcmoivre'i Theorem and the Binomial
Theortm for an Integral Index 27*-879
Gonerali.iAtion of the Addition Theorems for the Circular Fnnctiona 275
Expansion! of cos n0, sin nfl/nin S, Ac, in powcm of »in 9 or oos 9 276
Expna*lon of eoi^taia't in the form 2^coip9 or Za^tinpt 877
CONTENTS XVII
PAGE
Exercises XV 279
Expansiou of cos 0 and sin 9 in powers of 9 .... 280-283
Exercises XVI. 284
Binomial Theorem for a Complex Variable 285-288
Most general case of all (Abel) 287
Exponential and Logarithmic Series for a Complex Variable • 288-297
Definition of Exp 2 288
'Ei-i.'p(x + yi)-e'^(cosy + isvay) 290
Graphic Discussion of ie = Expz 290
Imaginary Period of Expz 292
LogU) = log I !0 l + i amp w; 293
Principal and other Branches of Log w 293
Definition of Exp^z 294
Addition Theorem for Log z 295
Expansion of ,Log (1 + z) 296
Generalisation of the Circular Functions 297-313
General Definitions of Cos z. Sin z, etc 297
Euler's Exponential FormulfB for Co&z and Sinz . . . 298
Properties of the Generalised Circular Functions . . . 299
Introduction of the Hi/perbolic Functions 300-313
Expressions for the Hyperbolic Functions .... 300
Graphs of the Hyperbolic Functions 301
Inverse Hyperbolic Functions 303
Properties of the General Hyperbolic Functions . . . 303-307
InequaUty and Limit Theorems 307
Geometrical Analogies between the Circular and the Hyperbolic
Functions 308
Gudermantdan Function 311
Historical Note 312
Exercises XVH 313
Graphical Discussion of the Generalised Circular Functions . 316-325
Cos(i + yi) 316
Sin(a; + i/0 319
1an(x + yi) 320
Graphs of /{.r + yi) and l//(.r + yi) 322
General Theorem regarding Orthomorphosis .... 323
Exercises XVIII 325
Special Applications to the Circular Functions .... 326-334
Series derived from the Binomial Theorem 327
Series for cosm^ and smm(t> (m not integral) .... 327
Expansion of sin"' x, Quadrature of the Circle . . . 329
Examples — Series from Abel, &o 330
Series derived from the Exponential Series 331
Series derived from the Logarithmic Series 331
Sine-i8in29 + Jsin3fl- . . . = ie 332
Remarkable Discontinuity of this last Series .... 332
xviii CONTF.VTS
TAdB
Series for tan-'x, OreRory's Qandraturo of the Circle . . 333
Kuto on the Aritlimetical Quadrature of the Circle . . . 833
Exeroiscs XiX., XX 334,886
CHAPTER XXX.
QENKIIAL TIIKOItEMS REOAnDIXO THE EXPANSION OF FUNCTIONS
IN INFINITE FOKMS.
KxpaosioD in Infinite Series 337-344
Expansion of a Function of a Function 337
Expansion of an Infiuitc Product in the form of an Infinite Scries 337
Examples — Tlicorems of Eulor and Caucby .... 339
Expansion of Scchx and Sec 2 341
Eulfr'$ Numbers 842
Expansion of Tanh z, x Coth x, Coscch i ; Tan x, x Cot x, Cosoc x 343
Exercises XXI. 344
Expression of Certain Functions as Infinite Products . . 34(>-357
General Theorem regarding the Limit of an Infinite Product . 846
Products for sinh^iu, sinhu; sinpd, sin 0 . . . . 348
Wallis's Theorem 351
Products for coshpu, cosh u ; cospO, cob0 .... 3S1
Products for cos ^ + sin ^ cot tf, cos ^-sin^tanff, 14- cosec # sin ^ 854
Product for cos <p - cos 0 856
Bemark regarding a Certain Fallacy 856
Exercises XXII 857
Expansion of Circular and Hyperbolic Functions in an Infinite
Series of Partial Fraction* 359-862
Expressions for tan $, $ cot $, 0 cosec 6, see S . . . . 860
Expressions for tanh u, ucothu, u cosech u, sechu . , . 863
Expressions for the Numbers of Bernoulli and Eolcr . . . 362-867
Series for B„ 868
Product for y?„ 864
Certain Properties of It^ 864
Budii of Convergence of the Power-Series for tan S,Scol9, ecusccO 364
Series and Product for £„ 866
Certain Properties of £„ 866
Radius of Convergence of the Power-Scries for sec 8 . , 866
Sums (if Certain Series involving Powers of Integers . . 867
Power-Series for log sin $, Ac. 867
Stirling's Theorem 868
Exercises XXIU 372
Beversion of Series — Expansion of an Algebraic Function . . 373-396
Oenoral Expansion-Theorem regarding 2 (m, n) x"y» = 0 . . 374
Reveriion of Srriei 878
Branch Point 878
CONTENTS xix
TABE
Expansion of the varioug Branches of an Algebraic Function . 379
IrreducihiUty, 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 Poiut 383
Cycles at a Branch Point 386
Newton's Parallelogram, Degree-Points, Effective Group of Degree-
Points 386
AH 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 XXIY 397
CHAPTER XXXI.
SUMMATION AND TKASSFORMATION OF SERIES IN GENERAL.
The Method of Finite Differences 398-409
Difference Notation 398
Two Fundamental Difference Theorems ..... 401
Summation by Differences 402
Examples — Factorial Series, S sin (a + njS) 403
n
Expression of 2«„ in terms of the Differences of «j . . 405
1
Montmort's Theorem 407
Euler's Theorem 408
Exercises XXV 409
Recurring Series 411-115
Scale of Relation . 411
Generating Futiction 412
To find the General Term 413
Solution of Linear Difierence Equation^iih Constant Coefficients 414
Summation of Kecurring Series 414
Exercises XXVI 415
Simpson's Method for Summation by taking every fcth 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
CHAPTEll XXXII.
SIMPLE CONTINUED FnACTIONS.
Nature and Origin of Continued Frnctinns
Terminating, Si n-Trrminating, Recurring or PerioiUe, Simple
Continued Fractions ....
Component Fraclioiu and Partial Quotientt
Every Number conrortible into a S.C.F. .
Every Commensurable Number convertible into
S.C.F
Conversion of a Surd into a S.C.F. .
Exercises XXVIII
Properties of the Convcrgents to a S.C.F. .
Complete Quotietitt and Convergentt .
liecurrence-Formultz for Couvergents .
Properties of p„ and q^ . . . .
Fundamental Properties of the Convergenls
Approximation to S.C.F
Condition that pjg^ be a Convergent to x.
Arithmetical Utility of S.C.FF. .
Convergence of S.C.F
Exercises XXIX
Closest Rational Approximation of Given Complc
Closeness of Approximation of pj<i„ .
Principal and Intermediate Convergent!
Historical Note
Examples — Calendar, Eclipses, Ac.
Exercises XXX.
ity
PAoa
423-429
423
423, 424
424
426
428
430
431-441
431
432
433
435
437
439
440
4tl
412
444-4S1
445
446
448
41<J
451
CHAPTER XXXIIT.
ON nECURRINO CONTINUED FRACTIONS.
Every Simple Quadratic Surd Number convertible into a Kccurring
S.C.F 453-468
Bccurrence-Formulni for P^ and Q, 454
Expressions for P^ and Q^ 455
Cycles of P,, (?,, a> 457
Every Recurring S.C.F. equal to a Simple Quadratic Surd Number 468-460
On the S.C.F. which represents ^{CjO) 460-469
Acyclic Quotient of ^Xjil 462
Cycle of the Partial Quotients of v'^V/J/ 463
Cycles of the national DividentU and Divi$ort of s/Slil . . 4G4
Tc«ts for the Middle of the Cycles 467
Examplnn— Rapid Calculation of High Couvorgonta, &c. . . 468
ExcrciMS XXXI 469
480
CONTENTS XXI
PAOE
Applications to the Solution of Biophanline Problems , . . 473^88
ax-by = c 4^4
ax + by=c ^"^
ax + by + cz = d, a'x + b'y + c'z = d' 47'^
Solutions of x—Cy==±H and x=-Ci/-=±l .... 478
General Solution of 12 _Cj/==±ir when if <VC ... 479
General Formulae for the Groups of Solutions of x' - Cy^= ± 1
and x'-Cy-=^n
Lagrange's Reduction of x^-Cy''=±E when H>^fC . . 482
Eemaining Cases of the Binomial Equation .... 486
General Equation of the Second Degree 486
Exercises XXXII 489
CHAPTER XXXIV.
GENERAL CONTINUED FRACTIONS.
Fundamental Formula; 491-494
Meaning of G.C.F 492
G.C.FF. of First and Second Class 492
Properties of the Convergents 492
Continuants 494-502
Continuant Notation — Simple Continuant 494, 495
Functional Nature of a Continuant 495
Euler's Construction 496
Euler's Continuant-Theorem 498
Henry Smith's Proof of Fermat's Theorem that a Prune of the
form 4X + 1 is the Sum of Two Integral Squares . . 499
Every Continuant reducible to a Simple Continuant . . 500
C.F. in terms of Continuants 501
Equivalent Continued Fractions 501
Reduction of G.C.F. to a form having Unit Numerators . . 502
Exercises XXXIII 502
Convergence of Infinite C.FF 505
Convergence, Divergence, Oscillation of C.F 505
Partial Criterion for C.F. of First Class 506
Complete Criterion for C.F. of First Class .... 507
Partial Criterion for C.F. of Second Class .... 510
Incommensurability of certain C.FF 512-514
Legendre's Propositions 612
Conversion of Series and Continued Products into C.FF. . . 514-524
Euler's Transformation of a Series into an equivalent C.F. . 514
Examples— Brouncker's Quadrature of the Circle, &c. . . 516
C.F. equivalent to a given Continued Product .... 517
Lambert's Transformation of an Infinite Series into an Infiuito
C.F 517
XXll CONTEVTS
PAflB
Example— after Legendre i20
C.FF. for tan z ami tanh x 5J2
IncommeuRurability of t and e 623
Gauss's Conversion of the Hypergeometrio Scries into a C.F. . 623
Exercises XXXIV 625
CHAPTER XXXV.
GENERAL PROPERTIES OP INTEGRAL NUMBERS.
Numbers vrhich are con;;ruent with respect to a given Modulai . 528-634
Modulus and Congruence 528
Periodicity of Integers 529
Examples of Properties dednced from Periodicitj — Integrality of
x{z + l) . . . {z+p- 1)IpU Pythagorean Problem, *c. . 529
Property of an Integral Function 532
Test for Divisibility o{ f{x) 632
/(x) represents an Infinity of Composites 532
Difference Test of Divisibility 633
Exercises XXXV 634
On the Divisors of a given Integer 536-646
Limit for the Least Factor of .V 638
Sum and Number of the Divisors of a Composite . . . 537
Examples — Perfect Number, Ac 638
Number of Integers <iV and prime to N, 41 {N) . . . 539
Euler's Theorems regarding 0(.V) 640
Gauss's Theorem ^<p{dJ = N 642
Properties of ml 543-646
Exercises XXXVL 546
On the Bosidues of a Series of Integers in Arithmetical Progresaioo 647-554
Periodicity of the Besidncs of an A. P 648, 549
Fermat's Theorem 650
Historical Not« 650
Euler's Generalisation of Fermat's Theorem .... 651
Wilson's Theorem 552
Historical Note 6(8
Theorem of Lagrange including the Theorems of Fermat and
Wilson 553
Exercises XXXVII 65|
Partition of Numhrrt 555-564
Notation for the Sumher of Partiliont 666
Expansions and Partitions 666
Euler's Table fox P(n\ »\>q) 568
Partition Problems solvable by nicaus of Kulcr'i Table . , 659-661
Cuantruclive Tlieory of Partitions 661-604
CONTENTS XXUI
PAaE
Graph of a Partition, JReffular Graplis, Conjuriate Partitions . 5G2
Frankliu's Proof that (1 - x) (1 - x-) (1 - a;') . . . = S ( - ))> a;i(»P'*Jj) 563
Exercises XXXVIII ". . . . 564
CHAPTER XXXVI.
PKOBABILITY, OR TOE THEORY OP AVERAGES.
fundameiital Notions, Event, Uiiiferse, Scries, &c. . . . 56G
Deliuition of Probability or Chance, and Remarks thereon . . 567
Corollaries on the Defiuitiou 569
Odds on or against an Event 570
Direct Calculation of Probabilities 571-575
Elementary Examples 571
Use of the Law of Distribution 573
Examples^Demoivre's Problem, &c. ...... 574
Addition aud Multiplication of Probabilities 575-581
Addition Rule for Mutually Exclusive Events .... 575
Multiplication Rule for Mutually Independent Events . . 576
Examples ........... 577-581
General Theorems regarding the Probability of Compound Events 581-586
Probability that an Event happen on exactly r out of n occasions 581
More General Theorem of a Similar Kind .... 582
Probability that an Event happen on at least r out of n occasions 583
Pascal's Problem 584
Some Generalisations of the Foregoing Problems . . . 585
The Recurrence Method for calculating Probabilities . . . 68G
"Duration of Play" 587
Evaluation of Probabilities involving Factorials of Large Numbers 589
Exercises XXXIX ' 590
Mathematical Measure of an Expectation 593-595
Value of an Expectation 594
Addition of Expectations 594
Life Contingencies 595-604
Mortality Table 596
Examples of the Use of a Mortality Table .... 597
Annuity Problems, Notation, and Terminolugy, Average Ac-
counting 598-601
Calculation of Life Insurance Premium 602
Recurrence-Method for calculating Annuities .... 603
Columnar or Commutation Method 603
Remarks, General aud Bibliographical 605
Exercises XL 605
RESULTS OF EXERCISES 609
LNDEX OF PROPER NAMES FOR PARTS 1. AXD II. . 014
SUGGESTION FOB THE COtJBSE OF A FIBST READING
OF PART II.
Chnp. mn., §§ 1-15. Chap, xxni., §J 1-1 Chap, mv., f5 1-9.
Chap. XXV. Chap, xxvi., §§ 1-5, 12-19, 32-,S."). Chap. xxvn. Chap, xxviii..
§§ 1-5, 8-15. Chap, iiix., §§ 1-19, 23-.S1. Chap. xxxi. Chap, xxxii.
Chap. XXXIII., §§10-14. Chap. xxxT. Chap, xxxvi., §§6-22.
CHAPTER XXIII.
Permutations and Combinations.
§ 1.] We have already seen the importance of the enume-
ration of combinations iu the elementary theory of integi'al
functions. It was foimd, for example, that the ]iroblem of finding'
the coethcieuts 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
different problems, distinctions which are not always ea.'<y to
ejcpress clearly in ordinary language. Close attention should
therefore be paid to the terminology we are now to introduce.
1. § 2.] For our present purpose we may represent individual
\things by letters.
By an r-jyer mutation of n letters we mean r of those letters
arranged in a certain order, say in a straight line. An M-permu-
tatiou, which means all the letters iu a certain order, is sometimes
called a. permutation simply.
Example. The 2-permutation3 of the three letters a, b, c are he, ch;
at, ca; ab, ba. The permutations of the thi'ee letters are abc, acb; hac, bca;
cab, cba.
By an r-combination 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. 11. 1
2 MODES OF PROOF CU. XXIII
Unless the contrary is stjited, 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 eacii permutation or comliiiiatiou couKl bo actually
selected and set down before our eyes.
Another point to be attended to is that in some prublcms
cortaiu sets of ti>c 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 iutcrcliauging the^o
letters.
§ 3.] The fundamental part of every demon.stration of a
theorem in the tlieory of permutations and combinations is an
enumeration. It is necessary that this enumeration be systematic
and exhaustive. If porwible it should also be siniple.it, that is,
eiu-h pcrnuitatiiiu 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 re:isoning step by step, called mathematical induction.
The results of the law of distribution, as applied both to
closed functions and to inlinite series, are often u.sed (after the
m.anner of chap, iv., §§5, 11, and exercise vi. 30) to lighten the
labour of cnuuienitiou.
All these methods of proof will be found illu.strated below.
We have called attention to them here, in order tliat the student
may know what tools are at his disposal.
I'EUMUTATIONS.
§ 4.] The numlier of r-per mutations of n letters (»Pr) m
H(n-l)(n-2) . . . (n-r+1).
I.s7 Prixf. — Sui)po.se that we have r bkink spares, the problem
is to find in how many dillereut ways we can till these with n
letters all dilTerent.
We can fill the first blank in w different w.-iya, namely, by
]i\ittiii;; into it any one of the n letters. Having put any one
letter into the first blank, we have » - 1 to choose from iu filling
§§ 2-4 r-PKRMUTATIONS 3
the second blank. Heuce we can fill the second blank in m - 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 7i--2 left to choose from in filling the third.
Hence we can fill the first three blanks in n («- 1) times (« — 2)
ways.
Reasoning in this way, we see that we can fill the r blanks in
«(m-1)(»-2) . . . (?*-?•+ 1) ways.
Hence „Pr = «(«-!) . . . (n-r+1).
2nd Proof. — We may enumerate, exhaustively and without
repetition, the „Pr ^'-permutations as follows : —
1st. All those in which the fir.st letter a^ stands first ;
2ud. All those in which «.> stands first : and so on.
There are as many permutations in which «i stands first as
there are (r— l)-permutations of the remaining »— 1 letters, tliat
is, there are n-iPr-i permutations in the first class. The same
is true of each of the other n classes.
Hence nPr = n„-J'r-j.
Now this relatif)n is true for any positive integral values of
n and r, so long, of course, as r :}> n. Hence we may write
successively
„_,P,., = («-1)„-..,/^ ,,
n-r^,P,= {n-r + 2)n-r.J\.
If now we multiply all these equations together, and observe
that all the Pa cancel each other except „Pr and „_r+iPi, and
observe further that the value of n-r-nPi is obviously n-r+l,
we see that
J>r = n{n-l) . . . {n-r+2){n-r+l) (1).
The second proof is not so simple as the first, but it illustrates
a kind of reasoning which is very useful iu questions regarding
permutations and combinations.
1—2
4 UNKAK AND CIRCULAB PERMUTATIONS CH. XXIIl
Cor. 1. The number of diff,rent vays in uhich a set qf n
letters can be airanged in linear order is
H(n-l) ... 3.2.1,
tliat is, the product of the first n integral numbers.
This follows at once from (1), for the number reqiiirod is tlio
number of w-pcrmulations of the n letters. Putting r = fi in (I),
we have
.P, = n(»-1) . . . 2.1 (2).
Tiie jiroduct of the first n consecutive integers may be re-
g.wled as a function of the integral variable n. It is called
factorial-n, and is denoted by «!*.
Cor. 2. ,/V = «!/(«-f)!.
For J\ = n(n-\) . . . (n-r+1),
_ »(n-l) . . . (H-r+ l)(»-r) ... 2.1
(n-r) ... 2.1
«!
-(n-ry:
Cor. ?>. The number o/trai/s of arranging n letters in circular
order is (h-I)!, or (»-l)!/2, acronling a.< clock order and
counti-r-cluck-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 kecj) it fixed. Wo
have thus as many diflercut arraugcmenta as there are («-l)-
permutations of the remaining n - 1 letters, that is (» - 1)1.
If, however, the letters written in any circular order clock-
wi.se 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, lleuce the number in this case 13
(h-1);/2.
§ 5.] When each of the n letters may be repeated, the number
of r-permutalions is »'.
• TbiB is Eramp's notntion. Formerly |n^wag u«rJ in Englinh worki, but
this is now being abanJoncd on account of tbe dillJcully in printing the (_.
Tho valuo of II is of course 1. Strictly speaking, 01 hu no nreaning. It is
convoiiirnt, bowpver. to use it, with tho nndcrstanding tliat its valne is 1 ; by
so doing WD avuid tlio (.'xu-^jlionol treatment ol initial terms in many scriva.
^5 -i-B CASE WHERE LETTERS ARE ALIKE 5
Suppose that we have r blanks before us. We may fill the
liist in n ways ; the second also in n ways, since there is now no
restriction on tlic choice of the letter. Hence the first two may
be filled in n x n, that is, h^ ways. With each of these ii' 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
ill n' X 71, that is, n' ways, and so on. Hence we can fill the ?•
blanks in 71'' ways.
§ 6.] The number of per77iutations of n letters of tvhich a
(jivup of a are all alike, a group of /3 all alike, a group of y all
(dike, <i;c., is
n!/a!/i!y! . . .
Let us suppose that x denotes the number in question. If
wo take any one of the x 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 xa! permutations from the x permutations.
If we now make all the /3 letters unlike, we derive a;a!/3!
permutations from the xa\.
Hence, if we make all the letters unlike, we derive xa\p\y\ . . .
permutations. But these must be exactly all pos.sible permuta-
tions of « letters all unlike, that is, we must have
a;a!^!y! , . . =7il.
Hence a; = «!/a!/3!y! . . .
Cor. The number of ways in ivhtch n things can be ptit into
r pigeon-holes, so that a shall go into the first, )3 into the second,
y into the third, and so on, is
m!/a!y8!y! . . .
N.B. — The order of the pigeon-holes is fixed, and must be at-
tended to, but the order 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 <?
6 EXAMPLES CH. XXII!
labels each marked 1, /3 each luarkcd 2, y each marked 3, and
so on.
The problem is now tc find in how many ways n lalH.'l», a of
which are alike, /3 alike, y alike, &c., can be distributed anion;;
n thingf- standing in a given order. The number in question is
n'.la'.fily'. . . ., by the above proposition.
Exaniplo 1. Id arranRing tbe crew of on cight-oarrd boat the captain haa
four men that can row onl; on the Btriike-side aud four (hut can row odI; on
the bow-side. In how many different wayn can he arrange his boat— l»t,
when the stroke is not fixed ; 2ud, when the litroke is fixed?
In the first case, the captain may arrange his stroke-side in aa many
ways ns there are 4-permntatious of 4 things, that is, in 4! ways, and he
may arrange the bow-side in jast as many ways. Since the arrangemtnts uf
the two sides arc independent, ho has, therefore, 4txiI(=STC) dillureut
ways of arranging the whole crow.
In the second case, since stroke is fixed, there are only 31 ways of
arranging the stroke-side. Henoc, in this case, there are 3! x 41 ( = 144)
different ways of arranging the crew.
Example 2. Find the naniber of permutations that can be made with the
letters of the word tratualpitu-.
The letters are traannilpic, there Ining two sets, each containing
two like letters. The number re<iuirt'd ia therefore (by § 6) lll/2!2! =
11. 10. 9. 8. 7. C. 5. 3 . 2 = 99711200.
Example 3. In how many different ways can n different liesds bo
formed into a bracelet?
Since merely turnmg the bracelet oTcr changes a clock-arrangement of the
stones into tlie corresponding counler-clock-arrangement, it follows, by { 4,
that tbe number nijuired is (n- 1)1/2.
C0MBINATION.S.
.5 7.] Thf numlirr of trai/s in vhick s things can be mhetei by
titkimj one out of a set o/ni , one out of a set ofn,, A:c., isn^n,. . .n^.
The first thing can Ik; selected in n, w.-iy.s ; the second in n,
ways ; and so on. Hence, since the selection of each of the
things docs not depend in any way on the selection of the others,
the number of ways in wliich the « tilings can be selected is
w, X fi, X . . . X n,.
§ 8.] The number of r-cnmhinatinns of n letters (.C,) m
rj(n-l) . . . (n-r-t-l)/! .2 . . . r.
SS 6-8 r-COMBINATIONS
ss
1st Proof. — We ma)' enumerate the conibiiiatioiis as follows: —
1st. All those that contain the letter «i;
2nd. „ „ „ r^;
wth. „ „ „ «„.
In each of these classes there is the same nnmhcr of
combinations ; namely, as many combinations as there are
(r- l)-combinations of m-1 letters; for we obviously form all
the r-cornbinations in which ai occurs by forming all possible
(r — l)-combinations of a^, a,, . . ., a„ and adding Wj 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. Plence
r„(7r= «n-lCr-l (1).
This relation holds for all values of n and r, so long as r^ii.
Hence we have successively—
■-1 W-l — _ . n-2' 1-2,
«-2
n-s'^r-2 — „ .1 "-3'-''— 3>
r — \i
^ n-r+2 „
n-r+2^2 ;^ fl-r+l -'!•
If we multiply these r—\ equations together, and observe that
the C's cancel, except „Cr and „_r+i(7i, and that the value of
,-r+iCi is obviously n-r-^\, we have
r - n{n-l) . . . (n-r+l) . .
" *■ 1.2 ... r ^''
2nd Proof. — Since every r-combination of n letters, i! permu-
tatod in every possible way, would give r! j'-permutations, and
all the /•-permutations of the n letters can be got once and only
8 PROPERTIES OF „C, CH. XXIII
once by dealing in this way with all the r-combinations, it follows
tliat.Crr!-./^. Hence
,Gr = nPr!r\^n(n-l) . . . (n-r+l)/1.2 . . . r.
Cor. 1. If we multiply both numerator and denominator of
the expression for ^C, by (n - r) (» - r - 1 ) . . . 2 . 1 , we deduce
,Cr=n\/rl{n-r)\ (3).
Cor. 2. nCr = nCn.r.
Tliis follows at once from (3). It may also be proved by
enumeration ; for it is obvious that for every r-conibination of
the « things we select we leave behind an (n - r)-combination ;
there are, therefore, just as many of the latter as of the former.
Cor. 3. ,C = „-,a + ,.,C,-. (4).
This can be proved by using the e.vpressions for ,C,, n-\Cr,
,_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 cla.ssifying the r-combinations of n letters a,, a, o,
into, 1st, those that contain Oi, n-iCy-i in number, and, 2nd,
those that do not contain a,, .-iCr in number.
Cor. 4. ,.,C. + „.,C. + n-,C. + . . . + .C. = ,C„, (5).
Since the order of letters in any combination is indifferent,
we may arrange them in alphabetical order, and enumerate the
(s+ l)-combination8 of n letters by counting, 1st, those in
which a, stands first ; 2nd, those in which a, stands first, &c.
This enumeration is clearly both exhaustive and simplex ; and
we observe that a, cannot occur in any of the combinations of
the 2nd cla,s,s, neither a, nor a, in any of the 3rd class, and so on.
Hence the number of combinations in the Ist class is ,-iC, ; in
the 2nd, n-tC, ; in the 3rd, «-iC, ; aud so ou. Thus the tiieorem
follows.
Cor. 5.
pC7, 4 pC7,_, j(7, +pC7,-j,C, + . , . +pt7i,C7,.i + ,C, =p«,C, (6).
If we divide p + g letters into two groups of p and 7 re-
spectively, the ,«,C, s-cond)inations of the p + q letters may i»
clas-sified exhaustively and simple.xly as follows : —
§ 8 vandermonde's theorem 9
s
1st. All the 5-combinations of the j» letters. The number of
these is j,C,.
2u(i. 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 f things with every one of the
2-combinations of the q things. The number of these is
p^s-2 ^ q^3
And so on. Thus the theorem follows.
It should be noticed that Cor. 4 and Cor. 5 furnisli proposi-
tions in the summation of series. For example, we may write
Cor. 5 thus —
■p{p-\) . . ■ (/j-g+l) ^ ;■>(/>- 1) ■ . . (j3-s+2) q_
1.2 .. .
s
+ q)(p-
1
.p(p-
1
.2 . .
-1). .
.2 . .
(s-1) -1
. {p-s+3) q
. (s-2) •
(q-l)
1.2
p q(q-l) .
. . {q-s + 2)
Ap
hy-i) .
1.2 .
-1)..
1.2 .
. . (P
. . (s-1)
. (q-s+1)
. . s
+ q~s+ 1)
(7).
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., § IG). 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 ... s, and
denote_p(/)— 1) . . . (;)-s+ 1) by />„ we deduce
(p + q).=p,+ .Gip.-iqi + .Gip.-iqi. + • ■ • + -y. (8),
which is often called Vandermonde' s tJieorem, although the result
was known before Vandermonde's day.
10 p I-ETTERS ALIKE CI1. XXIII
§ 9.] To find t/ie unmlirr of r-comOiiialioii.i '</'/» + (/ letters
p of which are alii-e.
1st. Witli tlie '/ unlike letters we can form ,C> r-com-
biuatiuiis.
2iid. Taking one of the p letters, and r- 1 of the q we can
fonn ,Cr-i r-coiiiliinations.
3rd Taking two of the p, and r - 2 of the q, wc can form
,Cr-j r-combinations; and so on, till at last we tiiko r of tho
p (supposinj; /? > r), and form one r combination.
We thus fmd for the number rciiuired
,Cr + ,Cr-i + qCr-t + . . . + ,C, + 1
, f 1 I 1 1^ 1
*lr!(7-;)!''(r- \y.O, - r + I)] '^ ' ' "^1!(./- iy.*qll'
Cor. The number of r-jhrniuUUions of p + q things p of which
are alike in
q\r
■\r\{q-
1
r)! i!(r-l)!(7-r+ 1)! 2!(r-2)!(7-r+2)!
■■• (r-l)ll\{q-l)\*r\qir
For, with the ,(7, combinations of the 1st class above we can form
jCr/-.' i)erni\itation.s ;
With tho ,Cv_, combinations of the 2nd cla.ss, ,(7,., r! per-
mutations ;
With the ,Cr-j combinations of the 3rd class (in each of
which two letters are alike), ,Cr_i r!/2! permutations : and
so on.
Hence the whole number of permutations is
,C;r!+,<7r-,r!/ll + ,Cr-,r!/2! + . . . + ,C,r!/(r- 1)1 + 1,
whence the rcstiit follow.s.
A similar process will give the number of r-conibinations,
or of r-pennutitions, when we have more than one group of
like letters ; but the general fonnula is very coniplirat<Hi.
§ 10.] The number of r-combinations of n letters (,//,), when
each Utter may be repeated any numlier of times up to r, is
n (h + 1) (h + 2) . . . (« + r - 1), 1 . 2 . 3 . . . r ( 1 ).
§§ 9, 10 COMBINATIONS WITH REPETITION 11
In the first place, we remark that tlie uuniber of (r+l)-coni-
binations, in each of whicli the letter a, occurs at least once, is
the same as the number of r-combinatians not subject to this
restriction. This is obvious if we reflect that every (r + 1)-
combination of the kind described leaves an r-combiuation when~
a, is removed, and, conversely, every r-combination of the n
letters gives, when ai is added to it, an (r+ l)-combiuation 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)-combi nations
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 peraiuta-
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 aaaabhhcc . . . With this notation
there will occur in each of the {n + r)-combi nations the n letters
fli, flj, . . ., rt„ 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, Oa, . . ., an is indifferent, we maj' fix a, in the first place.
We have now to consider the different arrangements of the n - 1
letters a^, ch, • • •, (in along ^vith r s's. In so doing we must
observe that notliing depends on the order of a^, a,, . . ., a„
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 + »• things, n - 1 of which are alike, and r
of which are alike. Hence we have
(n + r-l)\
" ■■ {n-iy.rl ^-■''
_«(»+!) . . . (n + r-l)
172 . . . r
12 THEOREMS REGAUDINO „//, CU. XXIII
Cor. 1. nffr=n.r-lCr.
This follows at once from (2).
Cor. 2. J/r .-Jfr + nf/r-,.
For the r-coinhinations consist, 1st, of tho.sc in which a, oceiint
at least once, the number of which we have seen to bo J7r-i ;
2nd, of those in which a, does not occur at all, the number of
which is n-i//r.
Cor. 3. .//r = «-l//r+,.-,//r-. +,-l//r-. +. . .+,-,//■ + 1.
This follows from tlic consideration that we may cla.ssify the
r-combinatinn.s into
1st Those in which a, does not occur at all, ,-,//r in
number ;
2nd. Tliose in wliich a, occurs once, ,-,//r-i iu number ;
3rd. Those in which a, occurs twice, ,-i/7r-i in number :
and 80 on.
Cor. 4. T/ie numbir of different r-ary jmxlucts that can be
made with It different U'tters w «(«+ 1) . . . (h -t-r- 1)/1 . 2 . . . r;
and the number of terms in a complete Integral /unction of the rth
degree in n variables is (n + !)(« + 2) . . . (n + 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 resi>cctivcly. Hence the number of its terms is
1+.//.+,//, + . . .+,//,.
But, by Cor. 3, this sum is «ti^r.
Wc have tliua obtftinod n general Boltition of the problpmo FHRgBStpd in
chap. IV., §§ 17, 19. As a verification, if wo put n = 2, wo have for the
iiunibcr of tcmiH in the poncral integral funelion nf the rth dogrco in two
variables 3.-1 . . . (r + 2)/l.'i . . . r, which reduces to (r + 1) (r + 2)/2, io
agreement with our former result.
EXERCIRRS I.
Combination* and Permviafionn.
(1.) How many different numbers can bo made with the digits
11122333t.')()7
(2.) How many different permutations oun bo mode of the letters of the
lontcnco (/( lensio lic vul
§ 10 EXERCISES I 13
(3.) How many different numbers of 4 dibits can be formed with 012345G?
(4.) How many odd uumbers can be formed with tUe diyits 3094?
(5.) If :„C._i/j„_5C„= 132/35, find n.
(6.) If 7n=„C2, show that m'^i: = '^n+i('f
(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 tlie
same holds of r-combinations.
(8.) lu how many ways can the major pieces of a set of chess-men be
arranged in a line ou 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 — 1st, if there be
no restriction on the choice ; 2ud, 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 Ueutenants, 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 G 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, a 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 G of the wife's ?
(16.) In how many ways can 7 ladies and 7 gentlemen be seated at a
romid 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 2ii guests bo aiTanged so that 2 particular guests do
not sit together?
(18.) In how many ways can a team of G horses be selected out of a stud
of 16, so that there shall always be 3 out of the 0 ABCA'B'C, but never AA',
BB', or CC together ?
(19.) With 9 consonants and 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 cords, all different, be dealt into 4 equal
14 BINOMIAL TllEOltEM
CU. XXIII
huDdx, tlio order of the hauJd, but uot of the cards in the haudn, to be
attended to?
In how many cuhcs will 13 particular cards fall in one hand?
(21.) In how many ways can a set of 12 black and 12 white draagbt-men
be placed ou 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-combinatiun3 and how many 3-permutatious can be
made with the letters of farabolal
(21.) With uu unlimited number of red, white, blue, and black balls at
disposal, in how many ways can a bapfnl 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 (/ seats ; (1) in how many ways can an elector Rive his votes; (2) if there
be r voters, how many different states of the poll arc there?
If there be 16 candidates and 10 scats, and a voter nivo 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 7
BINOMIAL AND MULTINOMIAL THEOREM.S.
§ 11.] It has already been sliown, in chap, iv., § 11, tliat
where ,C,, ,Cj, . . ., ,Cr . . . denote the numbers of 1-, 2-,
. . ., r-coinbinations of n tilings. Using the expressioua just
found for ^C,, .Cj, &c., we now have
(a + b)' = a" + na'-'b + "_^Llil a«-'i' -h . . .
This is the Binomial Theorem as Newton discovered it, proved,
of course, as yet for positive intcgnil indices only.
§ 12.] We may est^iblish the Binomial Theorem by a some-
what diflerent process of reasoning, which has the advantage of
being ajijilicible to the c.vjiansiou of an integral p<jwcr of any
mnltiuiiMiial.
Consider
(a, ■^ n, 4- . . . + aj)* (2).
We have to distribute the product of n factors, namely,
(o,-Ha,-H. . .-^•a>)(a, + a,■^. . .+a,) . . . (o,-Ki,-h. . .•fa.)(3);
^§ 10-12 MULTINOMIAL TUEOREM 15
ami tlic problem is to find the coefficient of any given terra, say
«i"'«,.'^ . . . «„»- (4),
wlicre of course ttj + a, + . . .+a,„ = ?j. In other words, we have to
liiid how often the partial product (4) occurs in tiie distribution^
of (;!).
We may write out (4) in a variety of ways, such as
a,rtirtoa2a2a3«jrt4 . . . (5),
there being always a, a/s, a„ a^s, &c.
Written as in (5) we may regard the partial product as
formed by taking Ui fi'om the 1st and -Jud brackets in (3) ; a.
from the 3rd, 4th, and 5th ; a^ from tlic Gth ; and so on. It
appears, therefore, that the partial product (4) will occur just as
ol'ten as we can make diflerent permutations of the n letters, such
as (5). Now, since a, of the letters are all alike, a^ all alike, &c.,
the number of diflerent permutations is, by § 6, nljaju^l . . . a,„!.
Hence we have
(<7, + «o + . . .+«„,)" = 2 -j — j — : «i°'a/» . . . a,„'~ (6):
ailoj! . . . a,„!
wherein a,, a„, . . . a„ assume all positive integral values con-
sistent with the relation
tti + aj + . . . + a„, = W (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 a, + 04 = n, and therefore uj = « - Oj,
= 2 n(n-l) . . (n-a,^X) ^^^^^ ^^^„_„^^
which agrees with (1).
Cor. To find the coefficient of of in the expansion of
(b, + b,x + . . .+ b^x'^-'T (8)
we have simply to pick out all the terms which contain .r^ The
general term is
— /) 'iIj "i I) »m J-«1 + "''I+- • • +("1-1)"-
16 EXAMPLES CU. .Will
Hence we have to take all the terms which are such tliiit
a, + -ia, + . . . + (7« - 1) a„ = r (9).
The coefficient of of in the exjiansion of (8) is tlierefore
^a.icu!."' . ^,^-^»-- ••<'-'- (10).
where a,, a, a„ have all positive integral values subject
to the restrictions (7) and (9).
Example 1. The coefficient of a'M in the expansion of (a + {> -|- e -|- J)* ig
51
31210! 01"
Example 2. To find the coefficient of i» in (1 + ar +!>)*.
Here we mubt Lava a, + a, -|- a, = 4,
a,+ 2o,=5.
Hence o, = a,-l, o, = 5-2oj.
Since o, and o, must both be positive, the only two admissible values of a,
are 1 and 2. We have tlierefore the following table of valuu8 :
«1
«s
»»
0
1
3
1
1
2
The required coefficient is therefore
41 41
0!3!11 ^1!1!2I ~'^''-
The correctness of the result may be easily verified in the preaent mm ;
for (l + 2x + r')«=(l + i)», the coefficieut of x»in which i«,C, = 5C.
Example 3. To find the f^catcst coefficient, or coefficients, in the
expansion of (a, + a3+. . .+n„)''.
This amounts to determining j, v, r, ... so that n!/x!y!xl . . . shall be a
maximum, where r + j/ + x+ . . . = n. This, again, amounts to dutcrmininR
X, y, I, ... so that
u = x!y!tl ... (1)
shall bo a minimum, subject to the condition
i + y + x + . . . = 11 (2).
Let ns first consider the case where there are only two variables, x and y.
We obtain oil possible values of jr!yl by giving y snccessivily the value*
0, 1.2 fi, J- taking in consequence the vulues n, n - 1, n- 2, . . ., 0. The
conseculivo value to zlyl is (jr- 1)! (y + 1)!, and the ratio of the Utter
to tlie fuiuier is (y + l)/x; that is (smco x^t-y^n), (h + 1-x)/x, that is.
i; 12 MAXIMUM COEFFICIENT 17
(«+l)/.c-l. Tliis ratio is less than uuity so long as (ii + l)/.r<2, that is, so
loiiK as x>-{k+1)/'2. Until x falls below this value the tonus in the series
above mentioned will decrease; and after x falls below this limit they will
be;;in to increase.
If n be odd, ='2k + l say, then (« + l)/'2 = ft + l. Hence, if we make
j = i + l, the ratio (tt + l)/x-l = l, and two consecutive values of x\y\, viz.
(/; 4- 1)! i! and kl (k + 1)! , are equal and less than any of the others.
If n be even, =2k say, then (7i + l)/2 = 4 + i. Hence, if we make x = k,
we obtain a single term of the series, viz. klk\, which is less than any of
the others.
Returning now to the general case, we see that, if u be a minimum for all
v:ilues of X, y, z, . . . subject to the restriction (2), it will also be a minimum
fur values such that x and y alone are variable, z, . . . being all constant.
In other words, the values of x and y for which xli/lz! . . . is a minimum
1st be such as render x\y'. a minimum. Hence, by what has just been
ved, X and y must either be equal or differ only by unity. The like
luws for every pair of the variables X, y, I, . . . Let us therefore suppose
it p of these are each equal to f ; then the remaining m-p must each be
"|iial to t + 1. Further, let q be the quotient and r the remainder when n is
divided by ;»; so that n = mq + r. We thus have
p^ + {'ii-p) {i + l) = mq + T.
Hence ni^ + (m-p) = mq + r;
so that i + {m-p)lm = q + rjm.
Now (m-p)lm and rjm are proper fractions ; heuce we must have
| = g, m-p = r.
It follows, therefore, that ;• of the variables are each equal to q + 1, and
tlio rest are each equal to q. The maximum coefficient is therefore
nll(qlr-'{{q + iy.}r;
tli:itis, n!/(5!)'»('; + l)'' (3).
This coefficient is, of course, common to all terms of the type
ui U2 ... «m-r "m-r+1 ■ * • "?n
As a special case, consider (aj + rto + flj)*. Here 4 = 3x1 + 1; q = l,T=l.
Ill lice the terms that have the greatest coefficient are those of the type
'I3-, and the coefficient in question is 4!/(l!)'2i = 12. This is right; for
1, c tind by distributing that
(<!] + a, + a.jY = Till* + *-"i'"3 + <}~<ii-<io- + i2Za{-a.^3.
Example 4. Show that
n 1+x n(n-l) l + 2x n(n-l)fa-2) l + 3x
1 l + nx"*" ^lT2~ (l + 7ixp~ 1.2.3 (I+nx)» ■^ ■ ■ ■ ~ ■
{WoUtenholme.)
The left-hand side may be written
n 1 n (k - 1) 1 n(n-l)(n-2) 1
ll + nx"*" 1.2 (l + nx)« 1.2.3 (1 + nxf'*" " ' '
n x^ n(K-l) 2x n (n - 1) (>i - 2) 3x
"Il + jix"*" 1.2 (i+;Ix)2~ X.2.b (l + ;ixj3+ ■ • •
C. II. 2
18 PROrEHTIES OF ,(7, Cll. XXIIl
n 1 n (n - 1) 1 _ _ ri(n-l)(n-a) 1
~ il + iu* 1.2 (l + fur)' 1.2.3 (l + nx)'"^ • • •
fix I (B-1) 1 (n-l)(n-2) 1 1
i + iix t 1 (l + nx)"^ 1.2 (1 + nx)' " * j '
■-I
= ii__Lr__i«_|i__i_i
\ l+»ijr( 1 + nx { l + nx|
- i "^ \ " JI^ \ — I ""'
~[l + fu:) ~l + nill + fixj '
- f *" 1 " ( nx 1*
~|l + nx( (l + nx) *
=0.
13.] The Binomial TLeorcni can be used in its turn to
est.-iMlsli idi'iitities in the theory of conibiuatioua ; as the two
following c.\;iiiii>los will show : —
Examplo 1. We have
= (l + x)'--,C,x(l + xr' + ,C,x'(l + x)'-'- . . . (-)VC,x'.
On the rigbt-liand Bide of this identity the corflioient of every |Hiwar of x
mast vanish. Hence, « being any positive iutcger \e»s than r, ne have
,C.xl-,_,C,.,x,C, + ^5C..,x,C,-. . . + (-r'_»4.,C,x,C.., + (-)VC. = 0
Example 2. Tu lind the sum of the squares of the binomial coefficient!).
•Wchave (1 +x)» = (l + x)"x (x + l)»
= (l + ,C,x + ,C^»+ . . . +.C,x-)
x(x- + ,C,x-' + ,C^»-»+ . . . +,rj.
If we imngine the product «n the right to bo distributed, »o see that tht
cocflicient u( x" is l'+,l'i' + ,C,'+ . . . +„CV ; 'he cucOicient of x" on the
left is ^C.. Hence
l' + -C,« + ,C,= + . . . +,C.'=^C.=2«l/«!nI.
Siocu
2n! = 2ri(2n-l)(2n-2) . . . 1 .3.2. 1 :=2". 1 . 2 . . . nxl.3 . . . (2rt-l),
we have 1» + «CV+,C,'+ . . . +,C,» = 2». 1 .3 ... (2n - l)/iil.
A Croat varii'ty of results can be obtained by the above procctu of equating
coGfliciints in identities derived from the binumial theorem; some specimuaa
arc given aiuuiig the cierciscs below.
EXERClSEii II.
(1.) Find the Uiird term in the expansion of (2 + 3x)*.
(2.) Find tlie coefficient uf x* in the expansion o( {I -t- x + x*) (I -x)".
(3.) Find tlic term which is iudc|>eudeiit uf x in the expansion of
(x + l/x)*
§ 12, 13 EXERCISES II 19
(4. ) Find the coefficient of ar* in the expansion of (x - 1/j-)^.
(5.) Find the ratio of the coefficients of x-" in (1 + .t)''" and (1 + x)-".
(6.) Find the middle term iu the exijansion of (2 + Jx)".
(7.) The product of the coefficients in (l + x)"+i : the product of the
ioeffioients iu (1 + x)'' = (k + 1)'' : nl.
(8.) The coefficient of x' in {(r-2) x- + «x-r} (x + 1)" is ii„C,^„.
(9.) If I denote the integral part and /'' the proper fractional part of
3 + ^/5)", and if p denote the rational part and a the irrational part of the
lame, show that
I=2{3» + „C'5 3»-=.o + „C'^3''-*.5=+ . . .}-!,
(10.) If (,,/2 + l)"-^'+' = J + i^, where Fis a positive proper fraction and 7 is
integral, show that F{I+F) = 1.
(11.) Find the integral parts of (-J^fi + S)"-"', and of (2^3 + 3)="'+'.
(12.) Show that the greatest term in the expansion of (a + x)" is the
(r+l)th, where r is the integral part of (« + l)/(«/x + l).
Exemplify with (2 + 3)"' and with (2 + ^j".
(13.) Find the condition that the greatest term in (a + x)" shall have the
gieatest coefficient. Find the luuits for x in order that this may be so
iii{l + x)>«'.
(14.) If the pth term be the greatest in (a + .c)"*, and the ijth the greatest
in (a + x)", theu either the {p + q)th or the {p + q -l]th or the [p + q - 2)ih is
the greatest in (a + x)*"*^.
(IS.) Sum the series
•£i+2»?»+3 2?'+ . . . +n-''r--
(16.) Sum the series
l + 2„C, + 3,.C.j + 4„C3+ . . .
(17.) U Pr denote the coefficient of x,. in (1 + x)", prove the following
relations : —
r. Pi-2p, + 3p^- . . . +;i(-l)"->y„ = 0.
a-. iP.-iP. + ■ ■ ■ +^^' Pn=^i-
30. l+p^+p^ +.-.+-^ =^^.
2 S n+l n+1
(18.) If pr have the same meaning as in last question, show that
(-1)--' ,11 1
Pi-iP'. + iPs- •••+—„ -i'» = l + 2 + 3+ • • • +,-,•
(19.) Show that
,C,xl + ^iC^iX,Ci + ,_5C,_„x,C3+ . . . +,._^,C,XrC,_, + lXrC, = rC,2'.
(20.) Show that
(i-„c,+„c,- . ..r+LC,-,fi,+ . . .)==i+„q + .A+ . . .
2—2
20 KXKUCISES U CIl. X.VllI
(21.) Show tlial
1 -.(-'• + -<''i'<.t'.+ • • • +.0,_,x.C. = (2«)V(n+2)!(fi-2)l.
(32.) Show that 1 -„.+ ("> -i>)' - (liili^-'j, <?-?>)•+ . . . =Oif
boodd, aud =(- l)*"(n + 2) (n+4) . . . 2h/J . 4 . . . n if n be even.
(23.) Show that
,.„(,.,,,, !;,„.,)„+''('^|i)(„_,,(„.i,+"i"_ti^)(iti.>(„_3,(„_2)
+ . . . =2(2n + l)!/(n + 2)!(n-l)I.
(24.) If IV stand for jf+ljx^, show tlmt
«r+l+r*|Cl"r-I + r+lC,Ur-3+ • ■ . = "i ("r + r^l "r-3 + r<^5 "r-** • • • )•
(25.) If <if Jcuoto the cocdicicnt of x'' iu (l+i)'i"-''i(l - j)» Bhow tliat
"t-fk^i^i + n^^j"!' • ■ • =^ '"f "" Tohics of p except p = n, in which casa
the rigbt-haud eido of the oiualiun is i*.
(20.) Show that
1 _ .^ ^ .("a _ , ( - ii V'. ^ n\
X x + 1 1 + 2 ■ ■ ■ i + n x(x+l) . . . (i + w)'
(27.) riu.ithccocnicicntof x'iu (l + j- + x'+ ...)».
(28.) Find the cocOicicut of x" in (1 fj* + z« + *»)♦.
(29.) Find the coefficient of x" in (l+x + 2x' + 3x'+ ...)».
(30.) If (I,, a "i, •"! the cucfBcicDls of the poweni of x in
(I+2x + 2x')», show thnt "."a - a,a„., + . . . +a^n„=0 if n bo od,l,
=.2^i!/{(Jn)!j» if It be even.
(31.) If a, be the cocllicicnt of x' in (l + i + x»+ . . . +x'')», iiliow tlmt
"r - .<^i "r-i + «Ci "r-j - ... =0, nnli'SH n 1)0 a niuUiple of p + 1. What
do(i the oqiiatiun bccuniu in the luttcr coac?
(32.) Find the coiflicicut of x" in (l + 2x + 3j-+ tx")".
(33.) Write out Ihu exjiauiauu of (a + 6 + c + <i)'.
(31.) Show that
v»''if . . . n» 1 Jn(m.l)l P
rl.I . . . il~p! I 2 ( •
whore r, » * have oU values between 0 and p, both inclusive, subject
to the rei4liirtion r + §+ . . . -^-k-p.
(35.) If ,;/, have the meaning of f 10 above, prove that
l"- -*«Wr = «"r + ."r-.X,//,-f«H,.,X.//,+ . . . +,/^,x,/V,.
r. l-.C,x,//, + ,C,x,i/,-,C,x,//,+ . . . +(-l)%C,,//,=0.
(M.) Ifx, = x(x4l) . . . (x + r-I), nhowlhat
(37.) Find the largoal coedieicnl in the expansion of(a + 6 + c + <l + <)".
i:>-15 LAW OF DISTRIBUTION USED 21
EXAMPLES OF THE APPLICATION OF THE LAW OF
DISTRIBUTION.
§14.] If we haver sets, consisting of Hi, n.,, . . . , nr dij/'erent
ktters respectivdij, the ivhole number of different ivat/s of making
combinations by taking l,2,Z, . . . iip to r of the ktters at a
time, but never more than one from each set, is
(«.+ l)(»,+ l) . . . (;/,.+ 1)-1.
Consider the product
(1 + Oj + 61 + . . . «i letters)
X {l + a.2 + b.i+ . . . n„ letters)
X (1 +rtr + /;,. + . . . tir letters).
In the distributed product there will occur every possible com-
bination of the letters taken 1, 2, 3, . . ., ?• at a time, with the
term 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 +«,)(!+«=) • • • (l + "r)-l
= 5«i + S»,Ho + . . . + ?;,?(.> . . . n,.
This result might have been obtained by repeated use of § 7.
§ 15.] If we have r sets of counters, marked ivith the fulluwing
numbers —
"i, Pu ■ ■ •. "u
a,, ji^, . . ., K.,,
Or, Pr, ■ • ■> Xrt
the number of counters not being necessarily the same for each set,
and the inscribed numbers not necesaarHy 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,
u tlie coefficient of x^ in the distribution of the product
22 nisTniniiTinN i-rohlkm cu. xxiii
(j"i + a^> + . . . + iC')
X {x^ + x^+ . . . + j-«)
X (t** + a^' + . . . + r''-).
Tliis lliporem is an obvious result of tlie principlas laid down
ill clm]i. IV.
Cor. 1. //* in the firft art there he a, couutiTS marked with
the nnmhiT ",, /», murked with /i,, <{r., in the second a^ marked
with a,, ft, marked with (i,, Xc, the number of wai/s in whieh r
rt'iinti'm can he dniiim so that the mim of the numbers on them it
It, is the coej/icient of 3^ in the dislrihiilion of
{atSf' + b,a^' + . . . 4 k,a:'')
X {a^t + b^' + . . . ^ /vf •)
X (a,J"r + i^r + . . . + k,JC*r).
Cor. 2. In a box there are a counters marked a, h marked ft,
<f-C. A couxter is drawn r times, and each time rephtci'd. The
iiumhcr ofwai/.t in whieh the sum of the drawimjs can amount to
n is the ooejjicient of j^ in the distribution of
(ttj- + bjfi+ . . .y.
ni.STRinUTION.S AND DERANGEMENTS.
§ 16.] Tlio variety of iirobloins that arise iu connection with
the 8ul)jecl of llio ]>ri'stMit chajitor is cndlc.s.s, and it would be
iliflirult within tlio. limits of a textbook to indicate all the
methods that have been used in solving such of these problems
as niathoinaticians have already discus.'^od. The followinjr have
l>ecn .selected jum typis of problems which are not, very readily at
least, reilucililo to the elementary ca.ses above discussed.*
§ 17.] To find the number of ways in whieh n different Iftten
can l/e distributed among r j>i{jeim-hohs, attention tn-intj paid to
tlui order of the j'liieon-holes, hnt not to thfl filler of the letters in
any one piffeon-hub', and no hole to contain less than one letter.
Let Dr denote the nund)er in ipiestion.
* For fuiUiur infuruuttiua ao* Wlutwurtli'i Clioiet and Chane*.
J
j;^ 1 ■ 17 DISTRIBUTION PROBLEM 23
It' we leave s specified holes vacant aud tlistribnte the letters
among the remaining r-s holes under the conditions of the
question, we shonld thus get Drs distributions. Hence, if ,(7,
liave its usual meaning, the number of distributions when s of
the holes are blank is rC^, D^-,.
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 ;•" ways.
Hence
Br + rC, Dr-, + rC, A-. ^ . . . + .(7.-1 A = »•" (A).
The equation (A) contains the solution of our problem, for, by
putting r = 2, /• = 3, &c., successively, we could calculate D^, D3,
&c., aud 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
A-l + r-xC\ Dr-,+ . . . + r-lCr-, A = (^ - l)" (B).
Prom (A) and (B), by subtraction, remembering (§ 8, Cor. 3)
that
we derive
A + r-i^iA-i + r-,tr,A-.+ . . . +r-,a-.A
= r''-(»--l)" (1).
From (1), putting r— 1 in place of r, we derive
A-l + r-iGl Dr-1 + . . . + r-oCr-a A
= {r-\Y-{r-2Y (1').
From (1) and (1'), by subtraction, we derive
A + r-3C',A-l+r-2C2A-2+ . . . +r-,Cr-2D.,
= r» - 2 (r -!)"+(»• -2)" (2).
Treating now (2) exactly as we treated (1) we derive
A + r-3C,A-l + ,-3C2A-5+ • • • +,-sCr-3A
= 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., § 11). We shall therefore
liuaUy obtain
24 DEHANOKMENTS CH. XXIII
D .,--,C7,(r-l)- + ,(7,(r-2r- . . . {-)'-\C,,\\
= f--^(r-l)« + ?^^(r-2)--. . .(-r'Jl- (4).
Cor. If the ordi'r ofthfipigeon-holeg he indifferent, the numher of
Jistriliutiitns is DJrl. In other worth, the uumber of partitions of
n diffrreiit letters into r lots, no vacant lots being allvwed, is Dr/rl
Wi- sliall discuss tlie cltwely-allicd problem to find tlie
uimibtT of r-iKirtitions of n— that is, to find the number of
waj-8 Id whicli n letters, all alike, may be distributed among
r iiijteon-hnlc's, the onler of the holes being indill'erent, and no
hole to be empt}' — when we take up the Theory of the Partition
of Numbers.
§ 18.] Giien a series nf n letters, to find in hmc many uai/s
the iirder may be diramjed so that no one out nf r assigned Liters
shall occupy its original jtosition.
Let ,A,. denote the number in (jncstion.
The number of dillerent deraiigementa in which the r assigned
letters do all occupy their original places is {n-r)]. Hence the
number of derangement.s in which the r assigned letters do not
all occupy their original places is til-{n-r)l Now, this last
numln-r is made up of —
iBt 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 occupie-s its original place, and no one of the remaining
r-1 does so; that is, ,C, ,-,A,.,.
3nl. The number of derangements in which any two of
the r letters occui)y their original places, and no one of the
remaining r-2 dws so; that i.s, ,C',,-^,.,. And so on.
Hence
+ rC>-l n-rtl-^l (A).
If we write in this equation n - 1 for n, and r - 1 for r, and
subtnwt the new oipiation thus dcrive<l fnim (A), we deduce
u'. - (n 1)1 -A,. ♦ , ,r,' ,A , + .^' _\ 4.
■t r-lC>.|..r,jA, (I)
^§17-19 SUBFACTORIAL M 25
We can now treat this eiiuatiou exactly as we treated
equation (1) of ^ 16. We thus deduce
„A, = „!-r(«-l)! + 1^Jll)(«-2)!-. . . (-)-(«-»•)! (2). ^
If we remember that (?» — ?•)!, above, stands for the number
of derangements in which the r letters all occupy their original
positions, we see that, when r = «, {n — r)\ must be replaced by 1.
Hence
Cor. The number of derangements of a series of n letters hi
which no one oftlie oi'iginal n occupies its original position is
The expression (3) may be written
n{ . . . (4(3(2(1-1) + 1)-1)+1) . . .-(-l)-) + (-l)-.
Hence it may be fonned as follows: — Set do\vn 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 braucli of mathematics, and has been called by
Whitworth suhfactorial n. He denotes it by ||m. 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 au order
of ideas which forms the foundation of that important branch of
modem algebra which is called the Theory of Substitutions.
Consider any two permutations, becda, bcade, 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 by a, d hy d, e by c. This process may be
represented by the operator ( i '? ) ! -ind we may write
/ebadc\
febadc\
\abcde)
becda = bcade :
20 THE SUIISTITUTION OPERATOR CII. XXIII
or, omitting tlie letters that are uualtercd, and thus reducing the
operator to ita simple^ /"im,
I ) Itecda = icade.
\acej
The operator \) , and the operation wliicli it cffcct«, are called
a Suhstitiitim ; and the operator is often denoted by a single
capitui letter, S, T, itc.
Since the number of different permutations of a group of n
letters is «!, it is obvious tliat the number of diflorent substitu-
tions is also ;i!, if we include among them the identical snl>Mi-
tution ("^','^^'' ' ' '), (denoted by S" or by 1), in which no letter
\abcde . . J
is altered.
We may effect two substitutions in succession upon the same
permutation, and represent the result by writinj,' the two symbols
representing the substitutions before tiie permutation in order
from right to left. Thus, if «S = (^'J^) , 7' a Q .
STaebcd - eaibd.
We may also effect the same substitution twice or three times
over, and denote SS by <Sf', SSS by S', &e. Thus, 6' being as
before,
S^aebcd = Sceabd = becad.
It should be observed that the nudtiplication of substitution
symbols is not in general comnuitative. For example, S and T
being as above, STaehcd - ecu/id, but TSiubcd = cacbd. If, when
reduced to tlioir sinijilest form, the symbols .S' and T have no
letter in common, they are obviously commutative. Tiiis con-
dition, although s>iflicient, is not necessary ; for we have
/dr(ih\ /Uidr\ , , ., /b<ldc\ fdniliX
[um) W) "''^''^ = "^''"'' = [aUdJ [ubcd) '''"•''^-
8 20.] Since the number of permutations of n letters is
limiti'd, it is obvious that if we repeat the same substitution, S,
sufficiently often we shall nltiniately rejirodnce the permutation
that we started with. The smallest numlier, /», of rej^titions
for which this happens is aillod the order of the suistitutiun S.
§ 19-22 ORDER AND GROUP 27
Hence we have S'^ = l, and S'"'=l, where p is any positive
integer.
TVe may define a negative index in the theory of substitu-
tions by means of the equation S''' = S""^'^, fi being the order of
S, and p such that p/i > q. From this definition we see that
S'tS-" = SXS'"'-^ = /S"" = 1. In other words, S' and S'^ are inverse
to each other ; in i)articuhir, if
„ _ (dahc\ , „.j _ (abcd\ _ fhcia\
~ \abcdj' \d(tbc) ~ \abcd)'
A set of substitutions which are sucli tluit tlie product of
an3' number of them is always one of the set is called a group;
and the number of distinct substitutions in the group is called
the 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 /)> where each letter
is replaced by the one that follows it, and the last by the first, is
called a Ci/clic Substitution, 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 eSect of a cyclic substitution may be represented by
writing the « letters at equal intervals round the circumference
of a circle, and shifting each tlirougli l/zjth of the circumference.
Thus, or otherwise, it is ob\'ious 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 (bcdtfa), {cdej'tib), Ao.
28 CYCLES CH. XXIII
g ^ fbj'dcgiuh\
\abcdefijh) '
This replarps ahy b, b l)y/, /liy a; theso top;ptlior constitute
the cyclic sulistitntion (nbf). Next, c is replaced by </, and d by
c\ this is equivalent to the cycle {cd). Afc'uiii, <? is rej)lace<l by
g, aud <7 by e ; this gives the cycle {eg). Finally, h is unaltered.
Hence wo have the following decomposition of the substitution
S into cycles —
S=(ubf)(cd)(eg){h).
The decomposition is obviously uniijue; 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 arc commutative, and it is indilVerent
in what order we write them.
§ 23.] Every cyclic subiifilutioii nj n letters can be dicom}>osed
into the product o/n— I tran.ij)ositwns.
For e-xamplo, we have {abed) = (ab)(bc){cd) ; and the process
is general.
Cor. Every substitution cun be de^u)mpised into n-r transpo-
sitions, where n is the number of letters which it displaces, and r
the number of its projitr cyclrs.
Tliis decomposition into transpositions is not unique, as will
be seen iircsently, but tiic above gives the minimum number.
§ 24.] The following ]iriiper(ies of a proiluct of two trans-
positions arc of fundanientjil ini|Hirt:incc.
I. The product of tiro tninsposifions which have two letters
in common is an identical sulistitiUion.
This is oi)vious from the meaning of {ali).
II. In the product of two tninsposilions, TT' , which hair a
Utter in common, 7" may he placed Jirst, pnirldul j/v rijilaee the
common Utter in T by the otli«r Utter in 7".
§§ 22-25 DECUMPOSITluN INTO TUANSPOSITIONS 29
For we have {ab){bc) = (J^) , («'c)(«c) = (Jj,'') .
therefore (ab){bc) = {bc){ac).
Cor. 1. ie/Ka/) = (ae){e/).
Cot. -2. (ae){af) = {af){e/).
III. 1/ two transpositions, T and T', have no letter in common,
thy are commutative.
This is a mere particular case of a remark already made
regarding two independent substitutions.
§ 25.] Ths 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 —
Tki number of the transpositions which represent a given sub-
stitution is alaai/s 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 hist
transposition, T, on the right that involves a, and proceed to
commutate this transposition successively with these to the left
of it. So long as we come across transpositions that luive 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 7" which is
not a, we see (§ 24, II., Cor. 1) that the « in T 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.
:iO iJliCOMl'OSmo.S into TKANSrOSlTIONS Cil. XXIU
Tlie result of this process, so far as a is concerned, will bo,
eitlier that all the tninsjHjsitinns containing a will have dis-
appeared, or that some even number (including 0) will have dono
80, and one only, say (ab), will remain on the extreme left.
Consider now 0. If among the reniaining factors b does not
occur, then wo have obtained a cycle (al>) 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 commutate 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 loft with, say {be), in the second place. We now defd
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, s-ay /, that di.sajipears from the factors not yet
finally arranged. Wc thus arrive at a product (ab){bc)(cd){de){^
on the left.
Now {ab){bc){cd)ide){ef) = (^^^^J)
= (abcd^').
We have, in fact, arrived at one of the independent cycles of
the sulxstitution. If we now take any other letter tliat 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 nundier of transpositions lost is even, the tnitli 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, etvn subnti tut ions,
which are equivalent to an even number of transjKwitions, and
odd siibstltiifioiis, which are eipiivalent to an odd number of
trans{)ositions.
Cor. 1 . 1/nbethe number qf letters altered by a stihstitutton, r
the nundier ii/it.t ri/rlt\i, and 'Js an tirbitniri/ eten intfijir, the num>>er
oj'/acturt in an ojiuivalent prudiict oj tranjtjMjsitiom w «-r + :i».
§§ 25-27 EVEN AND ODD SUBSTITUTIONS 31
Cor. 2. The number of the even is equal to the number of tlie
odd substitutions of a set of n letters.
For any oue transposition, applied in succession to all the
difl'erent odd substitutions, will give as many even substitutions,
all dilTereut. Hence tliere are at least as many even as there
are odd substitutions. In like manner we see that there arc 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
pa/rticidar, any power ivhatevei- 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 nl/2.
§ 26.] If we select arbitrarily any one, say P, of the n! per-
mutations of a set of n letters, and call it an even permutation,
then we can divide all the n\ permutations into two classes —
1st, ?i!/2 even permutations, derived by applying to P the nl/2
even substitutions ; '2nd, 7ilj2 odd permutations, derived by
applying to P all the «!/2 odd substitutions.
The student who is familiar with the theory of determinants
will observe that the above is preci.scly the classification of the
permutations of the indices (or umbrae) 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 ichat-
ever." Alternating functions, on the other 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 Ei^uations. The reader who desires
32 EXERCtSES (It CH. XXItl
to pursue tliis siilijoct farllier will fiiicl iiifurmatinn in tlic fidlow-
inp works: Scrret, Coiirs d'Alijihre Sup/rieure (I'aris, 1879);
Jordan, Traite di's Substiliifi'ms (I'aris, 1870); Netto, »S'«/«/iVm-
tioiu'ii-t/tcoru; (Leiiizig, 18S2) ; Burnsitle, Theory of Groups
(Cambridge, 1897).
ESERCIBES III.
(1.) There nrc 10 countora io a bux oiHrKuil 1, 2, . . .,10 rospcctively.
Throe drawings arc miule, the counter drnwn being ri-phtced cnch time. In
huw many ways cau the sum of the numbem drawn amount — Ist, to 9
exactly; 'ind, to 9 at least?
(2.) Out of the integers 1. 2, 3, . . .,10 bow many pairs can be selected
80 that their kuni liliall be even ?
(3.) How many diflorcnt throws can be made with n dice?
(-1.) In how many ways can 5 black, 5 white, o blue balls bo equally
distributed amon;; three bah's, the order of the b.'i^-s to bo attended to?
(5.) A Bclection of c tliinRs is to bo made partly from a Kroup of a, the
rest from u K™"P "f f>- Prove that the number of ways in which such a set
can bo made will never be ^renter than when the nnmber of things taken
from the croup of u is next less than (ii + 1) (t-(- !)/(« + 1 + 2).
(6.) In how many ways can p +'» and n - 's be placed in a row so that no
two - 's cumo together ?
(7.) In the Morse signalling system how many signals can be made
without exceeding 5 movements 7
(K.) In how many ways cm 3 pairs of subscribers be set to talk in a
telephone exchange having ii subseribers ?
("J.) There are 3 colours, and in balls of each. In how many ways can
they l>c arranged in 3 bags each containing m, the order of the bags to
be attended to?
(lU.) If of ;) + f + r things p be alike, q alike, and r different, the total
number of cunibinatiuMH will be (/> + l) ('/ + !) 'i'- 1.
(11.) In how many ways ctn 'in things bo divided into n pairs?
(12.) The numl>er of eombinatiuns of 3n things (n of which arc alike),
taken n at a time, is the cooflicieut of x* in (l+f)^/(l -x).
(13.) N boat clubs have n, 6, e, 1, 1 1 boats each. In how many
ways can the boats be arranged subject to the restriction that the Ist boat of
any club is to be always above its 2nd, its 2nd always above its 3rd, *c. 7
(14.) If there lie p things of unc sort, if of another, r of anotlicr. Sic, the
numtxT of combinations of the p + q + r^- . . . things, taken k at a time, ia
the coefficient of x» in (1 - x^>) (1 - jr»»->) . . . /(I - x) (1 - j-) . . .
(1.").) In hi>w many ways can an arrangement of n things in a row be
deranged sn that -1st, each thing is moved ono place; 2nd, no thing more
than one plaro?
(16.) Uiveu n things arranged in soooessioo, the number of sets of 8
§ 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) (k-3) (tt-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 + 7i-l).
(20.) The same as last, no empty holes being admitted, nl(n-l)!/(«-r)l
(r-l)l.
(21.) The same as last, the oidcr of the holes not being attended to,
nl(n-l)!/(n-r)lr!(r-l)!.
(22.) The number of ways in which n letters, all aUke, can be distributed
into T pigeon-holes, the order of the holes to be attended to, empty holes to
be excluded, is „_jC,_,.
(23.) Same as last, empty holes being admitted, „+,_jC,_i.
(24.) Same as last, no hole to contain less than q letters, „_j_,(,_ijC,_i.
(25.) The number of ways of deranging a row of « letters so that no letter
may be followed by the letter which originally followed it is n\ -(- (»i - 1) j .
(26.) The number of ways of deranging m + n terms so that m are dis-
placed and 11 not displaced is (m + n)\m\jm\n\.
(27.) The number of ways in which r different things can be distribated
among n +p persons so that certain n of those persons may each have one at
least is
S^=[n^pY-n(n+p-\Y + ''l^^^(n+p--2Y-. . .
Hence prove that
S, = S3=. . .=S„_, = 0, S„=nU S„+i=(^+i')(« + l)!.
( WoUtenholme.)
(28.) Fifteen schoolgirls walk out arranged in threes. How many times
can they go out so that no two are twice together? (See Cayley's Works, vol.
1., p. 4S1.)
EXEKCISES IV.
Topoloffical.
(1.) The number of sides of a complete n-point is Jn(«-1), and the
number of vertices of a complete 7!side is the same.
(2.) The number of triangles that can be formed with 2n lines of lengths
1,2 2n isn(n-l)(4n-5)/G.
(3.) There are n points in a plane, no three of which are coUinear, How
• Exercises 19-25 are solved in Whitworth'a Choice and Chance; q.v.
C. u. 3
;U KXKKCISKS IV CM. XXIll
many closod r-sidod figures can bo lurmt-d by joining tbc poiiila by straiglit
lincH?
(4.) If wi puiutB in ono KtraiKhl lino bo joined to n pointa in another in
every poiwiblu way, show that, oxcluHivo o( the m+n given pointK, there are
mil (in - 1) (II - l)/2 points of intersection.
(5.) On three striii^-lit lines, A, II, C, are taken I, m, n pointa respectively,
no on« iif which in n point ol intorsectiou. Show thnt the number of triangles
which can be formed by taking three of the f + m + ii points is i (in + n)(n + f)
{l + m)-mn- nl - Im.
(ti.) There are ii points in a plane, no three of which arc cullinear and no
four omcyclic. Through every two of the points is drawn a straight line and
through ev<Ty three a circle. Assuming each stniiglit hne to cut each circle
in two diiitinct (loiuts, find the number of the intersections of stniight lines
with circles.
(7.) In a convex polygon of n sidt's the number of exterior intersections of
diagonals is ,>jii (ii - 3) (n - 4) (n - 5), and the number of interior intersections
is ,'.n (II - 1) (n - -2) (n - 3).
(8.) There arc ii points in space, no three of which ore coUincar, and no
four coplaiiar. A plune is drawn llirough every three. Vind, 1st, the num-
ber of ilistiiict liius of iiiter.sectionH of these planes; "Jnd, the number of these
lines of iutersection which puss through one of the given ii points; 3rd, the
number of distiuct points of intersection exclusive of the original « points.
(9.) Out of II ^lraight lines 1, 2. . . . , ii inches long respectively, four can be
chosen to form a pericyclic iiuadriluleral in {2ii(ii- 2)(2ii- 5)-3 + 3( - 1)"|/18
ways.
( 10.) Show that n straight lines, no two of which are parallel and no throe
concurrent, divide a plane into J(ii' + n-l-2) regions. Hence, or ollierwise,
allow that ii pluueK through the centre of a sphere, no three of which arc
coaxial, divide its surface into ii'-ii + 2 regions.
(11.) Show that two ]icucils of straight lines lying in the same plane, one
containing m the other ii, divide the plane into iiiii + 2ni -t- 2n - 1 regions, it
bvnig supposed that no two of the lines arc i)araUcl or coincident
(13.) If any number of closed curves bu drawn in a plane each cutting all
tbc others, and if ii, be tlic number of jioints through which r curves pass,
the number of dibtiuct closed areas formed by the plexus ie
l + n, + 2H,+ . . . + rn,+,+ . . .
CHAPTEE 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 onlinary sense can
be made, because any such number is expressed in terms of two
absolutely heterogeneous units. Strictly speaking, tliere is a
similar difficulty in comparing real algebraical quantities which
have not the same sign ; but this difficulty is met (see chap,
xni., § 1) by an extension of the notion of inequality. It will
be remembered tliat a is defined to be algebraically gi-eater 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
36 ELEMENTARY THEOREMS CH. XXIV
parallel to tlic theory of coiiditioiiul eiiuations. In fact, the
approximate numerical Kuliition of eipiations depends, as we have
seen, on the establisiimeiit of a series of inequalities*.
TIic ti)llowing theorems will briiig 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 eipiation, 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 a]ip(\'irs.
The modifications necessary when the other sign appears are in
all cases obvious.
I. IfP>Q,Q>R,R>S,thenP>S.
Proof.— (P-Q) + {Q-J{) + {R-S)BP-S,hcoce,BmccP-Q,
Q- It, R-S are all positive, P-S'k positive, that is, P>S.
II. If P>Q, then P±R>Q±R.
For (P±R)-(Q±R)^P-Q\ hence the sign of tlie former
quantity is the same as the sign of the latter.
Cor. 1. I/P+Q>R + S, then
P+Q-R>S, -R-S>-P-Q, -P-Q<-n-S.
It thus appears that we may transfer a term from one side of
an ine4]uaUty to mmther, provided we change its slijn ; and ire
may change the signs of all tlie terms oti 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,>(^„ P.Xh PnXin.
then /^ + A +...+/',> Q. + Q, + ...+ ^. ;
for {P^P,+ . . . +Pn)-(Q, + (?,+ . . . * V.)
= (/^-(^.) + (/^-V=)+ . . . M/'. -<?.),
■ whence the theorem follows.
It should lie noticed that it does not follow that, if Pi>Q,,
P,>Q„t\lcnP^-P,>Q,-Q,.
* Soo, for example, the proof that eTtu7 cquntion has a root.
§ 3 ELEMENTARY THEOREMS 37
IV. If P>Q, then PE>QR, and P/B> Q/R, provided R
be positive; but PR<QR, P/R<Q/R, if R he negative.
For {P-Q)R aud {P - Q)/R have both the same sigu as
P-Q it R be positive, and both the opposite sign if ^ be"
negative.
Cor. 1. If P>QR, and R>S, then P>QS, provided Q be
positive.
Cor. 2. Every fractional inequality can he integralised.
For example, if P/(2>R/S, then, provided QS be positive,
we have, after multiplying by QS, PS>QR; but, if QS be
negative, PS<QR.
If there be any doubt about the sign of QS, then we may
multiply by Q-S\ which is certaiidy positive, and we have
QPS"->Q'RS
V. ifPi>QuPi>Q^..---, Pn > Qn , and all the quantities
he positive, then
PJ\ . . . P„> Q,Q. . . . &.
For PJ\P, . . . Pn>QiPJ\ . ■ . Pn.
since A>Qi and P.Pa . . . P„ is positive ;
>Q.Q.Pz . . ■ P.,
since Pi>Qi and Q1P3 . . . Pn is positive ; and so on. Hence,
finally, we have
P,P, . . . Pn> Q>Q. . . . Qn.
Cor. 1. If P>Q, and hoth he positive, then P"> Q", n being
amy positive integer.
Cor. 2. If P>Q, and hoth he positive, then P"">Q"", h
being any positive inte{/cr, and the real positive value of the nth
root being taken on hoth sides.
For, if P""5Q"", then, since both are real and positive,
(pi/i.)n=(Qi/n)n_ j^y (Jqj_ j . ^\^^^ jg^ p~ Q_ ^s\\\i:\\ coutradicts our
hypothesis.
Cor. 3. If P>Q, hoth being positive, and n be any positive
qitantity, then P-''<Q''', where, if the iiidices are fractional,
there is the tisual understanding as to the root to be taken.
Remark.— 'YXw 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 niiglit be removed in certain cases ; for
example, it follows from - 3 > - 4 that ( - 3)'>( - Af, in other
words, that - 27 > - 64 : but the imporbince of such {articular
cases docs not justify tlieir statement at length.
Cor. 4. An inequality may be rationalised i/ due attention h«
paid to tlie above-mentioned restrictions regarding sign.
g 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 ojiposite sign, and, in particular, that the s^piare of any
real quantity is positive, we can solve a great many questions
regarding inequalities.
The following are some examples of the direct investigation
nf inequalities ; the first four are chosen to illustrate the paral-
lelism and mutual connection between inequalities and equa-
tions : —
Example 1. Under wliat circumstances is
(3j-1)/(i-2) + (2z-3)/(x-6)> or <57
Ittt. Let us suppose thnt x does not lie between 2 and 5, and is not eqnol
to cither of these valaos. Then (x -2)(x- 5) is positive, and we may multiply
by this factor without reversing the signs of inequality.
Hence f= (3x - l)/(x - 2) + (2x - 3)/(x - 5) >< 6,
according as
(3x - 1) (x - 5) + (2x - 3) (x - 2) ><6 (i - 2) (i - 6),
according as 6x'-23x + ll><5x'-35i + 60,
according as \2x> <39,
according as x> <3^.
Under our present supposition, x cannot have the vnluo 3} ; but we con-
elude from the atrarc that if x^-S, /■'>6, and if x<2, F<6.
2nd. Suppose 2<x<5. In this case (x-2)(x-6) is negative, and we
must reverse all the signs of inequality after multiplying by it.
Wc therefore infer that if 2<x<3|, i''<5, and if Si<x<6, then
ii'<5.
The student shnnid observe that, as x varies from -a> to -f eo , the sign of
the inequality is thrice reversed, niimely, when x = 2, when x = 3J, ami when
1 = 5; the first ami Init revorwals occur because F changes sign by passagg
through an inlinito value; the second reversal occurs because F
§§ 3, 4 EXAMPLES 39
tlirough the value 5. The student should draw the graph of the func-
tion F.*
Example 2. Under what circumstances is
F=(3x-4)/(x-2)><l?
Multiplying by the positive quantity (x- 2)^, we have *
(3a:-4)/(x-2)><l,
according as (3x - 4) (x - 2) >< (a: - 2)',
according as { (3i - 4) - (x - 2) } (x - 2) > <0,
according as 2(x-l) (x-2)> <0.
Hence ■F>1, if x<:l or >2;
F<:1, if l<a:<2.
Example 3. Under what circumstances is x' + 25x > < 8x' + 2G ?
i' + 2.5x><8x2 + 2G,
according as x'-Sx- + 2ox-26> <0,
according as (x-2) (x'-6x + 13)> <0,
according as (x-2){(x-3)= + 4}> <0.
Now (x - 3)^ + 4 is positive for all real values of x ; hence
xS+25x> <8x2 + 2G,
according as x><2.
Example 4. If the positive values of the square roots be taken in all
ftfLRPS IS
V(2-i + 1) + N'(-t - 1) > < v/(3x) ?
Owing to the restriction as to sign, we may square without danger of
reversing the inequality. Hence
J(2x + 1) + V(x-1)>< v/(3x),
according as 2x + 1 + x - 1 + 2,^{ (2x + 1) (x - 1)} >< 3x,
according as 2;^{(2x + l) (x- 1)}> <0.
Now, provided x is such that the value of ^ { (2x + 1) (x - 1) } is real, that is,
provided x>l,
2V{(2x-|-l)(x-l)}>0,
therefore ,y(2x + 1) + ^(x - 1) > V(3x), if x > 1.
Negative values of x less than -J would also make ,^{(2x + l) (x- 1)}
real ; but such values would make ,^(2x + l), ^(x-1), and ^{3x) imaginary,
and, in that case, the original inequality would be meaningless.
Example 5. It x, rj, z . . . be n real quantities (n - 1) ^x^-j 22xy.
Since aU the quantities are real, 2 (x-i/)--tO.
Hence, since x will appear once along with each of the remaining n - 1
letters, and the same is true of :/, 2, . . ., we have
(n-l)2x2-22xy-tO,
that is, (n - 1) 2x= < 22xi/.
* The graphical study of inequalities involving only one variable will be
found to be a good exercise.
40 EXAMri.KS ClI. XXIV
In tliP case where x = ;/ = «=. . . . we have Si'r^njr', 2j;ry = 2,C^
= n(n-l)j', fo tliHt the inequality just becomes an equality.
When n=2, we have the theorem
x» + y'-t2ry;
or, if we put x = ,Ja, )/ = v">> " «nd 6 being real and positive,
a + b< 2s/{ab),
a theorem already establiebed, of which the preceding may be regarded ai a
genernlisfition. A more important generalisation of another kind will be
given jirraontly.
Example 0. If x, i/, r, . . . be n real positive quantities, and p and g any
two real quantities having the same sign, then
Ti£x''+«-«2x»'2:x«.
We have seen that xf-yf and r»-j/« will both have the Bome sign ai
« - y. or 'both opposite signs, according as p and q are both positive or both
negative. Hence, in either case, («'' - j/"") (J« - y«) has the positive sign.
Therefore
(xv-y'>){xf>-ii'i)<0,
whence z'*+« + i/''+^-<x''y' + i*y''.
If we write down the ,C, inequalities like the last, obtained by taking
every possible pair of the ii quantities x,y,z, . . ., and add, we obtain the
following result —
(n-l)2xP+«-«2:zV-
If we now add 2j»"+« to both siilcs, we deduce
N.H. — Up and q have opposite signs, then
nSxP*« i- 2x''i-r».
These theorems contain a good many others as partienlar caiM. For
Giample, if we put q= -p, we deduce
ZTP^x-f ■in'',
whic)i, when n = 3, p = l, gives
(x + y + i)(l/T + l/y + l/t)<9i
whence (x + y + t) (yi + ix + xy) -t 9xy» |
and so on.
Example 7. If r. y, t be real and not all equal, then Zx'> <3xy>,
according bm Xs> «0.
For 2r« - Sxi/i = 2x (Sx* - Zry),
siSx2(x^!/)«.
nencc the theorem, since 2 (x - y)* is osBcntially positive.
Example 8. To show that
i 1 . S ■ ■ ■ (2»i - 1) s'(" + 1)
s'(J.i-i 1)* 2.4 . . . iii " 2ii + l •
where n is any positive iutegcr.
^^ 4, .') EXAMPLES 41
From the inequality a + b>2^f{ah) we deduce
(2H-l) + (2tt + l)>2J{(2«-l)(2n + l)};
whence (2n- l)/2n<^{(2»- l)/(2n + l)} (1);
similarly (2n- 3)/2(n-l)<^{(2«-3)/(2n-l)j (2);
5/2.3<V{5/7} (n-2);
3/2.2<V<3/5} (K-1);
l/2.1<V{l/3} (n).
Multiplj'ing these inequalities together, we get
1.3.5 ... (2« - 1) 1
2.4.G . . . (2n) ^{2n + l)
Again, n+ (n + l)>2^{n(n + l)},
that is, 2n+^>2V^n()^ + l)}.
Hence wo have the following inequalities —
(A).
(2n+l)/2»>V{(n + l)/n}
(1)'.
(2»-l)/2(n-l)>V{H/(n-l)}
(2)'.
7/2.3>,^{4/3}
(n-2)',
S/2.2>J{3/2}
(n-1)'.
3/2.1>^/|2/l}
(n)'.
Multiplying these n inequalities together, we get
1.3.5.. .(2« + l)_ „
2.4 ... 2k
Hence 1 -3 .5 ■ . ■ (2»-l) ,An±l) (B).
2.4.6...2n 2n+l ^ '
(A) and (B) together establish the theorem in question.
Since J(n + l)/(2n + l)>^(ii + l)/(2n + 2)>l/2^(H + l), we may state the
above theorem more succinctly thus,
1 1.3 .. ■ (2«-l) 1
^(2;i + l)^ 2.4. . .2n ^2,^(71 + 1)"
DERIVED THEOREMS.
§ 5.] We now proceed to prove several tlieorems regarding
inequality which are important for their own sake, and will be
of use to ua in following chapters.
Ifbi,bi,. . ., bnhi! all positive, the fraction (ai + a. + . . . + a„)/
{b, + bi + . . . + b„) is not less than the least, and not greater t/ian
the greatest, of the n fractions a~,/b,, a^/bi, . . ., a„/b„.
Let / be the least, and /' the greatest of the n fi-actions,
theu
ajh-i^f a^lb^<if, .... a„/b„<if
42 MEANS AMONG RATIOS CH. XXIV
Hence, siuce 6,, 6,, . . ., i, are all positive,
a,<tA, (/,<//>„ . . ., «,<{://',.
Adding, we liave
{a,+aj + . . . + «,) «t/(i, + 1, + . . . + 6.) ;
whence
(rt, + n.j + . . . + «.)/(/>, + /'j + . . . + 6,) <t/.
In like manner, it may be shown that
(at + (h+. . . + a,)/(6. + ^, + . . .+6,)>/'.
liemark. — This theorem is only one among many of the same
kind*. The reader will tind no ditliculty in demonstrating the
following : —
lfai,<h, . . .,a„,bi,b,, . . .,b, l>c as htifore, and I,, I, U
be n positive quantities, then ^l,aiftlib,{x not /ess t/ian (he liiuit,
and not greater than the greatest, among the n fractions a,/<>, , a^bt,
Ifa\,<h,- ■ ■,a„,bi,b,,. . .,b„,l,,l^,. . ., l„ be all positive,
then {S/,«,"'/S/, /.,"•}"" and !«,«, . . . a.//>,6, . . . A,}"" are,
each of them, not less than the U-ast, and not greater than the
greatest, among the n fractions ajbi, a^Jb^, , . ., ajbn.
Example, to prove that
1 ya.3...(2H-i))
2 V I 2.4 ... 2n ] •
Since the fractions 1/2, 3/4, . . . (2n-l)/2n arc obvionsly in ascending
order of nmt;nitude, wo hare, in the second part of the last of tho thcoromi
just stated,
1 "/(I.S ■ . . (2n-l)) 2n-l
2 V ( 2 . 4 . . . 2n J 2n ■
Now, (2h-1)/2h = 1-1/2h<1, honco the theorem follows; and it holds, be it
observed, however great n may be.
§ 6.] If X, p,q be all jtositite, and p and q be integers, then
{jf - \)/p> <(a^ - I )lq affording as p> <q.
Since p ami q arc positive,
(j^-\)lp><{:if>-\)lq,
according as y (j* -!)></> (i^- 1),
• Sec the interesting remarks on Mean Values in Caticliy's Analyu
Algf.liriqut.
§§5-7 (;,P_l)/^>(^.,_l)/5 43
according as
(w-l){q{x^-' + .vP'^ + . . . + l)-p(af>-' + af'-' + . . . + 1)}><0.
If p>q, we have
X = {x-l){q{a^-' + .x''-- + . . . + i)-p(afl-' + af--+. . . + I)], "
= (^-l){g'(a*-' + irP-= + . . . + af)- {p-q){j^-' + af-"- + . . , + 1)|.
Now, if a:>l,
a^-' + .rP-= + . . . + .7^>{p-q)afl;
af-^ + of-- + . . . + 1 < qafl-^ ;
therefore,
X> {x-\){q{p-q)af'- (p - q) q:^-\
>q{p~q)afl-'{.v-l)-,
>0.
Again, if a'<l,
af-^ + a-P-' + . . . + af<(p- q) afl ;
ar«-' + afl-- + . . . + 1 > qx"-'^ ;
but, since a; - 1 is now negative, the rest of the above reasoning
remains as before.
Hence, in both cases,
{x^-\)lp>{afl-\)lq.
By the same reasoning, if q>p,
. {afl-\)lq>{o?-\)lp,
that IS, \ip<q,
{a^-l)lp<{af'-\)lq.
§ 7.] If X be positive, and =t= 1, tlien
mx'"-^{x-l)>x"'-l>m{x- 1),
unless m lie between 0 and + 1, in which case
maf-^ (x - 1) <.t'" -l<m{x- 1).
From § 6, we have
{^-l)><{p/q){i''-l) (1).
according as^Xg, where t is any positive quantity +1, and
p and q positive integers. In (1) we may put a^'" for ^, where x
is any positive quantity =# 1 (the real positive value of the yth
root to be taken), and we may put m for p/q, wliere m is any
positive commensurable quantity. (1) tlien becomes
af"-l><m{x-l) (2),
44 ni.r"— ' (.r - 1 ) ^ .r" -\>vi(x-\) Cn. XXIV
according as wxl, wliicli is part of the theorem to be
established.
In (2) we may replace x by l/x, where x is any positive
quantity +1, and the inequality will still hold.
Hence (l/ar)"- l><m(l/x- 1) (3),
according as »»> < 1.
If we multiply (3) by - x", we deduce
ar-l<>mxr-'(.r-l),
that is, wij^-'(jr-l)><j:"-l,
according as wj >< 1 .
We have thus established the theorem for positive values
of TO.
Next, let TO = -n where n is any positive commensurable
quantity. Then
a;-"-l><(-«)(-^-l).
according as l-afx-nx'ix-l),
according as x'-lonx'ix- I),
fix'^'-nj^xa:"-!.
Add af*^ - .r" to both sides, and we see tliat
a-"-l><(-«)(x-l),
according as
(h + 1)x"(j--1)><j'*'-1.
Now, since n is positive, h+1>1, therefore, by what we
have already proved,
(n+l)x"(x-l)>a:"*'-l.
Hence a— -!>(-»)(*- 1) (•*)•
In (4) we may write 1/x for x ; and then we have
(l/jr)-"-l>(-«)(l/x-l).
If we multiply by - x"", this last inequality becomes
a:--l<(-n)x--'(x-l).
that is, (-n)x— '(■r-l)>J— -1-
Hence, if m be negative,
my"-'(-r- l)>J--l>Hi(x - 1);
which completes the demonstration.
§ 7 w!,*™"' (x — y) 5 a'" - 2/*" < '» j/'""' (a; - y) 45
Cor. If .r and i/ bo any two unciiual positive quantities, we
may replace x in the above theorem by x/^. On multiplying
throughout by i/"', we thus deduce the following —
^x and y be jiositive and unequal, then
mx"'-'- {x-y)>x^- y" > mij'^-^ {x - y),
unless m lie between 0 and + 1, in which case
mx'"-^ {x - y)<x'"' - y'"<my'"-^ (x - y).
We have been carefid 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, (I + .t)"' always lies between
1 + mx and (l + x)/{l + (l-)ii)i}, provided 7hx<1 + x.
Suppose, for example, that m is positive and < 1. Then, by the theorem
of the present section,
wi{l + .T)'"-ix<(l+x)"*- l<mx.
Hence (1 + x)"'<1 + »hx.
Also, (l + x)'»-l>J7ix(l + x)'"/(l + x),
{l-mj/(l + x)}(l + x)"'>l.
If mx<l + x, 1 - 7nj/(l + x) is positive, and we deduce
(1 + X)'»>1/{1-»HX/(1+X)},
>(l + x)/{l+(l-m)x}.
The other cases may be established in like manner.
Remark. — It should be observed that
(1±X)"'> <1±)HX,
according as vi does not or does lie between 0 and + 1.
Example 2. Show that, iitt^,u^ . . . , «„ be all positive, then
(l + u,)(l + !(„) . . . (1 + «„)>1 + Ui + Ua+ . . .+«„;
also that, if Uj, u„ . . ., u„ be all positive and each less than 1, then
(l-K-,) (!-!/„) . . . {l-i(„)>l-«i-«a- • • ■ -«n-
The first part of the theorem is obvious from the identity
(l + U,)(l + «2) • ■ ■ (l + U„) = l + 2«i + :;«lWa+2UiHjH:,+ . . . +U^U„ . . . !(„.
The latter part may be proved, step by step, thus —
1 -iii = l -u,.
(1 - I<i) (1 - II.,) = 1 - Ui - tij + KiHj,
>l-Kl-«2.
* Several mathematical writers have noticed the unity introduced into
the elements of algebraical analysis by the use of this inequality. See
especially Schlomilch's Ilandbuch tier Alyebraischeii 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 Auirnsumc and ceometiuc means cii. xxiv
Heuco, giaco 1 - u, in positive,
11 - «,) (1 - uj (1 - u,) >(1 - «,) (1 - li, - ",).
>l-U,-li,-U,+ li,(ll, + uj,
>l-u,-u,-u,.
Aud 80 on.
These inequalities are a gcncrolisalion of (l±x)">l±«x (x<l and n a
positive integer). They are userul in the tbcor; of infinite products.
§ 8.] The arithnutlc mean of n positive quantities is not less
than their (jvomdric mean.
Let us suiipo.se tliis theorem to hold for » quiiutities
a, b, c, . . ., k, and let / be one more positive quantity. By
b}T)othcsis,
(« + 6 + c + . . . + k)ln^(al>c . . . X)"^,
that is,
a+b + c + . . . + A<t;« {abc . . . k)^.
Therefore
a + 0 + c + . . .+A + /<»! (a/jc . . . k)^' + L
Now,
« {aOc . . . X)"- +/<t(H + 1) (tt*c . . . X-0'i"+'),
provided
n{abc . . . X//"r"+K(H+ 1){«/'C . . . XV/f+'r'"*",
<t(H + l){«ic . . . X//-}"^"*'',
tluit is, provided
«f^' + l.«t("+«)s'",
where f«<-+'> = a^« . . . X//»,
that is, provided
{n+l)i'($-lHi'*'-l,
whicii is true by S 7.
Hence, if our theorem hold for « quuntitics, it will hold for
n+1. Now wu have seen that (<t +<>)/•_' -^Oi/')*, that i^s the
theorem holds for 2 quantitiej* ; therefore it holds for 3 ; there-
fore for 4 ; aud so on. Hence we have in general
(rt + 6 + c+. . . +X-)/«<(<i/t . . . *)"".
It is, of course, obvious that the inequality bccomc'> :iii
equality when a = 6 = c = . . . — X.
§§ 7, 8 ARITHMETIC AND GEOMETRIC MEANS 47
Tliere is another proof of this theorem so interesting and
fundamental in its character that it deserves mention here*.
Consider the geometric mean {ahc . . . X-)"". If «, b, c, . . .
be not all equal, replace the greatest and least of tiieni, say a
and /•, by {a + k)l'i; then, since {{a + k)l2\->uk, the result has
been to increase the geometric mean, while the arithmetic mean
of the n quantities (« + A-)/2, h, c, . . ., (a + /i)/2 is evidently tlie
same as the arithmetic mean of 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 eq\ial 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)1<(5 .4.4. 5)i<(4-5 . 4-5 . 4-5 . 4-5)l<4'5<(l + 3
+ 5 + 9)/4.
Cor. If a, b, . . ., k be n positive qiinntities, and 2^, q, . . . ,t be
n positive commcnsurtible quantities, then
pa + qb + . . • + ^/->fa,^, m.(,.M+. . .+n
p+q+ . . . +t ^
It is obvious that we are only concerned with the ratios
p : q : . . . : t. Hence we may replace p, q, ■ • ■, t by 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 tiieu becomes a mere particular case of
the theorem of the present paragrai)h, namely, that the aritlmietic
mean oi p + q + . . . + t positive quantities, p of which are equal
to a, q to b, . . ., t to k, 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.
♦8 1pa"'j'!ip^{1pa/1p)'' CH. xxiv
Example 1. Show that, if a, b, . . ., i be n positive qaanlities,
v. 0 + 6+ . . . +* J
/a + b+ . . ■+>c\»->**^- ■ •+»
The first part of the proposition folIowB from tbu above corollary bjr taking
p — a, q -b, . . ., k = c.
The second inequality is obvionsly equivalent to
{¥iy (fb)' • ■ • Q^y '■^'
which again is equivalent to
\npa) XnpbJ ' ' ' \npk ) '
where p is a positive intei^er which mity be so chosen that pa, pb pk are
all piisitive intvr;cr9. We shall therefore lose no generality by supposing
a, b, c A: to be positive integers.
Consider nuw <i positive quantities each equal to Zajna, b positive quantitia*
each equal to 2:(i/n6, Ac. The geometric mean of these is not greater than
their arithmetic mean. Hence
l/S<i\"/'i:ay /ri'Vi'^' a(Zalna) + baalnb) + . . ■+t(Sa/iit)
\\,uij \iib) ■ • ■ \nk)l " a-rl+.-.+i
mm-- - &)"-
Einmplc2. Prove that 1 . 3 . . . (2n-l)<n*
W'chave {1 + 3+ . . . +(2n- l)}/n>{l .3 . . . (2n-l)}"»,
that is, nVn> {1.3 . . . (2n-l)}'/».
Hence ti"> 1 .3 . . . (2fi-l).
§ 9.] I/a, b, . . ., k he n positive quantities, and p, q, . . ., t
be n positive quantities, then
pa" + qb" + . . • + ^^"* u -c /i^ ■*■?/'■>■. . . •♦• tiy . .
p+V+...+( \ p + g+ . . . +t ) ^ '•
according as m dofs not or does lie between 0 and + 1.
If we denote
PKP + 7 + . . . + /), q!(p^q+ . .
+ 1), &c,
by A, /i, . . ., T, ami
al{\a + /lA + . . . + tX), 6/(Aa *iib+ .
. . + rk), &C.,
by X, y, . . ., K", 60 that
A +/1 +...+T =1
(2).
Kt + fiy^. . . + TIC - 1
(3).
§§8,9 Sa™/w>(Sa/«)"' 49
then, dividing both sides of (1) by
{{pa + qb+ . . .+tk)J(p + q+ . . . + <)}"•,
we have to prove that
Aa;"' + /./" + . . . + ««"'<0.1 (4), ,
according as ?w does not or does lie between 0 and + 1.
Now, by g 7, if 7M does not lie between 0 and + 1, .r"'-l
t^:??; (.r - 1), y'" - 1 -^m {y - 1), &c. Tiieiefore, since A, ^, &c., are
positive,
2\(a;"'-l)<t:2X?w(a:-l),
<m{\-\),
by (2) and (3), that is,
2Aa;'»-2A.<!;0.
Hence 2Aa;"'<tl.
In like manner, we show that, if m lies between 0 and + 1,
2/U"'$>l.
Cor. If we make p = q = . . . = t, we have
n <>[ n ) (^^'
that is to say, the arithmetical mean of the mih powers of n positive
quantities isnot less or not (jreater than the mthpower of their arith-
metical mean, according as m doi-s not or does lie between 0 and + 1.
Bemark. — It is obvious that each of the inequalities (1), (4),
(5) becomes an equality if a = 6 = . . . = ^", if w = 0, or if m = 1.
Example. Show that SXx'", considered as a function of m, increases as m
increases when m>+l, and decreases as m increases when ni<-l,
X, jj., V, . . ., X, y, z, . . . being as above.
1st. Let m>l. We have to show that SXi'"+'":>2Xx'", where r is very
small and positive, that is,
2Xi"'(x'--l)>0.
Now, 2\i"' (^'" - 1) > SXx'^rx'-' (x - 1),
>ri;\x"'+'"-i(i-l).
• The earliest notice of this theorem with which we are acquainted is in
Eeynaud and Duharael's Problemcs et Developmens sur Dii'crses Parties des
Mathematiques (1B23), 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 seiies,
c. II. 4
60 EXERCISES V CH. XXIV
Since m>l, m + r>l, therefore (m + r) i"-"^' (r- l)>(m + r) (x- 1), tlinl
is, t-^^Mx-II^Cj:-!).
Hence l\x''{x'-l)>rZ\{z-\).
> r (rXx - DX),
>0.
Therefore Z\i^*^ > SXx".
2ad. Let m< -1.
SXi" (x' - 1) ^ rlXx" (x - 1).
Now (m + l)x"(i-l)>(ni + l)(x-l), eiuoe m + 1 is negative. ITence,
dividing by tlie negative quantity m + 1, wc have
x">(j-l)<(i-l).
Hence Z\x''{jf-l)<ry:\(z-l),
<r(^Xx-2\),
<0.
Tlierefore, 2Xx"*«'<2\a".
Exercises V.*
(1.) For what values of xji/ is (<i + fc) xy/(iix + by) > {az + by)l{a + 1) 7
(2.) H X, y, z bo uny real qiiautitics, and z>y>t, then x»y + y*i + r*x>
xj/' + !/J* + "*.
(3.) \t x,y, t be any real quantities, then 1(y - z)(t-x)>Q and i'j/i/
Sx«>l.
(4.) If x' + y' + »' + 2xi/« = l, then will all or none of the quantities x, y, t
lie between - 1 and +1.
(5.) If X and »i be positive integers, show that
jam+j< j: (^ ^ 1) (2X + 1) (8x» + 3x + Ij^/U . S" < (x + !)»"+».
(0.) (a»/l)> + ((<»*-« a* + l».
(7.) If x,,x,, . . ., X, all liave the same sign, audi -i-x,, 1 + x, l + «»
be all positive, then
ll(l+x,)>l + 2x,.
(8.) Trove that 8xi/j i-ll (y + r) i- JSx*.
(9.) If X, )/, : u, b, c . . . be two sets, each ooDlaining n real
quantities positive or negative, hIiow that
iu'^'-((l(ix)';
also that, if all the quantities be positive,
2(j/.i)/ix-«2x/2<ix;
and, if2:x = l, 21/x<n'.
(10.) If X,, X, X, and oUo y,, y,, .... y. bo positive and in
ascending or iu descending order of magnitude, then
Sx,Vi/i:x,y,>2>,'/£*,- (Laplact.)
* Unless the contrary is vlated, oil letters in Ibis set of exercises stand
fnr rtal positive quantitieo.
§ 9 EXERCISES V 61
(11.) Vn,h, . . .,( be in A. P., show that
a^J^ . . . P>aH''.
(12.) For what values of x is (j; - 3)/(x2 + x + 1) > (.r - 4)/(x= - 1 + 1) ?
(13.) Fiud the limits of x and y in order that
c>ax + by>-d,
a>cx +dy>b;
where ad- ic + 0.
(14.) x>'-x'y + ix'!/--2xh/ + ix-y*~xy'^ + y''>0, for all real values of
X and y.
(15.) Ib Wx- + ^y- + 13z-> = <Syz + 2xy + 18zx?
(16.) Up<2-^'2, then ^(x'' + y-)+p^{xy)>x + y.
(17.) Is ^/{a- + ab + b^) - J{a- - ab + 6=) > = < 2^{ab) 7
(18.) If X and a be positive, between what limits must x lie in order that
x + a>^{h{x' + xa + a"-)}+J{i{x''-xa + a')}7
(19.) If x<l, then {x+V(x--l)}i+ {i-^/(x'-- 1)}*<2.
(20.) If all the three quantities ^{a(b + c-a)], J{b{c + a-h)], ^{c{a +
h-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 |2xi:x- ^ 2x^ + xyz.
(22.) 21/x>2x8/x3y3.3_
(23.) If Pr denote the sum of the products r at a time of a, b, c, d (each
positive and <1), ihen p.^ + ip^^'ip-^.
(24.) 2x^<X!/.'Sx.
(25.) If s = a + i; + c+. . .7t terms, then 2s/(s- a) <7!-/(k- 1).
(26.) If ?K > 1, X < 1, and mx -c 1 + x, then 1/(1 =f mx) > (1 ± x)"' > 1 ± nil.
If m<l, x<l, 7nx<l + x, then (1 + x)/{l±(l -m)x} <(l±x)"'<
l±mx.
(27.) If z"=x" + ;/", then £"•:> <:x"' + i/'" according as m> <n.
(28.) If X and y be unequal, and x + y<<2u, then x'" + 1/"' > 2a'", m being a
positive integer.
(29.) )i{(n + l)i/»-l}<:l + l/2+. . . +l/H<:n{l-l/(/( + l)"" + l/(n + l)}.
(SchlomOch, Zeitschr.f. Math., vol. in. p. 25.)
(30.) IfXi.T2 . . . x„=i/», n(l + x,)<(l + 3/)".
(31.) If a, 6, . . . , kbe n positive quantities arranged in ascending order
of magnitude, and if M^={2,a'-lnyi^, W, = {2ai/f}7H, then
(ah . . . i)i/"<ilfj<J/j<. . .<A:,
{ab . . . &)'/»<. . .<Ar3<Nj<i^,.
(Schlomilch, Zeittchr.f. Math., vol. m. p. 301.)
(32.) If p, q, r be all unequal, and x + 1, then 2px«-'>2^.
(33.) If H be integral, and x and 7i each > 1, then
x»-l>7i(x(»+')'«-x (»-')/•-).
(34.) Prove for x, y, z that ('IZyz - •Zxr)'^'i (2x)S^II (2x- 2x)^
(35.) If« = ai + aj+. . . +a„, then H (s/a,-l)°'>-()i-l)'.
4—2
52 INEQUALITIES AND TUKNING VALUES CU. XXIV
(3G.) 3m(3m + l)»>4(3m!)"".
(37.) If <„ bo the Bum of the nth powers of a,, a,, . . . , a,, and p^ Die
enm of th.ir products m at a time, tlicn (n -!)!»„«« (ii - iii)!m!y„.
(38.) If a,>aj>. . . >a„, then
(<h -«»)"-' >(''-l)"-'(''i-<>i)K-''.) • • • (".-i-'O-
Hence, or otherwiac, show that {(ri- !)!['>«•-'.
(30.) Wliich is the greatest of the number* ^/2, ^/3, ^/l, . . . f
(40.) If there be n positive quantities j-, .x, x,, cach>l, and U
(i< fit ■ • ■ • {a l>e tl>e arithmetic means, or the geometric means, of all but
X,, all but X,, . . ., all but x„ then IIx/i j.n{,»i.
(41.) If u, 6, c be such that the sum of any two is greater than the third,
and X, y, z such that -x is positive, then, if £a*/x=0, show that xy: is
negative.
(12.) If A=ai + a^+ . . . +ii,, B = bf + li^+ . . . +6,, then Z{aJA-
b^lB) (ri J{i^)" has the same sign as u for all finite values of n.
(Math. Trip., 1870.)
APPLICATIONS TO THE TUEORY 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 illustnitcil as follows : — Let <^(j*, ;/, z),f(x, y, z) be any
two function.s of x, »/, z, and su[)i>(i,se that for all values con-
Bistent \vith the condition
f{T,y.z) = A (1).
we have the inequality
<i>(x,y,z)1('/(x,y,z) (2).
If we can find valuo.s of x, y, z, say a, b, c, which stati.'^fy the
equation (I) and at the same time make the inequality (2) an
equality, then <^ (a, b, c) is a maximum value of i/> (x, y, :). For,
by hyiMithc.sis, i^(a, b, c) = A and 'f>{x, y, z)'!^A ; therefore
<t>{x, y, z) cannot, for the values of x, y, z considered, be greater
than A , that i-s than </> {n, b, c).
Again, if we consiilcr all values of x, »/, z for which
,t>(x,y,z) = A (1).
if wo have /(x, y, z)<i<t> (x, y, z)
<A (2'),
it follows in like manner that, if a, b, c be such that <^(a. b, c)-A,
/((«, b, c) = -4, thcuy\a, b, c) ia u minimum value of/(x, y, z).
§^ 10-12 RECIPROCITY THEOREM 53
The reasoning is, of course, not restricted to the case of three
\ariables, although for the sake of brevity we have spoken of
niily 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 au inequality.
§ II.] 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 wLU
be made clear by the following useful general theorem : —
If for all values of x, y, z, consistent with the condition
f(pc,y,z) = A,
<i>{x,y, z) have a maximum value 4> {a, h, c)=Bsay {where B depends,
of course, upon A), and if when A iiicreases B also increases, and
vice versa, then for all values ofx, y, z, consistent with the condition
<t>(x,y,z)^B,
f(x, y, z) will have a minimum value f (a, b, c) = A.
Proof. — Let A' <A, then, by hj'pothesi.s, whcn/(.r, y, z) = A',
<i> {x, y, z)1^B' where B <B.
Hence, if </> {x, y, z) = B, f{x, y, z)<^A ; for .suppose if po.isible
that/(^, y, z) = A'<A, then we should have 'i>{x,y, z)1^B\ that
is, since B' <-B, ^ (x, y, z) could not be equal to B as required.
Hence, if a, b, c be such that i>{a, b, c) = B and /(a, b,c) = A,
f{a, b, c) is a minimum value of f(x, y, z).
By means of the two general theorems just proved, we can
deduce the solution of a large number of ma.xiuium 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 ILr has a maximum value, {k/n)", when x =
y = . . . = A/».
n. If X, y, z, . . . be n positive quantities subject to the
condition
Ux = k,
54 DEDUCTIONS FROM § 8 CU. XXIV
t/ii'H ffii'ir sum 2.r Aas a minimum value, wX;"", tphm x=y = , . .
= A"".
The second of these miglit be deduced from the 6rst by tlie
rcriprocity-tliciircm.
From the corollary in g 8 we deduce the following : —
III. If X, y, z, ... be n positive quantities subject to the
condition
1px=k,
where p, q, r, . . . are all positive constants, then U.r* has a
maximum value, [kjlp]'^, tvhen x = y = . . . - kj'S.v.
IV. 1/ X, y, z, . . . be n positive quantities subject to the
condition
Ux' = k,
whire p, q, r, . . . are all positive constantx, thni Ipx has a
minimum valw, ('^j>)k'-'', tch^n x-y = . . .=k''-''.
From the last pair we can deduce the following, which are
.•still more general : —
V. I/\, fi, V, . . ., I, m, n. . . ., p. q, r, . . . be all positive
constants, and x, y, z, . . . be all positive, then if
2A^ = X-.
rij'' is a maximum when
l\x'/p = tntiy^/q = nvz'jr = . . .
VI. And if Ux'^k,
SAx* is a minimum when
lKi*lp = mjiiflq = nvz'jr = . . .
Proof.— Denote pfl, q/m, r/n, . . . hy a, ft, y, . . . ;
and let A:r' = a^, fj/"' = /J'j, i':^ = yC,&c
So that X = (af/X)w &c. ; af = K/X)«, &c
We then have in the first case
^i = k (1).
nj* = n (a/x)«nf« (2).
Hence, since ("A)*, {Plii-Y, ... are all constant and all positive,
rij* is a ma.ximum when Ilf" is a maximum. Now, tinder the
condition (1), H^' is a maximum when f ^ij-. . . = X/2a.
§ 12 EXAMPLES 55
Hence Il.r'' is a maximum when \a^/a-iJLi/'"/P = . . ., tliat is,
when l\.r'/p = miJ.i/'"/q = . . .
The maximum vahie of Uaf is n (a/X)« (A-/Sa)=«, and the
corresponding vahies of x, y, z, . . . are given by
X = (p.l\\1aY, . . .
Applying the reciprocity-theorem, we sec that, if
n.i'' = n(a/\)«(A/2a)'«,
the minimum vahie of 2W is k, corresponding to
x^iaklX^ay . . .
Whence, putting i=n(a/X)"(^/2a)'», wc sec that, if Ux''=j,
the minimum value of 2-W is 2a {j/n (a/X)"}''-', corresponding
X = [a{j/U{a/\YY^-'/Xr ■ . .
Cor. If we put l = m = n= . . . =1, p = q = r= . . . =1,
we obtain the following particular cases, which are of frequent
occurrence : —
j[f IXx = k, Tlx is a maximum when \x = it.y = . . . ;
If TLx = k, 2Aj; is a minimum when Xo; = /«/ = . . .
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 parallelo-
piped by x, y, z, its smface is 2(yz + j.t + xi/) and its volume is x\jz. If we
put i = yz, ■n = zx, i=xy, the surface is 2(£ + ij + f) and the volume sliivi)-
Hence, analytically considered, the problem is to make frjf a maximum when
{ + ,, + f is given, and to make 4 + t; + f a minimum when f tis" is given. This,
by Th. I., is done in either case by making J=7) = f, 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= ^/s (s - a) (s - b) (s - c). Let x = s-a,y = s -b, z=s-c\
then i + »/ + z=s; and the area is Jsxyz. Since, in the first place, » is given,
we have merely to make xyz a maximum subject to the condition x + y + z=t.
This leads to x = !/ = j (by Th. I.).
Next, let A be given.
Then (x + y + z)xyz=A' (1);
« = A2/xi/2 (2).
If we put i = z'yz, r) = xy-z, f =ri/j', we have
£+,+f=A^ (IT;
»=A=/(«'>i)"* (-')■
SR nEnumnvs KfioM § 0 ni. xxvi
Ilcncc, to mnkc ( n niiniiuum when ^ is Kivon, we haro to mnke {ijC a
nKij-iRiiim, i>ul>ject to the coadition (I'). TliU liads to ( = i; = f, that U,
^y: = Ty*z=Tiji*\ wboiicc x=y = f.
Example 3. To coDE^truct a rigbt circular cylinder of given volamo and
minimum total surface.
Let X be Ibu radius of tbe ends, and y the hci(;bt of the cylinder. The
total surface ia "ir (x-^rzij), and the volume is xi'ij.
We have, tbcrcforc, to make u = x'-i-ry a minimum, subject to tbe
coadition x'ij = c. We hove
u=z- + Ty = ely-^elx (1);
xh, = c (2).
Let l/x = 2{, l/y = i,;
then u = c(2f + i,) (lO;
r-ij = !/*<: (2').
We have now to make 2{ + 1; (that is. { + ^ + ij) a miuimnm, subject to the
condition {tj = constant. This, by Tb. II., lends to {=t = ij, wbicb Rivca
2x = y. Hence the height of the cvlinder is equal to its diamiter.
By tbe reciprocity-theorem (applied to tbe problem as oriRinally stated in
terms of z and t/), it is obrious that a cylinder of this shape also has maximum
volume fur givtn total surface.
§ 13.] From the inequality of § 9 we infer the following : —
VII. I/m do not lie betu-cen 0 and + I, and i/p, q,r, . . . be
all constant and positive, then, for all jwsitive values of x, y, c, . . .
such that
Ipx^k,
Ipif (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 (ititing the reciprocal theorem it is neces.s!iry to notice
that, in the ineijnality, Ipx occurs raised to the with {Hiwer; so
that, if m be negative, a maximum of 2/>x corrpsptuid.s to a mini-
mum of (Syjj-)". Attending to this point, we .see that—
VIII. j[f m> + 1, and if p, q, r, ... be all constant and
positite, then, for all jMsititv talui-s of x,y,z, . . . sucA that
2/)r" - 1 (m unchanged),
2/jj- has a maximum inlue when x - y = z = . . .
Ifm<+1, tee have a minimum instead of a maximum.
Theorem VIII. might also be deduced from Theorem VII. hy
the substitution i = a^, v^!/", { = -". &c. . . .
5515 12-15 DEDUCTIONS FROM § 0 57
§ 14.] Theorem VII. may be generalised by a slight trans-
formation into the follo^nng : —
IX. If mill do not lie between 0 and + 1, and if p, q,r, . . .,
\, //., I', . . . be all constant and positive, then, for all positive ^
values of .v, y, z, . . . such that
SXr" = k {n unchanged),
'^/).v'" (m unchanged) has a minimum value tchen px^'/^af^
'nr/i^i/" = - ■ ■
Jfm/n lie between 0 and + 1, instead of a minimum we /tave a
nutaimum.
The transformation in question is as follows : —
Let \af' = pi, i^f'^crr,, . . . (1),
px'^ = p^, qy'^^o-r,^, . . . (2).
From tlic first two equations in (1) and (2) we deduce
t^-'=;j;r"'-7A, //-' = W"-'"//', &c. Hence, if we take fn=m,
that is, /= m/n, p, a; . . . will be all constant and obviou.sly all
positive ; we have, in fact,
|=0'a;'"-/\)W-", •r,= (?2/'"-"/,x)W-'), . . . (3),
p = Q/lpyV-^\ cr = (;a//g)'V-», ... (4);
and we have now to make Ipi^ a maximum or minimum, subject
to the condition
Now, by Th. VII., Spf' is a minimum or maximum, according
as /docs not or does lie between 0 and + 1, when ^ = i? = . . .
Thus the conditions for a turning value are
(;?a;'"-"/X)W-» = (yy"-7ft)"t^->' = . . .,
which lead at once to
paf/kx'' = qi/"'/ni/'' = . . .
Cor. A very common case is that where n = 1, \ = /t = . . .
= 1.
We then have, subject to the condition '2.a: = k, ^pi^, a
minimum or maximum when /)«*""' = g-y""' = . . ., according as
m does not or does lie between 0 and + 1.
§ 15.] We have hitherto restricted p, q, r, . . . in the in-
68 EXAMPLES CH. XXIV
equality of § 9 to 1)0 constant. 'I'tiis is unncccssjiry ; they may
be functions of tlie variables provide*! tliey be such that tliey
remain positive for all positive values of x, y, z.
We therefore have the following theorem and its reciprocal
(the last omittoil for brevity) : —
X. If p, q, r, . . . be functions of x, if, z, . . . ufilch are
real and positive for aJl real and jwnitire valw.f of r, y, s, . . .,
t/ien, for all jKisitive values of X, y, z, . . . which satisfy
^px = k,
(2;)x") (2/))""' {m unchanged) has a minimum or maximum value
wlien x = y-. . ., according as m does iwt or does lie between
0 and + 1.
For example, we may obviously pat p=Xa*. q-=l»^, • • •
We thus deduce that if m^ +1 or <0. then, for all positiro valne* of
x,y,z, . . . consistent with 2X.r«+' = *, (ZiVr'"*') (ZXx-)~-> is s minimum
when x=y= . . .
Theorem X. may again be transformed into others in appear-
ance more general, by methods which the student will rea<lily
divine after the illustrations already given.
Also the inoiiualities of § 8 may l>e u.sed to deduce ma.xima
and minima theorems in the same way as those of § 9 were uaed
in the proof of Theorem X.
Example 1. To find the minimnm Talne of u = j- + y + i, snbjcct to tha
conditions a/z + 6/y + c/i=l, x>0, y>0, r>0, a.b.e being positive oonstant*.
Let x=^. y = <V. ' = Tf;
alx = f4, b/!/ = <r7, c/« = rf.
Hence (/"' = a//j/+'. If we take /= - 1, we therefore get
x=^ar\ »=v'fr'7-'. »=s''-r';
Tlic problem now is to make u = 2^/<if-" a minimnm snbject to the con-
dition 2^/<i{ = l. By Th. vn. thin is accomplishrd by making f = i| = f.
Uence i = i} = f = IjZyJa. The minimnm value required is thcrvfora
(Z^/a)'; the corresponding values of *, y, x are ^/al.^la, sJI>Z^/a, ^cZy/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 bo a minimum (m>l).
Lot a, b, c be the sldi'*, *, y, i the three distances; then we have to mal •
tt^Xr" a minimum, subject to the condition S<u=2A, whoro A is the an .
of the triangle.
§§ 15, 16 grillet's method 59
If p^ = x'", pi = ax, then /5'"-' = a'", p=«'":'('"-i).
Heuce, if we put ai = tt"'("'-i)f, 61/ = 6"'/(">-i) ,,, cs = c"' '("'-') f, we have
The solution is therefore given by t = 7, = f=2A/So'»/('"-i).
Whence a: = 2Aa'/(">->)/2a'»/("'-i), y = (S:c., z = &c.
Example 3. Show that, if x' + 7/* + r'=3, then (.t« + j/O + s") {»' + y' + ?»)
has a minimum value for all positive values of x, y, 2 when x=y=z = l.
This foUows from Th. X., if we put m=2, p=x', q=y\ r=z*, which is
legitimate since x, y, z are all positive.
Example 4. If x, y, z, . . . be n positive quautilics, and m do not lie
between 0 and 1, show that the least possible value of (Zx^-^) (21/x)"'-' is 71*".
This follows at once from the inequality of § 9, if we put p = l/x,
J = l/i/
§ 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 GriUct*.
Suppose, for example, it were required to discuss the turuiug
values of the function
u = {ax+pf{hx + qT{cx + rY (1),
where I, m, n are all positive.
We may write
u = {\ax + XpY {ixhx + iiqY {vex + vrfjk'ix'^v'' (2),
where \ /x, v are three arbitrary quantities, which we may sub-
ject to any three conditions we please.
Let the first condition be
l\a + mixb + nvc = 0 (3) ;
then we have
l{Kax + \p) + m {nbx + ixq) + n {vex + vr)
= l\p + miJ.q + nvr = k (4),
where k is au arbitrarj' positive constant.
This being so, we see by Th. III. that n(\ax + >^j>)' is a
maximum when
Xax + A/> = fibx + it-q = vex + vr
= k/ll (5).
Houvelles Annalei de Math., ser. i., tt. 9, 10.
60 EXAMPLKS CIl. XXIV
The four oiiii.itiiiiis (3) and (ri) arc not more than sntFicient
to exhaust the tliree conditions on A, i^, r, and to determine x.
We can easily deteruiine x by itself. In fact, from (3) and
(5) we deduce at once
i»/((i,r + p) + mbldbx + 7) + ncl{cx + r) = 0 (6).
This quadratic gives two values for x, say a:, and x, ; and the
equations (5) give two corresponding sets of values for X, /x, v,
in terms of /-, say X„ ft,, v, and A,, /i,, i-,.
If, tlicn, AiV,""!-," be positive, x^ will correspond to a maxi-
mum v;duc of u ; if XiVi"'''i" be negative, a-, will correspond to
a minimum value of m ; and the like for a*,.
Example 1. To discuss 11 = (x + 3)' (x - 3).
Wc liav6 ti = (Xx + 3X)' 0*x - 3^)/X'/i.
Now 2 (Xx + 3X) + (mx - 3/i) = t,
proviJcd 2X + /i = 0 (1),
lA-3,i = i (2).
Therefore (\x + 3\)'(aix-3m) will bo a mnximmii, provided
\x + 3X = Au:-3^ (3).
Hence, by (1),
2/(x + 3) + l/(x-3)=0;
which Rives x= 1. From (2) and (8) we deduce X = J.712, m= - */6 ; "O th«t
XV 's nopativc.
We therefore conclude that 11 \* a minimuiu when x = l.
The student should trace the Rraph of the function u ; he will thus find
that it has also a maximum value, corresponding to x= -3, of which this
method gives no account.
Example 2. For what values of x and y is
It = (a,i + fc,y + (-,)* + {<i:,i + b,!/ + <•-)' + . . • + (".x + 6,0/ + c J'
a minimum?
Let X. , X,, . . ., X, be undetermined multipliers. Then wo may write
„ = i;x,'{('>,x+fc,!/ + c,)/X,l' (1);
and * = »,M('>i' + ''iy + '-i)/M (2).
where k is an arbitrary positive constant, that is, independent of x and y.
provided
S<i,X,=0, Z6,X, = 0, S(-,X, = » (3).
This being so, by Th. VII., u is a minimum when
(<iii + 6,y + c,)/X, = (nft + h^ + <-,)/X, = . . . = kiZ\* (4).
The n + 2 equations, (3) and (4), juot sufhco for tbo determination of
X,.X, X,, X, y.
From the tirst two of (3), and from (4), wo doduoa
§§ 16, 17 METHOD OF INCREMENTS 61
S<Zi(n]X + 6i»/ + Cj)=0,
:i:;)i(<iix+ti!/+<;i)=o.
Hence the values of x and y corresponding to the mininmm value of n are
given by the system
^Oj^x + Sflifti!/ + SajCj = 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 fuuctiou of one variable, we may
define
I=4>{x + h,y + k,z + l)-<j^{x,y, z)
as the increment of <^(a?, y, z). If, when x = a, y = h, z = c, the
value of / be negative for all small values of h, k, I, then
<^ (o, b, c) is a maximum value of 4' (^. V, -) \ ii"d if, under like
circumstauce.s, / be positive, <^ (a, b, c) is a minimum value of
^(«, y, z).
Owing to the greater manifoldness of the variation, the ex-
amiuation 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 establislied
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*.— 7/" ^ (.r, 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),
t/ien^{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 Muthematict, vol. i. (ISOi^J.
62 PURKISS'S THEOREM CII XXIV
<f>(x, y, z) ami /(x, y, z) to be iiitej,'nil functions, and this Bup-
positiou, 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 = s and /(x, y, 2) = 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, zha a, then a will be a root
of the equation /(a, a, a) = 0.
Consider the functions
I=<f>(a + h, a + k, a + l)-<t>{a, a, a), and/(o + A, a-*-t, a + l).
Each of them is evidently a synnnetric function of A, t, I, and
can therefore be expanded as an intojrnil function of the
elementary symmetric functions 2A, 2^/, /lAl. We observe also
that, since each of the functiims vanishes when A = 0, 1 = 0, l-O,
there will be no term inde[iendont of A, k, I.
Let us now suppose h, k, / to be finite multiples of the same
very small quantity r, say h = ar, k^ fir, 1 = yr. Then 2/< = r5o
= r«say, :i/ik = t^la^ = r^v, hkl = i^w. Exjjanding as above in-
dicated, and remembering that by the conditions of our problem
/(a + h, a + I-, a + /) = 0, we have, if we arrange according to
powers of r,
/=^lH/+(/yM'+Cr)r' + &c. (I),
0 = i^ur + ((y + liv) r' + &c. (•.>),
where the &c. stands for terms involving r" and hii,'her powers.
From (2) we liave
«r = -(V"'+A'y)r'//' + &c.,
ttV = 0 + S:c.,
22n/?r' = - SaV + &c.,
&c- as before including powers of r not under the 3rd.
Hence, substituting in (1) and writing out only such t«rma
as contain uo higher power of r than r", we have
I=(0-AH/P)vi' + &c.,
= - Jr" (C-yl /?//') 2«' + Ac.
Now (see chap, xv., § 10), by Uking r sufficiently small, wo
may cause the tirst term on the ri^^ht to dominate the sign of /.
§ 17 EXERCISES VI 63
Hence /will be negative or positive according as {CP-AIi)jP
is positive or negative ; that is, <^(a, a, a) will bo a maximum or
miuimum according as (GF-AR)/P is positive or negative.
Kxaruple. Discuss the turniug values of 0 (x, y, z) = xijz + b{yz+zx + xy), ,
subject to the condition x- + y- + z-=3a'.
The system
x=;/ = z, x- + y- + z--5a-=0
has the two solutions x=y = z= ±a.
If we take x = y = z= +a, we find, after expanding as above indicated,
/= (a^+2ab) ur + {a + b) vr- + &c.,
0=2aur + {w'-2i')r-.
In this case, therefore, /i = a= + 2a6, C = a + b, P='2a, Ii=-2; and (CP-AR)I
P=2a + Sb.
Hence, when x = y = z= +a, <^ is a maximum or a minimum according as
2a + '6b is positive or negative.
In like manner, we see that, when x = y = z= -a, 0 is a maximmn or a
minimum according as -2a + 'ib is positive or negative.
Exercises VI.*
(1.) Find the minimum value of bcx + cay + abz when xyz = abc.
(2.) Find the maximum value o( xyz when x-ja- + y-jb'^ + z-lc-=l.
(3.) If 2j- = c, Zlx is a maximum wlien x : y : z : . . . =1 : m : n : . . .
(4.) Find the turning values of X-t'"" + /i!/""* + vz™, subject to the condition
■p3^ + qy'>-\-r~' = d.
(5.) Find the turning values of ax'' + iy' + m' when xyz = (P.
(6.) It xyz = a-{x + y + z), then yz + zt + xy is a minimum when x = y = 2 =
J3a.
(7.) Find the turning values of {x + l){y + m) (z + n) where n==6»c»=d.
(8.) Find the minimum value of iix"' + blx".
(9.) Fmd the turning values of (3x - 2) (x - 2)= (x - 3)=.
(10.) If cx{b-y) = ay{c-z) = bz (a - x), find the maximum value of each.
(11.) Find the turning values of x"'/i/" ('"="')! subject to the condition
x-y = c. (Bonnet, Nouv. Ann., ser. i., t. 2.)
(12.) If x''!/' + xiyf = a, then x''+« + J/''^ has a minimum value when x=y =
(al'2)V{i^i) ; and, in general, if Zxi'yi=a, Xx"^ has a minimum value, «/(n - 1),
when x — y = z= . . . = {<i/(n-l) n}'A"+«). Discuss specially the case where
p and q have opposite signs.
(13.) If x''y'' + x''y'=c, then x'y" is a maximum when xJ'-^C™ "*')=!''"'/
{qt-pu), the denominators, ru-st and qt-pii, being assumed to have the
same sign. (Desboves, Questions d'AUjibre, p. 455. Paris, 1878.)
• Here, unless the contrary is indicated, all letters denote positiTe
quantities.
64 EXERCISES VI CU. XXIV
(II.) If p>q. and x'' + y'' = aP, then «« + y« is a mioimani when x = y=
afi^f. State tho reciprocal Ihiorcm.
(15.) I'iud the turning values of (ai' + J'j/')/v/(aV + !<»!/») when i« + y'=l.
(IG.) If Xi.i , X, bo each >a, and such that (4-,- a) (j-,- a) . . .
(x,-a) = fc", tho least value of r,r, ... i, is (a + (<)», a and 6 hcing both
positive.
(17.) If /(m) denote the greatest product that can be formed with n
integers whcie sum is m, show that /(m + l)//(>n) = l + V? "hero g is the
integral part of m/n.
(18.) ABCD is a rectangle, APQ meets DC in P, and DC produced in Q.
Find the position of APQ when the sum of the areaa AliP, PCQ is a
minimum.
(19.) 0 is a given point within a circle, and POQ and ROS are two per-
pendicular chords. Find the position of the chords when the area of the
quadrilateriil PltQS is a maximum or a minimum.
(20.) Two given circles meet orthogonally ot A. PAQ meets the circles
in P and Q respectively. Fiud the petition 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
miuiiiium volimio.
(23.) Given one of the parallel sides and also the nonp.irallcl sides of an
isosceles trapezium, to lind the fourth side in order that its area may be a
maximum.
(21.) To draw a line throngh the vertex of a given triangle, gnch that tho
sum of the projections upon it of the two sides which meet in that vertex
shall be a maximum.
CHAPTER XXV.
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 dt-al
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 DifJerential 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 dilferential
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.
r.
C. lU J
66 MEAMN(i <>y A MMITlNd VAUK CII. XXV
5) 2.] The cliaractifii.stic ditliculty ami tlie way of nieetiiif; it
will be best explained by disc\is.sing a simple example. If in
tlie function {x'-\)/(x-l) wo put j: = 2, there is no ilifficulty
in i-irrying out successively all the operations indicat^jd by tlie
synthesis of the function ; tlie 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 cxcluiletl from the fundamental
laws; not included even under the case a/0 (rt + 0) alre;uly dis-
cussed in chap, xv., § 6. The first impulse of the learner is to
assume that 0/0=1, in analo^^^y with a'a=l ; but for this he
has no warrant in the laws of algebra.
Strictly speaking, the function (x*- l)/(a:- 1) has no definite
value when x= I ; that is to say, it ha.s no value that can bo
deduced from the principles hitherto laid down. This being so,
and it being obviously desinibJo to make as genend as po.ssible
the law that a function has a definite value corre.sponding to
every value of its argument, we proceed to define the value of
{ar'-l)l{x- 1) when x=l. In so doing we are naturally guided
by the principle of continuity, which leads us to deline the
value of {x'-l)l(x-\) when 3;= I, so that it shall dilVer in-
finitely little from v.-dues of (x' — l)/{x - I), corresponding to
values of x that diller infinitely little from 1. Now, so long as
ar* 1, no matter how little it differs from 1, we can jjcrform tho
indicated division; and we have the identity (j*- l)/(x- 1) =
x+l. The evaluation of a:+ 1 pre-sents no difficulty; and we
now see that for values of x differing infinitely little from 1, the
value of (jr- l)/{x- 1) differs infinitely little from 2. Ift l/icre-
fore define the value of (jr- l)l(x- 1) w/wii x=\ to be 2 ; and we
see that its value is 2 in the useful and ])crfectly intelligible
sense that, /'// briuijiuij x sujficieutlij tuur to 1, we can caust
{x'- \)l(x- 1) to differ from 2 by as little as we pl^fose*. Tho
value of (j;*— l)l(x- 1) thus specially defined is spoken of as the
limitinff value, or the limit of{x' - l)/(x - 1) fir jr = 1 ; and it is
symboli.seil by writing
* Tho reader shoald obwrro iliat tho definition of the critical value joit
(rivi'n has anothvr odvantaxv. namvljr, it cimbli'« u« to owiort the truth uf the
i<Iciitity (j'- l)/(x- l)3x+ 1 without cxcojitiuu in the cudv whtic x-L
§ 2 FOlUtfAL DEFINITION OF A LIJIIT 67
1=1 a:- 1
where L is the initial of the word "limit." The subscript x=l
may be omitted when the value of tlie 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 value
indicated by its synthesis is indeterminate : —
W/ien, by causing x to differ sufficientli/ little from a, ive 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,
of fix) when x = a; and we write
L fix) = I.
I=a
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 tvhich 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 \vill not be
strictly true unless the phrase "differ infinitely little from" mean
"differ either in excess or in defect infinitely 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
value. In this last case, the graph of the function in the neighbour-
hood of X = « would have the peculiarity figured iu 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 .r up to a from the process of taking a limit
by decreasing x down to a, we may use the symbol L for the
former, and the symbol L for the latter.
i— a+O
Cor. 2. If L f(x) = I, then f{a + h)=l + d, whe-re d is a
function of a and h, whose value may be made as small as we
please by sufficiently diminishing A.
5—2
68 CONSllyUENCES OK THK UEKIMTUJN til. XXV
'Diis is simply a ro-tiit:iteiucnt of tliu dctiiiitiuu of a liiuit froui
auotlRT point of view.
Cor. 3. Any ordinary value of a Junction sal'uifies the
definition of a litnitimj value.
For e.xaiiipif, L (r*- l)/(j:- l) = (i."- l)/(2 - 1) = 3. Tlii.s re-
I— a
iiiarii would lie superfluous, were it not that attcntiou to the
point onalilcs u.s to abbreviate deinon.strations of limit thcorunis,
by u.sing tlie .symbol L where tliere is no peculiarity in the
evaluation of the fuuctinii to which it is prefi.xciL
§ 3.] It may happen that the critical value a, instt-ad 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 f{x), or L f{x),
according as the quantity in question is positive or negative.
For e.xaiuple,
L{x+\)lx= L(i^■\|x) = l.
In this cane, we can, strictly spcuking, approach the limit from one side
only ; and the ciuu.-ition of cuntiniiity on both sides of the liiuit dues not
arise. If, however, we, as it were, join the series of alKcbrnical quantity
-CC...-1...0. ..+!.. .+00 through infinity, by considerinR
+ oD and - 00 as consecutive values; then wo say that /(x) is, or is not, con-
tinuous (or the critical value x=ao , according as L/(x)and L /(x) have,
x—» x^ — m
or have not, the same value. For example, (z-l- l)/x is continuous for 2 = ao ,
for wc have L (x+l)/i = l= /, (x + l)/x; but (x'-f l)/x is not coDlinuoas
for X := CD .
S 4.] The value 0 may of course otcur as a limiting value ;
for e-xamjile, /> jr(a;- l)=/(.r'- 1) = 0. It may also happen, even
X — 1
for a finite value of a, that /(.r) can be nnulo greater than any
finite (juantity, however groat, by bringing x sutlicienf ly near to o.
In thi.s case we write L f{x) -^- «. In thus admitting 0 and ■»
X— «
as limiting valuej<, the student must not forget that the general
ndes for evaluating limits are, as will bo shown presently, sub-
ject in certain cases to exception when these particular limits
occur.
§§ 2-6 CLASSIFICATION OF INDETERMINATE FORMS 69
ENUMEKATION 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 u x v
becomes indeterminate if one of the factors become zero and
the other intiuite ; and that u^v becomes indeterminate if u
and V become both zero, or botli infinite. We thus have
the indeterminate forms — (I.) co— so, (II.) 0 x <», (III.) 0-^0,
(IV.) 00-00.
It is interesting to observe that all these really reduce to (TIL). Take
00-00 for example. Since u + t' = (l + i;/«)/(l/u), and Ll/u = l/ao =0, this
function will not be really indeterminate unless Li'/ii= -1. The evaluation
of the form oo - oo therefore reduce.'! to a consideration of eases (IV.) and (III.)
at most. Now, since «-M) = (1/i')-=-(1/h), case (IV.) can be reduced to (HI.);
and finally, since u x ii = uH-(l/i'), case (II.) can be reduced to (HI.).
To exhaust the category of elementary algebraical operations
we have to discuss the critical values of u'. This is most simply
done by nTiting u" = 0."^°^'" where a is positive and >1. We
thus see that «" is determinate so long as ^•loga« is determinate.
The only cases where v loga u cesuses to be determinate are those
where — (V.) v = 0, logo u = + oo, that is v = 0, m = oo ; (VI.) » = 0,
log<,a = -oD, that is ^ = 0, m = 0; (VII.) v = +oo, logaM = 0,
that is ij = + 00 , M = 1. There thus arise the indeterminate
forms— (V.) 00°, (VI.) 0», (VII.) I*"*.
All these depend on a'x'o . gr, if we choose, upon a"!"; so that it may
be said that there is really only one fundamental case of indetermination,
namely, O-r-O.
EXTENSION OF THE FUNDAMENTAL OPERATIONS TO LIJIITINQ
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 O"*"", 0-", co +», to— give 0, ±oo, ±oo, 0 respectively, uo
matter what their origin.
70 FUND AMENTA I, OPERATIONS WTTII LIMITS Cn. XXV
operations exactly like ordinary operands. Tliis is established
by means of tlie follovriug theorems : —
I. Thf limit of a sum of functions of x i» the fum of their limits,
provided the latter does not take the indeterminate form ao - oo.
Consider tlie sum f {x) ~ <t>{x) ■¥ x{t) for the critical value
x = a; and let IJ{x) =/', L4>{x) = <^', Lx{r) = x'. Then, by § 2,
(."or. 2,
where a, (i, y can each be made as small as we please by
bringing x .-iutliciently near to a.
Now, f{x) - <l>(x) + x(-r) =/' - «^' + X' + (« - /J + y)-
But, obviously, a-fi + y can be made as small as we please by
bringing x sufficiently near t<^) a. Hence
L\f{x)-4>{x) + x{z)]=f-<f>' + x'.
that is, = Lf(x) - L^{x) + Lx {r) ( 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 /' - <^' + x and L \f(x) - </>(j-) + x(j')! 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
opiK)site, f -4t + x' ceases to be intorpretjible in any definite
sense ; and the projiosition becomes meaningless.
II. Tlie limit of a product of functions of x is the product qf
their limits, provided the latter does not take the indelerminaU
form 0 X oo.
Using the same notation as before, we have
/(.i) .^(.r) X(x) = (/'+ a)(,^'+ ^)(X'+ y)
= />'x'+ 2a<^'x' + 2a/3x' + a/3y.
Now, provided none of the limits /', <^', x' be infinite, since o, /5,
y can all 1h' ni;ulo as small as we plea.'sc by bringing x sufficiently
near to a, the .same is trvie of 2iu</>'x', "S.afix , and o/3y. Hence
JJ{x) ^{x)x(t) =/>'x' = I/{x) I^(x) Lx(x) (2).
If one or more of the limits/', <f>, x' be infinite, providcil none
of the r&it be zero, the two sides of (2) will still bo equal in the
I
§§ 6, 7 LF [fix), <\> {x\ ...]=F [Lf{m), Lcj> (x), . . .} 71
sense that both are infinite ; but, if there occur at the same time
a zero aiul au infinite value, then the right-hand side assumes
the indeterminate form 0 x so ; and the equation (2) ceases to
have any meaning.
III. T/i*i limit of the quotient of two futictions of x is the
quotient of their limits, provided the latter does not take one of the
indeterminate forms QjO or oo /=o . We have
f{x) z' + g j^ .r±^ _r ji . °<^'-^/'
<t>{x) <^' + /3 <#.' <^' + /3 <!>■ <!>' ^'(<^' + /3)-
From this equation, reasoning as above, we see at once that, if
neither/' nor <j>' be infinite, and <^' be not zero,
^<f>{x)-<i>''LHxy ^''>-
It is further obvious that if /'=oo, <^'# », both .sides of (3)
will be infinite ; if <^' = oo , /' 4= oo , both sides will be zero ; and
if <^' = 0, /' + 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, <t>' = 0, the right hand of
(3) takes the form 0/0 ; if /' = oo , <^' = » , the form x. /go ; and
then 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 particnlar cases : —
JfF{u, V, w, . . .) be 0)11/ function ofu,v,w, . . . , which is
deterviinate, and finite in value, and also continuous when
u = Lf{x), v = L<t>(x), w = Lx{x), . . .,
then
LF{f(x), 4>(x), x(A ■ • .} = F{Lf{x), L^{x), Lx{.r) }.
The reader will easily prove this theorem by combining § 2,
Cor. 2, with the definition of a continuous fimction given in
chap. XV., §§5, 14.
72 LIMITS OK UATIONAL FUNCTIONS CU. XXV
The most important case of Ibia proposition wbiob we shall have ooooaion
to use is that where we have a funotion of a single funotion. For example,
L l(*»-l)/(x -!)}'={ L (r»-l)/(x-l)l« = 4.
»-i i-i
/, log{(x'-l)/(x-l)}=log{ L (x«-l)/(x-l)} = lo«2.
ON THE FORMS 0/0 AND W /» IN CONNF.CTION WITH
UATIONAL FUNCTIONS.
§ 8.] The form 0/0 will occur with a rational function for
the value a; = 0 if the absolute terms in the numerator and
denominator vanish. Tlie 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 tliat occurs
in nuiiu'r;it<ir or denominator, and then put x = 0. Tlie limit
will be finite, and +0, if the lowest terms in numerator and
denominator be of the same degree ; 0 if the term of lowest
degree come from the deniiminator ; oo if the term of lowest
degree come from tiie nuiiiemtor. All this will be best seen
from the following examples : —
Example 1.
Example 2.
Example 3.
2x' + 3x' + j«_ 2 + 3x + i'_2
,., 3*» + x*+x» ~^3 + x«+*«~3*
2x« + 3x< + x»_ 2z + 9x' + i*_0
,.0 Sx' + x' + x' ~.li 3 + x» + i« 8 "•
2x« + x«_ 2 + x^_2_
JiU ««+x» "i;,x' + x*~0~*'
§ 9.] The form oo/oo can arise from a rational function when,
and only wlien, j- = x . The limit cjin he found by dividing
numerator and denominator by the iiighcst power of r that
occurs in either. If this highest power occur in both, the limit
is finite ; if it icuni' from tlic dcnoniinntor alone, the limit is 0 ;
if from the numerator alone, the limit is or.'.
Example I.
8x« + x« __ , 3/' + l "(1 1
,_.ar' + j* + :Jj-«~,_.2/x'+l/x + 3~0 + 0 + 3" 3"
,7-10
USE OF THE REMAINDER-THEOREM
73
Example 2.
i_„ •ix + a^ + lix' '
l/x< + 3/x3 + 4/x° 0
cJi 2/i= + l/i3 + 6 ~6~""
Example 3.
x2 + 3x3 + 4x»_ l/j;< + 3/z»+4 _4_
x_„ 2x + 3.t- + x3 ~ti«2/a:° + 3/x^ + l/i3 ~ 0 ~ °° *
§ 10.] If the rational function /(a;)/<^ (a;) take llic form 0,u fur
a finite value of x, =1= 0, say for x = a, then, since /(a) = 0, </>(a) = 0,
it follows from tiie remainder-theorem that x — a is a common
factor in f{x) and <^ {x). If we transform the function by re-
moving this factor, the result of putting a: = a in the transformed
function will iu general be determinate ; if not, it must be of
tiie 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)j<ii {x) when x = a.
Example. Evalu.ite (3x«- 10.1-3 + 3x2+ 12x-4)/(x^ + 2x'-22x'+32i- 8)
when x = 2. The value is, iu the first instance, indeterminate, and of the
form 0/0 ; hence x - 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 ag.iiu. We
then have a determinate result. The work may be arranged thus (see chap.
v.. §13):-
3 - 10 + 3+12-4
0+6-8-10+4 2
1 + 2 -22 +32 -8
0 + 2+ 8 -28+8
3-4-5+2+0
0+ 6+ 4- 2
1 + 4 -14+ 4 +0
0 + 2+12- 4
3+2-1
0+ 6+16
+ 0
1 + 6 - 2+ 0
0 + 2+16|
3+ 8| + 15
1 + 81 + 14
The process of division is to be continued until we have two remainders
which are not both zero. The cjuotieut of these, 15/11 in the present case, is
the limit required.
The evaluation of the limit in the present case may also be
eflFected by clumging the rariahle, 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 + z)l<i> (a + z) when 2 = 0. Since /(a + s) and
<^ (a + c) are obviously integral functions of z, we can now apply
the rule of § 8. It will save trouble iu applying tlii.s method if
it be remembered — 1st, that iu arranging f(a + z) and <^ (a + s)
according to powers of z wc need not calculate the absolute
74 CUANGi; OK V.UUAlihE en. XXV
terms, since they imist, if tlio fonii to be evaluated be 0/0, be
zero iu eacli CibjC ; 'Jnd, tliat we are only concerned wjtli the
lowest powers of z that occur in the numerator and denominator
respectively.
3jr« - lOx^ + St' + 12j - 4_ a(2 + z)' - mC-' + 1)' + HCi -t- .•)' + l'-M2 + i) - 4
rl, ^F+it* ^ir' + 3-Jj - 8 ~ ^(2 + *)« + 2l2+7)> - 22(2 + »)' + 32 (2 + 1) - «
ISt' + Pt' + Ae.
IS + Pf + ito.
\i + Qt + &e.'
_ '^
~ 14"
This mptliod is of coufrc at buttom iilrntical with the forrncr; for, since
z = x- a, the division by z' corresponds to the rejection of the («ctor (x - a)'.
§ II.] The methods which are applicable to the quotient of
two intoj,Tnl fiinction.s apply to the iiunticnt of two algebraic
.sums of constant multiples of fractional powers of x. Ilach of
the two sums might, in fjict, be transformed into an integral
function of y by putting ^• = y', 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
j^^ xl+xi + 3jc!
«-«*i+2x4+«
II wo divide by x», the lowest power of x that occurs, we Imre
,_ . iJ+jl + axA
I— J J ,
i-o l + 2x»+xl
§ 12.] The following theorem, although p.irtly a special case
under the present head, is of great importance, bocAu.sc it givea
the fundamental limit on which deiM^uds the "differentiation" of
algebmic functinns : —
If mbe any real ct>mmengwal>k quantity, positive or nrgativt,
A(x--l)/(x-l) = fli (1).
I
§? 10-12 L{x'^-l)l(x-l) = m 75
First, let ?» be a positive integer. Then we have
(x"'-l)l(x-i) = x"'-' + .v""- + . . . + X+1.
Hence
L (jf" - l)l{x-l) = 1 + 1+. . . + 1 + 1 (to terms),
= m.
Next, let m be a positive fraction, say/i/y, where ]j and q are
positive integers. Then the limit to be evaluated i.s L (.»'''' - 1 )/
(x—1)*. If we put x = z'', and observe that to a; = l corresponds
c ^ 1, the limit to be evaluated becomes L (z'' - l)/(3«— 1). This
2=1
may be evaluated by removing the common factor z—l ; or thus
£<-"/'->-.a-f)/(f^)'
—pjq = m.
Finally, suppo.se m to have any negative value, say - 7i, where
n is po.sitive. Then
L {x-''-l)l(x- \)=L (l-af)lx''(x- 1),
I— 1 x=l
= -L(x"-l)l{x-l)x'',
= - { X (a;» - 1 )l(x - 1)} X i l/x".
1=1 x=l
Now, by the last two cases, since n is positive, L (x" - 1 )/
(x-l)=n. Aho L ljaf = l. Hence
2=1
L{x-''-l)/{x-l) = -n;
that is, in this case also,
L{x'"-l)/{x-l) = m.
I— I
Second Demonstration. — The atove theorem might also be deduced at once
from the inequality of chap, xxiv., § 7, aa follows: — For all positive values of
a, and all positive or negative values of m, x™-! lies between )hx'""'(x-1)
and m(»-l). Hence (x™ - l)/(x - 1) lies between ni-r™-' and vt. Now, by
• There is here of course the usual understanding (seo chap, x., § 2) oa
to the meaning of xM.
76 EXAMPLES CM. XXV
bringing x snflTiciently near to 1, nur"^' can be made to differ an little from m
as w<> pleaw. Tbe Baiiic is tlicrcforc true of (j:* - l)/(x - 1) ; that is to say,
L(j:"-l)/(x-I) = m
(or all real values of m.
Example 1. Fin.l the limit of (z" - a'')l{x9 - a*j when x=a. Wo hare
L (xP-a'')/(r«-a«)= L a'^{(*/a)''- l)/{(i/a)«- 1|,
.mii':^)-
»-ix
where y=xla. Hence we have, by the theorem of the present paragraph
L {z' - ai')l{x^ - a^) = a'^plti.
Exauiplu 2. Evaluate log(r'- l)-log(j:' - 1) when x = l.
L{log(x!-l)-loR(Ti-l)}=Llog{(x'-l)/(xi-lll,
= 1ok{L(x»-1)/(x4-1)}, l.yS7.
-n'{i-.')A(#r)l'
= logH/il.
= log3.
Example 3. If Iz. Px, . . . denote logx, log (log x), . . . respectively,
th^n, whon x = », W{z + l)irz = l.
In the first place, we have
/(x+l)/lr={l(x + l)-;x + tr}/lr,
= J(l + l/j-)/li + l.
Now, when x = a>, J(l + l/x) = ;i = 0 and lx = «. Ilenoe LI (x + l)/ti = l.
"> If wc asMimo thiit /.f(x + l)/rx= 1, we have
r+M-t + i)/r"x={f^' (x + 1) - r+'x + r+'x}/r«'x,
=/{r(x+i)/rx}/r*'x + i.
nence
Lr+'(x + l)/J"'x = a/x +1,
= 1;
that is, the theorem holds for r + l if it holds for r. But it holds for r=l,M
wo have aevtt, therefore for r = 2. iVc. It is obvious th:kt this theorem holds
fur any lognrithmic base fur which In = <o.
Example i. If f have the same meaning as befon mai X have a similar
meaning for the ba«e a, then
L X'x/rx = l/logo.
Let M= 1/1"K<>. Since Xx = nU, the theorem clearly holds when r=l. It is
therefore sullicicnt to hIiow that, if it is true for r, it is true for r + l. Now
X'^'x/I'^'x = X (\'xV/"'x,
^^{Vz)lt^'x,
=M {' (x'x) - r*'x+ JM-'xi/f^'x,
=M{'(Vx/rx)//"'x+i).
Bence, if we assnrao that J.VzlFz = /i, we have
/.X'^'x/r»'x = M('W»+ M.
= »■•
^12,13 Z(l + i/a.f = e 77
EXPONENTIAL LIMITS.
§ 13.] Tlie most important tlieoieiii in this part of the .sub-
ject is the following, on which is founded the ditl'erentiatiou of
exponential functions generally : —
The limit of {\. + IjxY when x is increased tmthoiU limit either
positively or negatively is a finite number (denoted by e) lyin<j
between 2 and 3.
The following proof is due- to Fort*.
"We have seen (chap, xxiv., § 7) that, if a and b he positive
^quantities, and m any positive quantity numerically greater
than 1, then
»?«"'-' (ft - 6) >«'" - b'" > mb'"-' {a -b) (1).
In this inequality we may put a = {y <r \)ly, b= 1, in=yjx, where
y > .1- > 1 . We thus have
\ y J X
Hence
(1+i) >l + i,
\ y) x'
that is,
where y>ir.
Again, if in
before), we have
(1) we put « = 1, b^ (y- 1
i>i-(?^--iy".
X \ y J
)I>J, {>»,
(2),
y, X being as
Hence
('-J)'>(-,i)"
and therefore {\--\ <fl —
where ^>ar.
We see from (2) and (3) that, if we
give a
(3).
series of in-
• ZeiUchrift fiir Mathematik, vu., p. 46 (1862).
creasing positive values to x, tlie function (1 + 1/j-)* continually
increases, ami the function (1 - l/x)'' continually decreases.
Moreover, since a^>a^-l, we have
x x+l
that is, A--y'>l + -.
Hence (l - i)->(l . '/ (4).
TIic values of (1 - l/.r)"' and (1 + i/j-y cannot, therefore,
pass each other. Hence, when x is incrcivsed without limit,
(1 — lfx)~' must diniiui.sh down to a finite limit A, and
(1 + IfxY must iucrea.se up to a finite limit B. The two limits
A and B must be equal, for the dilTerence (1 - l/x)''-{l + l/xY
may bo written {x/{x - 1)\' - {{x + l)/x)' ; and by (1) we have
1 / X \' ( X \' (x+\\' 1_ (x-^\\' ,,.
x\x-\) ^\x-\) \ x ) ^a:(l-l/^)V X ) ^•'^•
But, since, as has already been shown, {x/(a;-l)}* and
\{x-i-\)lx]' remain finite when x = fa, 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 i (1 + l/x)" and
// (1 - 1/x)"' have a common finite limit, which we may denote
by the letter c.
Since (I + l/6)«=2r.21 . . . and (1 - 1/6)"' = 2-985 . . .,
e lies between 2"5 and 29. A closer approximation mi^'ht bo
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
6 = 27182818285 . . .
Tlie constant e is usually called Napier's Base*; and it is the
logarithmic or exjmnential base u.sed in most analytical calcula-
tions. In future, when no ba.«o is indicated, and mere arith-
* III honour of Napier, and nut because ho ciplicill^ uned this or indeed
any utUcr baiic.
§13 Z(rt»-l)/.r = loga 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 logcZ and
expjo; (that is, e*) respectively. »
Cor. 1. i(l+a;p = e.
1=0
For L {l + l/zY = e; and, if we put z=l/x, so that x = 0
corresponds to 2=00, we have Z (1 +ar)''^ = e.
Cor. 2. L\oga{{l + l/x)'} = 2/ log„ ((1 + a;)''^} = log„ e.
I=» 1=0
For, since logai/ is a continuous function of 1/ for Unite values of
y, we have, by g 7,
L log, {(1 + l/xT) = log, { Z (1 + l/x)%
= loga e.
The other part of the corollary follows in like manner.
Cor. 3. X (1 + y/xf = Z (1 + xi/y'" = e".
If we put \lz=ylx, then to a; = 00 corresponds z= 00 ; hence
L{\+ylxf=L{l + llz)^,
= z{(i+i/c)r,
= {Z(l + l/c)T, by §7,
Cor. 4. Z(«^-l)/a;=loga.
1=0
If we put y = a'—\, so that a; = loga(l +y), and to a; = 0 corre-
sponds ^ = 0, we have
Z(«^-l)/*-=Z2//loga(l+3^),
1=0 V=0
= Zl/log,(l+y)''>',
y-dt
=i/iog„{Z(i+y)n
11=0
= 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 /, (a' - l)/x ia
jt— 0
so I X ION KM I. \ I, AM) l.oCAKrrilMlC INKQUALITIES CII. XXV
tiiiitti; and tLence, b; transfurinatiun, to pruve the leading tbeoiem of Ihii
paragraph*.
Cor. 5. If X be any positive quant it if .
and, i/ .r be positive and less than 1,
«•"'>! -^, -Iog(l -j-)>x.
Since «>(1 + 1/n)", wlieu « way be as great as we please,
e'-l>(l + l/«)«-l,
>n.r {(1 + 1/h) - \\>x, by chap. XXiv., § 7,
for, however small j-, we cau by suthcieiitly iiicreasiug n make
n.r>l.
Hence e'> \ +x.
It follows at ouce that log «'>Iog(l +;r), that is, a:> log (I +x).
Agaiu, since e<(l - 1/h)'" and d-'>(l - l/«)",
«-'-1>{(»i-1)/hJ"-1,
>nj-{(«-l)/n-l},
>-r.
Hence «"'>! -x, and therefore 1/(1 - x)>«<".
It follows at once tliat log j 1/(1 - x)], that i.s, -log(l -t)>x.
Cor. Gt. If {.r, Px, . . . denote Inijx, log (litijx), . . . rv.ytect-
iivlif, x>y> 1, and r be any positive inttyer, then
(x-y)hM^y • ■ • ry>r*'x-r*'y
>(x-y)lxLi'x . . . I'x.
This may be [)roved by induction as follows.
Hy Cor. 5,
lx-ly = l{x/y) = / {1 + {x-y)ly\<{x-y)ly,
whicli jirove.-i the first ineqn.ility when r-O.
Assume that it is true for r, i.e. that
r*'x-f*'y<{x-y)ly/yry . . . Ti/, then
r*'x-r"'yi(r*'x/r-'y),
= / II +(/-'*- /'♦»//'*'yl,
< (/'•*'x - l'*'y)!l'*'y, by Cor. 5.
Hence the induction is complete.
• See SchlOiiiilch, ZtiUchrift far Mnlhfmatik, vol. in., p. 387 (1868).
f Maluutuu, Onintri'i Archiv, viii. (181G).
§ 13 euler's constant 81
Again, we have by Cor. 5,
lx-hj = -l{!/lx) = -l{l-{x-y)lx}>{x-y)lx.
Using this result, and proceeding by induction exactly as before,
we establish the second inequality.
If we put X +\ and x for x aud y respectively we get the
important particular result
Ijxlxfx . . . l'x>l'-*'{x + V)-l'-*'x
>\l{x+l)l{x+l)P{x+\) . . . l'-(x+l).
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)-rx] = 0,
L {;■■+' {x+l)- Vx] xlxl-x . . . Vx = 1.
3:=
Example 1. Show that the limit when n is infinite of 1 + 1/2+ . . .
+ \jn - log « is a finite quantity, usually denoted by 7, lying between 0 and 1.
(Euler, Comm. Ac. Pet. (1734-5).)
Since, by Cor. 5,
-Iog(l-l/«)>l/K >log(l + l/H).
We have log {«/(«- 1)} >l/« >log{(n + l)/H},
log{(«-l)/(K-2)}>l/(«-l)>logW(«-l)},
log {.3/2} > 1/3 >log{4/3},
log {2/1} > 1/2 > log {3/2},
1=1 > log {2/1}.
Hence l + logn>21/n>log(n + l).
Therefore 1 > 21/n - log n > log (1 + 1/n).
Now, when n=<», log(l + l/K)=0. Thus, for all values of «, however
great, 21/h - 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 IG places, namely, 7= •o77215004901532(,';). (See Inst. Calc. Dif.,
chap. VI.)*
• Euler's Constant was calculated to 32 places by Masoheroni in his
Adnotatio)ies ad Hukri Calculum Integralcm. It is therefore sometimes
called Mascheroni's Constant. His calculation, which was erroneous in the
20th place, was verified and corrected by Gauss and Nicolni. See (iaus-i,
Werke, Bd. m., p. 134. For an interesting historical account of the whole
matter, see Glaisher, Mas. Math., vol. i. (1H7L').
c. II. G
82 CAUCIIV'S TUEOKKMS CU. XXV
Examples. Show tliat ;, {1/1 + 1/2+ . . . + l/n}/logn = l.
Tliia follows ut once from the iuequality of last vxnmplo.
From this result, or from Example 1, we see that L {1/1 + 1/2 + . . . + l/n}
-co ; aud also that L {llk + ll{k + l} + . , , + l/n} = x , where k ia any finite
positive integer.
GENERAL THEOREMS.
§ 14.] Before proceeding further with the theory of tlie limits
of exponential forms, it will be convenient to introduce a few
general theorems, chiefly due to Cauchy. .Vlthough these theorems
are not indispensable in an elenientar}' treatment of limits, the
student will find that occasional reference to them will tend to
introduce brevity and coherence into the subject.
I. For any critical value of x,L{f(x)\ = {//(x)} , pro-
viJeJ the latter form be not indeterminate.
This is in reality a particular ca.se of the general theorem of
§ 7. The only que.stion that arises is as to the continuity of the
functi(liis of the limits. We may write
{/{x)} = e
Now w = Iog u is a continuous function of u, so long, at least, as
u lies between + 1 and + » ; and e** is a continuous function
of V and w. Hence, so long as L<f> (x) and L log/(j-) are neither
of them infinite, we have
L {/(j-)i = Le
i*(i)Zlog/(»)
= « ,
Z4(«)lo((L/(x)
= e
Hence Zl/W^' HVW}^'^ (1).
An oxaniination of the special cases where either L<f>(x) or
^'"K/i-*'). '"■ both, become infinite, shows that, so long aa
{//(.r)} * does not nssunio one of the indeterminate forms 0,
oc , 1"°°, both sides of (1) become 0, or both « ; so tliat the
theorem may be Blate<l us true for all cases where its sense ia
dotunuinuttf.
§§ 13, 14 caucht's theorems 83
II. L [/(a- + 1) -f{x)} = Lf{x)lx, provided L \f{x^\) -f{x)\
X='«> X=aO X=3D
be not indeterminate*. (Cauchy's Theorem.)
Since x is ultimately to be made a.s 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) -f{x)\ is not infinite, = k say.
Since L\/{x+ \)—f{x)\=k, we can always choose for h a
definite value, so large that for x=h and all greater values
f{x+ l)—/{x)- k 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,
/{h + 2)-/(h+l)-k<a,
fili + n) -f{k + n- l)-k<a;
and, by addition, f{h + n) ~/{k) - nk<na ;
that is, f{x) -/{//) -{x-h)k< (x - h) a.
Hence ^-^ ^-^ - (l-^^ k<(l-^)a,
X X \ xj \ xj
f{^) ;.^„|/W h{k + a)^
X X .r
Since /(h), h, k, and a are, for the pre.sent, fi.xcd, it results
that, by making x sufficiently large, we can make f(x)/x — k
numerically less than a. Now a can be made as small as we
please by properly choosing k ; hence the theorem follows.
Ne.x't, suppose that L {/(x + l) -/(x)) = + cc ; then, by
taking h sufficiently large, we can assume that
/{h+l)-/{h)>l,
/{h + 2)-/{h + l)>l,
f{h + n)-f{h + n-\)>l,
where i is a definite quantity as large as we please.
* Tlieorems II. and III. are piven by Cauchy in his Analyse Algehriqur
(which is Part I. of his Cours iVAnalyse de I'icole Royale Polytechnique).
Paris, 1611.
6—2
84 CAUCUY'S THEOUEM3 Cll. XXV
Heuce /(h + n) -/(A) > nl,
that is /(j-) -/(A) >{x-h)l.
Hence -^— ' > / +"i-^— ' .
Since /(A), /', / are all definito, we can, by siilTicicntly in-
crejisiug x, kwAct f(h)lx— hljx as suiali aa wc ple;ise, therefure
/{x)/x>l. Now, by properly clionsiug /», / can be made as largo
as we please ; hence L/{x)/x = oo .
The case where L{/(x+ l)-/{x))=- oo can be included in
the last by observing that (-/(«+ l))-{-/(,x)) lias in this case
+ 00 for its limiting value.
III. L f(x + l)//(x)=Ly{x)]'^, provided Lj\x^\)lf(x)
be not indeterminate.
This theorem can be deduced from the \s\»i by transformation,
as follows* : —
We have L \<i>{x-\-\)-<i>{x)}^ L'^^,
where i/' {x) is any function such that L {^(a; + 1) - f {x)\ i.s not
indeterminate. liCt now tf/ (x) = log/(j;) ; so that i/i (j- + 1 ) - «/' (x) =
log / (a: + 1) -log /(a:) = log {/(a: +!)//• (J-)}; and ^ (j-)/* =
\\og/(,x)\lx = log {/{x)]"'. Then we have
Hence h.g [^Jy^ } = log [£{/Win
provided L/{x+ l)//{x) be not indeterminate. Hence, finally,
Cauchy makes the imjwirtant remark that the dpmon.<itration8
of his two tlieorems evidently apply to functions of an iutogral
variable such as x'., where only positive integral values of z am
admissible.
* The reader will GdiI it a good exercise to oatablisli tbit theorem directly
from Gritt principlen, UK Cuuchj' Joui.
I 14. 15 La^/x, Lhgax/x, Lxlog^^ f^6
For example, we have L (« + 1)!/.eI = L (.t + 1) = co. Hence L (xl)V*=oo,
and consequently L (l/j;!)'/»=0.
EXPONENTIAL LIMITS RESUMED.
§15.] I/a>l,theii L w'/x^co; X log^.r/a; = 0 ; L x \og„x ■-- 0.
The first of these follows at once from Cauchy's Theorem
(§ 14, 11.) for we have
L («•'+' - a'} = ia.^ (rt - 1) = a, .
Hence La'/x = <x> .
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
X«=a) X t=^J + Z
r f z a'
»=» y T *.. ^
= a/L%, (1),
where we have to deal merely with La-jz, z being a positive
integer.
Let Mj = a'jz, then «,+,/«, = azl{z + 1) = a/(I + 1/-). Now,
since L a/{l ■i-l/z) = a>l, we can always assign an integral
value of z, say z = r, such that, for that and all greater values of z,
^»-¥ihh>b, where 6>1. We therefore have
Ur = «r,
Ur+ijUr>b,
Ur+i/Ur+l>b,
86 Lx"/n\, L,„C„ cii. xxv
Honce, by multiitlyiug sill tliesc inequalities together, we deduce
U:,>l/-''Ur>b'Ur/b''.
Now Mr/A' is finite, and, since 6>1, 0' can be nioiie as great as
we please by sulliciently increasing z. Hence L u,^ <x>, on the
supposition that z is always integral. But, since a^ is finito, it
follows at once from (1) that L a'lx= <x>, when x is unrestricted.
The latter parts of the theorem follow by transformation.
If we put a' = y, so that x = loga^, and to x = « corresponds
y = an , we have
cc = i c^lx = L yfiog^y.
Ileuce L loga^/y = l/oo = 0.
»-"
If we put a' = Ijy, so that x = - logay, and to x- <» corre-
sponds y = 0, we have
00 = i o7a; = - /y Xjy log^y.
Hence L y log^y = - l/a> = 0.
ExaiDple 1. Show that, if a>l and n be positive, then L a'/x*=co ;
L logai/x»=0; L a;"log„i=0.
«— « r— -H)
I- aVx"= i {a*A>/j}i»,
= { L (a'/")'/x}-,
X— «
= ao" = oo ;
for, sincca>l ami n is positive, we have a"*>l, bo thai L((J "")'/* = » und
The two remaining results can be established in like manner, i( we put
y = \of^^x in tlie one case, and i/= - log, i iu the other.
It should be noticed that if n be negative we see at once that L a*jx' = aB;
L log„ x/x" = X ; L x"log„x= - 00 .
JT— ■> X— 0
Example 2. If x be any fixed finite qoantity, L z*/nl=0.
Since n is to be made infinite, and x is finite, we ma; select some flniU
positive integer k such that x < % < n. Then we have
nl (t-l)l ■ * ■ »+l ■ ■ ■ n'
*(k-l)'\k)
Now, since x<.k, A(x/*)'>-**' = 0, htncc the theorem.
I
l»<^n—
§§ 15, 16 EXAMPLES 87
Example 3. Lm(m-1) . . . (hi - k + 1)/h! = 0 or oc , according as m>
or <: - 1.
First, let m> -1, then m + 1 is positive. We can alwaj-s find a finite
positive integer k such that 7(H-l<A;<re. Therefore we may write
m(m-l). ■ . (m-« + l)_, ,„.j-+, , fi _'!^\ (i _"'±1\
■••(-^).
= (-)'-*+',„C^_,P,say.
Now
lo,l/.= -lo.(l-'^)-lo.(l--^)-...-.o.(l-"LL^),
>{m + l)lk + (m + l)l{k + l)+ . . .+(m + l)/n,
by § 13, Cor. 5. Also, by § 13, Example 2, the limit of (m + l)//j + (hi + 1)/(/; + 1)
+ . . . + (Hi + l)/rt is infinite when n = co . It follows, therefore, that LP = 0,
and therefore that i,„(7„ = 0.
Next, let m< - 1, say m= - (1 + a), where a is a positive finite iiuantity.
We may now write
„c„= ( - )" ^' + "J(; ,^-^:^i"i'" = ( - rp. say.
Now
lo,P=_,o,(l.^)-lo,(l-.^y-...-log(l-,-^-J.
>a/(l+a)+a/(2 + a)+ . . . + a/(H + a),
>al{l+p) + al{2+p)+ . . .+al{n+p),
where p is the least integer which exceeds a. But the limit of 0/(1+7))
+ a/(2 + p) + . . . +a/(n+y) is infinite. Hence LP = oa.
When m=-l, m<^i. = {-l)"i ''"'^ ^^^ question regarding the limiting
value does not arise.
§ 16.] T/ie fundamental theorem for the form 0° is that
*=+0
This follows at once from last paragraph ; for we have
Laf = i(f ""^ = e^^'^' = e» = 1.
Example 1. L (x»)"==l.
X-+0
For i(x»)'^ = Ls'" = I-(j;'^)" = (Lr')'' = l" = l.
Example 2. L x'" = l (n positive).
I-+0
For Li*"=Le=="i'>it=:=«""i<>B^=«'' = l, by § 1.5, Example 1.
iV.C— If n be negative, L x'"=0" = 0.
I-+0
88 THE KOUM 0" CM. XXV
§ 17.] If a unit V ho functions ff r, hith if wh'irh I'limsh wlimi
x = a, and art such tluU L vju" - 1, where n h posit ivn and neither 0
nor oo, and I is not infinite, then L »"= 1, provided the limit be so
X— «
approached that u is positive*.
For Lu' = L («"")""■ = (Lu"')'-'^'.
Now, by § 16, E.\anii)le 2, since n is positive, L m""= 1. Ilcnce
«-- M
i4t''= l'=l.
If L r/«''=ao, this traiisfonnatioii leiuls to tlie furui 1";
x-a
aud tlicrcfore becomes illusory.
The above tlieoreni iiichnlcs a very large number of parti-
cular cases. We see, for cxauiplo, that, if Lvju be determinate and
not infinite, then Lu' = I. Af^aiii, since, as we shall prove in
chapter xxx., every algebraic function vanishes in a finite ratio
to a positive finit.e j)0\ver of .r-a, it follows that every such
function vanishes in a finite ratio to a positive finite power of
every other sucli function. Hence Z«'= 1 whenever u aud v
are algebraic functions of art.
Example. Evaluate L{x-1 + v'(x' - l)}'5'l»-'l when x = 1.
Here u = ^{x-l){^l{x- 1) + ^/{x> + x + l)}, v=*l{x-l), uI^/b = {^(x- 1)
+ V(x» + i + l)}"'.
Hcuce Lu^lv = i/X Therefore LW = L (u«*')'/»'^= l"^'= 1.
Jj 18] In cases where the la.st theorem doe.s not ajjply, the
evaluation of the limit can very often be efi"ected by writing «•
in the form e"'"", and then socking by transformation to deduce
the limit of tjlogM from some combination of standard cascs^.
Example. Evaluate x'''°«<«'"'l when x = 0.
It is obviously puKgootcd to attempt to make thin dcpcod on
L {{e*-l)lx} = l. This may bo effected as follows. Wo have
• Sec Franklin, Amrriran Journnl of ifathematict, 187ft.
t Sec Spragiic, Proe. F.dinb. Math. S»r., vol. in., p. 71 (188J5).
J At one lime an crrnneous imprpHKion prevailed that the iDdctcrminato
form 0° lia« alnayn the value 1. Sec Cretlt't Jour., Dd. xii.
§§ 17-21 THE FORMS oo », 1 89
log X log X
Now
log(c^-l) log{(e'-l)/x} + iogx'
1
log{(e=-l)/x}/logx + l'
Since ilog {{e'^-l)/x} = 0, by §13, Cor. 4, and Llogx=-oo, we see that.
L log xjlog (^-1) = 1.
Hence ixi/>'>B(«'-i) = e.
§ 19.] Since u'=l/{l/u)'', indetermiiiates of the form »"
can always be made to depend on otliers of the form 0°, and
treated by the methods already explained.
Example. Evaluate (1 + x)''^ when x = oo .
Let l + x = l/j/, so that y = 0 when x = oo ; then we have
L (l+x)V^=I, {l/j/!'/(i-»)} = l/L()/!')i/a-w.
NowLi/» = l andLl/(l-t/) = l; hence L (l+x)V»:=l.
X=ao
§ 20.1 The fundamental case for the form T is i (1 + l/xY
X=50
= Z (1 +j:y'' = e, already discii.ssed in .§ 13. A great variety of
x=0
other cases can be reduced to this by means of the following
theorem.
1/ u and V be functions of x such that m = 1 and « = oo when
x = a, then Iai' = e^"'"-'', provided Lv{u-l)be determinate.
We have in fact
«» = {(! + M-1)''''-";"'"-''.
Hence, by § 7,
provided Lv{u-l)he determinate.
Example 1. L x'/l':-it= i (1 + jri)i/(x-i) = <;.
Example 2. Evaluate (1 + log x)'/(*-J) when x= 1.
We have
Z = L (1 + log x)V(»-') = L { (1 + log a;)i/l«!X}i<«*/(x-il,
= jLlogx/(i-l).
Now L log j/(x - 1) = i log xi'l^-'l = log LiV|x-i) = log e = X. Hence I = e.
TKIGONOMETRICAL 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 TRIGON'OMETUICAL INEQUALITIES CH. XXV
We shnll ro<iuire the fnllowiiif^ iiiiM|uality tlicoreniR: —
1/ X bt' thf numlx'r of rddians {circular units) in any pusitive
anijle lens than a right angle, then
1. tanx>x>siux;
II. z>a'inx>z- \j';
III. l>co.sar>l-A.r'.
If PQ be tlie arc of a circle of nvliiis r, whicli subtends the
central au^'Ie 2.r, ami if PT QT be the tangents at P and (},
then we ji.ssuuie as an axiom that
P7'+ 7'(^>arc Pq>c\\w\ PQ.
Hence, as the reader will easily see from the geometric defini-
tion of the trigonometrical functions, we have
2r t;in .r>'irx>2r sin x;
that is, tan.r> x> sin j-,
wliich is I.
To prove II., we remark that sin x = 2 sin \x cos Jj-
= 2 tan Jj: cos' ia: = 2 tan J.r (1 - sin' Ax). Hence, since, by 1.,
tan }tx>\x and sin ix<ix, we have
8inx>2.ix{l -(Ax)'l,
>x-\j*.
The first part of III. is obvious from the geometric
definition of co.-fx. To prove the latter part, we notice that
cosx= 1 -2sin'ix; licnce, by I.,
cosx>l -2(ix)'
% 22.] The fundamental theorem regarding trigonometrical
limits is as follows: —
If X be the radian mctisure* of an angle, then L (sin xjx) = 1.
a— ♦
This follows at once from the first inequality of last para-
graph. For, if x<Jir, we have
tanx>x>8inx;
therefore sec x > x/si n x > 1 .
* In all that foUowi, and, in fact, in all analytical treatment of the trigimo-
metrical (uuclions, the aroumoDt ii auumod to douot« radian meaaora.
§§21-23 Lsinx/x, Lta,nx/x 91
If we diminish x sufficiently, sec x can be made to dillcT from
1 by as little as we please. Hence, by making x sufficiently
small, we can make a'/siu x lie between 1 and a quantity differing
from 1 as little as we please. Therefore
Lx/smx= 1.
Hence also L sin x/x = 1.
Cor. I. L tan x/x = 1.
X=Q
For L tan x/x = L{sm x/x)/cos x^L sin x/x x L 1/cos a; = 1 x 1 = 1.
Cor. 2. L sin - / - = i tan - /- = 1 provided a is eitlier a con-
i=„ X/ X x=~ X/ X
slant, or afimction of x which does not become infinite whenx= <x> .
This is merely a transforuiatimi of tlie preceding theorems.
It should also be remarked that
provided a and /3 are constants, or else functions of x which
do not become infinite when ar= oo.
If, however, a were constant, and /? a function of x wliich
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-
grajih 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 which is not infinite when
x= oo , then
xftanV-y=l-
x-.\ X/ Xj
To prove the first of these, we observe that for all values of
ty/x less than Jir we have, by § 21, II.,
'>(''» i/ir>{-'e)T
92 LUiul/iy. L[r<»iy en. XXV
Now
L (1 - a.y.l.T')' -L !(1 -a'/.lx')-*^'-'}-'''*',
=Ti (I - aV4j!')-*'''"'}-^''*'.
= «»=1, by S7 and 13.
llcnce L (sin -/-) = !•
In exactly the same way we can prove tliat // ( cos -j 1.
Filially, since
the thirrl result follows as a combination of the first two.
Kxaniplo. Evaluate (cos r)''^ when x = 0. By § 20, we have L (cosi)'"^
^eKmi-iifl". Now (coflJ--l)/i' = -2Bin'Jx/x'= -i(ainix/Jx)». Ueuce
L(coax-l)/x»= -4.
We therefore have Z, (co8i)'/^=e-'.
SUM OF AN INFINITE NUMHER OF INFINITELY
SMALL TEUMS.
§ 24.] If we consider the sum of n torm.s .«ay, «, + «,+ . . .
+ M,, each of which depends on n in such a way that it becomes
infinitely small when n becomes infinitely j^at, 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 3c ; for we cannot tell a prvtri whether the smallncvs of the
individual terms, or the infiniteness of their number, will ulti-
mately predominate. We shall have more to do with such cases
in our next chapter; but the fnllomnp instance is so famous in
the history of the Infinitesimal Calculus before Newton and
Leibnitz that it deserves a place here.
//*r+ 1 bf pofltivc, then
z. (r + 'J"- + . . . +«•■)/"'*' = ^Rr + 1).
In the case where r is an integer this theorem may be
deduced from the formula of chap, xx., § 9.
§§ 23, 24 i (l"- + 2'- + . . . + «'•)/«'■+' 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'''-ir'%{r+\)f(.v-y) (1), -
which we have already used so often.
If we put first x = 2}, y=p - 1, and then x =jj + 1,2/ =P, "'G
deduce
(p + !)'■+' - p'--'' 5 (?• + l)p'' >;?'"+' - (p - ly-"' (2)
where the upper or tlie lower signs of inequality are to be taken
according as the positive number r + 1 is > or < 1.
If in (2) we put for j9 in succession 1, 2, 3, . . ., n and add
all the resulting inequalities we deduce
(?j +!)'■+> -I2(r+l){r + 2'"+. . .+if)%n'-'\
Hence
{(1 + !/»)'•+' - !/«'■+'}/(?• + 1) 2 (r + 2-- + . , . + «'•)/«'■""
>l/(r+l).
That is to say, (F + 2*" + . . . + «'')/«''*' always lies between !/(»•+ 1)
and ((1 + !/«)'■+' - l/n'"+'}/(r + 1). But Z (1 + !/«)'■+' = 1 ;
and L Ijrf'^^ = 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 !/«''■'■' = oo .
Cor. 1. Ifsbe any finite integer, and r + \ be positive,
L{v + '2r+ . . . + (« - «)'•}/»'•+' = l/('- + !)■
This is obvious, since L{V+2''+ . . . + (n - sY]/n''+^ differs
from L(r + 2''+ . . . +«'')/«''*' luy a, finite number of infinitely
small terms.
Cor. 2. I/abe any constant, and r + 1 be positive,
L {{a + l)-- + (a + 2)'- + . . . + (a + »)'}/«•■+' = l/(r + 1).
This may be proveil by a slight generalisation of the method
used in the proof of the original theorem.
94 DiRTcm.Frr's limit ch. xxv
Cor. 3. If a and c be constants, and r + 1 + 0,
L {{na + cY + (na + 2cY + . . . + («a + ncy\ln^*^
= {(a+c)'-+'-o'-+'}/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.
8 25.] Clo.sely coniiectoil with the re.sulLs of the foregoing
paragraph i.s the following Limit Theorem, to which attention
has been drawn by the researches of Dirichlet: —
If a, h, p he all positive, the limit, when m = x, of the sum of u
tei-ms of the serif's
1 1 _ 1__ 1
o'+p "^ (o +'6)'+o "^ (a + 2by*i' + ' • • + (« + „/,)<+p "^ ' • • < ' '•
is finite for all finite values of p, howeeer small; and, {/'
2 l/(o + «i)'+<' denote this limit, then
»-<
Zp 2 l/(o + «/»)'+''= 1/6 (2).
By means of the inequality (1) of last paragraph, we readily
establish that
{a+ (j»-l)ij-p- \a+pb\-i'>i>b 'o+;>i["''"'>{a + pb]-'
-{a + (/>+l)t}-P (3).
Putting, in (3), 0, 1, 2, . . ., n successively in place of />,
adding the resulting incijualitieii, and dividing by bp, we deduce
1/ -A 1 l>v 1 >ifl L_ 1
^plla-tl" {a + «/*!"/ p_<,{a+;/6}'+p *p In^ |a + (» + I)*!'/
(4).
Since Ll/{a + nb\i' =■ 0, and L\/[a + {n+l) b\' = 0, when
n=cc, we deduce from (4),
1 -. 1 1
pb{a-by^^(a+pby*''^Pj^ ^^)-
From (5) the first part of the above theorem follows at
once; and we .see that \/pb (a- by and l/p6»+' are finite upper
and lower limits fur the sum in quuHtiou.
§§ 24-26 GEOMETRICAL APPLICATIONS 95
We also have
1 _ s 1 1
b{a- by '^pZ {a + pby+f bw '
wliPiice it follows, since L l/b (a- by = L l/baf = 1/b, when p = 0,>
tliat
p=o''p-o(a+j»i)'+p b'
From the theorem thus proved it is not difficult to deduce
the following more general one, also given by Dirichlet: —
1/ ^'i, L, . . ., k„, . . .be a series of positive quantities, no one
of which is less than any following one, and if they be such that
L T/t = a, where T is the number of the k's that do not exceed t,
1=1,
then ^1/kn+i' is finite fur all positive finite values of p, fiowever
small; and L p'S.Xjkn^i' = a*.
Cor. It follows from (5) that
an inequality which we shall have occasion to use hereafter.
GEOMETRICAL APPLICATIONS OF THE TUEORV OF LIMITS.
§ 26.] The reader will find that there is no better way of
strengthening his grasp of the Anal}tical 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{x) can be solved by evaluating the
limit when ^ = 0 of {f(x + A)-f{x)]/h; for, as will readily be
seen by drawing a figure, the expression just written is the
tangent of the inclination to the a.vis of x of the secant drawn
through the two points on the graph whose abscissae are x and
x + h; and the tangent at the former point is the limit of the
* Sop Dirichlet, Crelle's Jour., Ld. 19 (1839) and 63 (1857) ; also Heine,
ibid., Bd. 31.
96 GEOMETRICAL APPLICATIONS CH. XXV
Hccant wlicii tlic lattor point is made to approach infinitely close
to the foruier*.
Example. To find the inclination of the tangent to the graph o( y = r'
at the point where thist graph crosses the axis of y.
If 0 be the inclination of the tangent to the z-axis, we have
tantf = i:,(<»+»-e»)/h,
= L(e»-l)/;i,
= log<=l.
Ucnce 0 = Jir.
§ 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 = af/l^~\ 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 = ei: Dirirle AH into n cqaal parts; draw the ordinatea
through A, B, and the n - 1 points of division ; and construct — 1st, the scries
of rectangles whose bases arc the n parts, and whnsc altitudes are the Ist,
2nd, . . ., nth ordinates respectively; 'iud, the series of rectangles whose
bases are as before, but whose altitudes are the 3nil, 3rd, .... (n4-l)th
ordinates. If/, and </, be the sums of the areas of the first and second scries
of rectangles, and A the .irca enclosed between the curve, the axis of x and
the ordinate^ through A and B, then obviously I^<.A-<^J^.
Now
/. = c{a' + (a+f/n)' + {a + 2c/n)'-+. . . + (a + nTTc/i.)'-} /nr-> ;
J,=c{(a + c/ri)'- + (a + 2<:/n)'-+. . . +(a + nc/n)'-}/n<'^'.
Since J, -/,= c(fc'-a'')/»i/'^', which vanishes when n = m , Z./.= Z^,,aud
therefore A = LJ„ when n = x> . Hence
c {na + Ic)'' + (na + 'ie)' ^■ . . . + (iia ■(■ tu)'
^-,T:i^ ^l .
-M"-i^ir^\-'''^"'^'-
Hence X = (fc'+>-a'+')/(r + l)r-'.
This gives, when r = }, and a = 0, the Archimcdian mlc for the quadrature
of a i>arabolic segment.
* We would earnestly reoomnnmd the learner at this stage to begin (if
he has not aln'ody done so) the study of Frost's Curre Tracing, a work which
should be in the hands of every one who aims at becoming a mathematician,
cither practical or aciontiflo.
t The reader shuuld draw the Qgure for hiuisclC
§§ 20-28 THEOuy of irrationals 07
NOTION OF A LIMIT IN GENERAL. ABSTRACT
THEORY OF IRRATIONAL NUMBERS.
§ 28.] lu tlie earlier part of this chapter liuiitiiig values have ,
beeu associated with the supply of values for a functiou in speei;il
cases where its definitiou iails owing to the operations indicated
becoming algebraically illegitimate. This view naturally sug-
gested itself in the tii'st instance, because we have been mt)re
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 he
observed that no general criterion was given for the existence of
a finite definite limit. AH 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 aritlmiotical difficidty, viz. the
question as to the definition of irrational number. For examjile,
in proving the existence of a finite definite limit for (1 + ]/:»)'
when X is increased iudefinitelj', what we really proved was not
that there exists a quantity e such that \e-{l + llxY\ 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 + Xlx'Y will lie between them if only x be made
sufficiently large. From this we infer without farthur 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 remaius 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. II. 7
98 THEORY OK lltllATIONALS CH. XXV
rntioiiiil iiumbur wliich can claim to bo the value of the limit.
The iiitroiluctiou of a definite quantity e as the value of the
limit under these circumstmccs is justified by the fact that wo
thus cause no algebraic contradiction. Such ijuautitics aa J'i,
XJi, &c. have already been ailniittcd as algebraic operands on
similar grounds.
§ 29.] The greater refinement and rigour of modern mathe-
matics, especially in its latest develoi)ment — the Theory of
FunctioiLs — have led mathematicians to meet directly the logical
dilHcultius above referred to by giving (i j>rit>ri an abstract defi-
nition of iiTational real quantity and building thereon a purely
arithmetic theory. There are tliree distinct methods, commonly
spoken of as the theories of Weierstras.s, Dedekiud and Cantor*.
A mi.xturc of the two last, although perhaps not the most elegant
method of exposition, apiwars to us best suited to bring the issucij
clearly before the mind of a beginner. We shall omit demon-
strations, except where they are neces.sary to show the sequence
of ideas, the fact being that the initial difficulties in the Theory
lie not in framing demonstratiuns, but in seeing where new
definitions and where demonstrations are really necessjiry. For
a similar reason wo shall at once a.ssume the properties of the
onefold of Kational Numbers as known ; and also tho theory of
• The theory of Wcierstrass, earliest in point of time, waa given in his
lectures. Lilt not pultlishod by himself. An account of it will bo fouml in
Bicrmauu, Tlitorie Jtr Analytitchtii Fuitctioncn (Leipzig, 184J7), pp. I'J — 33.
A brief but excellent account of DcJekiuJ's theory is given by Weber,
Lehrbuch dcr Aljcbra (Braunschweig, 18'J5, 1898), pp. 4 — 10 : 8c« also
Dedukind's two tracts, Stetigktit und irrationuU Zithltn (Braunschweig,
1872, 1892) ; and H'of tind und mat $olUn die Zahlrn t (liraunschweig,
18S8, 1893). For ujiiKisitionB of Cantor's theory sec Math. Ann., Ud. 5
(1872), p. rjs, and lb. Ud. 21 (18K3), p. OOu; ako Heine, CrelU'$ Jour.,
Bd. 7-J (1872): and Stolz, Atlgemeiw Arithmetik, I. Th. (Leipzig, 1n8.",),
pp. 97—124.
Meray, in his Nouvetiu I'ricit d'Aiuily$e Infinil^timalt (Paris, 1872),
published indi'|itndcntly a tliojry very similar to Gintor's, which will b«
found set forth in tho lint volume of his Le^oiu KuufelUi tur I'Analyu
IitfinilitinuiU (Parin, IS'.H).
A good general sketch of tho whole subject is given by Priiigshcim in his
article on Irtatioualzahlcn, Ac, Kncycluyiidit dcr ilalhematiichen fl'iueit-
icha/ten (Leipzig, 1898), ltd. 1., p. 47.
^ 28-31 THE RATIONAL ONEFOLD 99
terminating and rei)eating decimals, wliich depends merely on the
existence of rational limits.
§ 30.J Starting with 1 and confining our operations to the
four species +, — , x, ^, we are led to the onefold of Kationaf
Quantity
. . ., - min, . . . - 1, . . . 0, . . . + 1, . . . + w/n, . . . {li)
in which every number is of the form + mIn, where m and n are
finite integral numbers.
The onefold R possesses tlic following properties.
(i) It is an wdered 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, h- in which the operands are
rational numbers (excepting always division by 0) leads to a
number in R.
(iii) a and b being any two positive quantities in R, such
that 0<a<b, we can always find a positive integer h so that
na>b*; and consequently bjiKn.
(iv) Between any two unequal quantities in R, however
nearly equal, we can insert as many otiier quantities belonging
to i? 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
o, a + {b-a)/n, a+ 2 {b-a)/n, . . ., a + {7i-l){b - a)/n, b
are obviously in order of magnitude, and the integer n may be
chosen as large as we please.
§ 31.] DedekiiicUs Thmry of Sections. Any arrangement of
all the rational numbers into two classes A and B, such that
every number in .4 is le.ss than every number in B, we may call
a section t 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 Schnilt.
7—2
100 THKOUY OF SECTIONS CH. XXV
section of R ; for wo may t;ike A to inclinlc all tlio rational
numljLT.s wliich are not greater than o, and D to inciiule tlie rest,
viz. all that iire greater than a. Conversely, if in the class A
thoro be a number a which is not exceeded by any of the others
in A, then the section may be rej^arded jw j,'enerated by o. The
same is true if in the class B there be a number a which is
not greater than any of the others in li ; for we mi^ht 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 ft in H is obviously impossible.
For a and ft must be dillercnt, since the two chusses A and JJ are
exhaustive and mutually exclusive ; but, if o and ft were different,
we could, since R is compact, iu.scrt numbers between them which
must belong either to A or to B \ .so that o and ft could not lie
greatest and least in their respective classes as snpjMxsed.
But it may happen that there is no greatest rational n\imber
in A, and no lea.st rational number in B. There is theti no
rational number which can be said to generate the section. Such
a section is called an «mpty or imttimial section. It is not
difficult to prove that, if m/n be any positive rational number
which is not the quotient of two integral square numbers, and A
denote all the rational numbers whose sipiares are less than mf»,
and B all tho.sc whose squares are greater than m/n, then the
section (A, B) is empty.
§ 32.] An ordered onefold which has no empty sections is
said to be coiitinwHis. It will be observetl that the onefold of
rational numbers is discontinuous although it is compact.
Starting with the discontinuous onefoM of nitional numbers
//, we construct another ouefohi iS by as.signing to every empty
or irrational section a symbol which we shall call by anticipation
a numlter, adding the adjective irnitional to show that it is not a
number in R. As the section and the number are coordinated,
we may use the synibol {A, B) to denote the number as well as
the section. We can also without contra<liction re-name all the
rational numbers by atbichiug to each the corresponding sectional
symbol.
Matundly wo detinu the number (.1, 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' aud moie besiiles ; and consequently B' coutains all the
numbers in B aud 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 R and B.
0 is the section iu which A consists of all the negative and
J5 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 - ^ 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 R is compact. It is also
continuous, i.e. every section iu S is generated by a number in S ;
for, if a, /3 be a classification of all the numbers (or sections) of <S'
such that every number in a is less than every number in /3, then
(o, ;8) determines a section in <S' of the most general kind. But,
if -4 contain all the rational sections in o and B all the rational
sections in fi, then {A, B) is a section in R, i.e. a number in S;
and it is obvious that every number in <S<(^, B) is a number in
a, and every number in S>{A, B) a number in ji. Hence (a, p)
corresponds to the number (^i, B), which is a number in &
§ 33.] Si/stemattc representation of a number, rational or
irrational. Consider any number defined by means of a section
{A , B) of the rational onefold R. We are supposed to have the
means, direct or indirect, of settling whether any ratiiuial number
belongs to the class A or to the class B. Suppose (^l, B) positive.
Consider the succession of positive integers 0, 1, 2, ... ; and
select the greatest of these which belongs to A, say <h- Then
fco = ao+l belongs to B. The two ration.al numbers «„, b^ de-
termine two sections in R between whicli there is a gap of
width 1. Within this gap the section {A, B) lies, i.e. <h<{A, B)
<bo.
Next divide the unit gap into ten parts by means of the
rational numbers a, + 1/10, Oo+2/lO, . . ., Oo + 9/10, and select
the greatest of tliese numbers, say a, =a„ +/j,/10, which belongs
to A ; then b^ = aj + 1/10 belongs to B. We Lave now a gap in
102 SYSTEMATIC RKI'RESENTATION OF A SECTION" CU. XXV
R of wiiltli 1/10, determined by tlie numbers a,, 6, within which
(A, li) lies.
Wc next divide the gap of 1/10 into ton parts by means of
the numbers Oj + 1/10*, «, + 2/10*, .... «, + 9/U>'; and so on.
Proceeding in this w.iy, we run detcriiiine two rational numbers
(termiualiii^ docinials in fact),
o, = o,+/),/10 + . . .+Pnl\0\ t, = a.+ l/lu' (I)
between wliich {A, li) lies, the width of the gap between a, and /»,
being 1/10". It is obvious that a,, a,, . . ., a, are a non-decreas-
ing 8ucces.sion of positive rational numbers; and it can easily
be proved that b^, b,, . . ., b, are a non-increasing succes-sion.
1". At any stage of the process it m.ay haiii>en tliat a, is the
gre.itest possible number in A, in other words that p,+,, and all
successive pa are zero. The .section (A, B) is then determined
by the number a, ; and (A, li) is the rafion:d number a,.
If the process does not stop in this way, two things may
happen.
2". The digits jw,, jt>„ . ..,/»,,... may form an endless
succession but rei)eat, say in the cycle /v, Prti, • ■ ■, P»- In this
case there exists a rational number a to which a^^a,-*- pJlO + . . .
+/>,/I0" approximates more and more closely as we incrca.se w ;
and, since 6, = a« + 1/10", A, also approaches the same limit It
follows that the rational numbers of class A might he defined as
the numbers none of which exceeds every numWr of the succession
ft,, «,,..., o,, however large u be taken. Hence, if we agree to
att-ich the number a to the class y|, it will be the greatest number
of that class, and the .section {A, li) is genoratod by a.
3°. The digits p,, ;»i, . . ., p» may form an endless non-
repeating succRssion. Since the gap 6, -a, --= 1/10" can be made
as small as we plen.se, it follows as before that the rational
numliers of clitss A may l)0 defined as all the rational numbers
none of which exceeds every number in the endless succession
0,, a,, . . ., a, This sl:it'Oment does not as in last case
cnabli! us to identify (A, li) with iuiy rational numl>or; hut, since
n m.ay be as large as wc ple^ase, we can by cnlrulating a Hiilhcient
niimlicr of tiie digitus j>,, p,, . . . separ.ito (A, Ji) from every otlier
§§ nS, S4 CONVERGENT SEQUENCES 108
uumber, 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
R, and therefore a rational number ; and every non-terminating
non-repeating decimal an irrational section in It, 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 <ir irrational, as we please.
§34.] Caritor's Theory. The rational numbers flo, «i, • • •,
ffi„, . . . in § 33 evidently possess the following property. Given
any positive rational number «, however small, we can always find
an integer v such that la„-a„+r|<e when m<|:v, r being any
positive integer whatever.
We are tlius naturally led to consider an infinite sequence of
rational numbers
Ui, lu, . . ., «„, ... (2)
which has the property that for every positive rational value of £,
however small, there is an integer v such that \ u„ - «„+,. | < « w/ien
w^;^, r beinj any positive integer wliatever.
Such a sequence is called a convergent sequence; and )/i, 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-dccrcasing or a non-increasing (monoclinic) scipu-nce.
To every convergent sequence corresponds a definite section of
the onefold of ra/ional numhers (E) : so that every suck sequence
defines a real number, rational or irrational.
We may prove this important theorem as follows.
Let e, be any ])ositive national nuudjcr whatever; then we can
find 1', such tliat, when H<t:i'i, | ?«„- «„+r |<ei. lu particular, we
shall have, if )«>i',, | w^,-«,„|<ei, whence
W,, -€i< i; ,„<«,,, + €i (2).
In other words, the two rational numbers «, -- «,., - «i , ^i = «.-, + «i
determine two sections in R such that :dl the numbers of the
101 CONVERGKNT SEQUENCES CH. XXV
se<]ncn(-c 2 on and after w,., lie in the gap of width 2(| bctwceu
those two sections.
Next choose any rational number <.<*,. We can then es-
tablish a gap of width Sc,, whopc boumiing sections are given by
0,= «r,- *j, ^j = Ml, + «j. The niiiidnT v, will in general be greater
than I', ; but it nii;;ht be less. Also the gap a.h, might partly
overlap tlie gap (ij>,. But, since all the convor^'ents on and
after m,, lie within the gap a-tli,, we can throw aside the part of
Ofl),, if any, that lies outside Oi^i, and detormiDe a number i','<^i'i
such that
when m-^v,. Then, all the convergents on and after u^ He
within the gap Hili^, whose width 1^ •.•«,<•-'€,. This process may
be repeated as often as we please; and the numbers tj, €,, . . .
may be made to decrease according to any law we like to choose.
The numbers Oi, (u, . . . form a non-decrc:ising and the numbers
bi, A,, ... a non-increasing sequence : and each successive gap
lies within the preceding, although it may be contcrminoua with
the ])receding at one of the two ends. Since «i, tj, . . . can bo
made as small a.s 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) uuiubers which
do not exceed every one of the numbers ii,„ »/«+,, . , . when m is
taken suthciently large, (B) numbers which exceed any of the
numbers u„, «m+i, • • • when m is taken sufficiently large.
Hence every convergent .sequence detennines a section of R ;
and therefore defines a numlwr, rati0n.1l or irrational
Conversely, as wo have seen in § 33, every number, rational or
irrational, may be defined by means of a convergent sequence. If
the sdinence is 11,, «,,...,«,,... we shall often denote both
the sequence and the corresponding numln'r by («,). Since it is
only the ultimate convergent^ that determine the section, it is
clear that we may omit any finite nund>er of terms from a con-
vergent sequenc c without all'ecting the uumber which it defines.
In particular, the sequences M,, u,, . . . tir, . . ., Vn, ■ ■ ■ luid
Mr, . . ., «„, . . . ilefine the same nniidjcr. It f^hould be notire<l
that in the c^u^e of nitinuul numbers the convuigenU on and alter
^1 34-3(J AIllTIIMETICITY OF IRRATIONAL ONEFOLD 105
a particular rauk may be all equal : in fact we may define any
rational number a by the sequence a, a, . . ., a, . . ., and call
it («).
Since each gap in the above process lies within all preceding
gaps, and the section in E which is finally determiued within
them all, we have, if v he suck that |7/„ -?<„+,!<£ when n-^v,
«„-£:}>(«,.):}>«■- + « (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, xu, . . ., u,,,
§ 35.] Null-sequence. If by taking n sufficiently great we
can make | m„ | less than any given positive quantity c, 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 the sequence ?<,, u«, ...,«„,...
corresponds to 0 ; and we call it a null-sequence.
§ 36.] Definition ofthefowr species for tlie generalised onefold
of real numhirs S.
If (m„) (r„) be any two numbers, rational or irrational, defined
by convergent sequences, it is ea.sy to prove that the sequences
(«n + v,.)i («»-»'"). ("nO, (un/vn), are convergent sequences*,
provided in the case of («„/i'„) that (y„) is not a null-sequence.
We may therefore define these to mean («„) + (vn), («„) - (vn),
(tin) X (^^n), («n) "^ (*n) respectively. For it is easy to verify that,
if we give these meanings to the sjTnbols +, -, x, -^ in connection
with the numbers («„) and (i'„), then the Fundamental Laws of
Algebra set forth in chap. i. § 28 ^TilI all be satisfied.
For exaniplet,
(«-.) - (««) + (v„) = («n - ■«•„) + («„), by definitions
= {\ii„-Vn] + Vn), by def.
= («„), by laws of operation for K
* The reasoning is iinioli the same as in § fi above.
t The pLaiii bracket ( ) is nppropriated to the definition of the number by
a sequence ; the ciooUeJ Li nckct Las reference to operations in li.
lOG AniTHMETICITV OF FRRATrONAI. ONEFOLD CH. XXV
Again,
(m.) X !(«'«) + ((''-)1 = (iin) X («•, + w'.), ^y dcf.
s (m» {". + if.}), by (lof.
= («««'• •'■M.w'ii), by laws nf operation for (//),
= («»«») + (".w",), by def.
= («»)(«•,) + (",.)(»,), by. Kf.
and so on.
In order that two nunibors («„) and («,) may be equal it is
formallij necessary and sufficient tliat (u,)-(»,) = 0, in other
words, that («,-t)„) = 0, that is, that u, -d,, ri,-t',, . . ., u,-t",,
. . . shall be a null-sei[ueucc. This from the point of view of
our exposition might also be deduced from the fact that («,) and
((•,) must correspond to the same section in R. We can also
readily show that all nuU-se.iuences are ecjual, as they ought to
l>e, since they all correspond to 0.
M'e have now shown that tiie onefold of real quantity {S)
built upon II by the introduction of irrational numbers is on
arithmetic manifold. The proof that 6' lias the property iii. of
§ 30 is so simple that it may be left to the reader. Heno^fortli,
then, we may operate with the numbers of iS e.XiU'tly a.s we do
with rational numbers.
S 37.] It is worthy of remark that the properties of the
rational onefold It can, by means of ai>priipriat« abstract defini-
tions, be estal)lished on a jiurcly aritiunetic.-d basis. It is not
even necc.*«ary to introduce tJic idea of measurement in terms of
a unit. The numbers may be reganled as ordinal ; and addition
ami subtraction, gre;itemess and Icisne.ss, &c. int4'riireted merely
as progress backwards and for^vards among objects in a row, which
are not nece.s.<<arily placed at equal or at any detenninatc distances
apart*.
Following the older m.ithematicians since Descartes, we have
in the earlier part of this work assumed that, if we choo,se any
point on a straight line a,s origin, every other point on it has for
* S«!, r<ii cxftiiipli>, linrkncMK nnd Alorlc;, Inlrodiirlion U> the Theory oj
Analytic Functiniu. (Mariiiillnn, 1m iH)
§§36-39 GENERAL CONVERGENT SKQUENCE lU7
its coordinate a definite real quantity : and conversely that every
real (juantity, rational or irratinnal, can be represented iu this way
by a delinite point. The latter part of this statement, viz. tliat
to every irrational number in general* tlicre corresponds a definite^
point on a straight lino, is regarded by the majority of recent
mathematicians who have 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.] GeneraUsatlnn of the notion of a Convergent Seqmnce.
It is now open to us to generalise our definition of a convergent
sequence by removing the restriction that £ and Wj, u,
«„,... 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 tlieorem that
ever;/ convergent sequence oi real numbers Ui, u^, . . ., u„, . . .
defines a real number {u„).
Also we can show that, if e be any real positive quantity,
however small, we can always determine v so that
U,n-e<{Un)<U,„ + e (4),
when ni'iv.
For we have merely, as in § 34, to determine v so that
|Uro-M„,+r|<«'<e, when ?«<}:i'.
Then we have
and therefore
«,»-« <(«„)<«». + «,
when OT<ti'.
§ 39.] General Definition of a Limit and Criterion for its
Existence.
Returning now to the point from which this discussion
started, tve define the limit of the infinite sequence of real
quantities
i(\, n. Un, . . . (2),
as a quantity u such that, if e be any real quantity however small.
• We do not speftl? nf spfoial irrntionaliticg, such as ,^/2, wliicli ari.se in
clcmmitmy geometrical consuucliona.
108 LIMIT OK A SEQUENCE Cll. XXV
then there exists alwat/s a positive int-egir v such that | «, - « | <€
w/n'ii H<^r. And we prove the following fundamental theorem.
The neri'.isurif and finllicient condition thut the seAimnc^, 2, luive
a finite definite limit is that it be a converijent sequ, nee; and titt
tiiiil/ is the real number whirh is then definetl hij the setjtwncs.
The condition is necessiiry ; for, if a limit u exist, then
I «, - U,+r 1 5 I tt, - « + U - U^^r I ,
>l«»-«l + |M»+r-«|.
Now, since « is the limit of tlie sequence, we can find v such
that |?<»-Ml<At when n-^v; and, it fortiori, |k,+, -m|<J€
when v-^v. Hence we can always find v so that |M,-«,^r|<«,
where € is any positive quantity as small as we choose. Hence S
is convergent.
Also tiie conrlition is sullicicnt In fact, we can show that
{u,^, the numlxT defined by the sequence when it is convergent,
satisfies tlie dilinition of a limit. Vnr, given c, we have seen that
we can find v t>o that
«„-«<((/.)<«„ + «
when m-^v : whence it follows that |«„-(«t,)|<« when w-f-
Moroovpr there cannot be more than one finite limit; for, if
there were two such, say u and e, we should have
|u-t>| = |«-«, + M,-«|,
>|«.-tt|+|H,-tj|.
But, since bntli u and v are limits we could, by sufiiciently
increasing n, make |f/, -«| and |f«, — r| each le.s.s tlian Jc, and
therefore | « - e | < <, i.e. as small as wo plojwe. Hence u aiid r
ciinnot be unocpial.
The reader will readily prove that, ifih, «!,..., «,, • . . b«
a non-decreasing (non-inrrt'osing) infinite setjiiettr.e, no numlwr qf
which is greater than (/ess than) the finit,- number I, then this
sequence has a finite limit not gnatir tlutn (not L'ss than) I.
§ 40.] Let us now consider any function of x, say f(j-), which
is well defined in the sense that, for all v.ilues of j- that have to
be considered, with the possible exception of a finite numl>er of
i.solated criliml values, the value of /(j-) is dekTuiined when the
value of J- is given. We define tlw limiting viilue, i, <fj(x) when
S;1; rV.l H CONDITION FOR EXISTENCE OF A MMIT I'lO
X is increased up to the value a, by the property that, w/ien any
positive quantity t is given, there exists a finite quantity ^<a such
that
\/{x)-l\<e
when i:!f>.r<a.
This obviously includes our former definition of a limiting
value ; and we may denote I hy L f{x)-
Let a,, «a, ...,«»,■• • 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 ai1^x<a. Then, if we consider the sequence th =/(at),
rh=f(a-^, • • •> ««=/(«»). • • •> the results of last paragraph
lead us at once to the following theorem.
The necessary and sufficient condition that L f{x) be finite
a=a-0
and d^'finite is that it be possible to find a finite quantity i<a
such that, ichen i^x<x'<a,
where e is any finite positive quantity however small.
The reader will easily formulate the corresponding proposition
regarding L f{x).
§ 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) when
x = a. For example, i (ar-l)/(a;- 1) = 2 ; but (a^ -!)/(»- 1)
1=1 iO
has no value when x=\.
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
fix) = L {sin xjl - sin 2aT/2 + . . . + (- 1 )""' sin nx/n},
n=oo
then it is shown m chap, xxix., § 40, that L f{.r) = + 7r/2,
I-tT-O
L /(a;) = -7r/2; whereas /(t) = 0.
no EXEKCISES VII CU. XXV
EXEIICIBEU VII.
Limitt.
Fiuit tho limiting valuei of tho followin); (anctions for tbo givuu valuui of
the variubles : —
(1.) (3ji + 2x' + 8xi)/(xi+ii + i4), 1 = 0, and 1=00.
(2.) (x<-x'-9i'+16x-4)/(x'-2x'-4x + 8), i=2.
(3.) log(x'-2x'-2j-3)-loK(i*-4x»+tr-3), x = 8.
(4.) {x-(n + l)i"+' + iix"+^l/(l-x)», x=l (11 nposilivB integer). (Eulcr,
liijr. GaU.)
(5.) {^(x-l)-(x-l)(/{.y{x-l)-v/(x:-l)}. x = l.
(G.) (x'"+"-a"'x»)/(x'^«-.i''n), x=a.
(7.) {(a + x)"'-(a-x)'»}/{(ci+x)"-(a-x)-}, x = 0.
(8.) {(x-'-l)P-{x»-l)n/{(x-l)''-(x-l)»}. x = l.
(x"-l)»-(x"-l)(x'-l) + (x«-l)'
* ' (x"-l)= + (x"'-l)(x»-l) + (x"-l)«' '
(10.) {u - v'(a' - *')}/*'. a^=0- (Euler. DiJ. Calc.)
(U.) {i/(.<+x)-</(.i-x)l/{^(« + x)-^(a-x)}. x = 0.
(12.) {(a' + <ix + x'-)'-(u«-ax + x')i}/{(a + x)4-(o-x)i}, x=0. (Eulor,
Z)!^. Calc.)
(13.) {(2a'x-x«)4-a(a'x)^}/{o-(<ix*)i(, x = a. (Gixeory, fixdnip/.-. in
Di/r. Calc.)
(14.) {a + ^/(2a>-2ax)-^/(2ux-i»)}/{a-x + ^(a>-x')}, x = a. (Eulcr,
Dif. Cale.)
(IS.) X - ^'(x* - !/'), wlicn X = 00 , 1/ = (» , but y'/x finite = 2p.
(l(i.) ix«(y-')/n (!/-«), x=!,=z.
(17.) 2x"'(!/»-i»)/2x<'(y«-i''), x=y = i = a,
(18.) nx»-'/(-r"-a")-l/(x-<i), x = <i.
(19.) 2'((i"»'-l), x = «. (20.) x"», i=x.
(21.) (l + l/x")'. x=». (22.) x^/(l+x»)', x = a).
(23.) (1 + 1/x)', x = 0. (24.) (l + l/x)»", x = oo.
(25.) xVH-'l', x=l. (26.) xM»"-H, ,= 1.
(27.) a«*/x, x = «. (2«.) (l..gx)"«, x = ».
(29.) OoR'W^ x = ao. (30.) log" x/Ior" x, x=t>.
(31.) <i'/(x), x = ao, where /(x) ia a rational fiiuutiun of r, and a a
constant.
(32.) ((ix»+tx»-'+ . . . )", x = ac. (Caucby.)
(83.) xiAi+«i««n x = 0.
(31.) {(x» + x + l)/(x'-x + l)}», z=a>.
(35.) {4("' + «^)}"'. * = 0.
(30.) {l + 2/^(x'+l))v'<«'*'i, x = oo. (LoDgchampi.)
§ 41 EXERCISES Vll 111
(38.) {l/(f»-l)}'/^ ;r = a>.
(39.) {l0g(l + x)}l"tll+*\ 35=0.
(10.) log (1 + «.T)/log (1 + 6x), j; = 0.
(41.) (c'^-«-^)/log(l + x), x = 0. (Eu\eT, Dif. Cidc.)
(12.) (4 IT -x) tan a;, x = ^. (13.) tun'' xjx, x = 0.
(U.) (l-sinx + cos3-)/(sinj: + co8x-l), x=J?r. {Eu\eT, iJiJf. Ciik.)
(•45.) Bin x/(l - x=/7r=), x = jr. (4G.) x {cos («/x) - 1}, x = co.
(47.) (sinx-sina)/(.i;-a), x = a. (18.) seox-taux, x = l7r.
(49.) (sin-'x-tan*x)/(l + oosx)(l-cosx)3, x=0.
(50.)* 6iuhx/x, x = 0. (51.) (cosh x - l)/x=, x = 0.
(52.) t;iuh-'x/x, x = 0. (53.) siu i/log (1 + x), x = 0.
(54.) sin X log X, x = 0. (55.) cos x log tan x, x=4?r.
(56.) log tan 7nx/log tan H.T, x = 0.
(.57.) (logsinmx-logx)/(logsm;ix-log.i-), x = 0.
(.58.) siux•l"^ x = 0. (59.) siux'""^ x = 0.
(CO.) (ainhx)'*"^ x = 0.
(CI.) {(x/a)6in(a/.T)}":"(m.c2), x = co.
(62.) (cosmx)""', x=0. (C3.) (cosm.!-)™"''"', .•;-=0.
(64.) (2 -x/o) '«"''■'/■-'', x = rt.
(C5.) log, (log, x)/ cos ^, x = c.
(CC.) Show that sin x cot («/.c) log (1 + t!in («/.t)) has no iletei niinato limit
when X =: cc .
(C7.) If l^'x stand for log„ (log„:r), l/x for log„ (log„(log„x)), etc., show
that L [1- {l^''xll^'>{x + l)}"']xl^xl,;\c . . .lj'x^m(l^c)i'. (SchlOmilch,
Alijehidische Analysis, chap, ii.)
(68.) Show that L S (a + «)'/»/(( = 1.
(C',1.) Show that L i; {(« + s)/h}" lies between t" ami t"+'.
(70.) Show that L 2 {(a + sc/K)/(a + c)}»isfinitoif a + cbcnuinLiically
greater than a, and that L S {(a + sc/H)/''}"is finite if a + cbe numerically
lees than a. »=" "=i
(71.) Trace the graph of y = (u'^- l)/x, when a>l, and when a-il.
(72.) Trace the graph of j/ = x''» for positive values of x ; and liud the
direction in which the graph approaches the origin.
* For the definition and elementary properties of the hyperbolic fuuctiona
ooshx, sinhx, tanh x, &c., see cha)). xxix. All that is really wanted here ia
CQshx—i^(c' + e~-'}, 6iuhx = ^{(r'- (,• •').
112 EXERCISES VII CH. XXV
(73.) Trace tlio craph of y = {I + \ Jr)' ; and find thu angle at which it
cro«K<'8 tht> a\i8 of y.
(74.) Find the orders uf the zero and iiilinity vuliies ofy when detorminod
u a function of z by the foUovini; equations* : —
(a) i(x»-(iy)'-y»=0. (Frost's CurF« Tracinp, § 155, Ex. 3.)
ifi) iV + '»V-**i^' + '»*V-<»*-«^'=0- (76., Ex. 7.)
(>) (x-l)!/« + (x»-l)y'-(x-2)'y + z(x-2)=0.
(73.) If u and v be funclioiiB of the inte^^ral .variable n determined by the
equations ti, = ii,_, + i„_,, r,=«,_,, bUow that t ''iJ''i,=(l*V6)/2. How
ou);ht the nnibii^uuus si^n to be Kc-ttled when Ug nml u, ore both ponilivo?
(76.) Show that
«-»(...)-(;)-("i-T'- ••(.-.)■(;)'•
ei.) sbo.u,.i;. |l»±!L<-J:aiJ^t*_"l)'".,,
,,-« I 1 . 2 . . . « I
(78.) Llog(l-i)logx = 0, when 1=0.
* Fui a general method for dealing with such problems, Bee 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 tiie 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, wliicli arc in the first instance Inid down for
chains of operations wlierein the number of links is finite. Tiuit
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 sunimands
and multiplicands.
§ 2.] One very simple case of an infinite series, namely, a
geometric series, has already been discussed in chap, xx., § 15.
Tiie fact that the geometric series can be summed considerably
simplifies the first of the two difficulties just mentioned*; never-
theless the leading features of tiie 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 csee.
* The second vraa not considered.
0. IL 8
1 14 CONVEKGENCY, UIVEUUENCy, OSCILLATION CH. XXVI
'I'lic consi'lcrHtion of tliu iiilinite geometric 8oric8 suggests
the I'olluwiiig deliuitioiis.
Consider a succession of finite real sumniands M, , m,, m,, . . .,
?/,, .... unliiniti^d in nuinbiT, foniicd accordiiif,' t-o a ^ivon law,
so tliat the rttli term ii, is a fniitc oue-volucd fiinctinn of n ; and
consider tlie successive sums
Si = Ui, /Sf, = U,+M,, <S', = u, + «j + 1/3,
(Sii = u, + «j + . . . + M, .
When « is increased more and more, one of three things must
hapiien : —
l.st. Sn may apprcxtcb a fixed finite quantity S in such a tniy
that by increasing n sujficiently we can make S^ differ from S by as
little as tee please ; that is, in the notation of last chajiler, L &', = 6'.
In this case tlie series
«1 + J«j + ttj + . . . + H, + . . .
is said to be convehoent, and to converge to the value S, which is
spoken of as the sum to infinity.
Example, l + r, + T+ • ■ • + .,,+ • • • IIcreS= L S, = 2.
- '* - «— •
2nd. /S', may increase with n in siirh a way that liy increasing
n sufficiently we can make the numerical value of .S, exceed any
quantity, however large; that is, L /S', = +«. In this case th«
aeries is said to be divkroent.
Example. 1+2 + 3+ . . . Here L S, = a).
3rd. IS„ may neither become injinil-e nor approach a definite
limit, but oscillate between a numbrr if finite values the selection
among which is determined by the integral character of w, that is,
by such considerations as whether n is odd or even ; of the form .'Im,
'6m + 1, 3w + 2, itc. In this rase the s^-ries is said to i>s«ULLatk.
N.B. If all the terms of the series have the same sign, then iS',
continually increases {or at least never decretu^s) in numerical value
as n increases: and the series cannot oscillate.
Kxample. 8 - 1 - 2 + ."1 - I - '.' + 3 - 1 - 2+ . . . IJiro L .S.^O, 3, or 8,
ftccirjiiitj an >i in o( Uie funu 3ui, !im + 1, or Urn + 2. ""
§§ 2, 3 CRITERION FOR CONVERGENCy 115
lu cases 2 and 3 the scries
Ui + 1U + Us+ . . . +?<„+. . .
is also said to be non-conven/ent*. In many important senses
iioii-convergent series cannot bo said to have a sum ; and it is
obvious that infinite series of tliis description cannot, except in
special cases, and under special precautions, be emploj-ed 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
U^ + 11-2+ U3+ . . . + ?<„ + . . .
involves the supposition that the terms are taken successively in
a given order. In other words, the sum to inlinity 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 wTitten.
§ 3.] Two essential conditions are involved in the definition
of a convergent series — 1st, that S„ shall not become infinite
for any value of 71, however great ; 2nd, that, as n increases,
there shall be continual approacli to a definite limit S. If we
introduce the S3-mbol m^„ to denote ?/„+, + !<„+,+ . . . +M„+m.
that is, the sum of m terms following the 71th, following Cauch)'
we may state the following criterion : —
T/ie necessary and sufficient condition for the convergence of a
series of real terms is that, by taking n sufftcieiitly great, it be
possible to make the absolute value of ^Jln as small as we please, no
matter what the value of m may be.
* Some writers use divergent as equivalent to non-convergent. On the
wLole, especially in elementary exposition, this practice is inconvenient.
8—2
llf) CRITERION FOR CONVEHOENCY CH. XXVI
This condition may bo amplified iiitx) tlie following; form.
(Jiven in a«.lvauce any positive ([uautity », however small, it nuist
be possible to assign an integer v such that for n = v and all
greater vabies lm/^»|<« : or it may be contracted into the form
L„Nn = 0 wlien n= -j: , for all values of m.
The condition is necessary ; for, by the definition of con-
vergency, we have L S^ = S, where <S is a finite definite quantity ;
therefore also, wliatevor m, L S„t„-S. Hence
that is, L „7?„ = 0.
It— •
Also the condition is suftu-ipnt : for, if we assign any positive
quantity «, it is possible to find a finite integer r such that, when
n-^ v,\ m/i„ I < «, that is | <S',+„ - /S„ | < «. In particular, therefore,
|<Si,t«-'S'.-|<«. Since S,, being the sum of a finite number of
finite terms, is finite, and ?« may have any value we please, it
follows that for no value of n exceeding v can S, become infinite.
Hence L S„ cannot be infinite.
Also the limit of <S', cannot have one finite value when n has
any particular inti-gral character, and another value when n has
a dilTerent integral character ; for any such result would involve
that for certain values of m L S, and L <*>'„,„ should have
different values ; but this cnnnot be the case, since for all values
n— to il-«
It should be noticed that, when all the terms of a series have
the .sjune sign, there is no possibility of o.scillation ; and the
condition that <S', be finite for all values of « however great
is .sufficient In ciise the subtlety of Cauchy's single criterion
should puMlc the beginner, he should notice that the proof which
shows that L„ 11^ = 0 can usually be readily modified so as to
show that LS^ is not infinite. In fact some of our earlier
• A more riRoroiis deinonslrnlion of tlio nKovo criterion in obtAinod
by npplyiDK the rosult of §39, chap. xiv. to the BO<nicnc«> .S, , .S',. . . .,
S , . . . Wo Imrp K>vcu tho above dcmoniitriitioD fur the Mtko of roadan
who have uul lUiutoruJ Iho Xiioury ijivvu iu chap. xxv. , §j 2S— 10,
§3 RESIDUE AND PARTIAL RESIDUE 117
demonstrations are purposely made redundant, by proving both
i'mlin = 0, and LS'n not iufiuite.
Cor. 1. In any convergent swies L k,i = 0.
For ?<„ = )S'„-jS„_i = ii?„_i, and, by tlie criterion for con-
vergency, we must have L Ji„.i = 0. This condition, altliough
necessary, is not of itself sufficient, as will presently appear in
many examples.
Cor. 2. 1/ Rn= L mRn, and S and /b'„ kave tlie meanings
above assigned to them, then Sn = S- Rn.
For Sn+m^'Sn + mRft, therefore L <S'„+„, = <S'„ + L mR<i', aud
L S„+m = S, hence the theorem.
Rn is usually called the residue of the series, and „,7?„ a
partial residue.
Obviously, the smaller R„/Sn is for a ^iven value of n, the
more convergent is the series : for R,, is the ditlerence between
Sn and the limit of S„ when n is infinitely great.
Rn is, of course, the sum of the infinite series
Mn+l + «n+2 + Un+3 + . ■ .]
and it is an obvious remark that the residue of a convergent series
is itael/a convergent series.
Cor. 3. Tlie 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 S„ 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. Consiclor the series 1/1 + 1/2 + 1/3+ . . . +I/1+ . . .
Here„i?„=l/(n + l) + l/(«+2)+ . . . +l/(„ + m),
>l/(n + 7n) + l/(n + m)+ . . . +l/(n + m),
>ll(nlm + l).
Now, however great n may be, we can always choose m so much greater that
n/m shall be less than any quantity, however small. Hence we cannot cause
^R, to vani.sh for all values of m by sufficiently increasing n. We therefore
conclude that ihe series is not convergent; hence since all the terms are
•+-H
118 EXAMPLES Cn. XXVI
positive it mufit diverge, iiotwitlittandiiiR the fact that the temii ultimately
become intiuitvly email. We chall give beluw a direct proof that /^,= s .
Example 2.
1, V 1, 3» 1, (n + 1)'
i'°8l .3 + 2'°^— 4+ • • • +n''"'Mi^^)•
Smee(»+l)Xn+2)=(l+l/n)/^l+l/(n+l)}. wehave
» -_LiocLLH.(^»)+_L i„-1->-1/("^2)
"^~« + l^l + l/(n + 2) 11 + 2^1 + 1/(11 + 3)
1 , l + l/(n + iit)
i + m "^l+lMn + m+l)*
1 I, l + l/(n + l) , l + l/(« + 2) . l + l/(n + ni) 1
"=„Ti h'i+ii(nT2i+'°«iTi?(irr3)+ • ■ ■^'°8i-+i7(.m,rTTy[ ■
1 , i+i/(«+i) ,,^
*nTi'°«rT-i/(™T.;rfT) <»>•
Now, whatever m may be, by making n large cDough we can make l/(n + l),
and, a fortiori, l/(n + in + l), as email as we please, thcrerore L „R, = 0 (or
all values of m. *~*
If in (1) we put 0 in place of n, and n in place of m, and observe that
1 + 1/1
^■"'°gl + l/(n + l)'
BO that S. can never exceed log 2 whatever n may be.
Both conditions of conver;.'incy are therefore satisfied.
Pntting in = aci in (1), we find for the residue of the series
«.<[log{l + l/(n + l)}]/(n + l);
a result which would enable us to estimate the rapidity of the convci^cncy,
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 la.st p.nragrap]i. Some of
them will be found very u.«cful in discussing questions regarding
convergence. We sh.ill use 2«, as an abbreviation for «*i + u,
+ . . . + u, + . . . , that is, " the series whoso nth term is «,."
I. If u. and V, be positive, tt,<v, /or all value« of n, and
2r, cnmrrffent, th' :t 2u, is convergent.
1/ M, and r, be positive, u,>i', /t all value* of n, and 2r,
divergent, then In, is divergent.
For, under the first set of conditions, the values of N, and
„/t, belonging to 2«, arc less than the values of the corre.<pond-
ing functions tS'„ and „/f, Wonging to 5r,. Hence we have
0<jS',<6"„ 0<»/^,<mA*',. But, by hyixithesis, 6", is finite for
§§:>,!• ELEMENTARY COMPARISON THEOREMS 119
all values of n, and L mli'n = 0 ; hence Sn is finite for all values
of «, and L ,„Bn = 0 ; tiiat is, 2«„ is convergent.
n=ao
Under the second set of conditions, ;S'„ > (S"„. Heiiee>
since L (S"„ = co , we must also have L Sn=^; that is, 2m„ is
divergent.
II. If , for all values of n, Vn>0, and ?«„/v„ i:^ finite, t/ieti
2m„ is convergent if 2y„ is convergent, and divergent if 2v„ is
divergent.
B}' chap, x.xiv., § 5, if .4 be the least, and B the greatest of
the fractions, i<„+,/'«„+i, M„+2/»n+2, • • •, «n+ra/v,.+,„, then
'fn+l + '«n+2 + . . . + i'„+m
Now, since ?<„/»„ is finite for all value.s of n, A and B are
finite. Hence we must have in all cases „Jhi= C„Ji „, where C
is a finite quantity whatever values we assign to /« and «.
Hence S,, (that is, „/?o) will be finite or infinite according as
S'n is finite or infinite ; and if L mli'n = ^, '^^'^ must also
have L m.S„ = 0.
n=oo
III. If Un and Vn be jjositive, and if, for all values of n,
^n+il^n<'»n+il'Vn, «"'^ 2c„ is convergent, then 2i/„ is convergent ; and
if Un+i/Un>'Bn+i/Vn, and 2v„ is divergent, then Sms is divergent.
We have, if m„+i/«„ <■?;„+,/■«„,
( Ui U., th )
I Vi «a Vi
< — S'n.
Now, by hypothesis, Z*S"„ is finite : hence LSn must be finite.
Also, since all the terms of 2«„ are positive, the series cannot
oscillate, therefore 2«„ must be convergent.
In like manner, we can show that, if u„+i/un>v„+i/v„, and
2y„ be divergent, then 2«„ is divergent.
A\B. — In Theorems I., II., HI. we have, for siuii)licity,
stated that the conditions must hold for all values of n ; but
120 AnSOLUTE CONVEnOENCE CM. XXVI
wc see from § 3, Cor. 'A, tliat it is siifTuii'iit if tlipy liolii for all
values of 11 exceeding a certain Jinite value r ; for all the tonus up
to the rth in both series may be neglected.
Also, when all the terms of a series have the same sipn, 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 «„+,/«„ as the Ilatio of Con-
vergence of S«„. Thus we might express Theorem HI. as
follows : — Any series is convergent (divergent) if its ratio of
coijiver^nce is .always less (greater) than the ratio of convergence
of a cojivcrgent (divergent) series.
IV. If a series which contains negative terms be convergent
when all the mgative terms hai-e their sigtis changed, it will be
convergent as it stood originally.
For the effect of restoring the negative signs will bo to
liimini.sh the numerical value both of jS', and of ^„.
Dertnition. — A scries which is convergent irhen all its terms are
taJcen positively is said to be absolutkly converoknt.
It will be seen immediately that there are series who.se
convergency depends on the i)resence of negative signs, and
which become divergent when all the t<;rm8 are taken positively.
Such series are said to be semi-convergent. In §§ A and 6, unless
the contrary is indicated, we suppose any series of real terms to
consist of positive terms oidy, and convergence to mean absolute
convergence.
SPECIAL TESI^ OF CONVEROENCY FOR SERIE.S WHOSE TERMS
ARE ULTI.MATELV ALL POSITIVE.
§ 5.] If we tike for standard series a geometric progrcs-sion,
say 2r", which will be convergent or divergent arronling as
r< or >1, and apjily § -J, Th. 1., we see that 2m, will be con-
vergent if, on and after a certain finite value of n, tt„<r",
where r<\ ; divergent if, on and after a certain finite value of
H, H,>r", where r>\. Hence
I. 1u, M convergent or divergent according as m,"" it
ultimately less or greater than unity.
§§4,5 GEOMETRIC STANDARD 121
This te^t settles nothhuj in the case ivhere ?<„''" is iiltimaleljj
unit//, or w/cere L m„"" Jluctxiates between limits which include
unity.
Example. 21/(1 + 1/h)" is a convergent series ; for
L u„'"' = l/L(l + l/n)" = l/e,
by chap, xxv., § 13, where e > 2, and therefore 1/e < 1.
If, with the series Sr" for standard of comparison, we apply
§ 4, Th. III., we see that 2«„ is coiiveri^ent or divergent according
as Un+i/u„ is, on and after a certain finite vakie of n, always < 1
or always >1. Hence
II. 2m,i is convergent or divergent according as its ratio of
convergence/ is ultima telij < or>l.
Nothing is settled in the case where the ratio of convergenct/
is ultimatel;/ equal to 1, or where L Un+Jun fluctuates between
limits which include unity.
The examination of the ratio m„+i/m„ is the most useful of
all the tests of convergence*. It is sufficient for all the series
that occur in elementary mathematics, e.xcept 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
fundamentaUy the same when L ?«„+,/«„ is not indeterminate.
This will be readily seen by recalling the theorem of Cauchy, given
in chap, xxv., § 14, which shows that L «„+,/?*„= L «„"". 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 'Zii''x", where r and x are
oonstauts. We have in this case
"n+i/" » = (" + !)'' ^"■'"'/"''•"^"i
= (\ + llnYx.
Hence Lti^^Ju^ = x. The series is tlierefore couvergrat if x < 1, and divergent
ifi>l.
* We here use (as is often convenient) "convergence" to mean " the quality
of the series as regards couvurgtucy or divergency."
122 EXAMPLES CH. XXVI
It z = l, wo oannot settle the question hy means of the {ircscnt test.
Example 2. If <fi (n) be any algebraical fanction of n, Ztp (n) x* is con-
Tergent if i< 1, divergent if j > 1.
This hardly needs proof if L <p{n) be finite. If L 0(n) be infinite, we
know (see chap, xxx.) that ve can always find a positive value of r, suck
that L ^ («)/«'■ is finite, =A say. We therefore have
_ I »(n+l) / *(n)l (n + 1)'-
=x{AIA]xl,
This very general theorem includes, among other important oases, the
integro-geomctric scries
^(l)j + 0(2)i'+ . . . +0(n)x»+ . . .
where ^ (n) is an integral function of n ; and the series
X x* I"
j+_+. ..+_+.. . (1).
which, as we shall sec in chap, ixviii., represents (when it is oonvergenl)
-log(l -x). It follows, by § 4, Th. IV., that, since the series (1) is con-
vergent when x<I, the series
X X* X*
is also convergent when x< 1.
When (2) is convergent, it represents log(t-hx).
Example 3. 2Xc*/ii! (the Exponential Series) is convergent for all value*
orx.
= -r/('i + l).
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, zl(n -t- 1)<1. Since the limit
of the ratio of convergence is zero in this caw, we should ex|iect the con-
vergcncy for moderate values of x to be vcrj- rapid ; and thi8 is so, as wo
shall show by examining the residue in a later chapter. We have tuppotrd
X to bo poKJtive ; if x be negative the scricii is convergent a fortiori ; the
convergence is in fart absolute, § 4, Th. IV'.
Example 4. S ( - )* m (m - 1) . . . (m - n -f 1) x*/ti! (x positive), where m
has any real value*, is oonvergeut if x< 1, divergent if x> 1.
* If m were a positive integer, the series wonid terminate, and the
qnestioD of convergenoy would not arise.
§§ 5, 6 cauchy's condensation test 123
T. T t m-n
I'or Lu^+Jii„=-xL —^,
_ mill - I
Hence the theorem.
The series just examined is the expaiiiion of (l-z)*" when a!<l. It
follows, by § 4, Th. IV., that the series Xm{m-l) . . . (i»-71 + 1)x"/k!,
whose terms are ultimately alternately positive and negative, is convergent
if x«:l; this series is, as we shall see hereafter, the expansion of (l+i)"*
when a!<l.
g 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 2«„ be a series of positive terms which constantly
decrease in value from the tirst onwards. Without altering the
order of these, we may associate them in groups according to
some law. If Vi, v^_, . . . v^, ... be the 1st, 2nd, . . . with, ... of
these groups, the series 2«„. wiU contain all the terms of 2m„ ;
and it is obvious from the definition of convergency that 2«„
is convergent or divergent according as 2t',„ is convergent or
divergent ; we have in fact L 'S,Un= L Sv^. It is clear that the
convergency or divergency of S(',„ will be more apparent tiiaii
that of 2m„, because in 2i',„ we proceed by longer steps towards
the limit, the sum of n terms of tv^ being nearer the common
limit than the sum of n terms of 2m„. Finally, if 2u'„ be a new
convGr'^Gnt
series such that «'„5z^,„ then obviously 2m„ is j. J' . if 2y'„
. convergent
divergent
We shall first apply this process of reasoning to the following
case : —
Example. The series 1/1 + 1/2+ . . . +l/n+ . . . is divergent.
Arrange the given series in groups, the initial terras in which arc of the
following orders, 1, 2, 2-, . . . 2'", 2™+', . . . The numbers of terms in tlie
successive groups will be 2 - 1, 2* - 2, 23 - 2-', . . . 2'"+i - 2"', V^- - 2'"+>
respectively. Since the terms constantly decrease in value, if 2'"+' be the
greatest power of 2 which does not exceed n, then
124 CAUCUV'S CONDENSATION TEST CH. XXVI
„ 1 /I l\ /I 1 1 1\ /I 1 I \
^-^l + (-2 + 3) + (•2' + .5-^0 + 7)+ • • • +(2^ + ---. + 1 + • • -+2-— ,}
> 1 + (2' - 2) |, + (2' - -J') .],+ ...+ (J""' - 2") ._, j^, .
,11 1
, m
>l+2.
Ilciiw, by making n Euflicicntly grrat, we ran make S^ n't large aa wo pleaie.
The 8erie8 1/1 + 1/2 + 1/3+ . . . ie tlivrcrorc (li%'<.'ri.-viit. This might alio be
deduced from the inequality (6) of chap, xiv., § 25.
duchy's Conclcnsaticm Te^st, of which the example just
discussed is a particular case, is as follows : —
If f(n) be pomtive for all values of n, and corufantltf den-fOM
as n increases, then -f(n) is convergent or divenjcnt ncmrding
as ^(i''/{a') is convenjent or divenjent, where a is any positive
integer -^ 2.
The series ^/"(n) may he arranged as follows : —
[/(l)+. . .+/{„-!)]+ {/(a) +/(« + !)+. . .+/(a'-l)}
+ i/(a')+/(a'+l)+. . .+/(«'-!)}
+ {/(a")+/(a-+l)+. . .+/(<»-♦' -1)}
Hence, neglecting the finite number of terras in the square
brackets, we see that ^'(«) is convergent or divergent accord-
ing as
2 {/(«-)+/(«"+ 1) + . • .+/('•■"' -1)1 (I)
is convergent or divergent. Now, since /(a")>/(a" + 1)>. . .
>/{a""- 1 )>/((«"'•'), we liave
(o"+'-u")/(a-) >/(a")+/(a"'+ 1)+ . . . +/(«"*'- 1)
>(«-+' -a")/((r+'),
that is,
(a - l)a-/(rt-)> /•(«-) +/(a"' + 1) * . . . +/(a-+« - 1)
>{(a-l)/o}a-*'/(a"*')-
Hence, by § 4, Th. I., the series ( 1 ) is convergent if 2 (a - 1)
«"'/('*'") ><* convergent, divergent if 5 |(a- iVaja-^'y^a"*') is
§ 6 CRITERIA OF DE MORGAN AND RERTRAND 125
divergent. Now, by § 4, Tli. II., 2 («-!)«'"/(«"') is convergent
if 2a'"/(a"') is convergent, and 2 {(as- !)/«}«'"+'/(«"'+') is
divergent if 2a'"+'/(a"'+') is divergent ; and for our present
purpose 2(f"'/(a"') and 2a"'+'/(a'"+') are practically the sanio
series, say 2a7'(a''). Hence Cauchy's Theorem is establi-shed.
N.B. — It is obviously siij/icieiit that the function /(«) be
positive and cmistantly decrease for all values qf n greater tlum
a certain finite value r.
Cor. 1. Tlte theorem will still hold if a have any positive
value not less than 2*.
Let a lie between the positive integers b and 6 + 1, (i <t 2).
If SaVCa") be convergent, then L a''/(a")=0, thatis, L Tf(x)=0.
Hence, on and after some finite value of x, the function xf{x) will
begin to decrease constantly t as x increases. We must therefore
have (6 + l)"/{(6+l)"} <«"/(«"). on and after some finite value
of n. If, therefore, 2a'/(«") is convergent, afm-tiori, will 2 (6 + 1)"
/{(i+1)"} be convergent, and therefore, by Cauchy's Theorem,
2/(?i) will be convergent.
If ^a^fia") be divergent, xf{x) 1° may, or 2° may not decrease
as X increases.
In ease 1°, b'f{b") > a'f{a"). Hence the divergence of 2a'/(a")
involves the divergence of 2i/"/(i") ; and the divergence of 2/(?*)
follows by the main theorem.
In case 2°, the divergence of lf{n) is at once obvious ; for,
if L xf{x)=¥0, then ultimately xf{.t)>A, where A>0. Hence
f(x)>A/x. Now %A/n is divergent, since 21/?* is divergent;
therefore "Sfin) is divergent.
In what follows we shall use fX, e'x, . . . to denote a',
a"', . . ., a being any positive quantity <^2 ; and \a; \-x, . . .
Ix, Px,... to denote loga^:, loga(loga.T), . . . log.a;, loge(log,a,-), . . .,
where e is Napier's Base.
* Also if l<a<2, see Kohn, Grunerl's Arcldv, Bd. 67 (1882) and HUl,
Mess. Math., N. S., 307 (180G).
+ This assumes that xf(x) has not an infinite number of turning valneB;
so that we can take x so great that we are past the last tuiuing value, which
must be a maximum.
126 nilTERIA OK DE MORllAN AND BERTRAND CH. XXVI
Cor. 2. -/(ii) is contertjent or divfrgmt accordimj us
2c«e'« . . . i'^iij{i'ii) is convergent or divergent.
Tliis follows, for integral values of the base a, by ro|)eato«l
applicatiiiii of Catichy's Comleiisjitinn Test; ami, for nou-iiitei^J
values of a, by repeated applications of Cor. 1. Tiius %/{n) is
convergent or divergent according its 2««/(€n) is convergent or
divergent. Again, '^tn/(in) is convergent or divergent acconling
as icH€(tH)/{c(en)}, tliat is 2« «€'«/(«'■'«), is convergent or divergent;
and so on.
Cor. 3. 5/(h) is convergent or divergent according us the first
of tlie functions
T, = \f{x)lx,
T, = \{xf(x))/\x,
T, = \{xkrf(x)]/\'x,
Tr = \{xXxyx. . . X.'-'xf{x)\/X'-x,
trhlrh does not ranish ir/ien x = oo , fiiis a tiegalive or a jwsitir,^ limit.
By Cor. '2, V(") is convergent or divergent according :is
Scfirn . . . €';;/(«'«) is convergent or divergent.
Now the latter series is (by § 5, Th. I.) convergent or
divergent acconling iis
L {tncn . . . *■•;'/(«'«)}""< or >1 ;
that is, according as
L log.lcnc'H . . . <V('''»0}'*<>0;
that is, L logJcwf'H . . . c'-;i/"(«'«)}/"<>0.
If we put x = ^n, so that \x = €'''n, k*x = ^'*n, . . .
y~'x = €n, yx = n, and ;r=ao when M=ao, the condition for
convcrgency or divergency becomes
L XjxXxX'j: . . . y-'j-f{x)\/yx<>0 (1).
If, on the strength of Cor. 1, we tjiko e for the exponential
ba.sc, the condition may be written
L I'xM'x . . . l"'xf(x)\/l'x<>0 (2),
where all the logaritluus iuvolveil arc Napioriau logarithiua.
i^ 6 DE morgan's LOGAKITIIMIC «CALE 127
We could establish tlie criterion (2) witliout the iuterveutioii
of Cor. 1 by first establishing (1) for integral values of a,
and then using the tlimrem of chap, xxv., g 12, E.\:iniple 4,
that L k^xjl'-x = 1/la.
Cor. 4. Each of the sci'ies
21///'+" (1),
21/H{/»r+- (2),
21/k/h{P«}'+" (3),
tXlnlnl-n . . . /'•-'« {r»}'+'' (r+1),
is convergent if a>0, and dirergcnt i/a = or<0.
As the function nlnl-n . . . fn frequently occurs in what
follows, we shall denote it by Fr{n) ; so that P„(«) = w, i^i(«)="
nln, &c.
1st Prao/— Apply the criterion that 2/"(») is convergent or
divergent according as LI {Pr(.i-)/{.r)}/l''''a;<>0. In the pre-
sent case, f{.v) = l/Fr (.r) (f^f. Pleiice
= — a.
It follows that (r+1) is convergent if a>0, and divergent
if a<0. If a = 0, the (pie.stion is not decided. In this case,
we must use the test function one order higher, namely,
/ {l^.^ {■r)/{x)\ll'^"-x. Since f{x) = l/P^ (•»), we have
I {Pr« {a^/(x)Wx = I {l'^'w\ll^^"-x,
= 1>0.
Hence, when a = 0, (r + 1) is divergent.
2nd Proo/— By the direct application of Cauchy's Condensa-
tion Test, the convergence of (1) is the same as the convergence
of 2rt7(a")'+", tliat is, 2(l/rt")". Now the last series ia a geo-
metrical progression whose common ratio is l/«" ; it is tlierefore
convergent if a>0, and divergent if a = or <0. Hence (1) is
convergent if a>0, aTid divergent if a= or <0.
Again, the convergence of (2) is by Cauchy's rule the same
as the couvergenco of Sa'/a" {/«"}'+«, that is, 2l/(/a)'+»H'+° ;
128 DE morgan's LOCiAUlTllMIC SCALE CIL XXVI
anrl the conver],'enco of this liLst the same as that of 21/n'+".
Hence our theorem is proved for (2).
Let us now assume that the theorem holds up to the series
(r). We can then show tliat it holds for (r+ 1). In fact, the
convergence of (r+l) is the same as that of 2i<i"/a"^"/'a" , . .
/'-'a«{/'-o"}'+*, that is, 2l/(H/a)/(n/a) . . . r-'(«/a){/'-'(»/ci)}'+«.
First suppose o>0, and o>«. 'I'Ucn la>l, ula>n. Hence
l/{,ila)l{nla) . . . l'-' (nla) [l'-' (nla)]'*'
<\ji>hi . . . r-'nj/'-'n}'^'.
But, since o>0, ^l/Pr.j{n) {/*■"'«[• is convergent, a fortiori,
^l/Pr(n) {/"■«}• is convergent
Next suppose a^O, and 2<a<«. Tlien nla<n\ and, pro-
cee<liug as hefore, we prove SI//', (n) {/■■«[" more divergent than
the divergent series ^l/Fr-i(H) {/'"'hJ".
Logarithmic Scale of Convergency. — The series just discussed
are of great importance, inasmuch jis thoj' 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, «„ and «',, of the rth and (r+l)th series. We Imve
«'■/". = {'''"' "!*/{''"}'**. where o is very .small but >0, and
o' is very large.
If we put x = l'~^n, we may write L u'Ju,= L {j*^'+*Y
»-• »— •
irl'**'. Hence, however small a, so long as it is greater than 0,
and however large o', Lu'Jti^ = oo .
If we suppose the character of the logarithmic scale estab-
lished by means of the second demonstration given above, we
may, by comparing liu with the various series in the scale, and
using § 4, Th. I., obtain a fresh demonstration of the criterion
of Cor. 3. Wo leave the detaii.s as an exercise for the student
This is perhaps the best demonstration, because, apart from the
criterion itself, nothing is presupixjsed rcganling /(x), except
that it is positive nheu x is greater than a certain huite value.
§ t) DE MORGAN AND LEUTKAND'S SECOND CKlTEltlON 120
By following the same cour.sc, auJ using § 4, Th. III., \vc
can establish a new criterion for series whose ratio of con-
vergence is ultimately unity, as follows, where Px=f{x+ 1)1 f{x).
Cor. 5. If f{x) be always positive v-lien x exceeds a certain^
finite value, 'S.f{ii) is convenjent or dicenjeiit according as the first
of the following functions —
To=pi-i ;
T, = P(,(.c+l)p^-P„(a;);
T. = P,{x+l),>,-l\{x);
T, = p,_,(,i' + i)p,-iVi(.r);
which does not vanish when a; = oo has a negative or a positive limit.
Comparing 2/(«) with 2l/Pr(«){^''«l". we see that 5/(?i)
will be convergent if, for all values of x greater than a certain
finite value,
Px<Pr (^) {l'x]'^/Fr (X + 1) {/- {X + 1)1« (1),
where a>0.
Now (1) is equivalent to
Fr{x + l)p,-l\(x)<Pr{x) [{l^XJl^ix + 1)|« - l].
Also LPr {x) [{I'xll^ (x + 1)1' - 1]
= - LPr-, (./•) {r (.. + 1) - / X] . ^,. ^ — -^ . J7r.^.;^r(^^l)j_l .
= — Ixlxa- — a,
by chap, xxv., .^.^ 12 and 13.
Hence a sufficient condition for the convergency of 2/(?«) is
L {Pr {x +l)px- Pr (x)] < - a (a positive),
X=QO
<0.
lu like manner, the condition for divergency is shown to be
£ {Pr{x+1) i>j, - Pr (x)} > - a (a uegati ve),
X=OD
>0.
Example 1. Discuss tbe convergence of ^c~^~'P~ — ~'/"/ii''.
Here 2'„ = J {/(»)[/«,
1 + 1/2+. . ■ + l/» + Wn
~ n
Now, by cliap. xxv., § 13, Example 1,
1 + (i+1)(k>1 + 1/2t. . . + l/yi + ri»^rI/i + Un + l).
c. II. 9
130 EXAMPLES CU. XXVI
UcncoL7, = 0. Wo must tbcroforo examine r,. Now
T, = l{nf{»)\lln,
= -{1 + 1/2+. . . + l/n + (r-l)Iril/:n,
= -{1 + 1/2+. . . + l/nl/Jii-(r-l).
By chap, ixv., § 13, Kxamjile 2, L(l + l/2+. . . + l/n)/In = l. Ilcnoe
LT^= -l-r+l= -r. The giveu series is therefore convergent or divergent
according as r> or <0.
If r=0, Lr,=0, and /.7',=0. But wo have
T^ = l{nlnf{n)\IPn,
= l-{l + l/2 + . . . + l/n-tn}/Pn.
Now, wh(>n n is very large, the value of 1 + 1/2 + . . . + 1/n - In approaches
Enler's Constant. Hence X.7']=l:>0. In this case, therefore, the series
under divcussion is divergent.
Example 2. To discuss the convergence of the hypergcomctric series,
g./S a(a + l)./808 + l)
7« y{y + l).d(S + l) "^" • •
The general term of this series is
//n>-°(''-'-l) • • • (a + n-l)./i(^ + l) ■ ■ ■ (/3 + n-l)^
•'* ' 7(7 + 1) . . . (-y + n-lj.JCa + l) . . . (i + n-l)*^-
Tlio form of /(n) renders the application of the first form of criterion
somewhat troublcbume. Wo shall therefore use the second. We have
_(a + n){fi + n)
'^' {y + „){S + n)''
_(a + n){fi + n) _
^•-(7 + «)(« + «)' ^•
Lt, = x-1.
Hence the scries is convergent if x<l, divergent if x>l.
If z=l, Lr, = 0, and we have
_(H + l)(a + n)(j3 + fi)
^' (7 + n)(« + n) ""•
_ {a+p-y-S + l)n''+An + B ,
n' + Cn + D •
LT, = a + ^~y-i + l.
If, therefore, i = l, the hypcrgcometric series is oonvorgcot or divergent
according a«o + /S->-4 + l< or >0.
I(o + /S-7-« + l = 0, i;,r, = 0. But wo have
= [n{J(n + l)-/nl + (a + ^ + l){J(n + l)-/n} + {/»/(n + l) + Wri}/n
+ CHn + l)/fi'J/[ 1 + E/n + Fln'l
Hence, since Ln{f (.i + l)-/n} = l, t |((n + l) -Jn} =0. L/(n + l)/n'=0,
Llnln'=0 (<>0), &o., wo have
Z,T,= 1>0.
in this case, tlicrcfure, the scries is divergent.
^ 6 HYPERGEOMETKIC AND BINOMIAL SERIES 131
Example 3. Consider the series
m m{m-l) ,,„ "'("'-!) • ■ • jm-n + l) ,
^"r+~r:2~ +• • •+*"^) 1.2 . . .n + •••
This may be written
- m (-?«)(-;» + :) (-w)(-m + l) ■ . . (-ro + n-l)
^+^"*" 172 +■ • •+ 1.2 ... » +• • •
It is therefore a hypergeometrio series, in which a= -m, p=y, 3 = 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 = 1.
Example 4. Consider the series
m m(m-l) m{ni-l) . . . (m-n + \)
l + y + -y^2 +...+ 172 .. . „ +••• <^'-
In this series the terms are ultimately alternatively positive and negative
in sign. Hence the rules we have been using are not directly applicable.
1st. Let m be positive ; and let m - r be the first negative quantity among
m, m-1, m-2, . . . etc., then, neglecting all the terms of the series before
the (r + l)tb, we have to consider
m(m-l) ■ ■ . (m-r+1) i m-r (m-r)(m-r-l) 1
1.2...r r"^r + l+ (r + l)(r + 2) +•••]• H-
If we change the signs of the alternate terms of the series within brackets,
it becomes
, , r-m , (r-m)(r-m+l) ,
^+7Tr+ (r + l)(r + 2) +••• (^'■
Now (3) is a hypergeometrio series, in which a = r-m, fi = y, 5 = r + l,
x=l. Hence a + /3-7-a + l=r-7tt-(r + l) + l= -m<0. Therefore (3) is
convergent. Hence (2), and therefore (1), is absolutely convergent.
2nd. Let m be negative, = -/x say. The series (1) then becomes
■, ^ m(m + 1) I I i;.m(m+1) . . . (^ + 11-1) ,,
l"^ 1.2 • • -"^V ^^ 1.2 ... n ^ ''
Since /i is positive, the hypergeometrio series
1 a. ^ J M (m+JI) , , ;^(m + 1) ■ ■ • (iL+n-1)
■^l"^' 1.2 +■ • •+ 1.2 ...» ^- • ■ * '•
is divergent.
Hence (4) cannot bo absolutely convergent in the present case.
Since p„= - (/i + «)/{n + l), the terms will constantly increase in numerical
value if /ii>l. Hence the series cannot be even semi-convergent unless /u-cl.
If ytt be loss than 1, p„<:l, and the series will be semi-convergent provided
iu.=0.
Now log„„=21o3^ = Slogjl+^[.
Since Llog ^l-^(/x- l)/{n-H)}/{(^- l)/(;i-l-l)} = l (see chap, xxv., § 13),
the series 2;log{l-f(^- l)/(K-f 1)} and S (/i - l)/(n -»- 1) both diverge to an
infinity of the same sigu. But the latter series diverges to - oo or -I- oo ,
according as /i< or >1. Hence i«„ = 0 or oo , according as /i< or >1.
9—2
132 lllSTOBia*L NOTE CU. XWI
Ham the aarie* (1) isdirciiseDt if v^-I. «rmi-uimiiniwii if ^^l.
UokniouMj otiTlilM if ^=1. Heoee^ to toB ap, the aenet (1)
b afaaotatalj «aai«(sait, if O^B<-fx:
if -1<><0:
,if -1=-:
diiageBtiif -x<m<-1*.
SBKIBS WII06B TOUtS OATB PEXIOIMCALLT SBCCSUSG SHUTITB
EIGXS, OB OOXTACr A PESKHMC FACTOR SDCH MS StS m9.
§ 7.] Sefies vhich contain an infinh^ number of n^atiTe
tenus may or mar not be absolntelr oooTOgent. Tike Conner
class &lls under the cases alneadj discussed. We fropoae now
to give a few thecHems regariing the fauter daas of aeries, vbaae
conreqgencj depends on the distribatioD of negatiTe signs
throogbont the series.
The only cases of much practical importance are those — l^t,
where the infinity of negative signs has a periodic arrangement ;
* niticrictl Si*e. — If «• empt a aiiab« of teaiUeni thaanaa, pveo
cluc4r bj Wanng ia hit Jtfditatitmn Amalftic^, and OaaM ia kia graal
■^Beir OB the 'HrpapHmetiie Scnea, it nay be aaid that Gaacky «a* tha
faanfcr of the Bodem throty of eoamrsaat aariea ; aad noat of the gcBaial
|ai»i|ih« of the aabjeel woe pvea ia hia AfoaiA ^aaJgrtifwa umi ia
Am^fte dlffirifme, la his Extrtiea dt illhfmatifnrt, t. n. (1837), he ea«B
the ioUovine iBaagnl critaooa frmt which Boat of the highs cnteria have
: :— If, for lai^e laloea of s. /<■) be poaitiic aad deetaaae aa ■ i
3^(a) ii oooTBSea* i' ^ f ^(x)=0 (aartitnzy).
The aeeood step of the r<nlena «aa fiiat gi«an by Baahe, CTrlW§ •'•v.,
B.l.xiii.(l'^>. I>eMoi8aB.iahisI>>/<na«MlCaJfWu.liLS:3<ti»f. (ISS9I.
fint gave the LoeaiithBie Soala of FaaetiaaBl DiBenaoe. i
Lagahthsie Scale of Can*a;geac7 of Cor. 4. and atatad <
to. bat aot idmticai ia tooa *ith, ihoae of Coc. 3 aad Oor. S. CoarKawital
viitera. acterthdeaa. ahaoat iaratiafelT attiibala the vholc theofy to Bartiaad.
Bertnad. Liaw. J«ar. (ISU). fDOte* be Mo^saa, atatiac that he hal <
iadcyeodeatty |ian of Da Moc^a'a raaalta. Hia Miaanir ia wj i
heeaoae it ecataiaa a diaeaaaioo of vaiioo* toaaa of Ae flritariaaad I
tHBof thoreqaivaiaaea: we have iheiefuie attarhad hia aaaaa. aht with Da
MoacaaX to the two ln^rith»ir etitcna. Poaart. Lumr. Jma. (IMSy. pk««
alaBcataiy diiai'aii«iati<aia of Battnad'a tnraaaha : aad MalaaliB. Grwatrft
JrrUr (t$IC|.gaT« aa •
§§ C, 7 SEMI-CONVF.nr.ENTT SERIES l;}3
2iid, where tlio occurrence of negative signs is caused by tlio
presence in the nth. term of a factor, such as sin 7i9, whicli is a
periodic function of 7i.
lu the former case (whicli niiglit be regarded as a particuhxr^
insfcmce of the latter) we can always associate into a single term
every succession of positive terms and every succession of negative
terms. Since the recun'ence 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 grouping is really a ditl'ereut series from the original
one, because the sum of n terms of the original series does not
always correspond to the sum of in terms of the derived series.
The difference between the two sums will, however, never exceed
on the inequcolity of cliap. xxv., § i;5. Cor. 6, that 21/P,.(m + 7i) {/'■{;« + »)}<'
(where I'm is positive) is convergent or divergent, according as a< or «t 0; and
thence deduces Cor. 3. Paucker, Crclh'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 (18G0). Du Bois-Eeymond, Crelle'a
Jour., Bd. Lxxvr. (1873), gives an elegant general theory embracing all the
above oiiteria, and also those of Kummer, Crelle's Jour., xui. (1835). Abel
had shown that, however shghtly divergent -i(„ may be, it is ahv.ay.s possible
to find 7i, 72, . . ., 7„, . . . such that /^7„ = 0 and yet ^y„u„ shall be
divergent. Du Bois-Eeymond shows that, however slowly 2;'„ converge, we
can always find 7,, 72, . . .,7„, . . . such that Z,7„ = oo and 27„!i„ neverthe-
le.ss 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 tliat there is a domain of convcrgency on
whose borders the logarithmic criteria entirely fail — a point left doubtful by
bis predecessors. Finally, Kohn, Grunert's Archiv (1S82), continuing Du liois-
Eeymond's researches, gave a new criterion of a mixed character; and
Pringsheim (Mutli. 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 SKMI-CONVEUGENT SERIES CH. XXVI
the sum of a finite number of terms of the original scries ; and
this diflTercnce must vanish for n = oo , if the terms of the original
scries uitiiuateiy become iiiliuitcly small.
Einmple. Consider tho series
1 11
(1).
1 2 3^4 6 0^
3ii-2 8b-1
1
pare this vilh t)ic series
1 /I 1\ 1 /I 1\
i-U + 3)n-(5+6)+-
^3n-2 \-in-
i+i)
• (2),
that is, the scries whose (2n - l)th term is l/(3n - 2), and whose (2n)l'i term
is -(l/(3ii-l) + l/3n).
If S, S^' denote the siims of n terms of (1) and (2) respectively, then
•^3,-«=-V-i. ■S«-i = 'V-i-l/(3"-l). •> = Sj,'. Since Ll/(3»-l) = 0, we
have in all cases Z..S', = LS,'. Hence (I) is convergent or divergent according
as (2) is convergent or divergent. That (1) is really divergent may be shown
by comparing it with the scries
2{l/(3n-2)-l/(3.i-l)-l/3n| (3).
If .*>„" denote the snm of n terms of this lost series, we can show as before
that T.S," = LS,. But the nth term of (3) can bo written in the form
( - 9 + 12/11 - 2/n»)/(3 - 2/h) (3 - l/n) 3n ; and therefore bears to the nth terra of
21/h a ratio which is never infinite. But 2^1/n 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 seri^,
where Lu^ + 0, the grouping of terms may convert a non-convergent
into a convergent series; so that ire cannot in this case infer the
convergiuci) of the original from tho cvitcergency of ihc lUrirt^d
series*.
Example.
is obviously a non-convergent oscillating scries. But
i(" ;)•■(■ •;)M(-:)'-(-n)i*--!(-.9'-
whose nth tennis (8n' + Rn + l)/(lH»+2n)», i.e. (8 + 8/n + l/M»)/lfi(l + 1/2b)V,
is convergent, being comparable in tho scale of conTergeucy with Zl/n*.
• This remark i" all the more important because the converse prociMis of
splitting up the nth term of a scries into a group of terms with alteniatinx
signs, and using the rulen of § 8, often gives a simple nieiuis of deciding ai to
itsconvergeucy. The series 1/1.2-1 1/3. J t I/ii .C-f 1/7.8 -t- . . . may be lastad
in tlii* way.
^7-9 III— u.,+ u^ — Ui+ . . . 135
§ 8.] The following rule is frequently of use in the discus-
sion of semi-converging series : —
Ifui>Un>U3>. . . >«„>... and all be positive, then
U1-U.+ U3-. . . (-)"-'«„ + (-)"«n+l + . . • (1) "
converges or oscillates accoi-ding as i m„ = or 4= 0.
Using the notation of § 3, we have
„.^'» = ± {I'n+l - «,.+!! + . . . ± lln+m),
= ±{Mn+l-(«B+2-Mn+s)-- • •}.
= ±{(«,i+l- «n+2) + ("u+S-«n+4)+- • •}•
Hence we have
M„+j>„,72„>M„+l-M„+2 (2),
numerical values being alone in question. If, therefore, Zi/„ = 0,
we have Lun+i = ii<,.+3 = 0 ; and it follows that L „/t„ = 0 for all
n=oD
values of m. Also
Ui>„Ro = S„>2h-th,
so that <S'n is finite for all values of n. The series (1) is there-
fore convergent if Lu„ = 0.
If Lu„ = a*0, then L „ff„ = a or =0 according as m is odd
or even. Hence the series is not convergent. We have, in fact,
LiS^+i-Si„) = Lu,n+i = <^, ^Thich shows that the sum of the
series oscillates between S and S + a, where S=LSia-
Cor. The series
(2*1 - U.) + {ih - «4) + • • • + («M-1 - "2") + • • •
where u^ "n, • • • are as be/ore, is convergent.
Example 1. The scries S ( - l)"-'/n is convergent, notwithetanding the
fact, already proved, that 21/n is divergent.
Example 2. 2(- 1)""' (« + !)/« is an oscUlating series; but 2(-l)''-'
{(n + l)/n - (n + 2)/(n + 1)} is convergent.
§ 9.] The most important case of periodic series is 2«„cos
{n6 + <^), where a„ is a function of n, 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.
in(T ABFF.'S nfEQPALITY CH. XXVI
We observe in the first place that
I. If 2»», hi an absolutely converging neries thm 1a^rai{n0+4>)
w contvrgiiit.
Tliis follows from § 4, I.
II. If 6-0 or 2Xt (/• hfing an integer), 1ft^cns{n6 + if)) m
convergent or dirergnit according as ^, is ct>nvrrgtnt or divergiiit.
This is olivious, since tlie series reduces to Sa, cos </>.
III. If 6^0 or Ih-rr, then 1a^ cos (n6 + <^) is ronrerrieiit if. fur
all raliifS of n greater than a certain finite value, a, //«.< the stinw
sign and never increases as n increases, and if i a, = 0.
■—•
This is a particular ca.>ie of the following general theorem,
which is founded on an inequality ffiven by Abel : —
IV. If^ii„l>ecoinYrgnitorimillal<rri/,nndaj, «,, . . .,a„ . . .
be a series qf positive quantities, which never increase as n increases,
and if Z a, = 0, then Sa.M, is convergent.
Almfs Inequality is as follows : — If, for all values of n,
yl > «, + tt, + . . .■*- u^>B,
where Hi, h,, ...,»», are any real quantities whatever, and
if O], A,, . . ., a. be a series of positive quantities which never
increase as n increa.ses, then
OiA >a,K, + a,M, + . . . + a,ii„>a-,D.
This may be proved as follows: — Let 5, = «, + «, + . . . + »/.,
»Si' = a,i/, + ff;sH, + . . .+a,f/,. Then «, = <S'i, u, S^-zS,, &c. ;
and
<S.' = o,.Sr, + a, (S, - S.) + . . . + a. (S, - 5,.,),
= S,{ai-aj) + S,(a,-at) + . . .+S,.,{a^-,-a,)^■f!,n,.
Hence, since Si, S, .*>', are each < A and > li, and (a, - oj,
(oi-Ot), . . ., (a,_, -<T,), a, are all po.sitivc or zero,
{(rt,-«,) + («5-'»i) + . • . + ('/,-,- a,) + a,M
>iS','>{(ai-ai) + («»-a3) + - • . + (". i -«,) + ".1 /J :
that i.'*,
a,A>S,'>a,/J (1).
Tlienrriu IV. follows at oiico, for, since iu, is not divergent,
§§ 9, 10 TRir.OXOMETRICAL SERIES 137
Sn is not infinite for :my value of n. Hence, by (1), S^' is not
infinite. Also, by Abel's Inequality,
= Sn'+m-Sn'>an+tD (2),
where 0 and D are the greatest and least of the values of
„Rn (= Un+i + «„+= + . . . + M„-Hn = S„+n, - S„) for all different
positive values of ni. Now, since lii„ is convergent or oscillatory,
S„+m - Sn is either zero or finite, and L a„+i = 0, by b}-po-
n=«>
thesis. Therefore, it follows fi'om (2), that imK„' = 0 for all
values of m. Hence 2rt„«„ is convergent.
We shall prove in a later chapter tliat, when
u„ = cos {n6 + <^),
,S'„ = sin hie cos {h (n + 1)6 + <f>}lsm i^.
If, therefore, we exclude the cases where 6 = 0 or 2Z;r, we see
that S„ cannot be infinite. Theorem III. is thus seen to be a
particular case of Theorem IV.
Cor. Ifa„ be as uhove, 2(- !)"-'«„ cos (iiO + ^), Sa„ sin (h0+ </>),
avd 2 ( - l)"~Vt„ Sin {n9 + </>) are all convergent.
CONVEEGENCE OF A SERIES OF COMPLEX TERMS.
§ 10.] If the ?(th term of a series be of the form Xn + yJ,
where i is the imaginary unit, and .t„ and y„ are functions of /(,
we may write the sum of n terms in the form >S„ + TJ, where
Sn = 0Ci + X. + . . . + .r„,
T„ = y,+2/.2 + . . .+%■
By the sum of the infinite series 2 {.r„+yj) is meaut the limit
when M = 00 of ,S„ + T„i ; that is, (LS„) + (LTn) l
The vecessary and sufficient condition for the convergency of
^(.Xn + y„i) i erefore that 2.t„ and ly„ be both convergent.
For, if the series 2a:„ and ly„ converge to the values S and
T respectively, %(.r„ + yj) will converge to the value <S'+ Ti;
and, if either of the series 2.r„, 2_y„ diverge or oscillate, then
(Z/.S'„) -t {LT„) i will not have a finite definite value.
138 CONVKKCJENCE OK COMl'I.KX SKIIIF-S CH. XXVI
§ II.] Lot £, denote x, + y,i ; and let |s, | bo the modulus
of z*\ so that |«,i' = |x,|» + |y,|'. We have tlio following
theorems t, which arc sulBcient for most elementary purposes : —
I. The complex series 2r, is convergent \f the real series 2 1 s, |
is convergent.
For, since 2 1 s, ] is convergent, and | tr^ \ and | y, | are each less
than |«„|, it follows from § 4, I., that 2|a", | and 2|y, | are both
convergent ; that is, 5.r, and 2y, are both ccmvergent Hence,
by § 10, 2;:, is convergent
It should be noticed that the condition thus established,
although sujjicient, is not necessary. I-'or example, the scries
(l-t)/l-(l-i')/2 + (l-t)/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, ^/2/l + J'2/2 + J^/S + . . .,
is divergent.
When 2c, is such that 2 | s, | is conrcrgnif, 2r, i.i said to be
absolutely convergent. Since the modulus of a real ([uantity m, is
simply M„ 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. ]fw,Rn denote z^^i + «,+, . . . + z,+„, then the necessary
and sufficient condition that the compUjr series 2;, converge is that
it be possible, by taking n sufficiently great, to niuke ImB-J^ as small
as toe please, %chatever the value of m.
Cor. 2. If \^be real or complex, and z^ a complex; number
whose modulus is not infinite for any value of n, hoicever great, then
2(\ii,) will be aiisolutely convergent if 2\, is absolutely conivrgent.
For I X,2. 1 = I \i 1 1 2. 1 ; and, since 2X. is absolutely con-
vergent, 2 1 X, I is convergent Hence, since | s, | is always
finite, 2 1 A, 1 1 r, I is convergent by § 4, II. ; that is, 2 | A,r, | is
convergent Hence 2 (X„£,) is absolutely convergent
Riainpio 1. Tho acrica Zt'jnl ia absolutely convergent for all finite
values of I.
Example 2. Tlio norieii 2:<*/n ia absolutely ooDvcrgeut proviilocl | < | < 1.
* Soo chap, zn., { 13.
t Caucliy, lUium/i Annlyliqutt, § xiT,
§§ 11, 12 LAW OF ASSOCIATION FOR SERIES 139
Example 3. The series S (cos 0 + i sin fl)"/ft is convert;ent if 6 + 0 on 2kir.
For the series S cos nOjn and S sin nOjn are convergent by § 9, III.
Example 4. The series (cos S + i sin d)"jn^ is absohitely convergent. For
the series of moduli is 21/ii^, which is convergent.
II. Let n he the fixed limit or the greatest of the limits* to
lo/iirh |s„|"" tends when n is increased indefiniteli/, or a fixed limit
to which \Sn+-Jz„\ tends when n is increased indefinitely ; then the
series 2j:„ will be convergent if Q,<1 and divergent ifVi>l.
For, if 0<1, tlieu, by §5, I. and II., the series 2|jr„| is
convergent; and therefore, by § 11, I., 22;„ is convergent.
If n > 1 , tlien either some or all of the terms of the series
5 I Sa I ultimately increase without limit. In any case, it will be
possible to find values of n for which | 2,, | exceeds any value
however great ; and, since | »» | = (| -^n I" + | »/n |')"^, the same must
be true of one at least of \xn\ and \yn\- Hence one at least of
the series 2a;„, 2y„ must diverge ; and consequently 2 (.r„ + yj),
i.e. 23„, 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 theend of § 7.
It can, however, be applied without limitation provided the series
is convergent. For let jS'^' denote the sum of m terms of the new
series obtained by associating the terms of the original series into
groups in any way whatever. Then, if (S'„ denote the sum of n
terms of the original series, we can always assume m so gi-eat that
Sm includes at least all the terms in S,^. Hence 6'm' — Sn = plint
where p is a certain positive integer. Now, since the original
* It will be noticed that this includes the case where L |«„i''" baa
different values according to the integral character of n : but the corre-
gpondmg case where L \z„+Jz„\ oscillates is not included. We have
n— «
retained Cauchy'n original enunciation ; but it is easy to see that some
additions might be made to the theorem ui the latter case.
140 I.WV OK rOMMUTATinV FOR SERIES CH. XXVI
scries is conv('r},'ent, hy UikiiiK ii siiflicii'iitly liir;;i; wc can make pit,
as small as we pleJise. It follows therefore that L S„' =^^ L S,.
Hence the association of terms produces nn effect on the sum of the
infinite convergent series.
§ 13.] Jmw of Commutation. — The hiw of coiMinutation is even
more restricted in its application than the law of association.
We may however prove that the law of commutation can be
applied to absolufeli/ convergent series.
We shall consider here merely the cuso where each term of
the series is di.splaccd a finite numher of steps*. Let 2ii, be
the original series, 2)/„' the new series obtained by commnta-
tion of the terms of 2«,. Since each term is only displa«'eil by
a finite niiniher of steps, we can, whatever n may be, by taking
m sufficiently great always secure that &'„' contains all the
terms of (S„ at least. Under these circumstances SJ-S^ con-
tains fewer terms than ,,f{„, where p is finite, since m is finite.
Now, since 2m„ is absolutely convergent, even if we take the
most unfavouralde case and supjiose all the terras of tiie same
sign, we shall have L p//, = 0 ; and, a fortiori, L SJ - L .S', = 0.
Hence L S^ = LS^; which establishes our theorem.
7'he above reasoning trould not apply to a semi-convergent series
because the vanishing of L y/'„ does not dejjend .solely on the
individual magnitude of the terms, but partially on the alterna-
tion of positive and negative .signs.
Cauchy, in his IW.inme's Aunli/tiijues, § vii. (18.33), seems to
h.ave been the first to call explicit attention to the fact (hat the
c<mvergence of a .semi-convergent series is es.senti;dly dependent on
the order of its lenns. Dirichlet and Ohm gave e.xamples of the
effect of the orilcr of the tvrnis upun the sum.
Fin.ally Kieniann, in his famous memoir on Fourier's Soriest,
showed that the .scries S (- 1 )"''«, , where A»/, = 0, nn<l Sm^^ , and
lutn arc both divergent, can, by proper commiitation of its tonns,
• 8oo below, ; 3.3, Cor. 2.
\ Written ill iM.'il and puliliihol in lur>7. Sec liii OttammeUe Math
Werkf, p. 21 L
§§ 12, L'5 LAW OF COMMUTATIUN KOR SERIES 141
be made to converge to any sum \vc please ; and Dirifhlet has
shown tliat commutation may render a semi-convergent series
divergent.
When the sum of au infinite series is independent of the order"
of its terms it is said to converge itnmnditionally. It is obvious
from what has been said that unconditional convergence and
absolute convergence are practically synonymous.
Example 1. The series
J^ J^ J^ 1_ ,__J 1
Vl~^/2"'■^/3 ^i'^ ' ' '"''^/(■J/i- 1) ~ VP'j ' * * '
is convergent by § 8 ; but the serios
"*■ W(4m + l)"*'V(lm + 3)~V(2"t+2)j'^" * * ^^''
Trhieh is evidently derivable from (1) by commutation (and an association
which is permissible since the terms ultimately vanish), is divergent. For,
if «„ = l/^/(4m + l) + iy(4m + 3) -l/V(2m + 2), and r„ = l/^m, then
LuJ,;, = L {1/^(4 + 1/m) + 1/^(4 + 3/m) - 1/^/(2 + 2/m)}= 1/2 + 1/2 - 1/^2 =
1 - j^2. Hence «„Ji',„ is always finite ; and i;ii„ is divergent, by § G, Cor. 4.
Hence 2h,„ is divergent. (Dirichlet.)
Example 2. The scries
11111 1 1 ,^,
I~2 + 3"4"*"5"- • •■*'(2«-l) (2") * ''
(i+§)-^ + (i + y -5+- • • + Csrn+s;rf3)-2„.V2-^- • ■ '-'•
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 117G.
Addition of two infinite series. Jj' 2?<„ and 2d„ he loth con-
vergent, and converge to the values S and T respectively, then
2(m„ + i?„) is convergent and converges to the value S+ T.
We may, to secure complete generality, suppose •«„ and Vn to
be complex quantities. Let Sn, 7'n, U^ represent the sums of
n terms of 5w„, 2i'„, 2 («„ + v„) respectively ; then we have, how-
ever great n may be, U^ - ^n + T„. Hence, when « = oc ,
LUu = LSn + LT„, which proves the proposition.
142 LAW (IK DISTUIBUTION FOR SERIES Cll. XXVI
§ 14.] Imw of Distribution. — The application of the hiw of
distribution will he imlicateil hy the followiiifj theorems : —
Jf a be aril/ Jinite quautiti/, and 5h„ anurnje to the value S,
tlwn Saw, converges to aS.
The proof of this is so siinplo tliat it may be kfi to the
reader.
// 2tt« and 2j', converge to the values S and T respectively, and
at least one of tlie two series be absolutely convergent, then the series
H,f, + (Mil', + «.^',) + . . . + («iV, + «.jC,_, + . . . + «,ri) + . . . (1)
converges to the value ST*.
Let S^, Tn, Un denote the Ruins of n terms of 2«„ 2t',,
2(«it'» + «-jV»-i + . • . + M,t"i) respectively; and let us suppose that
2»» is absolutely convergent. We have
S,T,= U, + L„
where /.» = u.ji', + Ma»',-i + . . . + ",.»"»
+ Hj('„ + . . . + u,r,
= UiVn + «, (v„ + f.-i) + . . . + «/„(€■, + . . . + f,) (2).
If therefore « be even, = '2m say,
+ [«m+l ('"sm + • • ■+V„ ,.,) + . . . + «»(fsm + . • • + t^)] (3).
If n bo odd, ^ 2»i + 1 say,
2/« = [«j«'j».+i + «.(t'sm+i + n-..) + . . . + a«(tv„+, + . . . + tv,j)J
+ [«m + I (»»+! + • . .+»'~*j) + . . . + «»+l (t':«+l + . . .+t',)] (4).
Now, since 2r, is converjjent, it is po.ssible, by making m
sufiiciently great, to make each of the quantities |fiu,|, |«i.i-i+tij«|,
• . .,|Vm+> + . ■ .+t?2,„|, |f,„+,|, |t'» + t)»„|, . . .,!».+, + . . .
* Tbo oriKinal dcmoiiRtration of this tbeorom givi-n by Caachy in hii
Analyte Algibrique n-iniirod that both tho scrice -u,, Hr, bo Bbsolutrly con-
TorKont. Abi I'll dpinmiRtration in subject to the name restriction. The more
(:;«nenil form was givrn by McrtviiB, CreUr't Jour., i.xxix. (187.''). AIm'I had,
however, proved n more ni'tiiTal theorem (see % 20, Cor.), which partly io-
eludes the lenult in question.
^ 14, 15 THEOREM OF CAUCHY AND MERTENS 143
+ V2,„+i I as small as we please. Also, since \Ti\, \T«\,\T3\, . . .,
\Tn\, ... are all finite, and |7;- 7'.|<|7;| + \T.\, therefore
I r„,+i + . . . + l\m I, . . ., k^ + . . . + «2m i,
I V„+j + . . . + l-o„,+i I, , . ., I IV + . . . + V2m+1 I,
are all finite. Hence, if e„ be a quantity which can be made as
small as we please by sufficiently incrca.sing m, and fi a certain
finite quantity, we have, from (3) and (4), by chap, xn., § 11,
l-Z'»|<«m(iw.| + |«3|+. . . + |«„,|)
+ /3(|m„+,| + |m„+,| + . . . + \u„\).
If, therefore, we make n infinite, and observe that, since
2«,„ is absolutely convergent, Iw^l + lu^] + . . . + |?<„| is finite, and
X(|«m+i| + |",n+2| + . . . + |w»|) = 0, we have (seeing that i£„ = 0)
i I Z„ I = 0. Hence X^S'„ T„ = LV'„, that is, LUn = ST.
Cauchy has shown that, if both the series involved be semi-
convergent, the multiplication rule does not necessarily apply.
Suppose, for example, H„ = r„=(-l)»-'/\/". Then both Zh„ and i:i„ arc
Bemi-convergent series. The general term of (1) is
■""= ^ im ■" v/{(«- 1) 2} + ■ • • + ^{2 [l - 1)} + 7F}) (')•
Now, since r(n-r + l) = J («+l)-- {J (n + 1) -r}', therefore, for .ill values
of r, r(H-r + l)<J(ii + l)-, except in the case where r=^(n + Vj, and then
there is equality. It follows that i w"„ | > «/i (n + 1) > 2/(1 + 1/n). The terms of
2w„ are therefore ultimately numerically greater than a quantity which is
infinitely nearly equal to 2, Hence ^w„ cannot be a convergent series.
UNIFORMITY AND NON-UNIFORMITT IN THE CONVEKGENCE
OF SERIES WHOSE TERMS ARE FUNCTIONS OF A VARIABLE.
§ 15.] Ijct X for the present denote a real variable. If the
fith term of an infinite series be f{n, x), where/(??, 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/(n, x) will, if convergent, be a single valued
finite function of x, say '^(ir). At first sight, it might be
supposed that <t> {x) must necessarily be continuous, seeing that
each term of J\n, x) is so. Cauchy took this view ; but, as
144 UNIFORM AND NON-UNIFOUM COSVEKUENCE CU. XXVI
Abel* first pointed out, <t>{-f) is not necessarily continuous.
No doubt 2/(h, j: + /() and ~/(ii. x), beinj; each convergent, have
each definite finite values, and therefore 2 {/(», x + h)-/{n, x)\
is convergent, and has a definite finite value ; but this value is
not tu'cesgiirlfi/ zero when h-0 for all vultii's of x. Suppose, for
example, following l)u Bois-Keyniond, that /(«, x) = a-/(/(j- + 1)
(hjt -x+\). Since /(«, x) = nx/{nx + 1) - (n - l)r/{n - Ix + 1},
we have, in this case, S^- iix/(iix + I). Hence, provided x*0,
X<S', = 1. If, however, x = 0 then 8^ = 0, however great n may
be. The function <f> (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 discontiuuit)'. The Residue of the series, when
x^O, is given by
i?. = l-5'.= l/(nx+l).
Now, when x has any given positive value, we cAn by making n
large enough make l/(iix+l) smaller than any given jmsitivo
quantity e. But, on the other hand, the smaller x is, the larger
must we take n in order that l/{iix + 1) may fall under «; and,
in general, when x is variable, there is no finite lower limit for w,
independent of x, say v, such that if n>v then 7i',<c. Owing
to this peculiarity of the residue, the series is sjiid to bo non-
uni/brmly convergent in any interval which includes 0 ; and,
since, when x approaches 0, the nundier of t<'rms required to
secure a given degree of ajipro.ximation to the limit becomes
infinite, the series is said to Converge Infinitely Hlouhj 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. Jj', for all values ef z within
a ijiven reijion 11 in Anjand's Diagram, we can for every
positive value of <, however snutll, assign a jiosltive integer v
iNUEi'KMiENT OF z, nuch tluit, ichfii n>v, \J(,\<t, tiun the seriei
Rochcrchus bur U S6rio 1 + t- ' + — , — tt - *' + • • • CrtUe't Jour,
X X.J
Ud. 1. (1H-.'C).
i
§ 15 UNIFORM CONVERGENCE 145
2/(«, x) is said to be Uniformly Convergent within the region
in question.
Stokes*, who in his classical paper on the Critical Values of
the Sums of Periodic Scries was the first to make clear the
fundamental principle underlpng the matter now under dis-
cussion, has pointed out that the question of uniformity or
non-uniformity of convergence always arises when we consider
the limiting value of a function of more than one variable.
Consider, for example, the function /(x, y) ; and let us suppose
that, for all values of ?/ in a given region R, f(x, y) approaches
a finite definite limit when x approaches the value a ; and let us
call this Hmit/(a, y). Then if we assign in advance any positive
quantity t, however small, we can always find a positive quantity
A, such that, when |^-aI<X, \f{x,y)-f{a,y)\<€. If it be
possible to determine X so that the inequality
\f{x,y)-f{a,y)\<^
shall hold for all values of y contained in R, then the approach
or convergence to the limit is said to be uniform within R. If,
on the other hand, X depends on y, the convergence to the limit
is said to be non-uniform.
Example 1. Consider the serieg 1 + 2 + 2^+. . .+z»+. . .; an,j igj
I* I <p< 1. We have | i?„ | = | «''+V(l - z) | <p»+V(I - p). Hence, in order to
secure that i?„ < e, we have merely to choose n > - 1 + log (e - ep)/log p.
Since - 1 + log (e - cp)/log p is independent of z, we sec 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 2z" is uniformly convergent.
On the other hand, as p approaches unity log (e-cp)/logp becomes larger
and larger. Hence the convergence of Zz" becomes infinitely slow when | z
approaches unity. We infer that the convergence of 22" is not uniform
within and upon the circle of radius unity. And, in fact, when the upper
hmit 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 - 2) | < 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, Abhl. d. Bayerischen
Ak,id. d. Wiss. Bd. v. (1850). For exceptions, see Keiff, Gescliichte der
unendticheti Eeiheii, p. 207 (1889); and Pringsheim. Enc. d. Math. Wiss.
Bd. II. p. 93 (1899). In his first edition the writer, although well acquainted
with Stokes' great paper, by an unfortunate hipse of memory, fell into the
same mistake. The question of uniformity of convergence is now a
fundnmeutttl point in the Theory of Functions.
C. II. 10
Ii6 coNTiNunr and uniform convergence ch. xxvi
Example 2. Osgood* has shown that, if
<t>n W = \'(2<) n pinVi . ^--'i-'",
the infinite serioa which hag 0, (x) + 0,(21 x)/2I-)- . . . -t- 0, (nl x)/nl for the
Eum of n terms converges non-anilormly in over; intorval.
From the definition of Uniform Convergence we can at once
draw the foUowini; conclusions.
Cor. 1. If the terms of 2|/(n, z)\ are ultimately less than
the terms of a converging series of positive terms irfiose values are
independent of z, then '%f(n^ z) converges uniformly.
For, if 2h, be the series of positive terms in question, and R^
the residue of ^(n, z), then
Ii?,|>|/(« + l.z)| + |/(« + 2,c)|+. ...
< «,+, + u,+, + . . .
Since 2m, is convergent, we can find an integer v so that, when
n>v, Un+i + «„+j+ . . • <<; and v will be independent of c, .since
Mii+i, "n+s. • • • •ii'e independent of z. Hence we can find v
independent of s so tliat |/if, |<c, when n>v, t having the usual
meaning.
Cor. 2. If 2|/(n, z)\ is uniformly convergent, then V(". *)
is uniformly convergent.
§ 16.] We now proceed to est.ibli.sh a fundamental theorem
regarding the Continuity of a Uniformly Converging Series.
Let fill, z) be a finite single valued function of the complex
variahh z and the integral variaUe n, which is continuous a$
regards z for all values of n, however large, and for all values of
z within a region II in Argand's Diagram. Farther, let 'Sf{n, t)
converge uniformly within It, say to <^(s). Then 0(c) is a con-
tinuous function of z at all j'oints vAthin the region 11.
Let the sum to n terms and the residue after n terras of
2/(n, ;) be <S', and 7/, ; and let iS",' and 7/,' be the like for
^(n, z), where z and z are any two points within the region R.
Then wo have
4>(:) = S, + Il„ <^(r') = 6V + 7C (1).
* Hull. Am. M.ilh. Soe., Si-r. 2, m. (lJ-9fi). Tliin paiior in well wortliv ol
study on account ut lh« inlcrusliug (jcoiuulricul tuuUiuJji which Uiu aulhur
OHea.
4
§§ 15, IG CONTINUITY AND UNIFORM CONVERGENCE 147
Since 2/(/;, s) is uuiformly couvergent within R, given anj'
positive quantity «, however small, we can find a finite integer v,
independent of s, such that for all values of z within Il,Itn<f and
En<f, 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 |«-s'| sufficiently small cause the absolute value
of each of the difFerences/(l, s)-/(l, z), . . .,/(n, z)-f{n, z)
to become as small as we please, and, therefore, since « is Unite
we can choose 1 2 - 2' | so small that | Sn - S^ \ , which is not greater
n
than 2 \/{n, z) -f{n, z')\, shall be less than «.
1
Now
\4>{z)~<1>{z')\ = \S„-S: + R„-R:\
>l>s',.-5^„'l+|ii;„l + |7?„'l
<3e,
which proves our theorem ; for e, and therefore 3«, can be made as
small as we please.
It follows from what has been proved that discontinuity of
^(n, z) is necessarily accompanied by non-uuiformity of con-
vergence ; but it does not follow that non-uniformity of con-
vergence is necessarily accompanied by discontinuity. In fact,
Du Bois-lleymond has shown by means of the example
'%{xlii{nx + \){nx-x+ \)- arl{nar+ l){nx'-x-^ 1)}
that infinitely slow convergence may not involve discontinuity.
The sum of this series is always zero even when x = Q; and )'et,
near x=0, 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
^x/{nx + 1) (nx - X +1)
converges at x = 0; but, owing to the infinite slowness of con-
vergency at x=0, the sura is discontinuous there, being in fact
0 at a: = 0, and 1 for points infinitely near to x = 0. In such
eases it is necessary to state the region of uniform convergence
with some care. The fact is that the series in question is
oonvergeut in the real interval p:!(>x:^b, where b is any finite
10—2
148 DV bois-reymond's theorem ch. XXVI
positive quantity and p is a positive quantity as small as we
please but not evanescent. Tiiis is usually ex]>rc8sed 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*.
Eiamplct. If /i, be independent of t, and if, (j) be a single valacd
fanction of n and i, finite for all valncs of n, bowevcr great, and finite and
continnoos as regards t nitliin a region li, then, if ^/i^ be absolutely con-
vergent, SMn^a (<) is a continuous function of t within li.
It will be sullicicnt to prove that the series i:^,w, (i) is uniformly
convergent within if.
Since u>, (z) is finite for all points within li, we can assign a finite
positive quantity, g, independent of z, such that, for all points within 1!,
Consider 7J„, the re.<:idue of S^ir^ (t) after n terms. We have
„ •B.=^,+i«',+iW+M«+j«',+i(«)+ • • •
Uenco
|i?-l>lM.+ill«'.+,(i)| + |M.+,I|"',+,WI+. . ..
Since 2^ is ab.'Jolutely convergent, S | /i, | is convergent, and wc can a)t.<!ign
an integer y such that, wlicn n>r, | >«„+i | + 1 /:i„+5 1 + . . . <</?. where < is a
positive quantity as small as we please.
Both ;i„ and g being independent of z, it is clear that r is inde-
pendent of z. Hence we have, when nx', |7?„|<e, » being independent
of z. The scries is therefore uniformly convergent : and it follows from the
main theorem of this paragraph that its sum is a continuous function of i.
SPECIAL DISCUS.SION OF THE POWER SERIES 2a„r".
§ 17.] As the series 2fl(„c" is of prcat importance in Algebraic
Analysis and in the Theory of Functions, we shall give a special
discussion of its i)roperties as regards both convergence and
continuity. We may speak of it for shortness as the Power
Series ; and we shall consider both a, and s to be couiple.T
numbers, say a„ = r, (cos a, -n' sin oj, z = p (cos fi + i sin 6), where
r, and a, arc functions of the integral variable », but p and 6 are
iudependeut of «.
• Harkncsfl and Morley use these convenient words in their Intrnduetion
tn the Theory nf Analytic Functioiu. Macmillun (IS'JS).
t l^u Uuis-Ileymoud, Math. Aim. iv. (lt>71).
§§ IG, 17 CIRCLE AND RADIUS OF CONVERGENCE 149
The leading property of the Power Series is that it has wliat
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 vahies 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 2a„«" be at least semi-convergent when z = z„,
then it is absolutely and 'uniformly convergent at all points within
a circle whose radius < 1 2o I ■
Since Srt.jCo" 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 | a„»o" I < 9> for all values of n however large.
Hence | «„«"! = |a„s„"(5/s„)»|,
= l«„^o"|l(s/«o)"|,
<o\{zlzoT\.
Now, since z is within the circle |«o|, \zlZii\<l. Hence the
series g'S.iz/zo)" is absolutely convergent. Therefore (§ 4, I.)
2 1 a„s" I is absolutely convergent.
The convergence is uniform. For, since |2|<|2;o|, we can
find z such that |3|<|3'|<|so|- Now, by the theorem just
established, 2 | a,,-'" \ will be convergent, and its terms are inde-
pendent of z. But, since | s | < 1 2' | , | «„«" | < | a„s'" |. Hence, by
§ 15, Cor. 1, 2a„«" is uniformly convergent.
Circle of Convergence. Thi-ee cases are in general possible.
1st. It may not be possible to find any value Zo of z for which
the series 2a„5'' 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 (It), we mean a (•jrcle of
radius B whose centre is the origin in Argand's Diagram.
1.10 RADIUS OK CONVERGENCE, OAUCHY'S RULES CII. XXVI
however large. We sliall then say that the circle and the rwliiis
of coiivergenco are infinite. An example of this very important
class of series is 2y/«!.
3rd. There may be finite values of z for which 2a, j* con-
verges, and other finit.e values for which it does not converge.
In this case there must be a definite upper limit to the value
of \z^\ such that the scries converges for all points within the
circle l^,! and diverges for all points without. For the series
converges when |2:|<|?o|> n'ld it must diverge when |z|>|;,|;
for, if it converged even conditionally for |2'|>|2;,|, then it
would converge when |r]<|5'|. We could, therefore, replace
the circle |so| by the greater circle |s'|, 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-
vergeuce, or as it is often aiUed 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,
the points on its circumference are not to be held as being within
it. Thus, for example, nothing is proved a.s 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 2(i,i;" is uniformly convergent
within any circle whose radius is less than It by however little.
§ 18.] Cattchy's Ruks for determining the Radius qf Con-
vergence qf a Power Stries.
I. Let <D be the fixed limit or the greatest qf the limits to
which IobI"" tends when n is increased indefinitely, then l/«
is the radius qf convergence qf 2a,s".
For, a.«i we have seen in § 11, II., 2rt,s" is convergent or
divergent according as i|a,s"|''"<or>l ; that ig, according as
o)|c|<or> 1 ; that is, according as |r|<or>l/ci».
II. fyet u be a fired limit to which |a,+,/a, | tends when n it
increased indefinitely; then 1/<d is tits radius qf oonvergenc« qf
v., .»
§§ 17-19 CONVERGENCE ON CIRCLE OF CONVERGENCE 151
Tlie proof is as before. The second of tliese rules is often
easier of application than the first ; but it is subject to failure in
the case where L \ a-n+i/an \ is not definite.
Example 1. l + j/l + i'/2+ . . .
Here, by the first rule, w= L {l/n)''"= L m"'^! (chap, xxv., §16).
Hence iJ = l.
By the second rule, u= L {l/(n + l)}/{l/n} = L n/(n + l) = l. Hence
i? = l, as before.
Example 2. « + 2s» + z' + 2s<+ . . .
Here if n = 2m, L |a„»/»|= L VI" = 1,
if n = 2m + l, L |o„J/»i= i 2>/" = l.
Hence u = 1, and iJ = 1. The second rule would fail.
§ 19.] Convergence of a Power Sei-ies 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 (7„ be real, such, that ultimatehj (in has the same sign
and never increases; also that ZrT„ = 0, and La„+,/a„=l, when
n=co. Then the radius of convergence of SanS" is unity ; and
1st. If 2rt„ is convergent, 2rt„5" converges absolutely at every
point on its circle of convergence.
2nd. If 2«„ is divergent, 2a„s* is semi-convergent at every
point on its circle of convergence, except s=l, where it is
divergent.
If we notice that on the circle of convergence 2a„s" reduces
to 2a„ (cos nd + i sin n6) = Sa„ cos nO + «2a„ sin nO, we deduce the
above conclusions at once from § 9.
Cor. Obviously the above conclusions hold equally for
2 (-!)"»„«", except that the point z- — l takes the place of
the point z=l.
The folloiving Rule, given by Weierstrass in his well-known
memoir Ueber die Thcorie der Analytlschen Facultaten* , applies
♦ CrelU'$ Jour., Bd. 51 (18.50).
152 AlJEb's CONTINUITY THKOREMS CII. XXVI
to tlie innro poiieral case wlicre tlic coefficients of tlie jxiwer series
may bo comiilex. By § 6, Cor. 5, it is ea.sy to sliow lluit it
includes as a particular case the greater part of the rule already
given.
II. If on and after a certain value qf n ice can expand
o.+i/«» t» the form
0(.+i - o + hi a,
a, n «'
ich're g and h are real, then the beharioitr of 2rt,c" on its
circle of converf/ence, the radius of which is obviously unity, is
as follows : —
1st. If d'i^Q the series diverges.
2iid. If g<-l the series converges absolutely.
3rd. If - l^g<0 the series is semi-convergent, except at the
point 5=1, where it oscillates i/' g = -l and A = 0, and diverges
if g>-l.
For the somewhat lengthy demonstration we refer to the
original memoir.
§ 20.] Abel's Theorems* regarding tlte continuity of a power
series.
Since (§ 18) Sa.s" converges uniformly at every point within
its circle of convergence, we infer at once that
I. The sum of the power series Sa,;" is a cnntinuotis function
of z, say <^(s), at all points xnthin its circle of convergence.
This theorem tells us nothing as to what happens when we
pa.ss from within to points on the circumference of the circle of
convergence, or when we pass from jwint to point on the circum-
ference. Much, although not all, of the remaining iuformation
required is given by the following theorem.
II. If the power series -«,£" be convergent at a point z, on
its circle of convergence, and s be any point within the circle, then
•-•, 1 1
provided the order qf the terms in 2rt„i|," be not deranged in cases
where it /.« only semi-con vergent.
■ CrelWs Jour., Bd. i. (1826).
^ 19, 20 Abel's continuity theorems 153
In the first place, we can show that, in provinrr tliis thoniein
we need only consider the case where s and Zo lie on the same
radius of the circle of convergence. For, if z and «„ he 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 [s-^ol
sufficiently small, we can make | ~ - a, | and | c, - ;r„ | each smaller
than any assigned positive quantity however small.
Since z and Zi are both within the circle of convergence, we
can, by making |« — 2i| sufficiently small, make | <^ (s) - <^ (;,) |
less than any assigned positive quantity e, however small. But
\4>{z)-4>{z,)\ = \<i>{z)-^{z,) + <i.{z,)-^{z,)\,
>|<^(«)-<^(«,)l + l<^(~i)-'/'(-c,)|.
<£+|.^(c,)-<^(c„)|.
If, therefore, we could prove that by making 1 2^ - ^o I sufficiently
small we could make | <^ (s,) - <^ (so) | as small as we please, it
would follow that by making 1 2 - So I sufficiently small we could
make | ^ (5) - <^ (so) | as small as we please.
Let us suppose then that z and Zq have the same amplitude 6,
then we may put z = p (cos 6 + i sin 6), Zo = Po (cos 6 + isin 6), and
we take «„ = r„ (cos a„ + i sin a„). Hence
a„z"' = r„ (cos a„ + i sin a„) p" (cos nO + i sin nO),
= (f.J
'"nPo" {cos (nO + a„) + i sin {?id + a,,)}.
^X^U^ + iVn),
where iB = p/p„, and becomes unity when z = Z(,; 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 ^afU„ remains convergent when x='l,
then L 2a:"?7„ = 2f7'„, or, what is practically the same thing,
I-l-O 1 1
to prove that, if 2 f^„ be a convergent series, then
1=1-0 1
Let S„ = {1 - x) ni + {I - x") U^ + . . . + (1 - X") Un,
= {l-x'')U„ + {l-af-')U„., + . . .+{l-x)Ur.
154 ABEL'S CONTINUITY TUEOHEMS CH. XXVI
Since *:^ 1 , 1 — j^, 1 — .r""', . . ., I - x satisfy the conditions
imposed ou a,, a., a, in Abel's ludiiiality (S 9). Also,
since 2^7, is convergent, l/„, Z7«_,, . . ., Ux satisfy the con-
ditions imposed on «,, u, »,. Hence, A and B being two
finite quantities, we have
(\-a')A>S^>{\-3f)D.
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 a: = 1 - 1/h'. Then we hiive, for all values
of K, however great,
{1 -(1 -l/«')-}^>'S.>{l -(1 -1/"')"} A
But L {\- 1/h')" = X {(1 - l/tt')--'}-"" = «"• = 1.
Therefore, since A and B are finite, L <S„ = 0 ; that is,
r-l-o 1
It will be observed that, in the above proof, each term of
2j:"f7, is coordinated with the term of the same order in 2f7,.
Hence the order of the terms in 2Cr, must not be deranged, if it
converges conditionally.
It follows from the above, by considering paths of variation
within the circle of convergence and along its circumference, that,
if a power serie.? 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 regardt-d as a closed region. This
does not exclude the possibility of point-s 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 conver;;ence 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 iufmitrly near to P, it must conver>;e to the same
values.
It would tlius appear t<} be impossible tli;vt a ixjwer series
§20 CONVERGENCE OF S ("ni'i + Wn-i^'a + ■ • • + «it'n) 155
should converge infinitely near any point P of the circumforence
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 SwnS", S-Pnz", and the series
5 («„i>i + «„-, I's + . . . + ihVn) a''+',
wliich 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 %Un and 2v„ converge, say to limits
u and V respectively, then, if the series 2 («„ri + «„-ii'a + . . . + z<,»„)
be convergent, it will converge to uv ; and this will hold even if
all the three series be only semi-convergent.
Example 1. The series l+z+ . . .+j"+. . . has for circle of oon-
Tcrgence the circle of radius unity. Witliin this circle the series converges
to 1/(1-2). On the circumference the series becomes 2(co8»ifl + isinn9),
which oscillates for all values of 6, except 0 = 0 for which it diverges. At
points within and infinitely near to the circle of convergence the series
converges to J + icotK.
Example 2. The radius of convergence for s/l+ . . . +j"/n+ ... is
nnity. Within the unit circle, as we shall prove later on, the series con-
verges to - Log (1 - 2). On the circumference of the unit circle the series
reduces to S(cosn9-|-»sinji^)/n. This series (see § 9, UI.) is convergent
when 9=t=0 ; but only semi-convorgent, since 21/n is divergent. Wheu e = 0,
the series diverges. The sum is therefore continuous everywhere at and on
the circle of convergence, except when d = 0. At points within the circle
infinitely near to 2 = 1 the series converges to a definite limit, which is very
great; but at 2 = 1 the series diverges to +00.
Example 3. 22"/n' converges absolutely at every point on the circum-
ference of its circle of convergence (iJ = l): and consequently represents a
function of 2 which is continuous everywhere within that circle and npon
its circumference.
Example 4. 2n2" is divergent at every point on its circle of convergence
(P = l); and its sum is a continuous function at all points within its circle
of convergence, but not at points npon the circumfijreuco.
156 INDETERMINATE COEFFICIENTS CU. XXVI
Exnnipio 6. PriiiKiihcim * has cstablinhpcl the existence of n liirgo cl&M
of series which are tcmi-convcrgunt at every point on Ibc circumfcreuco of
their circle of coDvorgcnce : a particular case ia the series S ( - l)*<i*/n log ii,
>
where X, = l when 2«"j.n<2*»", X,=0 when 2=^'m<2»»+'.
§ 21.] I'r'iiiriple of Imletermiixite Coeffirlrnts.
If 0,4-0, tin re is a circle of non-ecaiw^cent raJius within
which the convergent power series 2a,z" cannot vanish.
.Since the evanescence of the series implies a„ = - a,c- tfjC* - • • •,
it will be sufficient to show th:it there exists a fiuitc po.-itive
quantity X svich that, if p = lsl<A, then
|-a,c-rt,s»-. . . |<lflro|.
Now, since the series 2</„c" is absolutely conver;;pnt at any
point Zo within its circle of convergenceT there exists a 6nit€
positive quantity j such that for all values of «, | a,V | = a^p'<g.
Hence |a,|<<;/p,".
Now
|-«.z-«,c*-. . .l>|a.c|+|a,z'| + . . .
>l«i|p + l«i|p' + - • •
<9{(pM + (j>ipoy+- • •}
<jpHp*-p)-
Hcnco, if we choose X so that g^/(p, - X) = | a, |, that is X = | a, | p^
(^ -•- 1 Og I ), we shall have
\-a,z-a.,z'-. . .|<|a,|
for all values of s within the circle X.
Cor. 1. J/ rt„ + 0, there is a circle of non-evanescent rwUus
within whiih the convtrgent power siriis tf„s" + a«+,c"+' + . . .
Vitnijyhes only when 2 = 0.
For a«s" + o«4-iS"*' + • • •
= «"(a„ + a«+,r + . . . ).
Now, since (i„+0, by the theorem just provod there is a cin-lc
of non-evanescent radius within which </,„ + <««+iC •••... cannot
vanish : and 2" cannot vanish unless a = 0.
• Uatk. A>M., Bd. UT. (llWo).
§§20-22 INFINITE PRODUCTS 157
Cor. 2. If ao + ttiS + Oos' + . . . vanish at least once at some
point distinct from s = 0 within every circle, however small, then
must ao = 0, ai = 0, a.,-0, . . ., that is, the series vanishes
identically/.
Cor. 3. ff for one value of z at least, differing from 0, the
series ^a^z" and 2^„3" converge to the same sum within every
circle, however small, then must a(,= ho, ai = bi, , . ., that is, tlw
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 i^resently see, it is only when the
factors ultimately become unity that the most important case
arises, we shall write the nth factor in the form 1 + m„.
By the value of the infinite product is meant the limit of
(1 + «i) (1 + 11.) . . . (1 + «,.),
(which may be denoted by 11(1 + ii„), or simply by P„), when n
is increased without limit.
It is obvious that if lAt„ were numerically greater than unify,
then LPn would be either zero or infinite. As neither of these
ca^es is of any importance, we shall, in what follows, suppose
I Ma I to be always less than unity. Any finite number of factors
at iJie 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. LF„ may be 0.
2nd. LFn may be a finite definite quantity, which we may
denote by n (1 + «„), or simply by P.
3rd. LP„ may be infinite.
4th. LP„ 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. XXVI
lu ca^es 1 and 2 the infiuite product mi>,'ht be said to be
convergent; it is, liowever, usual to confine tlie term convergent
to the 2nd case, and to this convenient usage we sliiill adhere ;
in case 3 divergent ; in case 4 oscillatory.
§ 23.] If, instead of considiTing /*,, we consider ite logarithm,
we reduce the whole theory of inlinite products (so far as real
positive factors are concerned*) t« the theory of infinite series ;
for we have
logP, = log(l + u,) + log(l + a,) + . . . + log(l + «.)
= 2 log (1 + J/») ;
and we see at once that
ft
1st. If 21og(l + M„) is divcrgnnt, and Z.2log(l +h,) = - «,
then n (1 + M„) = 0 ; and conversely.
2nd. If 2 log (1 + u.) be convergent, then n (1 + ii,) convcrgee
to a limit which is finite both ways ; and conversely.
It
3rd. If 21og(l +»<,) is divergent, and Z21ng(l +t<„) = + oo,
then n (1 + I/,) is divergout; and conversely.
4th. If 2 log (! + «„) oscillates, then n(l+u,) oscillates;
and conversely.
§ 24.] If we confine ourselves to the case where m, has
ultimately always the same sign, it is ea-sy to deduce a simple
criterion for the convergencj' of n (!+«,).
If iK,<0, then 2 log (1 + u,) = - oo , and n (1 + «,) = 0.
If Z,H,>0, 2 log (1 + f/„) = + oc , and n (1 + «,) is divergent
It is tlienj'itre a nece^ary con Jit inn for the convergence qf
n (1 + u,) that Lun = 0.
Since Lu^ = 0, /. (1 + «,)""" = e ; hence L log (1 + «,)/»*, = 1,
It therefore follows from § 4 that 2log(l + m,) is convergent or
divergent acconling as 2tt, is convergent or divergent. More-
over, if «, be ultimately negative, the last and infinite part* of
Stt. and Slog(l -f u.) will be negative ; and if u. be ultimately
• Tlie logarithm of a coiij|)lcx number bim not yet been drfmrd, much
Imb diitcudiuad. Given, however, the theory of the loi^'arithm of n pmiiplex
vai-uible there id iiothinR illotiioal in mnkin^ it the hasiii of the tbourjr of
inQoito produuli, a( the foraiur dooa not pnuuppuM tho Utt49r.
§^ 22-24 AND OSCILLATING INFINITE PRODUCTS 159
positive, the last and infinite parts of S«„ and 2 log (1 + «„) will
be positive. Hence the following conclusions —
If the terms of 2?<„ become ultimately infinitely small, and
have vltimatcly the same sign, t/ien
1st. n (1 + «„) is convergent, if 2m„ be convergent; and con-
versely.
2nd. II (1 + ?(„) = 0, if 2m„ diverge to - co ; and conversely.
3rd. n (1 + M„) diverges to +ai,if 2m„ diverge to +<x>; and
conversely.
Since in the case contemplated, where «<„ is ultimately of
invariable sign, the convergency of n (I +?i„) 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 pivduct in
which the sign of m„ is not ultimately invariable, but which is
convergent when the signs of u„ are made all alike, will be,
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 n (1 + u„) is absolutely
convergent when 2«„ is absolutely convergent; and conversely.
Cor. If either of the two infinite jjroducts n (1 + m„), n (1 - «„)
be absolutely convergent, the other is absolutely converge)^.
For, if 2?«„ is absolutely convergent, so is 2 (-;/„); and
conversely.
Example 1. (1 + 1/1=) (1 + 1/2=) • . • (l + l/n=) ... is absolutely conver-
gent Bince 21/h= is absolutely convergent.
Example 2. (1 - 1/2) (1 - 1/3) ... (1 - l/») . . . baa zero for its value
since S ( - Ijii) diverges to - a> .
Example 3. (1 + 1/^2) (1 + 1/v/-) • • • (l + l/v'") • • • diverges to +co
Bince 2{l/v/n) diverges to +co.
Example 4. (l + l/^/l) (1 - l/v/2) {1 + 1/^/3) {l-l/v/4) . • . Since the
sign of ]/„ is not ultimately invariable, and since the series 2 ( - 1)" '/V" '^
not absolutely convergent, the rules of the present paragraph do not apply.
We must therefore examine the series S log (1 + ( - l)»-'/V«)- ^he terms of
this series become ultimately infinitely small ; therefore we may (see § 1'2)
associate every odd term with the following even term. We thus replace the
series by the equivalent series
Slog il + ll.J{2n - 1) - l/v/(2n) - Wl*"' " 2'')}-
160 INUKPENUKNT CUITElilA Cll. XXVI
It is eesy to show t)iat
1/^/(2,. - 1) - 1/VC2«) - l/v't-in' - 2n) <0.
for all values of n > 1. Hence the terms of the serieii in qaestion altimately
become noRative. Moreover, l/^(2ii - 1) -l/ij(2ii) - l/VH"'-2n) is ulti-
mately comparable with - l/2ri. Hence Slog(l + (- 1)"-'/%/") diverges to
- 00 . The value of (1 + 1/v'l) (1 - l/^/2) (1 + l/v/3) (1 - 1/^/4) ... in there-
fore 0. This is ail example of a seiiii-convcrgont product.
Examples. <'+'«-l"ie' + l<r-*~* . . . The scries 2 log (1 -fuj in this
case becomes
(l-H)-{l-^i)-Kl-^4)-(l-^i)-^...
which oscillati'S. The iiiGiiitc product therefore oscillates also.
Example 6. n (1 - i»"'/n) is absolutely convergent if x < 1, and baa 0 for
its value when x = l.
§ 25.] We have deduced the theory of the convergence of
infinite products of real factors from tlie 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 tlie
properties of 0 (that is, +a- a) and 1 (that is, x a -^ a).
It is quite easy, however, by means of a few simple inequality
theorems*, to deduce all the above results directly from tlie
definition of the value of n (1 -i- «,).
If r„ have the meaning of g 22, then we see, by exactly the
same reasoning as we used in dealing with infinite series, that
the neces-sary and suflicicnt conditions for the convergency of
H (1 -I- «,) are that 1\ be not infinite for any value of n, however
large, and that L (/*,+« - I'n) = 0 ; and that the latter condition
includes the former.
If we exclude the exceptional case where L J\ = 0, then,
»—•
since P, is always finite, the condition L (/',+■ - P^) = 0 is
equivalent to L (P^+JP,- i) = 0, that is, i/\+«//',= l.
* 8po Wcionitraiui, Abhandlungcn aiu d. FuiutionttUehre, p. 203 ; or
CrtlU't Jour., Bd. 51.
§§ 24 -26 COMPLEX PUODUCTS IGl
It', tlierolbre, wc deuote (1 + m„+i) (1 + (',,+j) . . . (1 + «„+„,)
by mQm we may state the criterion iu tlic following form, where
M„ may be complex : —
T/w necessary and sufficient condition that n (1 + u„) coiivenjo.
to a finite limit differing from zero is that L \ ,„Qh - 1 | = 0, for
all values of m.
For, since L \ „,Q„ - 1 1 = 0, given any qviautity e however
n=ao
small, we can determine a finite integer v such that, if u-iv
UQ.i-l|<«- Therefore, since mQn = Pn+mll\, we have in
particular
l-£</',+„,/P.<l + e.
Since V is finite, P^ is finite both ways by hypothesis. Therefore
(l-e)P,</',+„<(l + .)P,.
Since m may be as largo as we please, the last inequality shows
that Pn is finite for all values of n however large.
Again, since P„ is not infinite, however large n, the con-
dition L I ,„Q„ - 1 1 = 0, which is equivalent to L „Q„ = 1, leads
to L Pn+m = L P„. The possibility of oscillation is thus ex-
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 + «„) /a" convergent if 11 (1 + 1 ?<„ |) is convergent.
Let p„ = I M„|, so that p„ is positive for all values of n, then,
since n (1 + p„) is convergent,
ii(l+P-.+.)(l+P"+.;)- • .(l + P.+,«)-l} = 0 (1).
Now
JU - 1 = (I + ^^.+,) (1 + u„ ,,) . . . (1 + «„+,„) - 1,
= .««„+l + -<«„+iW„-(..i + . . . + ?<„+! W„+.j . . . M„4,„.
Hence, by chap, xii., g§ 9, 11, we have
0^|mQ»-l I ^^Pn+I +2p„+,p„+o+ . . .+p„+,p,H2. . ■ Pn+m,
>(! + p„h) (1 + Pn+=) . . . (1 + p„+,„) - 1.
C. II. 11
1G2 ASSOCIATION AND COMStUTATION CO. XXVI
Hence, l.y(l),XUf^-l 1 = 0.
Ilemark. — Tlie converse of tliis tlieorcin is not tnie ; as may
be seen at once by considering the product (1 + 1) (1 - i) (1 + \)
{\-\) . . ., wliicli converges to the finite limit 1; although
(1 + 1)(1 + J)(l +i) (l + i) . . . is not convergent.
When n (!+</„) is such that n(l + |«„|) is convergent,
n (1 + j/„) is said to be nbsnlulely convergent. //" n (1 + ?<„) be
convergent, but n (1 + 1 «, |) non-conrergent, U (1 + «,) is said to be
semi-convergent. The present use of these terms includes as a
particular ca.se the use formerly made in § 24.
§27.] 1/ 'S,\u„\ be convergent, thin n (!+«,) is absolutely
convergent; and conversely.
For, if 2 I «„ I be convergent, it i.s absolutely convergent, seeing
that I «/n I is l)y its nature positive. Hence, by § 24, 11 (1 + 1 u„ |)
is convergent. Therefore, by § 26, n (1 + «,) is absolutely con-
vergent.
Again, if 11(1 + ?/,) be absolutely convergent, n(l + |«,|)
is convergent; that is, since |k,| is positive, 11(1 + 1 a, |) is
absolutely convergent. Therefore, by § 24, 2 1 ?<, | is absolutely
convergent.
Cor. If lun be absolutely convergent, 11 (1 + u„x) is absolutely
convergent, u-here xis either independent of n or is such a function
of n tliat i I a: I =*= ao when n= <x>.
Example 1. IT (1 -z*/n) is absolatclv convergent for all complex valaes
each that | x | < 1, but is not abRolut«1y convcrRcat when | z | = 1.
Example 2. 11(1 -z/n'), where z is indcpcmlcnt of n, ig absolutely
canvergcnt.
§ 28.] After what ha.s been done for infinite scries it is not
necessary to discuss in full detail the a])i>lication of the laws of
algebra to infinite products. We have the following results —
I. The law of association may be sa,fehi applied to tite f acton
o/' n (1 + w„) provided Iai„ = 0 ; but not otherwise.
H. The necessary and sufficient condition that n (1 + «,) shall
converge to the same limit {finite both ways), whatever the order of
if s factors, is that the .•'rries 2//, be nJistilutrly convergent .
When w, is real, this result foUow.s at once by considering the
series 2 log (1 + u„) ; and the .same method of proof ajiplies when
§§ 26-28 UNIFORM CONVERGENCE 1G3
«„ 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 unmnditionally convergent. '
Tlie theorem just stated shows that unconditional convergence and
absolute convei-gence 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. 1/ both n (1 + M„) and n (1 + Vn) be absolutely/ convergent,
then n {(1 +M„) (1 +i'n)} is absolutely convergent, and has for its
limit {n (1 -H w,,)} X {n (1 + V,,)] ; also U {(1 -i- «„)/(! 4- v„)\ is abso-
lutely convergent, and has for its limit {13 (1 + M„)}/{n (1 -^ v„)},
provided none of the factors of U {I + v„) vanish.
If Qn denote (l + Un+i) (1 + 11^+2) • • •, the total residue of
the infinite product n (1 + «<„) 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 \Q„— l\<e, when n<^v.
If M„ be a function of any variable z, then, when « is given,
V will in general depend on z.
If, however, for all values of z within a given region li in
ArgandJs diagram an integer v independent of z can be assigned
such that
\Qn-l\<^,
when nJs^v, then the infinite product is said to bo UNiroUMLY
CONVERGENT tvithiu R.
IV. Tff{n, 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 II {1 +f(n, z)} converges uniformly for all valves of z
tvithin R to a finite limit <^ (s), then <f> (s) is a continuous function
of z within R.
Let z and z' be any two points within R, then, since
* See Harkness aiul Morley, Treatise on the Theory of Functions (1893),
§ 79 ; or Stolz, AWjenieine Arithmctik, ThI. u. (laSG), p. 238.
11—2
IGi
CONTINUITY OK INFINITE PHODUCT CH. XXVI
if>{z) and 0(c') are each iiiiito both ways, it is sufficient to prove
that L \4>{z)/'f>{:)\ = l.
Let
where P., (?», &c. have the usual meanings.
Since tlie product is uniformly convergent, it is possihle to
determine a finite integer v (independent of z or z) such that,
when n-d(.v, wo have
|Q.-1|<€, and IQ-.-lKc,
where € is any assigned positive quantity however small. Hence,
in particular, we nmst have
\Q.\ = i + e., |Q'.| = i+x«;
where 0 and x are real quantities each lying between - 1 and + 1.
Now
*(^)
*(«)
=
Also, since L \P',IP,\ = \, v being a finite integer, and, e
t-t
being at our disposal, we can without disturbing v choose |s -a'l
80 small that ! P',IP, | = 1 + i/'f, where - l<i/'< + 1.
Hence
<^{z)
-1
(1
+ ^<) (1 +
xO
1+tfe
(-^
+ X - fi) « + iZ-X**
1 +<>€
< €
3 + e
Since €(3+e)/(l— e) can, by sulliciently diminishing €, be
maile as small as we please, it follows that L | ^ (s')/^ («) | = 1.
Cor. 1. //" /i„ nntl u\ (z) i><it!.i/i/ the rnmlitions o/tAs ejcampU
in § IC, thfii n {1 + /i„jf„(j)I /.< a continwnts J'unctiim of z within
the region H.
For, if wo use da.shea to demite ahsoiuto values, we have
|f^- 1|<(1 +/i',^,W.m)(1 +/^'«f»«''.t,i). . .-1.
§ 28 CONTINUITY OF INFINITE PRODUCT 1G5
Since ;/;„ (-) is finite for all values of n aud :, we can find a finite
upper limit, g, for !«'„+,, w'„+2, . . . Therefore
I (?„ - 1 I < (1 + !7/„+,) (1 + ffn'n+n) ... - 1.
Since 2/i'„ is absolutely convergent, 2.r//i'„ is absolutely con-
vergent. Hence n (1 + ^7/,,) is absolutely convergent ; and we
can determine a finite integer v (evidently independent of z,
since g and /a'„ do not depend on z), such that, when 7i<t''>
(1 +gr|x'„+,)(l +giJ.'n+2) . . . -1<£. Hence we can determine v,
independent of z, so that | Q„ - 1 1 < £, where e is a positive
quantity as small as we please. It follows that n {1 +/i„w„(«)}
is tiniformly convergent, and therefore a continuous function of
z within li.
Cor. 2. ^ ^a^z" be convergent when \z\ = B, then n (1 + a„s")
converges to ^ (z), ichere <f> {z) is a finite continuaus function of z
for all values of z such that |s|<ii!.
Cor. 3. If f(n, y) he finite and single-valued as regards w,
and finite, single-valued, and continuous as regards 1/ within the
region T, and if ^'{n, y) z" he ahsolutcly convei-gent wJien \z\ = Ii;
then, so long as\z\<li,'0.{\ +f(n, y) s") cotiverges to tp (y), where
'^(y) is a finite continuous function of y ivithin T.
Cor. 4. If 2a„ be absolutely convergent, then EI (1 + a„z)
cotiverges to i/f (z), where ip (s) 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, far a con tinuum of values of z including 0, II ( 1 + a^z")
and n (1 + ftnc") be both absolutely convergent, and n (1 + a„s") =
n (1 + i„3"), then rti = bi, a^ = b,, . . ., a„ = i„, . . .
For we have
21og(l+a„2») = 21og(l + 6„c'').
both the series being convergent.
Hence for any value of z, however small, we have, after
dividing by z,
la^z"-' log (1 + a^z'-f^'' = 26,s"-' log (1 + />„=")"'-^.
Ififi nnOTS OF A\ INFINITE PRODUCT Cll. XXVI
yince i log (1 +a,2")""^'= 1, wo have, for very small
values of z,
(hAi + aiAiZ + a,At:^ + . . .=biBi + l>.iBjZ + b,B,^+ . . . (1),
where A,, At, . . ., /?,, //j difler very little from unity, and all
have unity for tlioir limit wliun c = 0.
ITcnee, since in,;""' and 2<<,c""' are, liy virtue of our
li)'l)otheses, absolutely convergent, we have
L {cUiA«: + a-,A,!^ + . . .) = 0
L {b.,BtZ + tj/^.r" + . . . ) = 0.
Hence, if in (1) we put z = Q, we must have
a, L ^ 1 = 6| L yy, .
But LAx = LBi = \ ; therefore a, = bi. Removing now the
common factor l+rt,2 from both products, and proceeding as
before, we can show^ that «, = /*,; and so on.
§ 29.] The f<jllowing theorem gives an extension of the
Factorisation Theorem of chap, v., § 15, to Infinite Products.
Jf i/' (2) = n (1 + a,j) be contergont for all i-ulii^i of z, in the
sense that L \ ^Q, — 1 1 = 0, when n = « , no matter itftat value m
may hate, then <p{z) will vaiii.<h if z hmv one of the tYi/;/<',« - 1/a,,
- l/oa, . . ., - \/a, and, if i/'(s) = 0, tht-n z mujst have one
of the tallies - 1/a,, - l/oj, . . ., - l/or, . . .
In the first place, we remark that, by our conditions, the
vanisliing of LnQ„ when n = oc is precluded The exc«i)tioniJ
ca.se, mentioned in §23, where 21og(l+«,c) diverges to - »,
and n (1 + a,z) converges t^i 0 for all values of z, is thus excluded
Now, whatever « may be, we have
^i') = PnQ, (1).
Suppose that we cause z to appm.'ujh the value —l/or. We
can always in the equation (1) take n groat<'r than r; so that
1 +a,z will occur among the fact4)rs of the integral function P,.
Hence, when z = -l/ar, we have /\ = 0, and therefore, since
Again, suppose that iA(s) = 0. Then, hy (1), P,Q, = 0
But, since n may be us large as we please, and L(i» = 1 when
^ 2S, 29 FACTORS OF EQUAL PRODUCTS 1G7
m=oo, we can take n so large that <3„ + 0. Hence, if only »t
be large enough, the integral function F,, wiU vanish. Hence s
must have a value which will make some one of the factors of
P„ vanish ; that is to say, z must have some one of the values*
-1/ai, -l/oj, . . ., -1/ar, ...
It should be noticed that nothing in the above reasoning
prevents any finite number of the quantities Oi, a.i, . . ., Or, . ■ .
firom 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 + a„z, the product i/' (z) will take the form
n (1 + a^zY' ; 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. I/z lie within a continuum {£) which includes all the
values
-l/oi, -1/rto, . . ., -1/rt,., ... (A),
and -\lh, -l/b.„ . . ., - 1/6„, . . . (B),
(/■ n (1 + UnzY' and n (1 + InzY' he absolutely convergent for all
values of z in (z), iff{z) 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 + a,zY' = g(z)n(l+ b„zY' (1),
then must each factor in tlie one product occur in the other raised
to the same power ; and, for all the values of z in (z),
f(~)-9{z) (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 scries (A) and (B) are therefore identical.
Since the two infinite products are absolutely convergent, we
may now arrange them in such an order thatai = &j, a., = bi, . . .,
&c., so that we now have
/(s) (1 + a,zY' (1 + a,z)>^ ...=g{z){\ + a.s)"' (1 + a,zY^ . . . (3).
Suppose that /Xi4=v,, but tliat /^i, say, is the greater; then
we have, from (3),
f{z){l+a,zY'-'''{l+a,z)'^. . .=g{z){i + a.,zy'. . . (i).
168 FACTOIIS OF EQUAL PRODUCTS CH. XXVI
Now this is impossible, because the left-hand side tends to 0
as limit wlicn z=-l/at, whereas the richt-hand side does not
vaiiisli wlien 2 = — l/o,. We must thcrefure have /*] = >',; and,
in like manner, /tj = v, ; and so on.
We may therefore clear the first n factors out of each of the
products in (1), aud thus deduce the eiiuation
/(z)Q. = ff{=^)Qfn (5),
where Q, aiitl Q\ have the usual meaning. The equation (5) will
hold, however large n may be. Hence, since LQ^ = L(/n = 1, wo
must have
/(--)=i7(--).
Cor. 2. From t/ii.< it/nllows that a given /unction of z tchich
vanishes for any of the values (A) and for no others within thr
continuum {z), can be expressed within (z) as a convergent it\finite
product of the form f(z) U (1 +«,;)•'• {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 t<i
express a function, having given zero points, in the form described
in the corollary. On this subject the student should con.sult
AVeierstrass, Abhandlnngen am dcr Functlonenlehre, \i. 14 et saq.
K.STIMATION OF THE IlKSinLE OK A CONVERGING SE1UF:.S OR
iNKi.NiTE runnrcT.
§ 30.] For many theoretical, ancl for .some practical purposes,
it is often required to assign an upper limit to the residue of an
infinite series. This is ciisily done in what arc by far the two
nio.st important cases, namely: — (1) Where the ratio of converg-
ence (pn = Un+t/u») ultimately becomes less than unity, and the
terms are all ultimately of the same sign ; (2) Where the terms
ultimately continually diminish in numerical vaiuo. and altoni.'ite
in sign.
Cast (1). it is e.^>eutial to distiugui^ili two varieiics of series
§§ 29, 30 RESIDUE OF A SERIES 109
under this lioad, namely: — (a) That iu which p„ descends to its
limit p ; {b) That in whicli pn ascends to its limit p.
In case (a), let n be taken so large that, on and after n, p„ is
always numerically less than 1, and never increases in numericaP
value. Then
Rn = Un+1 + M«t!! + «'«+3 + . . . ,
= M„+i -^ 1 + + . + . . . K
I "u+l *'n+2 M„+i J
= M„+i {1 + p„+i + p„+i p„+2 + p„+i Pn+2pn+3 + ...}•
Therefore, if dashes be used to denote the numerical values,
or moduli, of the respective quantities, we have
R'n'ifu'n+l {1 + P'n+I + P'n+l' + • • •},
>«<'n+l/(l - P'n+l),
:t>M'„+,/(i - u'„+./u'„+,) (1).
And also, for a lower limit,
ii;'n<w'n+i/(l-p) (2).
In case (b), let n be so large that, after n, p„ is numerically
less than 1, and never decreases in numerical value. Then
Ji„ = «<„+! {1 + p„+i + P„+2P„+1 + . . •}•
.B'„>?«'n+i{l + p + p- + . . .},
:t>M'„+i/(i-p) (3);
and we have also
-K'„<t?t'„+l/(l - P'n+l),
<j: ?«'„+,/(! - u'n+,/u'„+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
il n = W n+i ~ U n+i + U ,,+3 — . . .
>f4+i (5);
<t«'n+l-«'«+J (6).
170 RESIDUE OF AN INFINITE PnODUCT CH. XXVI
§ 31.] Pesldue of an Infinite Product. Let us consider the
infinite products, n (1 + «,) aud n (1 - u,), in which m, becomes
ultimately positive and less than unity. If the series 2«, converge
in such a way that the limit of tlie 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\ denote the products of all
the factors after the «th in 11 (I + «.) and n (1 - u,) respectively,
so that Q,>1, and Q'»<1. We suppose n so great that, on
and after n, u, is positive, p, less than 1, and either (a) p, never
increases, or else {b) p, never decreases. In case (o), Su, falls
under case (1) (a) of last paragraph ; in case (b), isu, falls under
case (1) (b) of last paragraph. We shall, as usual, denote the
residue of 2«, by It, ; aud we shall suppose that n is so large
that |J?,|<1.
Now (by chap, xxiv., § 7, Example 2),
Q» = (1 + «.+,) (1 +«,+,) . . .,
>1 + »/.+ ,+ M.+J + . . .,
>1 + /^, (1).
Q'n^{l -«» + l)(l -«,+5) • • M
>1-/^ (3).
Also,
l/Qn = {1 - «.+l/(l + «.+.)} {1 - «.W(1 + «.«)} • • M
>1 - «»m/(1 + «.4l) - ««W(l + «*«+l) -• • •»
>1 -«, + ,-«,+,-. . .,
>l-/.V
Whence <?.-!< nj{l - /?.) (.3).
In like manner,
1/Q'. = {1 + ««,m/(1 - «.^i)} {1 + «.«'(! - ««-«>} • • ..
>1 +W,+,/(l -«.,,) + «.fj/(l -"»»:)+• • •»
>1 +W.,., + «,+, + . . .,
>1 + /.'..
Whence 1 - U:>I.'J(i + /■'.) (».
^31,32 DOUBLE SERIES DEKIXED 171
From (1), (2), (3), and (4) we have
/i„<Q,.-l<7?„/(l-70 (5);
E,J{l + R„)<l-Q\<E„ (G).
Since upper and lower limits for 11,,, can be calculated by
means of the inequalities of last paragraph, (5) and (G) enable us
to estimate the residues of the infinite products IT (1 + «„) and
n(i-«„).
Example. Find au uiiper limit to the residue of n (l-i"/H), x<l.
Here «„=x"/;i, p„=.T/(l + l/n), p=x. The series has an aseendinR con-
vergenoy -ratio ; and we have iJ„<«„^.J(l -p)<a.-"+i/(tt+l) (1-x). There-
fore, 1- Q'„<.r"+'/(H-(-l) (1 -x). Hence, if 7"„ be the ;ith approximation to
11(1 -x"/h), P'„ differs from the value of the whole product by less than
100x"+V(K-l-l)(l-a;) «/o of P'„ 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 Algebrique, and § 8 of the liesumes Ana/jjtiques.
Let us consider the doubly infinite series of terms repre-
sented in (1). We may take as the general, or specimen term,
Mm, n, where the first index indicates the row, and the second the
column, to which the term belongs. The assemldage of such
terms we may denote by 2«m, % ', and we .shall speak of this
assemblage as a Double Sei-ii's*.
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 convergcncy 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
DIFFEUENT DEFINITIONS OF THE
CII. XXVI
J^irst Way. — We may define tlie finite sum to be the sum of
all the mn terms witliin the rectnngiilar array OKMN. Tliis
wc denote by iS„, ,. Then we may t^ike tlic limit of this by
first making m and finally n infinite, or by first making w in-
finite and finally m infinite. If the res\dt of both these limit
operations is the same definite quantity S, then we say tliat
2«,i, a converges to S in the first way.
0
A
B 0 n
^
A'
13'
C
U'
"i.i
"l.t
»!.»
"1.4
«!.«
"l.M-1
"ii
"as
"2.3
"*4
"•.■..
"1..+1
"a.1
"3.5
"J.S
"*4
"l.
">.»+l
"4.1
"..J
"4. J
"4.«
"4..
"4..+1
•
K'
",..1
"...J
".UJ
"..4
"...
"■w.+l
•
•
•
•
L
"m.l
"m.l
"m..
"m.4
"m..
"m..+l
N
"nv+I.I
"m+l.J
"m+I.J
"in+1.4
"m+1..
"ll»+),»+l
- (1).
It ni:iy, Iiowcver, happen— Ist, that both these openitiona
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 SUiM OF A DOUBLE SERIES 173
other not ; 4tli, that oue leads to one detiuite finite vahie, and
the other to another definite finite vahie*. 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 Jj, „, Tj, „, . . ., ?'„,,«• Define
S'^,„=T,,,+ T.^,, + . . . + r„,„ (2)
as the finite sum.
Then, supposing each of the horizontal series to converge
to Ti, To, ■ . ., Tm respectively, and STm to be a convergent
series, define
S'='T^+T^ + . . . + T„ + . . . 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 Z7i, m, f^2, m, • ■ •, f^n, m-
Define
S",„,n=Ul.m+U,,„, + . . .+ U„,„, (4)
as the finite sum.
Then, supposing these vertical series to converge to Ui, U^,
. . ., Un respectively, and 'S.Un to be a convergent series,
define
S"= Z7i+ U. + . . .+ £/•„ + . . . ad CO (5)
as the sum to infinity in the third way.
So long as m and n are finite, it is obvious that we have
^ m,n~ ^ m, n *-'m, n ,
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
S'=L{LS,n,„\ (6);
m=-B n=«
and S"=L{LS„,.} (7);
A^ao m^s)
• Examples of some of these cases are giveo in § 35 below.
174 DOUBLE SKRIES OF POSITIVE TERMS CIL XXVI
BO that S' and S" will be c<ni!il t<> cjkIi otlicr and to 6' when the
two ways of taking tlio limit of A',,,, both lead to the Kimo
definite finite result*.
Fourth Wni/. — Sinn tlic tcnns which lie in the successive
diagonal lines of the array, namely, A A', BB', CC , . . ., KK';
and let these sums be />,, A, .... /),+, resi)ectively ; that is,
A = «1.I, A=Ki,. + «=, A+i = «,., + «,,,-, + . . . + «,,,.
Define
-5>";=A + A + . . . + />. (8)
as the finite sum ; and, supposing 2Z), to bo convergent, define
S" =LK + D, + . . . + />, + . . . ad « (9)
as the sum to iiijiniti/ in the fourth u-ai/.
The finite sum according to this last definition includes all
the terms in the triangle OKK'; it can therefore never (except
for m = n=l) coincide with the finite sum according to the
former definitions. Whether the sum to infinity {S'") accortling
to the fourth definition will coincide with S, S', or S", dei>ends
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 the terms are all ultimately of
the same sign. By f;ur the most important kind of double series
is that in which, for all values of m and n greater than certain
fi.vcd limits, «„, , 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 tonus of
the scries, we may, so far as questions regarding convergency
are concerned, suppose all the terms of 2«», , to have the same
(say positive) sign from the beginning. Suppose now (1) to
represent the array of terms under this l;ust su])position ; and let
us farther suppose that 2u,^ , is convergent in the first way.
Then, since L{S^^_n*^-S^») = 8-8=0, when TO=ao,
n = 00 whatever p and q may be, it follows that the sum of all
* For an ilUmtration of the caso whco tbia ia not to, tea below, { 35.
t S«o bvlow, § 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 tlie right of these lines respectively,
must become as small as we please when we remove NM suffi-
ciently for 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 and 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 w = oo .
2nd. The same is true for each of the vertical series.
3rd. The same is true for the series 2Z)„.
The last inference holds, since >S""„ obviously lies between
/S',,„-, and Sn-i,„-i.
Hence
Theorem I. If all the terms of %Um, „ be positive, and if the
series be convergent in the first sense, then each of the Imizontal
series, each of the 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 ;<„, „ becomes the term «„■. „-, 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 talue for every distinct pair of values of m and n (say
non-repeating functions), and each of which is finite for all finite
tallies ofm and n (Restriction A*), then we shall obviously leave
the convergency of the series unaffected. Hence
Cor. 1. If 2Mm, „ be a series of positive terms convei-gent in
the first way, then any commidation 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 tlie same limit S as before.
* No 6Uoh restriction is usually mentioned by writers on this subject;
but some such restriction is obviously implied when 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 DOUIlhE SKKIIS (»1 rosiTIVK TKlt.MS til. XXVI
Cor. 2. Jj titu terms (all jmsitict:) nf a coin\rijvnt siiigLi geries
iiMn (»> urraiKjid into a double svr'us 2h„._„., wlure m and n are
functions of n suhject to Restriction A, then Su.', „> will converge
in all four ways to the same limit a.i 2«,.
It shoiilil be noticed tluit tliis la.st corollary gives a, furtlier
extctLsion of tlio laws of coiDiinitivtioii and a.s.sociation to a series
of positive tonus ; and therefore, as we sliall see presently, to
any absolutely convergent series.
Let us ne.xt a.ssunic that the scries 2h„_ , is convergent in tho
second way. Then, since ^T„ is convergent, wc ciin, by suffi-
ciently increasing m, make the resiilue of this series, that is, the
sum of as many as we choose of the terms below the iufmite
horizontal line iVJ/, less than it, where t is as small as wo
please. Also, since each of the horizontal scries is, by our
hypothosi.s, convergent, we can, by sufficiently increasing n, make
the residue of e;K'h of them, less than c/'i;/* ; and therefore the
sum of their residues, that is, as many as we please of the terms
above iVJ/ producoil and right of J/A", less than J t. Hence, by
sufficiently increiusing both m and n, we can make the sum of
the terms outside OKM N, less than e, that is, as small as we
please. From this it follows that 2«„, „ is convergent in the
first way, and, therefore, by Theorem 1., in all the four ways.
In exactly the same way, we can show that, if 2m„, , is cou-
vergeut in the third way, it is convergent in all four ways.
Finally, let us assume that -u„^ , is convergent in the fourth
way. It follows that the residue of the diagonal series i/>,, can,
by making p largo enough, be made as small as we jilease.
Now, if only m and » be each largo enough, tho residue of <Si«,,,
that is, the sum of as many as wo jdejuse f(f the tt'rins outside
OKMN, will contain oidy t4.Tnis outside OKK', all of which are
terms in the residue of S"'p. Hence, since all the terms in the
army (1) arc positive, we can niake tlie sum of as many a-s wo
pleiuse of the terms outaido UKMN as snniU as we pltaac, by
^j 33, 3-i cauchy's test for aiisolute convergency 177
sufficiently increasing both 7« and 7i. Therefore S«„,_„ is con-
vergent in the first way, and consequently in all four ways.
Combining these results with Theorem I., we now arrive at
the following : —
Theorem II. 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
scries, horizontal, vertical, and diagonal, are all convergent.
Cor. Any single series 2m„ consisting of terms selected from
~t'm,n {nnder Restriction A) will be a convergent series, if 2!/„,,„
he convergent.
Restriction A will here take the form that n must be a
function of m and n whose values do not repeat, and wliich is
finite for finite values of ;« and n.
Example. The double series 2x"'y" is convergent for all values of x
and y, such that 0<x<: +1, 0-ci/< +1.
For the (m + l)th horizontal series is ^'"Si/", which converges to x"'/(l - y)
BinoeO^y < +1. Also Si"7(l-J/) converges to 1/(1 -x)(1-j/) since 0<x< +1.
§ 31.] Absolutely Convergent Double Se>-ies. — 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 idtimately 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
d<iuble series.
Theorem III. If u'^.n be the numerical or positive value of
i/„,n, and if all the horizontal series of 2M'm,n be convergent, and
t/ie sum of their sums to infinity also convergent, then
1st. 77/6 Horizontal Series of 2«„,,n are all absolutely con-
c. II. 12
17S EXAMPLES OF CAUCHT'S TEST CTl. XXVI
wrgetU, and the sum of their sums to infinity converges to a
definite finite limit S.
2nd. -«„,n converges to S in the first way.
3r(L AU the Vertical Series are 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 iM„,, {under
Restriction A) is absolutely convergent.
The like conclusiims also follow, if all the vertical series, or if
the diagonal series of -«'„.» be convergent.
Cor. Xf "StU^ and S», be each absolutely convergent, and con-
verge to u and v respectively, then 2 ((/,i'i + «,-it', + . . . + u,v„) is
absolutely convergent, and converges to uv.
For the scries in question is the diagonal series of the double
series 2«„rn, 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 scries 2 (-)"*, C,,!*""^'"
{n-t m, ,Cg:=l) lie absolutely convei');cnt; and find its sum.
The scries may be arranged thus : —
1 + x+ x'+. , . +i"+. . .
-y - 2yx- Syx'-... _(n + l)yi«- . . .
+ y»+ 3y'j-+ Ci/'i> + ... + }(n + l)(n + 2)y>i"+. . .
(-ry"' + (-r„+,C,y'»x + (-r„+,C,s,-x»+. . . + (-)-«+,C,y-x»+. . .
ir x' and y' bo the moduli, or positive values, of x and y, then the ecriei
2m'„,, correspoiidiuK to the above will bo
1 + z'+ x'»+. . . +x'"+. . .
+ y'+2y'x' + :tyV+. . . + (n + l)y'x*+. . .
In order that the horizontal series in this last may bo eonvcrf^nt, it is
BoooBsary and snlTicicnt that x'< 1.
Also !''„+, = !/'•"/( I -x')""*'; hence the necessary and rafBcient condition
that 27*„ be convergent is that y'<l-x', which implies, of course, that
The given series will thcrafnro satisfy Cauchy's conditions of absolute
eonvergi'ncy if |x|-;l, |x| + |t/|<l, and conseiiucntly also |y|«-L
These being fulflllod, we have Tim^i = ( - )"V"/(1 - x)"^' j
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 + y).
Example 2. If Ur=a;=' + a:-'*' + a;'^"+ ..., where i<l, show that
t/o Ui Ifc. - x'^ X-' x'^
2» 2' 2- " 2" 2' 2- ■■■
Let S denote the series on the left. Then S may be written as a double
series thus,
i(x=^+^' + x^V. . .+x^+. . .)
+ 2j(0 +ir+x--+. . . + x- +. . .)
+ 2j('' +0 + 1-'+. .. + !="+.. .)
Now each of the vertical series is absolutely convergent, and we have
l7,=ar''(l-l/2»+i)/(l-^) = x=" (2-1/2"). 2y„is of the same order of con-
vergence as 2x'", hence it is absolutely convergent. Also aU the terms of the
double series are positive. The double series therefore satisfies Cauchj-'s
conditions ; and its sum is the same as that of 2f7„, or of 2r„. Now
ZT^=u„l20 + u,l2> + u,l2-+. . .;
and 2l/„=2x="(2-l/2»),
= 22j;=''-2x-"/2",
= 2u„-i2°/2»-a:=V2'-. . .
Hence the theorem.
§ 35.] Examples of the exceptional cases thai 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
"....»=.'('«> n)-f{m-l, n)-/(m, n-l)+/(ni-l, n-1).
Hence we have only to Bive/(m, ;i) such a form that
L { Lf{m,n)]* L { Lf{m.v)],
12—2
ISO EXCEPTIONAL CASES CH. XXVI
and we shall liave a series whose sums of the second and third kind are not
alilie, and which cousciiucntly has no dclhiite gum of the first kind.
Suppose, for example, tbat/(m, fi) = (m + l)/(m + n + 2), then
u^,= (m + l)/(m + n + 2)-m/(m + n + l)-(m + l)/(m + n + l) + in/(m + n),
= (m-ii)/(m + n)(m + »i + l) (m-t-n + 2).
It is at uDcc obvious that the sums of the second, third, and fonrtb kind
for tliis series are all diffurent. For in tlio first pliioe we observe that
ij,^^= -u^„. Hence there is a "skew" arrangement of the tcnns in tlie
array (I), such that the terms equidistant from tlie dexter diagonal of the
array and on the same perpendicular to this dia;j;oual are equal and of opposite
sign, thoBe on the diogoual itself being zero. Each term of the diagonal series
SD, is therefore zero ; and the sum of the fourth kiud is 0.
Also, owing to the arrangement of signs, we have T,^^= - P,,.,; and.
since each of the horizontal and each of the vertical scries in tbis case is
convergent, T,„= - l'^, and therefore 8"= -S".
Now
r».,= 2 [(m + l){l/(m + n + 2)-l/(m + n + l)}-m{l/(m + n + l)-l/(m + ii)|].
= (m + l){l/(m + n + 2)-l/(m + 2)}-m{l/(m + n + l)-l/(m + l)>.
Hence
T„= -(m + li/(m + 2) + m/(m + l)= -l/(ni + l)(m + 2).
The series ST„ 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 sura it in the second, third, or fourth way.
At first si);ht, the reader might suppose (seeing that the horizontal series
are all nbsolutcly convergent, and that the sum of their actual snms is also
absolutely convergent) that this case is a violation of Cauchy's criterion.
13ut it is not so. For, if we take all the terms in the mth horizontal Berioa
positively, and notice that the terms begin to be negative after ni = n, then
we sec that T'„ the sum of the positive values of the terms in the mth scries
is given by
»■"! ii—t«-fl
= (Hi + l){l/(2m + 2)-l/(m + 2)}-m{l/(2m + l)-l/(m + l)}
-(in + l){0-l/(2m + 2)}+m{C-l/(2m + l)}.
= l-2ni/(-2m + l)-(m + l)/(m + 2) + m/(in+l).
= (m' + m + l)/(m + 1) (m + 2) (2m + 1).
Now the convergence of ~T'„ is of the same order as that of Dl/m, thut is
to say, 2I"„ is divergent. Hence Cauchy's conditions are not fully satisin- 1;
and tlie anomaly pointed out above reuses to be surprising. The present com
is an excellent example of the care required in dealing witli double series
which are wont to bo used somewhat recklessly by beginners in mathematics*.
* Doforc Cauchy the reckless use of double scries and riininqnwl
perplexity was not confined to beginncra. Bee a curious pojicr by Babbaf^
J'hiL Tram. It.S.L. (1819).
§§ 35, 36 COJU'LEX DOUBLE SERIES 181
Example 2. The double series Z{-y"*^llmii, whose horizontal aud
vertical series are each semi-convergent, converges to the sum (Ior 2)= in the
second, third, or fourth way (see chap, xiviii., § 9, and Exercises xiii. 14).
Bot 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 2(i„ and 26„ converge to a and b respectively,
and at least one of them be absolutely convergent, then it follows from § 14
that the double scries 2a„6„ converges to the same sum, namely ab, 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„, Zb,^, 2 (a„ii + a„_iia
+ . . . +ai6J be all convergent, even if no one of the three be absolutely
convergent*.
If, however, both 2a„ and 26„ 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/(m'i)'' where
o is a quantity ^ing between 0 and ^. This series obviously converges to the
finite limit (1 - l/2"-^l/J''— ...)'- in the second and third ways. For the
diagonal series we have
C„= 2 l/r«(ii-r)'».
r— 1
Now, since 0<o<l, we have, by chap, xxiv., § 9, r" + («-r)''<2i"'"{r
-f(K-r)}''<2i-''»«.
Therefore
1 2'-'«° 1 ^r°+(H-r)''
-'^n- 2i-a„<i ,-<i(n-,)a "* 2'-° ji" "^ r" (it - r)" '
2 » 1 2 n
< — 2 — <-
■i 2''7i>--''.
Hence, if a=i, LD„<2«; aud, if o<^, LD„ = a> , when h=od. Therefore
2D„ diverges if 0 < a > }.
IMAGINARY DOUBLE SERIES.
36.] After what has been laid down in § 10, it will be
obvious that, in the first instance, the couvergeucy 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,
2ot„ „ i/?m,„, will be absolutely convergent if the double series
• See Stolz, Allgemeine Arithmetik, Th. i., p. 248.
182 la^^n^y* CH. xxvj
2^/(o'„_, + ^,^,) is convergent. Hence, if «'^, denote the
nioduluB of u„., = a^, + I'/Sm,,, we see that *««., will converge
to the same liiuit in all four ways if 2m'„,. be convergent
In this case wo say that the imaginaty series is absolutely
convergent.
Since all the terms «'„,, are positive, we deduce from
Theorem II. tlie following: —
Theorem IV. J/ all the horizontal series in the double series
formed by the moduli of the terms ofZu^^^ be convergent, and the
sum of their sums to infinity be also convergmt, then the series
2«m,, 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«„., under Restriction A. Tlieorem IV.,
of course, includes Tlieorem III. as a particular case.
§ 37.] The following simple general theorem regarding the
convergency of the double series 2o„mX"y will be of use in a
later chapter.
If the moduli of the coefficients of the series 1an_^ify^ have a
finite upper limit X, tlien ^a^^^nX^y* is absolutely convergent for
all values of x and y such that |a:|<l, |y|<l.
For, if diLshes be used to indicate moduli, we have, by
hjiiothesis, a'„_n^K Hence the series 2a',^„x''y'' is, a fortiori,
convergent if the series 2Xa:'"y" is convergent ; tliat is, if
Vj.'«y'» ig convergent Now, as we have already seen (§ 33),
this last series is convergent provided x'<l audy<l. Hence
the theorem in question.
Exercises VIII.
Exnmiue the oonTcrgoncjr of tbo Korics wlioao nib tcnus aro ibo
followinK : —
(1.) (l + n)/(l + n«). (2.) nP/(n» + a).
(3.) «-«^ (4.) !/(«•* 1).
(5.) lMn^-n){^fi-^{n-l)\. (G.) «•/("" + '•).
(7.) (nl)»x«/{2n)l. (8.) n'/n!.
(9.) {(y + «-)/(• -«•))>/». (10.) nlo«{C.',. + l)/(2n-l)}-l.
(11.) 1.3.6 . . . (2n-l)/2.4.0. . . a™.
(12.) {l/l« + l/'.>« + . . . + l/n«|/«».
^ 36, 87 EXERCISES VIII 183
(13.) 1/(«K + /'). (14.) n/(aH= + fc).
(15.) m(m-l) . . . (m-n + l)/n". (10.) {(;i + l)/(n + 2)}''/n.
, „, , m )n(m + l) m(m+l)(;» + 2) . .
(17.) Show that - + ;' + V-nJI-T-oi + • • • 18 convergent or
* ' n 7i(n + l) 7t(ii + l) (n + 2) x
divergent according as n - ;«>■ or > 1.
(18.) Show that ai/" + aV"'+V('"+i) + aV™+i/(»>+i)+i/(m+=)+ ... is conver-
gent or divergent according as a < or «t 1/e. (Bourguct, Nmw. Ann. , ser.
II., t. 18.)
(19.) Examine the convergenoy of Sl/n'"*'"".
(20.) Show that 2n"'/(7i + l)'^" is convergent or divergent according as
o>or>l. (Bertrand.)
(21.) Show that 21/;i log n {log log nj" is convergent or divergent accord-
ing as o > or <: 1 .
(22.) Show that S1/(k + 1 + cos «jr)= is convergent. (Catalan, Traile El.
d. Series, p. 28.)
Examine the convergency of the following infinite products : —
(23.) II{1 +/(")'■"}, where/(K) is an integral function of n.
(24.) n{(x2»-H.r)/(x="-t-l)}. (25.) nK+>/(n-l)»(n-l-2)}.
(26.) If 2/(n) be convergent, show that, when »i = co ,
L{n(x+f{n))}^l»=x.
1
(27.) If p denote one of the series of primes 2, 3, 5, 7, 11, . . ., then
2/(p) is convergent if 2/(p)/logp is convergent. (Bonnet, Lioui-ille's Jour.,
Tin. (1843), and Tcbebichef, ib., xvii. (1852).)
(28.) If x<l, show that the remainder after n terms of the series
l'•x + 2'■x■-'-^3'■a:^-^ . . .
is <(n-|-l)'x»+V{l-(l-hl/")''a:}-
(29.) If Uj, «j, ...,!(„ be aU positive, and 2»„x'' be convergent for all
values of x- < a', then
2..-j«„-(« + l)««,,. + i^±;n';t^aX«-*e.[
will be convergent between the same limits of x.
(30.) Point out the fallacy of the following reasoning : —
Let S=l + i-Hj-l- . . . ad CO,
then log,2 = l-i + ^-i-t-. . .
= (l-l-}-l-i-)-. . .)-2(.J-l-UJ+. . •)
— 2 — 22/2 = 0.
(31.) If p and p' be the ratios of convergence of Sl/P^-i (") {'''"' "V*" i"^
Sl/P, (») {''•«}'+''' (see § 6), then L (p'„ - />„) Pr-i («) = a, when n = oo . What
conclusion follows regarding the convergence of the two series ?
(32.) If 2u„ is divergent, then 2m„/S„_,"» is divergent if o>l (where
S,= Ui-H(,+ . . . -h«„), and SiiJSn^+i is convergent if o>0. Hence show
ISi EXEKCISES VUI CU. XXVI
that there can be no function 0 (n) BQch that every Ecries £u, ia convergent
or divergent, according aa L <t>{n) u,= or +0. (Abel, CEuvrei, ii., p. 197.)
(33.) If 2u, be any convergent series whoso terms are nltinmti-ly positivo,
we can always find another convergent scries, -t\, whoso terms are ultimately
positive, and snch that Lvju^ = (z> .
If 2u, be any divergent ecries whose terms arc nltimatoly positive, we
can always find another divergent seriea whose terms are nltimatcly positive,
and such that I,i(Jr, = ao .
(ThcHo theorems are due to Dn Bois-Reymond and Abel respectively; for
concise demonstrations, see Thomae, Elementare Theorie der Analytitchtn
Funetionen. Halle, 1880.)
(34.) If u.+,/u. = (n« + ^n«->+. . . )/(n« + i4 'n«-> + , . .), then Su, wiU
bo convergent or divergent according as ^-.<l'>or >-l. (Gauss, tVerke,
Bd. III., p. 1.S9.)
(35.) If u,+,/M,=a-/3/n+7/n*+«/n'+ . . ., then 2u, is convergent or
divergent according as a< or >1. If a = l, Su, is convergent only if /S>1.
(Schliimilch, Zeitschr. f. Math., i., p. 74.)
(36.) 21/u, is convergent if u„+, - 2ii„+, + u, is constant or ultimately
increases with n. (Laurent, Nmiv. Ann., ser. ii., t. 8.)
(37.) If the terms of 2i/„ are ultimately positive, then —
(I.) If ^(n) can be fuund such that ^(ri)i9 positive, X,f(n)u, = 0, and
Xi {v^(") "«/",+! - ^ (n + l)} >0, -u, is convergent.
(U.) If ^f.(n) besueh that L^t(n)M,=0, I, i^(n)uJ«,+,-^(n + l)}=0,
and Lrf- (n) uj{<f' (ii) uju,^^ - ^ (ii + 1) | + 0, 2u, is divergent
(III.) If tiju„+, can bo expanded in descending powers of n, 2u, is
convergent or divergent according as I, {nii,Ju,+, - (n + 1)) > or >"0.
(IV.) If uji'„+, can be cxpauded in descending powers of n, 2u„ is
convergent or divergent according as Lnu^= or #0. (Kommer's Criteria,
Crelle't Jour., xiii. (1835) and ivi.)
(38.) If the terms of ^m, be ultimately positive, and if, on and after •
certain value of n, a,u,Ju,+i-'',»4.i>M, where a, is a function of n which
is always poHitive for values of n in question, and m is a positive constant,
then 2u„ ia convergent.
From this rule can bo deduced tho rules of Canchy, Do Morgan, and
Bcrtrand. (Jensen, Comptet liendiu, o. vi., p. 7*29. 1888.)
Discuss tho convergence of tho following double scries: —
(39.) i:(-)»-'r"/n. (40.) r (-l)»-'r-/nl.
(41.) 2 1 (H-l)'"/n"'+' -«•»/(« + 1)"^')-
(42.) Zx-^y'lim + tt). (43.) 2;i/(m + n)'.
(41.) Zll{m + u). (46.) 21/(m«-n').
(4)i.) Under what restrictions can 1/(1-1-2 + y) be expanded In a double
«eric» of the form l + ^A^^x'^*7
(47.) If -"„^,, converge to .V in the first way, and if its diagonal Mrie* be
convergent, sliow that tho diagonal series converges to S alio.
§ 37 EXERCISES Via 185
Deduce Abel's Tlieorein regarding tlic product of two semi-convergent
series. (See Stolz, ihith. Ann., xxrv.)
(18.) If i(„/u„_, can bo expanded in a series of the form l + a^jn-i-ajir + . . .,
show that
1°. If <r, = 0, a.,=0, . . ., a,i_i = 0, a^=t=0, then «„=u + v„/n, where u is u^
detinite constant +0 and +00, and Lv^ is finite when ji = oo.
2°. If Oj + O, and the real part of a^ be positive, then iu„=oo when
n=QO .
3°. If 0-1 + 0, and the real part of a^ = 0, then Li(„ is not infinite, but is
not definite.
i°. If (ii + O, and the real part of Oj be negative, then Z,«„=0.
Apply these results to the discussion of the convergency of SKni", and,
in particular, to the Hypergeomctric Series, and to the following series : —
^M-viGni^ + yi)", 2a-»/«''+'"-, X^GJ{m + n)>>, 2 ( - ) VCJ(nH- n)".
(See Weierstrass, Ueber die Theorie der Analylischen Facultdt. — Crclle's
Jour., LI.)
(19. ) Discuss the convergence of 2 „(7„ (a - n/S)"-' (x + n/S)".
(50.) If u„ and !'„ be positive for all values of n, never increase when n
increases, and be such that iu„=0, Lv„ = 0, when n = x, find the necessary
and sufficient condition that 2 (H„rj + «„_it'2 + . . . -n/irj = 2«„x 2t'„. (See
Pringsheim, Math. Ann., Ed. xxi.)
(51.) If 0 <: 3/„ <: il/„^.i and X..1/„=0 when n = oo, show that every diver-
gent series of real positive terms can be expressed iu the form 2 {M„^i - il/„) ;
and every convergent series of real positive terms iu the form 2 (J/„+i - J/,,)/
Also that the successions of series
S(il/„+,-]l/„)/P,(J/„), r=0,l,2, ...
S(3/„+,-il/„)/P,(il4«) (^.il/„«)^ r=0, 1, 2, . . .,
where 0<p<l, and Pr{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, Matli. Ann., Ddd. xxxv., xxxix.)
CHAPTER XXVII.
Binomial and Multinomial Series for any Index.
BINOMIAL SERIES.
§ 1.] We have already sbowa that, when m is a positive
int<^er,
(l+a:r=l+«C,x+„(7,:t' + . . . + «C.x» + . . .+»C,.r- (1).
where „C, = ffi(m-l) . . . {m-n + l)/n\ (2).
When m is not a positive integer, „C although it ha.s still a
definite analytical meauing, can no longer be taken to denote
the number of n-combinations of m things ; hence our former
demonstration is no longer applicable. Moreover, the right-hand
side of (1) then becomes an infinite scries, and h:LS, 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 lea.'sf
regarding the equivalence of the two sides of (1).
As has already been shown (pp. 122, 131), the series
l + ^CtX + ,Ctx' + . . .+.C,j' + . . . (3)
is convergent when x has any real value between - 1 and + 1 ;
also when x = +l, provided m>-l; and when ;r = — 1, pro-
vided m>0. We propose now to inquire, whether in these casc-
the series (3) still represents (1 + x)" in any legitimate seasc.
In wliat follows, wo suppose the numerical value of m to be
a commensurable number*; also, for the present, we consider
* If m be ioeommeniurnblc we moat snppoM it replaced hj • coauocnior-
able approzimatioD of aufiicicnt aoouraoy.
§§ 1, 2 FIRST PROOF 1S7
only real values of .r, and understand (1 +x)'" to be real and
positive.
§ 2.] If we assume that (1 +z)'" can be expanded in a con-
vergent series of ascending powers of x, then it is easily shown"
that the coefficient of *•" must be m (/»- 1) . . . {m-n+\)ln\.
For, let
{i+xy = (h + aiX + aia? + . . .+a„a:" + . . . (1)
where a„ + aiX + (ha? + . . .+«„a:" + . . . (2)
is convergent so long as |a:|<i? (it will ultimately appear that
B=\). Then, if h be so small that \x+h\<B, we have
{I + X + h)"' = a^+ ai,{x + h) + a.i{x + hf + . . . + a„(a; + /;)"+. . . (3),
the series in (3) being convergent by hypothesis.
Hence by the principles of last chapter, we have
(l+a; + ^)"'-(l+a;)°'_ (x + k)-^x {x + Kf-a?
(l+x-^h)-{\+x) ~ {x + li)-x ^ {x + h)-x
(x + hY-x''
{x + h)-x
(4).
the series in (4) being still convergent. Hence, if we take
the limit when h = 0, and observe that
{l+x + h)-{l+x) ^ ' ' {x + h)-x
by chap, xxv., § 12, we have
»j(l+ar)"'~' = ai+2a2a; + . . . + «a„a;"~' + . . . (5),
where the series on the right must still be convergent, since
L {n + 1) a„+i/«a„ = Lan+ila^ when m = go *. Hence, multiplying
by ].+ X, we deduce
m (1 + xy = Oi + (oj + 2«2) x + . . . + {w«„ + («+!) a„+i} a;" + . . . ,
that is,
ma„ + nutiX + . . . + ?wa„a:" + . . . = Oj + ((«i + 2a^ x + . . .
+ {nan + (n + 1) fln+i} a;" + . . . (G).
* We hero make the farther assumption that the limit of the sum of the
infinite number of terms on the right of (-1) is the sum of the limits of these
terms.
ISS EULER's I'KOOK CH. XXVII
V>y i-lia]i. XXVI., S 21, tlie cocflii'ipiits of tlio lowers of x on
both sides of (G) must Iw equal. Hence
o, = ma,. 2a.j - (m - l)a,, ...,(«+ l)a,+, = (;/» - «)a„, . . . (7).
From (7) we deduce at once
ai = mat, a, = »n(ni- l)a„/2!, . . .
a, = »j (;« - 1) . . . (n» - n + l)ajn], . . .
To dotcnniue «•» we may put x - 0. We tlien get from (1),
fl^ = 1*" = 1 (if we sujiposo, aa usual, the real positive value of
any root involved to be alone in question). We therefore have
(l+j-r=l+2„C,^ (8).
The thenrcm is tlierefore establi.shcd ; and we see tliat the
hypothe.fis under which we .'Started is not contratlicted providiMl
|ar|<l, tills being a suflicient condition for the couvergcncy of
§ 3.] Although the assumption that (1 + x)" can be expanded
in a series of ascending powers of x leads to no contradictirin in
the process of detenuiniug the coefficients, so long as |a:|<l ;
tiiis fact can scarcely be regarded .-is sufficient evidence for the
validity of a theorem so fundamentally importjiut. We proceetl,
therefore, to establish the following theorem, iu which we start
from the series in the first instance.
W/ienever the seru's 1 + 2;„6',a:* is convergent, its sum is the
real jmsitive value o/" (1 + x)"'.
The fundamental idea of the following demonstration is due
to Euler* ; but it involves important additions, due mainly to
Cauchy, which were necessary to make it accurate according to
the modem view of the nature of iulinito series.
Let us denote the series
\+^C^x + ^C^a* + . . . + ^C^a* + . . . (1)
by the symbol /(m).
So long as —\<x<+ 1, f{m) is an absolutely convergent
series, and (by ch.ap. XXVL, § 20) is a continuous function both
of m and of x.
• Nov. Comm. Petrop., t. m. (1775).
^ 2, 3 BINOMIAL ADDITION THEOREM 189
Hence, m^ and m. being any real values of m, we have
f{m,)f{m,) = {1 + ^^finx"\ {1 + ^^C„ar},
= 1 + 2 UGn + ™,^. mfin-, + ^C, ™,C„-, + . . .+^C:)af^ (2), ^
where the last written series is convergent (by chap, xxvi., § 14),
since the two series, 1 + 2„,(7„.«" and 1 + 2„^(7„a:'', are absolutely
convergent.
Now, by chap, xxm., § 8, Cor. 5,
hence /("'^/("O = 1 + 2„„+„^,C„a,-",
=/(???, + Mj) (3).
In like manner, we can sliow tliat
/(Wll + ?»o)/()»3) =/(»?, + Too + Wis).
Hence /{'mi)/{nh)/{iih) =/{m, + nh + m,) ;
and, in general, v being any positive integer,
/{m^)f{nh) . . . /(?«.) =/(»». + OTn + . . . + m„) (4).
This result may be called the Addttmi Theorem for the
Binomial Series.
If in (4) we put mi = rrh = . . . = 7«„=1, then we deduce
{/(I)}" =/('') (5).
where v is any positive integer.
K in (4) we put mi = m^ = . . . = »w„ =p/q, where p and <?
are any positive integers, and also put v = q, we deduce
{fiplq)V-f{p) (6)-
Hence, by (5), {/(W'i)}' = {fO)V (7)-
Again, if in (3) we put m^ = m,m^=- m, we deduce
/{m)/{-m)=f{m-m)=f{0) (8).
Hence /(- w) =/(0)//(ff») (9)-
These properties of the series (1) hold so long as -l<a;<+l,
and they are sufficient to determine its sum for all real com-
mensurable values of m.
190 SUMMATION OF S„C„«" CM. XXVIl
For, since ,(7,= 1, ,<7, = 0, . . ., ,C, = 0, . . . ,C7,=0, ,C, = 0,
• • • , oC", = 0, . . . we liave
/(1)=1 + ^. /(0) = 1.
Suppose, now, m to bo a positive integer. Then, by (5),
(1 + x)' =/(m) = 1 + ^C,x + ^C,a^ + . . . + .C.o- (10).
where the series terminates, since «C,i+i = 0, «C„+t = 0, . . .,
when m is a positive integer. This is another demonstration of
that part of the theorem with wliich we are already familiar.
Next, let m be any positive commensurable quantity, 8ay
p/q, where p and q are positive integers. Then, by (7),
{/(P/?)1' = (1+^)' (11)-
Hence/ip/q) is one of the yth roots of the positive* quantity
(1 + a-)*". But /{p/q) is necessarily real; hence, if (1 + x)*""
denote, as usual, the real positive 5th root of (1 + x)', we must
have
/0'/7)=±(l+^)"' (12).
The onl}' remaining question is the sign of the right-hand side
of (12).
Since /(p/y) is a continuous function both of p/q and ot x, its
equivalent ± (1 + a')" must be a continuous fiinction both of
p/q and of x. Now (1 + ar)'' does not vanish (or become in-
finite) for any values of p/q or of x atlmi.s.'iible under our present
hypothesis ; and being tiie 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 Ciise. Now, when x = 0, /(p/q) = + 1.
IIcDce the positive sign must be taken ; and we establish finally
that
/<J'/q) = + {i+^)'-',
that is,
(l+a:)- = l+«(7,x + ,C,x' + . . . + «C,jr»+. . . (13),
when m is any positive commensurable quantity.
• roaitivG, since -l-:i<:l, liy h^polliciis.
§ 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
/(-»0=/(0)//(«0 = !//('«')•
Hence, by (13),
/(- in) = 1/(1 + a:)'"-,
= (1+.^)-'"'.
that is,
(I + .r)"- = 1 + ,nCiX + rnC^ar + . . . + M.-r" +. . . (14),
where m is any commensurable negative quantity.
The results of (10), (13), and (14) establish the Binomial
Theorem for all values of a; such that -!<.»•<+ 1. It remains
to consider the extreme cases.
When x = +l, the series (1) reduces to
l + mCi + „,Cn + . . .+„,(7„ + . . .
This series is semi-convergent if - 1< ?« < 0, absolutely con-
vergent if m>Q. Hence, by Abel's Second Theorem, chap, xxvi.,
§20,
(1 + 1-0)"'= L {l+„(7,a; + „C:,.r-= + . . . + ,„a..T" + • • .},
1=1-0
that is,
2"'=l + ™(7, + „,C, + . . . + ,nC„ + . . . (15),
provided »n>-l, with the condition that, when -l<ffj<0, the
order of the terms in the series of (15) must not he altered.
If 0<a;< 1, we have, by the general case already established,
(1 -.?)"> = l-„(7ia;-H„,^,.i--. . .(-)"„.C„,r" + . . .
Hence, since the series
1— mCi + mCj — . . .(-)"mW + . . •
is convergent if 7»>0, we have, by Abel's Theorem,
(1-1^)'"= L {l-n,G,x + ^C,3r-. . .{-T^C„x'' + . . .),
I-l-O
that is,
0=l-„C, + ^C,-. . . (-r„.C„ + . . . (16),
provided m be positive.
The results of (15) and (16) complete the demonstration of
192 rAnrirtiLAR casks rn. xxvii
the Binomial Theorem in all cases where its validity is iu
question.
Cor. If x^y, it follows from the above result that we can
always expand (x + y)" in an absolutely convergent series. Wo
have in fact, if |d;|>|y|, that is, \ylx'\<\,
(x + y)"* = a-" ( 1 + yjx)",
= a- {1+ „<7, {ylx) + „(?, {ylxf + . . . + „C, {ylxf + ...},
= .r"' + „67,j-™->+„(7..r"-y + . . . + ,„C,.z-"'- "y- + • • • (17);
and if |a'|<|y|, that is, \xly\<\,
(a: + y)'"=2/"(l+a-/.y)"',
= y'"{l+-^.(-r/y) + «C;(a-/y)U. . . + „(7,(.r/y)" + . . .}.
= y'^ + ^C,y''-'x+„C,y"'-'j= + . . . + ^C^y-'-'x' + . . . (18).
If »» be a pcsitive integer, both the formula; (17) and (18) will
be jwlniissible because both series terminate. But, if w be not a
positive integer, only one of the two series will be convergent
§ 4.] The general formulas of last paragraph contain a vast
niimber of particular ca.<cs. 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 iu reality
iu most cases much smaller than they at first sight appear. We
assuiue in all cases that the values of the variables are such tlmt
the series are convergent.
Example 1.
(l+j-)-> = l-x + x'-. . . + (-)»j-" + . . .;
(l-i)-' = l + z + is + . . . + !" + . . .
For (l+i)-> = l+S_,C,i";
ami _,C,= -l(-l-l){-l-2) . . . (-1-w + ll/nl.
= (-)"!. 2. 3 . . . ii/iil,
= (-)"!.
(l-i)-> = l + 2.,C.(-x)-;
ana -,C, (-!)- = ( -)"(-)"«" = (-Px«
= x".
Example '2.
(l + x)-' = l-2i + 3i'-. . . + (-)''(n + l)x"4-. . .;
(l-i)-« = l + 2x + ar' + . . . + (n + l)i" + . . .
For _,C,= -2(-2-l) . . . (-2-n + l)/iil,
= (-)'(n + l).
§§ 3, 4 ULTIMATE SIGN OF THE TEUMS 193
Example 3.
(l+x)-s=l-3*+Gx'- . . . +(-)»i(« + l)(tt+2)x''+ . . .;
(l-x)-»=:l + 3x + ().c2+ . . . +i^(,i + l){H + '2)x«+ . . .
Example 4.
(l + x)l=l + Jx-ix' + ^x-3- . . . +(-)■'-' -^i • • • <^""^'^"+----.
2 . 4 . 6 . . . 2»
(l-x)l = l-ix-ix»-,Vx^- . . . _l-3-5-- • (2»-g)^._ . . .
^ . 4 . O . . . J/t
Example 5.
; (l + x)-t=l-ix + ix'-Ar'+. . . +(-)"^-^-f • • • ''•^""^^x"+...;
2 . 4 . b . . , 2n
(l-x)-»=l + Jx + ix» + A^»+ . . . + ^1^5 ■ • • ^^"-l);,n^. . . .
li . 4 . 6 . . . 2/1
Example 6.
nt(TO-2)(m-4) ■ . . (m-2ii + 2) /xV'
«l V2J + ■ • • '
_ m m{m-2) , m (m-2) (m-4) . . .(H>-2n + 2) „,
-^+2^+ 2.4 ^ +••• + 2.4.6 ■■ ■ 2« ^"+ • • •
n+x)'">r.^l + V , _ )n "'(m + 2)fH, + 4). . .(,„ + 2n-2)
Example 7.
(iir)""'-ii:!:^<P"^^'P"'-^'^>- • •(P-"? + g):t-
3 . 2(ji . 3(7 . . . 717 '
(1 - a:)-Prt= 1 + 2 P(P + 9)(P + '^1)- ■ ■(/' + '"/-?) ^„
7 . 25 . 3-7 . . . « J
Example 8.
(1 - x)-".= 1 + 2 '"("' + l)--^^-("' + "-l) ^„
It will be observed that the coefllcient of x" in this last expansion, when
Bi is integral, is (see chap, xxiv., § 10) the umaher (,„//„) of ii-corabiuations
of m things when repetition is allowed. It is therefore usual to denote this
eoefBcient by the symbol mli„, m being now unrestricted in value. We
shall return to this function later on.
Example 9.
i{{l + xr + {l-x)"'} = l + „C,x= + ,„C,x^+ . . . +„C„.x="+ . . .;
\{(\+xr-(i-xr}=„fi,x+^c,x^+ . . . +„c5„_ix--''-i+ . . .
Ultimate Sign of the, Terms. — Infinite Binomial Series belong
to one or other of two classes as regards the ultimate sign of
tlie 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 sigu.
c. n. 13
\
k
10-t INTEnnO-BINOMIAL SERIES CH. XXVII
If z and m deuote positive quantitica (m of ooanw not a positive integer),
Ist. The cxpausioDS of (1+x)" and (l + z)~** both belong to the fint
class. In (l + z)"" the first negative teru will bo that containing x*'*'', where
n is the least intci;er which exceeds m. In (l+x)~" the first negative term
is of course the second.
•Jnd. The expanhions of (1-x)", (I-i)-"", both belong to the second
class. Ill (1 - x)" the terms will have the same sign on and after the tenn
in x" n being the least integer which exceeds m, and this sign will bo -)■ or
- according as n is even or odd. In (1 - x)~* all the terms are positive
after the first.
§ 5.] A great variety of series suitable for various purposes
can be readily deduced froui 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 tlie methods by which the
summation is effected are tj'pical.
Consider the series 2<^r(")«i^»^. where </>,(«) is any integral
function of n of the rtli degree. Such a series stands in the
same relation to the simple Binomial St-ries as does the Integro-
Geometric to the simple Geometric Series. We may therefore
speak of it as an Iiitegro-fiinomial Scrifs.
We may always, by the process of chap, v., § 22, establish
an identity of the following kind,
<t>r(n)=.A,+Ain + Atn(ii-l) + . . . + ^r«(«-l)- • • («-r+l) (1),
where ^o, Au A,, . . . , ^r are constants, that is, are independent
of n.
We can therefore write the general term of the Integro-
Biuomial Series in the following form : —
+ Arn{n-\)... (n-r+l).(7,x*,
= At^Caf ■*■ mA,.r ^.,C,.,J*'^
+ III {in - 1) A,3^m->C,.,af~*+ . . . + TO (w - 1) . . .
(w -r + i;.lrJ:'.-,C.-,x— • (2).
§§ 4, 5 i;^, (n)„,C„x"/{n + a) {n + i) . . . (n + /.) 195
Hence, if the summation proceed from 0 to oo , we evidently
have
0 0 1
+ m{m-l) . . . {m-r+l)Ara;'-^,n-rC„-rX''-' (3),
r
= Ao{l+a;)'" + 7nAtx(l+x)'"-^+. . .
+ m{m-l) . . . {7n-r+l)ArX''{l+x)'"-';
since all the Binomial Series are evidently complete*. Hence
'S:'l>r{n)„,C„a;"' = {Ao + mAix/{l+x) + m{m-l)A.x-/{l+.cy + . . .
+ m{m-l)... (m -r+1) .4,u;7(l + xY} (1 + x)'" (4) ;
and the summation to infinity of the Integro-Binomial Series is
efiFectedt.
The formuhi will still apply when m 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 witliin the crooked bracket on the right-hand side
of (4) may become zero.
Cor. We can in general sum the series %'t>r{n)mC„af/(n + a) {n + 0)
...(« + k), where a, b, . . ., k are unequal positive integers,
in ascending order of magnitude.
For, by introducing the factors n + 1, n + 2, . . ., n + a - 1,
n + a+\.,n + a + 2, . . ., n + b-l, &c., we can reduce the general
term to the form
>!' {n),^^tGn^iX"*''/{m + 1) (m + 2) . . . (?« + k) x-^ (5) ;
where i// («) is an integral function of n, namely, <t>r (n) multiplied
by all the factors introduced which are not absorbed by m+kOn+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 ma.v be remarked that the series is evidently convergent when x<.l.
The examination of the convergence when x = l viiM form a good exercise on
chap. XS.VI.
13—2
19G EXAMPLES CH. XXVII
Hence
^<t>r (») ».C,a:"/(n + a) (ti + «.) . . . (n + k)
= {2^(n)m.»a^i^+»}/(»n+l)('» + 2)- . .('» + X-)^ (6;.
The suminiition of tlie series iii.siile tlie crooked bracket may
be effected ; for it is an Integro-Binomial Series. Hence the
suiumation originally uroposcd is always possible.
We have not indicated the lower limit of the summatiou,
and it is immaterial what it is. Even if the lower limit of
summatiou be 0, the Binomial Series into which the right-
hand side of (6) is decomposed will not all be complete (see
E.xample 6, below).
It should al.so 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 ca.ses the right-hand side of (6) would becomi
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. X.wi., § 20) all tin
above summations hold when a: = ±l, provided the scries in-
volved remain convergent.
Exuuiplu 1. To expand (x + >/)'" in a highly convergent seriea when x
ami y are nearly equal. Trom the obTious identities
|(i + y)/2xl"=12x/(x + !,)l-»={l + (x-y)/(i + y)l--,
((x + y)/2y}'»={2y/(x + y)}-«={l-(x-!,)/(x + y)|-",
(x + yr|l/(2xr±l/(2yr( = {l+(x-y)/{x4-y)} — ±{l-(x-y)/(x + j,)}--
wc deduce at uucc
(x + s,)"=2"x" jl + 2(-)V".(^-^||)") .
where „//, = m(m + l). . . (m + n-l)/n!,
_2"+'x"'!/'» I m(nn-l) /x-y\' m (m + I) {m + i) (m -t- S) /i-y\*
- xm^.ym jl+ 21 \,x + yj ■*" 4! U+W
2«.»ij».ym J „ /x_y \ m(m + l)(in + 2) /x- v\' 1
= ."-y" tiiU+y>''^ 3! \x + yj *■■■(■
All Ibesu scriua are hitjhly convergent, since (x - y |/(x -t- y) is smaU.
!■
§ 5 KXAMin.Ks 197
Example 2. To sum the series
2 2 /2y 2.5 /2\« 2.5.8 /'2\*
9 + 2! l^gy* + IT 1,9; "*■ ~li~ V9 j • • •
If we denote this series by !t,+H3 + «3+ . . ., we see that «
2.5. . .{2 + (n-2)3} 2»
"" n\ 3="'
_ i.j.l. . .(-j+m-l) /2\»
nl [3) '
_ (-^)(-4 + l)(-l + 2). • . (-^ + n-l)/2\"
nl \3) '
= _(_)» ia-i)(»-2). ..g-n+l) /2y_
Hence
l-(H, + »„ + Hs+ . . .) = (l-5)l/3,
= l/4/.-i.
Therefore, «i + W2 + i(3+ . . . =1-1/4/.?,
Example 3. To snm tlie series
»n(m-l) m (m - 1) (m - 2)
m+ J + j-2 + . . .,
whenever it is convergent.
Here we have
_m(m-l) (m-2) . . . (m-«)
m (m - 1) (»t - 1 - 1) . . . (m - 1 - n + 1)
~ ;n •
= m„_,C„.
Hence
iii + «j + «3+ . . . =ni{l+m-i<7,+„-iC5+ . . .}
= m{l + l}"'->=m2™->,
provided m- 1> - 1, that is ni>0.
It should be observed that we have at once from § 2 (5) the eqnatlon
m(l + x)"'-' = l„Ci + 2,„CjX+ . . . +n„C„i''->+ . . . (1),
from which the above result follows by putting x = l.
By repeating the process of § 2, we should doduce the equation
m(m-l). . . (ni-ft + l)(l + xr-* = 1.2. . . J:„Ci+2.3 . . . (fc + 1)
„C^,x+ . . . +{n-k + l){n-k + 2). . . n,„C,x»-t+ . . . (2),
whence it follows that
iB(m-l). . .(m-;; + l)2'»-* = 1.2. . . k^C^
+ 2.3. . .(A;+l)„Ct+,+ . , . (3),
provided m>k -1. These results might also be easily established by the
method Qrst used.
Example 4. To sum the series
1 ^ mCl^ . n.CjX'
l.i. . .k 2.d. . .(k + l) 3.4. . . (i-l-2)
+ .
198
Here we have
KXAMPLE3
,C.x«
cu. XXVII
Hence
(n+l)(n + 2) . . . (n + k)
"(in + l)(m + 2)
(1 + ^)"
(m + lt)!*'
1
(m + l)(m + 2) . . .{m + k)x* (m + 1) (m + 2) . . . (m + i) x* ^ ^ * "+*^' '
+m+*C,x»+ . , . +^tC4_,j*-'}+<u, + ti,+ u,+ . . . }.
Therefore
(l+i)"
• ^ ~ m4-t<^l» - m+t^^i^' -
— ..iiCt-l**"'
(m + l)(m + 2). . .{m + k)x*
If m> - i - 1, this gives as a particular case
2mCJ(n + l)(n + 2). . . {n + k) =
(1).
{a----*-!- 2 ^tC,}/(m + l)(m + 2). . .(m + t) (5).
1-1
The formula) (1), (2), (3), (4), and (5) contain of course a consiilerable
variety of particular cases.
Example 5. Evaluate Sn'^C.x".
0
Let n' = iJ|, + il,n + /l,n(n- 1) + /I,n (n-l)(n-2), then we have the follow-
ing calculation to determine A„. Aj, A,, A, (see chap, v., § 22).
1 +0 +0|+0 Af-0,
0 +1 +1
J, = l,
Hence
1 +1|+1
0 +2
'11+3
A, = 3. At = l.
2n'„C,i- = 0 . S„C.x- + l»ur2„_,C..,i«-> + 3m (m - 1) x'l „_,C^r-»
0 0 1 t
+ m (m - 1) (m- 2)x»S«-,C..,x"-»,
I
=mi (1 + x)"«-> + 3m (m - 1) i> (1 + x)"-« + m (m - 1) (m - 2) x> (1 + x)»-»,
= { m'x» + m (3m - 1 ) x" + mx ( ( 1 + x)*"-'.
Example 6. Evaluate 2„C,x»/(n + 2) (n + 4).
0
-4
+ 2)(n + 4)
+ l)(n + 3) =
1 +4 +3
0 -4 +0
x*(ra +
n'+ln
A,+ A
1)(,
+ 3,
(" +
ri + 2)(m + 8)(m + 4)'
4) + .l,(n + 4)(« + 3).
-8
1 +01+8
0 -8
A,=a,
l|-3
A, = -3. A,r
§ 5 EXERCISES IX 199
We therefore have
^(n + 2)"n + 4) = a* (m + 1) (m + 2) (m -TaUririT) *^f •»+«^>-m^"'^* - 3 (m + 1) x
0 0 »
+ (m + 4)(jn + 3)i2{(l + i)m+3_i_^^„C,x}],
=xM». + l)(,,. + -^)(,» + 3)(,» + 4)tU"' + ^)("'^'^)-^°-^('» + ^)^ + ^Hl+xr^'
+ {J('n + 3)("i + 4)i— 3}].
Exercises IX
Expand each of the following in ascending powers of x to 5 terms; and in
each case write down and simplify the coefficient of x''.
(1.) (1 + xyr-. (2.) (l-i)-!-^. (3.) {l-x)-V'.
(4.) {2-4xp. (5.) (a + 3i)i/3. (6.) i'ia'-x^).
(7.) ::/{l-nx). (8.) l/(l-3x=)i/3. (9.) (x-l/x)-»
(10.) Write down the first four terms in the expansion of { (a + x)/(a - x) } '/■■•
in ascending powers of x.
Determine the numerically greatest term in
(11.) (3 + xp, x<3. (12.) (2-3/2)11/2. (13.) (1 - 5/7)-"/».
(14.) Find the greatest term in (1 + x)"", when x = f, n = 4.
(15.) If 71 be a positive integer, find the greatest term in (n - l/n)***'.
(IG.) The sum of the middle terms of (l+x)"" for all even values of m
(including 0) is (1 - ix)-^P.
,„,,.. ,,„(..!),.Jlti)(..i)V...
(18.) Show that, if m exceed a certain value, then
om_i I ('» + l)»t , (m + l)TO(m-l)(>»-2) ,
2 -1 + — 2^ + jj +.,.
(19.) Sum the series
,> , .-.IV '"("'-1) / „,>»»('n- !)("'- 2)
a-(a + i)m+(a + 26)— >-2J— '-{a + 36)-^ ^^^ '+. . .,
for such values of m as render the series convergent.
(20.) V27 = 2 + A+y+...
,.,, V 23 2 1 1.3 1.3.-5
'^^•* 24 ~ 3^'~2^ " 2MI '^ 2^51 ' ' *
200 EXEKCISES IX CU. XX VU
('2'i.) Sam to infinity
1 Ui 1.4.7
6'''6.12''"6.r2.18"''* • *
(23.) Sum the series
, ,. ni(m-l){m-2) ni(in-l) . . . (m-r+l)
„.(„..:)+ A__U >^. . .+ K /_^_^^|
for such ^-alues of m as render the scries convergent.
(24.) If n be even, show that
n(n + 2) . . . (2n-2)/1..3 . . . (n-l) = 2»-'.
(25.) In the cxpuiiiiion of (1 -f)'" no coefhcicnt can be equal to the next
following unless all the coofTicients are equal
(20.) Prove by induction that
1 J.™ .!."'('" + ^) J. , m(m + l) . . . (m + r-l)_(m + r)l
l + m+ 2^— + ...+ — ^ —i^K'
where r is a positive integer. Hence show that, if x<l,
^ '' (m-l)lrl •
(27.) The sum of the first r cocfQcicnts in 1/^/(1 -*) : the coefDcient of
the rlh tenn = l + n{r- 1) : 1.
(28.) IfF(a) = l+^ + ^*'x» + ^ii±4j(i±^'x»+...,: . ■
being absolutely convergent, then
P(a)F[b) = F{a + h).
What is the condition for the convergency of the series?
(29.) Show tliat
I'-.C, j + .C,J-. . .=[l-{(f. + l)* + l}(l-x)-+']/(n+l)(n + 2).
Bum the following series, so far as they are convergent: —
(30.) Z(n-l)'m(ni-l) . . . (m-n + l)j-*/nl, from n = l to n = ao.
(31.) 2(-)»-'(H + l)(n + 2)1..3.5 . . . (2n - 5)i"/nl, from n = 0 to n=» .
(32.) 2:ni(ni + l) . . . (ni + n- l)x"/(n + 3)n!, from n = 0 to n = ii>.
(33.) 2(ri-l)'1.4.7 . . . (3n - 2)/(n + 2)(n + 3)nI, from n = l to n = » .
(34.) Wliy docs the method of snmmatioD given in § 5 not apply to
li«/(n + l)r
SEIUES DEDUCED BY EXPANSION OF RATIONAL FUNCTIONS OF x.
^ 6.] Since every rational function of x can be c.xprc8se<l in
the fnnn I+F, where / is an integral function of x, and Fa
proper ration.-il fraction, and since F can, by cliap. viil., § 7, be
§5 G, 7 EXPANSION OF (2 - px)/(l — px + qx') 201
expressed iu the form 2.4 (^ - a)-", where A is constant, it follows
that for certain values of a; a rational function of x can be ex-
panded in a serias of ascending powers of x, and for certain
other values of ar in a series of descending poweiis of x*. 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. Tliere are, however, certain particular cases
which may with advantage be studied here.
§ 7.] Series for expressing a" + /?" and (a"+> - /?"+')/(a - ft) in
terms of aft and a + ft, n being a positive integer.
If we denote the elementary s)nnmetric functions a + ft and
aft by p and q respectively, it follows from chap, xvni., § 2, that
we can express the symmetric functions a" + ft", (a"*' - ^''+')/
(a - ft) as follows : —
a" + yS" = a,p" + a^p^-^q + . . . + arP^'-'-'f + . . . (1),
(a"+' - ;8"+')/(a - ft) = h.p'' + hp'^-'q + . . . + brP"-"-' q'' + • • • (-2),
where both series terminate.
By the methods of chap, vin., § 8, or by direct verification
we can establish the identity
2 -px ^ 2-{a + ft)x ^ 1 ^ 1 /gv
1 -px + qx'~{l-ax){l-ftx)~l-ax l-ftx
Now if X be (as it obviously always may be) taken so small
^a.tpx-q.-i?<\, we have by the Binomial Theorem
+ {px - qirf + . . .+{j)X- qx')" + . . . } (4).
Now (by chap, xxvi., § 34) if x he taken between - a and + a,
a being such that the numerical value of ±pa±qa'<l, 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 ia as yet establislied only for cases where c
is real. The cases where o is imaginary will, however, be covered by the
extension of the Binomial Theorem given in chap. xxii.
202 o'l + zS" IN TERMS OF a^, a + 0 CH. XXVII
We thus find tliat
+ (-)\.rCrp'-''gr-^. . .)x-\ (5),
= 2{l + 2&c.}-j»j;{l+2&c.) (C).
Tlie coefficient of ar" on the right-hand side of (6) is
+ (-)Vr-C;i>"-''-'7'+- • •}•
Now
2,-,C;-„-r-i<7r = «(«-r- l)(«-r-2). . . (« -2r + l)/r!.
Hence
^-P^ = 2 + 2 /«" - -^ »•-»« + "("-^) „«-*^_
^^_^.«(»-r-l)(»-,--2). • -("-^^-^O^,.-.^^. . . j,, (7).
Again
-; + , a ={l+a^ + a'.r'+ . . . +a'x'+ . . . ] + \l + Bx
+ I3'x'+ . . . +^"j^+. . . },
= 2 + 2(a» + y3")a:« (8).
All the series involved in (8) will be absolutely convergent,
provided t be taken so small that \ax\ and \fix\ are each <1.
Now, by (3), the scries in (7) and (8) mnst be identical. Hence,
comparing the coefficients of x", we must have (by chap, xxvi.,
§21)
, ( ^^,»(»-r-l)(n-r-2). . . (n - 2r + 1)^..^^ ^
(9).
As we have indicated (by using h), tho equation (9) is an
algebraical identity, on the undcrst-'iuiling that p stands for o + /?
§7 SERIES FOR a" + /3", (««+>- /S"+i)/(a-/3) 20:1
and q for a/8. The last term ^vill or will not contain p according
as n is odd or even.
In like manner, from the identity
X X fl 111*
-^/_i L_l_
' W-ax 1 —fix] a-
X—px-vqa? l-{a + P)x + afix'
we deduce
subject to the same remarks as (9).
If we write the series (9) in the reverse order, and observe
that, when n is even, = 2m say, only even powers of p occur, and
that the term which contains p'" is
/ x^-. 2m{m + s-l){m + s-2). . . (2s + 1)
^ ' _ {m-sy. P ^ '
that is,
, ,„_,2?w(ot + s-1)(?w +S-2). ■ .{7)1+ l)m{m- 1) . . . (m-5+ 1)
^"^ (2s)I
i>"?"-'.
that is.
^"^ "^ (2s)! P 1
.st^m— •
then we have
tt'" + yy = (-)" 2 jg" - ^VV"' + "' ^"4," ^"^ jo^g""' - ■ . .
+ (-) (2s)! ^^^^ J ^ ^'
Similarly, we have
(»i + 2)»»(OT''-r) , „,_,
0!
■ / v-i("' + g-l)»'('»'-l')- . . (ffi'-g-2') „^,, „_.^i 1
^^ ' (2s- 1)1 ^ ^ ■ ■ ■/
(9").
204. SERIESFOR [a; + V(a:» + y'))"+{.r-v'(«* + y')l" CH. xxvii
^^_^.....K-n^^._.K-.-i')^,..^,..^ .} (10-).
„»^._^^. , (m4-l)ffi (m4-2)>n(w'-r)
a-p =("' 1'^ 2!~'"^ ■" 41
;*'7"-'-. ..+(-)'
.(»i + s)w>(>n''-l*). . .(m'-s-l*)
(2«)I
^"7"
}
(10").
Since o and ft jiro the mots of the quadratic function
:?-pz + q, we may replace a and /3 in tlie above identities by
h \P * Jip^ ~ 4?)}i <""! i {p ~ n/0»' ~ 4(/)[ respectively. If
this be done, and we at the s.-irae time put p = x and -iq = y',
we deduce the following : —
n(n-r-l)(n-r-2). ■ ■(»-2r+l) 1
fr2"" y "^- • • I •
= 2jy*+-a:»y"-' + — !-jj — ' jr'y-* + . . .
n' («' - 2') (n' - 4') . ■ . (»' - 27^2') .^ 1 I (g--).
(2s)! "^^ +-.-|. ^
if n be even ;
_„/ ,, «(«'-!'),, , n(n'- !')(«' -3')
Ji'y— + . . . +
H (/t* - 1') («' - 3') ■ . . (n' - 2.< -1')
jj.,y-«.-.+ . . j _ if „
(2.s-+l)l
be iiiliL
i 7, 8 SERIES FOR {x + ^/{x' + f)]" - {x - ^{x'' + f)}"' 205
1!2' -^ 2!2*
(w - r - 1) (?t - r - 2) ■ . . (n - 2/-)
r! 2-'"
= 2 VC*-" + y) (a^"-' + '-^fnjJ^ «"-'?/ + ^
= 2,/(ar' + y=)g,ry-U^l^.,y-^+. . .
^ (2s- 1)! "^ ^ +•••}.
if n be even ;
a.-*?/"-" + . . . +
(?i'- 1") («' - 3') ■ ■ ■ («' - 2s - 1")
ar»?/»-»'-' + . . . I , if «
(2s)!
bo odd.
(!(»"■)
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 lind expressions /or 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
\l{].-a,x){l-c^x). . .{\-a„x) = %A.(l-a,x)-' (1),
where A, = o.,''-^l{a,-a^{a,-a^ . . .{a,- On).
Also, .since (l-a,.r)-'=l + Sa/a:', 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 - ttja') ... (1 - a^x)
= (1 + 5a,'.?;'-) (1 + •S.a^'af) . . . (1 + :iaZ of) (2),
= 1 + I.KrOf (3),
where ^XV is obviously the sum of all the r-ary products of
a,, a,, . . . tt„. Since the coefficients of of on the right-hand
sides of (1) and (3) must be equal, we have
JT, = 2a."+'- V(a. - a,) (a. - a,) . . . (a. - a„) (4).
20G SUM ANU NUMllKU OK ;--AUY I'UUUUCTS Cll. XXVll
If, for example, there be tliree letters, <i,, a,, a,, we have
»"-r = 7- — w- — ^ + /„ „ \7- — \ +
(o, - a,) (o, - o,) (a,-o,)(a,-a,) (o, - a,) (a, - a,)
_ <^' (g, - g.) -t- g,-^' (a, - a.) -^ <^' (a. - g.)
(a, - o,) (a, - o,) (a, - a,)
If we put a, = a,= . . . =a„ = l, tlieu Biich of ths terms iu
n^r reduces to 1, aud .A'r becomes .iTr. Hence, from (3),
(i-a-)-"=i + :i„//,^ (6).
Equating coeflicieiits of a.' ou both sides of (6), we have
,//,= »(« + !). . . (M+r-l)/r!,
a result already found by another method in chap, xxiii., § 10.
§ 9.] Suiue interesting results can be obtained by expanding
l/(y + •t)(if + j: + 1) . . . (y + a: + «) in descending, and iu ascend-
ing powers of y.
If we wTite
l/(y + a?) (y + -r + 1) . . . {i/ + x + n)= i Ariy+x + r)-',
r— 0
then we find, by the method of chap, viu., § 6, that
l = Ar{-r){-r+l). . .(-1)1.2. . .(u-r).
Hence Ar={-)\Cr/n\.
Therefore
«!/(i/ + -r)(y + ^+l)-.(y + -r + «) = 2(-)'.C(j, + j: + r)-' (1).
Hence, if Pi, Pj, I\, . . . denote resi>ectively the sum of
X, X + 1, . . ., x + n, and of their products taken 2, 3, . . . at a
time (without repetition), we have
= 2(-)^6v(l + i (-)•(•" "^yi (2),
§§8,9 EXPANSIONSOF !/(?/ + a;) (?/ + a.' + !)...()/+ a' + h) 207
where we suppose y to have a vahie so large that all the series
involved are couvergent.
Since there is uo power of Xjy less than the nth. on the left
of (2). the coefficient of any such power on the right must'>
vanish. Therefore
(3 + «)" - „Cj (« + w - 1)' + „<72 {x + n-'2)'-. .. (-)'\if = 0 (3),
where .< is any positive integer <n.
Equating coefficients of 1/y", Ijy"*^, and l/y^'*'\ wo find
(x 4 n)" - „C, (x + n- 1)" + „Co {x + n- 2)" - . . .
(-)"a;" = 7j! (1);
(.r + ??)"''"' - nO, (^ + ?« - 1)"+' + „C, (x + n - 2)''+' - . . .
(_)«a;"+' = „!iJ„
= {n+\)\{x+hi) (5);
{x + «)"+' - u(7i (« + « - 1)"+- + „C, (*■ + M - 2)"*-- - , . .
(-)'U>"+==«!(Pr-P,),
= J (n + 2)! {x- + «^- + iV« (37J + 1)} (G) ;
and so on.
Again from (1) we have
x(x+ i) . . . {x + n)
T-a ' x + r\ x + r)
where Q„ Qo, Q^, . . . are respectively the sum of l/x, l/{x+ 1),
. . . , l/{x + n), and the sums of their products taken 2, 3, . . .
at a time. From (7), by expanding and equating coefficients of
y, we get
n\ f 1 1 ^ _^ 1 1
x{x+l) . . . (x + n)\x X + 1 ' ' ' (x + n)j
~.t' (x+if (x+2y •■•^ ' (x + ny ^°''
If we put x=l, we get the following curious relation between
the sum of the reciprocals of 1, 2, . . .,» + !, and the reciprocals
of their squares : —
208 EXAMl'LE-S CU. XXVU
1 /i 1 M - ^ »^'' I "^'
n+1 li ^2 • • • n+lj 1' 2' '3* ' ' '
§ 10.] We have now exeiuplilied most of tin; elementary
processes used in the transformation of Binomial Series. The
following additional exiiniiiles may be u.seful in helping; the
student to thread the intricai.ii-s of this favourite field of exercise
for the tyro in Mathematics.
Example 1. Find the oocQioient of x" iu the GxpansioD of (1 - z)'/(l + ')*'*
iu ascendini; powers of x.
If (l + xJ-»/= = l + i:.i„i», then (l-i)'/(l+x)V=(l_at + i»)(H-2<i.x").
Hence the coi'OJcient required ia ", - 2a,_, + a,_j . If we sabstitute the
actual values of a„ u„_|, a,_j, we tind that
°,-2...-. + ''..-. = (-)"("'"''-8"-l)^g^4;6 '■ ^^''.In-
Example 2. If /(j:)=Uj + u,x + (i,i-+ . . ., then the cooO'icient of x' in
the expansion of / (x)/(l - x)"" in ascending powei-s of x is n, „//, + o, „H,-j
+ 11, „Hr-4 + . . . + 0,. This follows at once from the equation
/(i)/(l-xr=K + 2,.,x'-)(l+i:„//,x'-).
In particular, if we put / (x) = (1 - x)"* and m = 1, we deduce tbkt
»+l^'r= J^r + »^^r-l + n^r-J + • ■ ■ + 1 !
and, if we put f{i) = (1 - x)~", we deduce that
results which have already appeared, in the particular case where m and n are
integral (xee chiip. xxtii., § IU).
Example 3. bhow that
.C J2 + «+,C'J2' + ,^C' J2' + . . . ad 00 = 1 + „C, + „C, + . . . + „r, (I).
The left-hand side of (1) is ubviously the oueQicicnt of x* iu
A' = (l + j)"'/2 + (l+x)"-t'/2' + (l + x)'»+'/2'+- . . adx.
Now j: = i(l + x)"'[l + {(l + x)/J} + ((l+i)/2l'+. . . adocl
= (l+x)"/2{l-(l + x)/2}, if we 8up|K>su«<l.
= (l + x)-/(l-x),
= 1 + 2(1+.C, + ,«C,+ . . .+„6'.)«-,
by last example. Ucnce the theorem follows.
Example 4. Sum the series
fi-3 (n-.«)(n-5) (n-6)(>.-6)(n-7).
5-1--^+ y, ^^ +....
n beiii{j a positive integer.
§§ 9, 10 KXAMPLES 209
The equations (9'") of § 7 being algebraical identities, we may substitute
therein any values of x and i/ we choose, so long as no ambiguity arises in
the determination of the functions involved. We may, for example, put
ir=-l and y = 2i. We thus find
Hence, If u and u- denote, as usual, the two imaginary cube roots of + 1,
we have
S={l + (-)"-i(a,» + <„'")}/n.
If we evaluate u^ + or" for the four cases where n has the forms Gm, 6m±l,
6m ±2, 6;« + 3 (remembering that u*"'=l, w~^ = (a', u-'=u), we find that
S has the values -1/n, 0, 2/«, and 3/n respectively.
Example 5. Sum the series
n (n - 1) ;t(n-l)(n-2)(»-3) «(n- l)(it-2) (n-3) (n-4) (n-5)
■'■2(2r + l)'*" 2.4(2r + l)(2r + 3) "*" 2.4.6(2r+l)(2r+3) (2f + 5)
+ . . .
n being a positive integer.
If we denote the series by 1 + «j + «j + Uj + . . . , then
n(n-l) . . . (>i-2s + l)
"« — -, — . .-, ., n+Ir^sr+M • H-«^« •
' 2.4 .. . 2s(2)- + l)(2r + 3) . . . (2)+2s-l)'
_»il(2r)!(r + l)(r + 2) . . . (r + »)
~ (n-2s)!(2r + 2«)lsl '
restricting r for the present to bo a positive integer. We may therefore write
nl (2r);
'••"(n + 2r)!'
Now ^,C, is the coefficient of x^ in the expansion of a:*+=« (1 + Ijx')'*' ; that
is, in the expansion of x'^^^{J{l + ljx-]l^'^^, Hence 2u, ia one part of the
coefficient of x-'' in the expansion of
(SSj'l "^ + '^'^^^ + l/x=)}»+-^ + { 1 - xj{l + l/x») }»+>].
Hence 2S is the whole coefficient of x^ in the expansion of
g5^,[{i+v(i+x»)}-+-^-+{i-V(i+^=)}"^n
Now, by § 7,
{l + ^(l + i')}-+» + {l-v/(l + x=)}"+=''
= 2»+* |l + S (" + 2'-)(» + 2r-»-l)(/. + 2r-»-2). . . (n + 2r-a» + l) i^i _
the coefficient of x" in which is
(n + 2r)(n + r-l)(« + r-2) . ■ ■ (n + 1)
rl2»
a II. 14
210 EXKRCISKS X CH. XXVIl
Henoe
o _ ,..^^-1 »l(2r)!(n + 2r)(n+r-l)l
(ii + 'Jr)!rlH!'i*- *
_ (n+r-l)(n + r-2) . ■ ■ (r + 1)
~ (n + 2r-l)(n + 2r-2) . . . (2r + l)*
The snramation is thin offectoJ for all integral values of r. So far, how-
over, 08 r is conccrneJ, the formula arrived at might be reduced to an
identity between two integral fuuctions of r of finite degree. Since we have
ehown that this identity hold.-) for an intinite number of particular valui>8 of
r, it must (chap, v., § IC) hold for all values of r. The summation ia there-
fore general so far as r is concerned.
Exercises X.
Find the coefficient of f' in the expansion of the following in ascending
powers of X.
(1.) x/(x-a)(i-fc)(x-r). (2.) x^^^Hx - a) (x - b) {x - e).
(8.) x'^'l{x - a) (x - 6) (x - c), where m is a positive integer < r - 8.
(4.) {3-x)/(2-x)(l-x)«. (5.) 2x'/(x-l)»{x' + l).
(6.) (1-pxni-qx)-^.
(7.) If (1 - 3x)"/(l - 2x)' be expanded in a.^cending powers of x, the co-
efficient of 1"+^' is (- l)"(r-2n) 2''-', n and r being jKisitive integers.
(8.) Find the numerically greatest term in the expansion of (a - x)y(b -f c)*
in ascending powers of x.
(9.) Show that
(x+ff)(x + 2/j) . . . (x+ufi)
(x-^)(x-2;3) . . .(x-n/3)
-14.'^/ 1^ "(" + ■•)("'- ^'j ("•- •■^•) ••• ("'-'•-'•) ^P ,
~ rT, *■' (rl)' ,.rfi*
and hence show that
r-I C')
(10.) If n be a positive integer, show that
l-mCl+™C,-. . . (-)"„C, = (-)"..,C,.
(11.) If n be an even positive integer,
(12.) If m and n be ]KiKitivo integers, show that
m^O • mtli^n + m^f (m-sl/>''ii-l +m^f (m-4)/»^i>-» + • • • + m^M • (m-lnyi^*
m»(m'-2») . . . (m'-aiTn;^
(2n)l '
_ m(m' -l»)(in«- 8^ . . . (m'-JgrTni)
(2n + l)!
(See Scblumilcb, Handb. d. Atg. Anal., jj M.)
§ 10 EXKRCISES X 211
(13.) Show, by equating coefficients in the expansion of (l-x~i)"'(l -a:)""',
wliero 7K is a positive integer, that
(1 J.) If n bo a positive multiple of 6, then
„C,-„C33 + „(7532-. . . =0;
(1.".) If {l + x)-^ = l + a^x + a.,x- + . . ., sum the series l-aj+a„-a^ + . . .
to n terms.
(16.) If (l + x)''' = l + a^x + a^x--i- . . ., then 1 - a^- + a.;^^ - . . .=
(-l)»2n(2»-l) . . . (n + l)/>il.
m\ jl_2M»jtl)l (-l)'-2-'-(2r)!_(-l)^
* ' r!ll (r-l)l:il 01(2r+l)l 2r + l*
(18.) ''z'llir (r!)2 (2« - 2r)l = (4?i)l/4''{ (2k)I (».
(19.) Sum to n terms S(2n-2)l/22"-'H {(k- 1)1 }^
(20.) Sum the series
, ,>1 , <^. 1.4 , „1.4.7 1.4 .. . (3n-5)
„ + (n-l)3 + (n-2)— + („-3)3-g-g+. . • + ^ . ■ . (3»-3) "
(21.) Find for what values of n the following series are convergent ; and
Bhow that when they are convergent their sums are as given below.
l_n_]_ n{n-l) 1 (m-l)l
m llm + l''" 21 m + 2 ■•" (ii + l) (j( + 2) . . . (n + m) '
m"^llm + l"^ 21 m + 2"*'- ' ■~(h+1)(« + 2) . . . (n + 7» )'"'+"'"-'
-m4.A.-22"+= + . . . + (-)--12''+™+(-)"'1},
tn in both cases being a positive integer.
(00 1 ' n" ('• + s)I ("' + "-'•-8-1)1 ^ (m + «)
* "■' ,=0 rl si (m-r- 1)1 (j!-s)! ml hI '
(23) r^m«y(,. + s)l(m + n-r-s)!^(m + n + l)l^
r=i)j«o J"! «!("!- rj^n-s)! mini
(24.) The number of the r-ary products of three letters, none of wliioh is
to be raised to a power greater than the »ith, where n<r<2n, is
r(3n-r) + l -]«(«-!).
(25.) Prove, for a, b, c, that 2a7(a - 6) (o - c) s 0, if r = 0, or r = 1 ; = 1 ,
if r=2 ; and generalise the theorem.
(26.) Show that
g (6 - f ) {be - aa') (a"* - a"") b{c-a) {ca - tfr') (ii"* - 6*")
a -a' ■*■ 6-^
c(a- b) (ab- ccf) (c"^ - c"^)
= {b-c)(c- a) {a - b) {hu - aa') {ca - 66') {ab - cd) S^_^\i\hc,
where aa' = bb' =cc' , and S„_3 is the sum of the (m-3)-ary products of
o, 6, c, a', 6', e. (Math. Trip., 188G.)
U— 2
212 EXEUCISKS X CII. XXVII
(27.) ir S^ be the siun of the r-ary products of the roota of the eijualioD
x" + a,x"-' + o,x* I-. . . + u,=0, then
0=S, + o„
0 = S, + S,<i, + a„
0 = S. + S.., a, + .?„.,«, + . . . + <!.,
0 = S,+ S^,a, + Srt(i, + . . . + .SV-1.''.- „
(Wronski.)
(29.) If .*?,. be the sum of the r-ary products of n letters, f, the sum of the
proilucUs r at a time, 2^ the sum of their rth powers, then
2, = ii5,-(n-l)/',S^,+ . . .+(-iy{ii-r)l\, if r<n-l.
= »S,-(fi-l)/'i.SVi+- • .+(-l)"-'iViS^«. if r>n-l.
(Math. Trip., 1B82.)
(29.) If 0= (1 - ttj)-' (1 - /3j-)-' . . . , the number of ways of dutribntiiiK n
things, X of which are of oue sort, n of another sort into p boxes
place<l in a row is the coeflicient of x"o*^ ... in the expansion of {v- 1}'
in ascending powers of x, namely,
ii,-pC,n,+pC,ii,-. . ..
where u,=(p + X-s)I(p + M-»)! • ■ • /(p-*)lX! (;>-«)! m' • • •
(Math. Trip., 1888.)
(30.) With the s.-ime 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
nl(ti-I)l/(n-p)l(p-l)IX!M!»'l • • •
(Math. Trip., 1888.)
Show that
(81.) 1 + 1/2 + . . . + VT=.C,-i,C, + l,r,-. . .
,o,v , {m + \)m {m + 2){m + l)m(m-l) „, J-l)"
(3^) 1- g, ■P+ ^— 6i - am+r
(34.) If m and n are both positive integer*, and m>n, then
a^* , (m-n)(w-n-l) (m-ii)(m-n-l) (m - n-8)(m-ii-3) .
nl ■*■ ll(n + l)! "^ iil(n + 2)I
1.8.5 . . . (2w-l)
■ ■ -^ (m + n)!
(85.) If r bo a positive integer,
= (X + ir' - ^C, (X + -i}r-* + ,_,C-, (X + 2)'^ - ^C, (X + a)--' + . . .
§ 11 CONVERGENCY OF MULTINOMIAL SERIES 213
MULTINOMIAL THEOKEM FOli ANY INDEX.
§ 11.] Consider the integral function aiX + a.,x'+ . . . +ar.r'',^
whose alDsolute term vanishes, the rest of the coefficients being
real quantities positive or negative. Confining ourselves in the
meantime to real values of a-, we see, since the function vanishes
when x = 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
I ai.r + a..v' + . . . + ar-r^ | < 1 (1).
In fact, it wiU be sufficient if p be such tliat
ap + ap^+ . . . +ap''<l
where a is the numerical value of the numerically greatest
among Oi, a., . . ., «r. That is, it will be sufficient if
ap(l-pO/(l-p)<l;
a/ortiari (supposing p<l) it will be sufficient if
ap/{l-p)<l;
that is, if P<l/(« + l)* (2).
p is, in fact, the numerically least among the roots of the
two equations
UrOf + . . . +a,a;±l = 0,
as may be seen by considering the graph of UrOf + . . . + a^x.
Therefore, whether m be integral or not, provided
-p<x< + p we can always expand (1 + Oia; + Wo.ir' + . . . + UrX'')'"
in the form
1 + 2„.C. (a,.r + a,a-'+ . . .+ ardf)' (3) ;
and the series (3) will be absolutely convergent whether m be
positive or negative. Hence, since aiX + a^a^-i- . . . +aT^is 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 13 merely a lower limit £or p ; iu any individual case it would in
gejiural be much greater.
214 MULTINOMIAI, COEFFICIENTS ClI. XXVII
of X, and the result will be a power scries wliidi will converge to
the sum (1 + a,j; + a.jj.^ + . . . + Oraf)" so long as -p<x< + p.
Since ,"! is a positive integer, we can expand „C,(rt,j + fi.^j:' +
. . . +arjf)' by the formula of chap, xxiii., § 12. The coelhcient
of of in this expansion will be
that is,
2a,°'"i°' . . • ar'^m{m-\) . . . (m -s+ l)/a,!a.,! ... a,! (.J).
where the summation ext-tnds over all positive integral values of
"^i. "i. • • •> "t> including 0, which are such that
a, + So, + . . . + ro, = n)
In order, tlicrefore, to tiud the coefficient of x* in (.3) «c have
merely to extend the summation in (4) so as to include all
values of s ; in other words, to drop the first of the two restric-
tions in (5).
Hence, whether m be integral or not, provided x be small
enough, we haw
(l+o,ar + o,a?+ . . . +«r^)"'
= 1 + 2 —5 '- — - — i -Ui 'a,^ . . . arx* (O),
O,! CL,! . . . u,!
the summation to be extended over all positive integral values qf'
"i, "-ii • ■ •. "ri including 0, such that
a, + 2a, + . . . 4 ra, - n.
The dct.-tils of the evaluation of the coillicient in any parti-
cular CAse are much the same as in chap, xxiil., .sj 12, Example 2,
and need not be farther illustntcd. It need scarcely be .added
that when n is very large the calculation is tedious. In some
cases it can be avoided by tmn.sforming 1 + «,x + a,3^ + . . . + Ortf
before applying the Binomial Kxpan-sion, but in most cases the
application nf the above formula is in the end both quickest and
most conducive to accuracy.
§§ 11-13 CONDITIONS FOR GOOD APPROXIMATION 215
Examj]lc. To find the coefficient of x" in (1 + .T + I-+ . . . +i'')"'.
We have
(l + a; + x=+. . .+x7"={(l-a:'-+')/(l-x)}"»,
= (1-x'^')'"(1-.t)-"',
Hence, if n<r + l, the coefficient of x" is siiniily
„H„=m(m+l) . . . (m + n-l)/Hl;
bnt, if n •* r+ 1, the coefficient of x" ia
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 +a;)'". According
as we retain the first two, tlie first three, . . . , the first n+1
terms of the series 1 + nCiX + nCnx' + . . ., we may be said to
take a first, a second, ... an 7(th approximation to (1 + j.-)'".
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 7i so great that the
numerically greatest term, at least, is included.
2nd, To take ii so gi-eat 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 uexi; 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.] T/ie order of the numericalli/ greatest term.
In the case of the Binomial Series (1 +.r)"', if * denote the
numerical value of x, so that 0<^<1, we have lor the numerical
value of the couvergency-ratio m„+,/"„
216 KUMEiaCALLY GREATEST TEK.M tU. XXVIl
(1).
m-n . n- m ,
<r« = r f , or = — — , t»
n+ 1 n + 1
according as m - n is positive or negative
Heuce it is obvious, in the first place, that, if - 1 S m<+ 1,
that is, if m be a positive or netjative proper fraction, the condi-
tion cr,<l is satisfied from the very beginning, and the first
term will bo the greatest
If «j>+ 1, the condition o-,<l is obviously satisfied for any
value of n which exceeds tn; in fact, the condition will be
satisfied as soon as
(m-n)i<M+l,
thatifl. n>(m^-l)/(l+0 (2),
the right-hand side of which is obviously less than m. Tliis
condition is satisfied from the beginning if f<2/(m-l).
If m be <-!=-/*, say, where /*>!, the condition <t,<1
will be satisfied as soon as
(/x + n)f<n + l,
that is, n>{i^-\W-i) (3).
This condition is satisfied from the beginning if ^<2/(^ + 1).
§ 14.] Upptr 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 cla-sses belong the expansions of (1 + J-)"
and (1 +;r)'"', where x and m are positive.
If n be greater than the order of the n\imcrically greatest
term, and in the c.ise of (1 +x)" (see § 4) also >m, tlien the
residue may be written in the form
■ff»-±(«<»+i-".*« + «*»+«-- • •) (')•
where «,+,, «,+,, »*,+•, • • • are the ntimerical values of the
various terms, and we have »*,+i>''i,+«>Uii+i> ■ • •
Hence, in the present ca.sc, the error committed by taking an
nth approximation is numerically less than «,+,, In other words.
§^5 13, 1-i UPl'Eli 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 lowei' limit for tlie error is obmously «„+i- (<„+j.
The expansions of (1-a;)"' and (l-a;)""" belong to the
second class of series, in which the terms are all ultimately of
the same sign. It will be conveuicut to consider these two
expansions separately.
In the case of (l-a')"", if we take n>m, then we shall
certainly include the numerically greatest term; and (r„, the
numerical value of the convergency-ratio, will be (« - m) x/{n + 1),
that is, {l — {m+l)/{n + l)}x. This continually increases as n
increases, and has for its limit x, when w = oo . Hence
Therefore, Mn+i. Mn+2, • • • having the same meaning as before,
^n = ± («'»+! + t<n+2 + «n+3 + • • •).
Therefore
\Ii„\<u„+i(l+x + ar + a^ + . . .),
<«„«/(! -^) (2).
Hence the error in this case is numerically less than M»+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 error is obviously Mn+i/(l — fn+j),
that is, ™C„+i«"+V{l - (« + 1 - »«) ^/i"' + 2)}.
In the expansion of (l-a:)"™, all the terms are positive;
and, in order to include the greatest term, we have merely
to take n> {mx - 1)/{1 - x).
We have, in this case,
<T„ = (re + m) x/{n + 1) = {1 - (1 - m)/(n + l)} x,
= {l + (m-l)/{n + l)\x.
Hence, if «i < 1
(r„+i<o-„+3<. . .<.<•< 1,
218
EXAMPLE
CH. XXVU
ami an upper limit of /;, will be «,+i/(l -x) as in last case, a
low, r limit being «,+,/(! -tr,+,), that is, „//,+,j:"+Y{l - (" + 1 +
m) x/(n + 2)}.
If «i>l,
l>tr„+,><r„+,>. . .>x,
and an upj>er limit of /.'„ will be »i,+,/(l - «r,+,), that m,
„//,+,a:"+V{l -(H + l + »i)a:/(H + 2)}, a loirer limit being u,+,/
The error for (1 - a-)'" is, of course, always in defect
Eiami>lo 1. To calculate the cube root of 29 to C places of decimals.
The nearest cube to 2'J is 27. We therefore write
4/2y=(3' + 2)'/>=3 (1 + 2/3')'".
= Ilj + U,-B, + U,-Uj . . . ,
The first term is here the greatest; and the terms alternate in si^ni after m,.
Also Uri written in the most convenient form fur calculating successive terms, is
«r=3(A)(rh)(,'A)(M)(A\). . -C^)-
Therefore
+
-
u, = «,2/81 =
u,= «,4/162 =
«j = ii,10/213 =
u, = «, 16/321 =
300O,000,i»0
74,074,07
75,27
■001,82<i,99
3,72
3074,149,34
•001,8.32,71
•001,832,71
u.=u, 22/405
8 072,316, r,3
20
Hence the error in defect, due to no,'lect of the residue, amonnto to loss
tbnn 2 in the Bcventli place. The error for neglect of di;nt» dotn not cioced
1 in the seventh place. Therefore, the best C-phkce a|>proximation to
•/29 is 3072,317. In Barlow's Tables wo find 3072,816.8 given as the
value to 7 placee.
Example 2. Tocalcnlato(l-x)"'/(l+»+i^"' to aseoondapproximatiiM.
X being small.
(i-irii+i+j^)--
55 14 KXERCISES XI 219
where we have already neylectetl all powers of x above the second in each of
the two series ;
( m(m-l) If m(m-l) „»
= n-m.z + — ^— ^x^V •( l-Hi:cH 5_^ — ^* f •
■, , ^ i m(m-l] . vi(m~ 1)1 „
= l + (-m-m).c+ i -^ — i + ma+-A_ — 'I ^^^
where higher powers of x than x- have again been neglected in distributine
the product ;
= 1 - 2mx + m {'2m — l)x-.
Exercises XI.
(1.) The general term in tlie expansion of {l + x + y + xi/)l(l + x + y) is
( - 1)'"+" (m + ?j - 2)1 x'"y''l{m - 1)! (n - 1)1.
Determine limits for x within which the following multinomials can bo
expanded in convergent series of ascending powers of x ; and find the
coeQicients of
(2.) a:* in (1 - 2x + x« - 3.t^)-V'. (3.) x» in (1 - 3x - 7x2 + jSj-a/s.
(4.) x» and x' in (x + Sx^ + Sx" + 7x' + . . . )-'■'.
(5.) x' in (1 - 3x + x3 - x')"'''. (0.) x' in (2 + 3x + x')-^.
(7.) Show that in (Oa* + 6ax + ix")-^ the coefficient of x'' is 2^ (3a)-*^';
and that the coefficient of every third term vanishes.
(8.) The coefficient of x"* in (1 + r -f x^)"* (m a positive integer) is
m(m-l) m(m-l) (m-2) (»»-3)
■^ (1I)= "*■ (2!p ■•"■■• •
(9.) The coefficient of r"^Mn (l + x)/(l+x + x=)^ is -(r + 1).
(10.) Evaluate 7(100/99), and 1^(1002/998), each to 10 places of deci-
mals ; and demonstrate in each case the accuracy of your approximation.
l>'ind a first approximation to each of the following, x being small: —
, {x + V(x2+1)}^-''-{x-^/(.t' + 1)1^"
^ ■' {x + ^{x' + l)]^'^i-{x~^(x- + l)Y--"'->-''
(12.) (l+x)(l + rx)(l + r-x). . ./(1-x)(1-x)'-(1-i)'*. . . .
(13.) ^(2- j(2-,,/(2- . . . -^(1 + x). . .))); where J is repeated
n times.
(14.) If X be small compared with N^, then J{N' + x) = N + xliN +
Nzji (2iV' + x), the error being of the order x'/W. For example, show that
v/(101) = 105VA. to 8 places of decimals.
(15.) If;) differ from N' by leps than 1 per cent, of either, then ^p differs
from iN+lpjN^ by less than jy/90000. (Math. Trip., 1882.)
220 EXKncisEs XI en. xxvn
(16.) II p=N* + x where x is small, then approximately
^^ = ^^'+^'''''^ + 13 ^''^''P + ^^'> '
show that whpn N = 10, x^l, this approximation is accurate to 10 places of
decimals. (Math. Trip., 1886.)
(17.) Show that L {l/,^n> + 1/V(n'+ 1)+ • • • + l/v/(ri' + 2n)| = 2.
(CatulttD, Nouv. Ami., sec. i., t. 17.)
(18.) Find an nppor limit for the robiduo in the uxpanuioo of (l-t-x)"*
when m is u puaitive int«^oi.
CHAPTER XXVIII.
Exponential and Logarithmic Series.
EXPONENTIAL SERIES.
§ 1.] Wc have already attached a definite meaning to the
s}Tnbol cf when a is a positive real quantity, and x any positive
or negative comuieusurable quantitj-. We propose now to discuss
the possibility of expanding aj' iu 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 its coefficients.
For, let
a' = Aa + AiX + A^ + . . .+A„x'' + . . . (1),
then, proceeding exactly as in chap, xxvii., § 2, we have
L{a'*''-a')/h = A, + 2A^ + . . . + 7iA„x"-' + . . .;
and the series on the right wih be convergent so long as x lies
within limits for which (1) is convergent. Now (by cliap. xxv., § 13)
L (rt^+* - a'yh = (fXL (e** - 1 )/M,
= X<
where X^log^rt, and e is Napier's Base, namely, the finite quantity
L {I + 1/71)". Hence
*"" Xa^=lAi + 2A^ + . . . + n^,3-"-' + . . . (2).
Therefore, by (1),
\{Ac + AiX+. . . + An-iHf'^ + . . .)
= \Ai + 2A^ + . . .+«^„a^-' . . . (3).
Since both the series in (3) are convergent, we must have
l^li = A^o, 2xlj = A.^i, . , ., Hj4, = \4,-i.
222 nimcuMiNATioN of the coefficients cii. xxviii
Usiug these equations, we find, successively,
^, = yl,X/l!, ^, = ^.\V2! J,= yUX"/«! (4).
Also, since, by the meaning atbichcd tf) «*, a* = + I, putting
x = 0 on both sides of (1), \vc have
+ 1=^1. (.U
Hence, finally,
a' = 1 + X^/1 ! + (\j-)V2! + . . . + (Xj-)"/«! + . . . (C).
We see, a posti'riori, that the expansion found is really con-
verj^ent for all values of x (chaji. xxvi., S ^}, a"d also that the
scries in (2) is convergent for all values of jr. Our hj^othcscs
are therefore justified.
This donionstration is subject to the siinie objection as the
corresponding one for the Binomial Scries : it is, however, interest-
ing, because it shows what the expansion of o* 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.] Di'dnrthm of the f'Jxpont'nfial/rom t/ii- li'mnmial Krpnn.oittn.
By the binomial tluorem*, we have, provided z be numeric-
ally greater than 1,
(••:)"=
1 zx{zx-l)l
^ zx{zx-\) . . . {zx-n^■\)\ ^
n\
= 1 + a: + ^ „,'+... + — ^ ' — ^— i—^ — '■
2! n!
+ -R. (1),
where
(n+l)l (n + 2)!
^ ^- • • (-0-
* In wliat frillowR we linvr rofitricted the value of the index tx. Bince
z is to be ultimaU'lv innde infinite, tlirre is no objxtion to our supposinR it
always so chosen that tz is a pofiitivo intei^cr. We then depend merely
on the binomial ez)>an»ian for positive intr^al indices, Thix will not affect
the value of 1.(1 -t- l/i)**, for it has Ixtn shown (chap, xzv., § 13) that this
has the same valu>' when x becomes + ur - cc , and whether i iucnoiies bjr
iuli.'ijral ur b> (rocliuual inctvuivut*.
^;j 1, 2 DEDUCTION FROM THE BINOMIAL THEOREM 223
Suppose now a: to be a given qiiantity ; 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 l/zx<2/zx<. . .<n/zx . . .
<(j}- l)/zx<l. With this understanding, it follows that
<;?;"+'/(« + 1)!{1 - xl{n + -2)] (3) ;
and we have
(\ I M"-l i.r|-^'^^~^/-^^ ^.T"(l-l/p)...(l-»-l/p)
\ z) ' 2! ■ ■ ■ w!
+ /-■,. (4),
where R,, satisfies the condition (3).
Now let z, and therefore also jj, increase without limit («
remaining fixed as before). Then, since
L(\-\Ip) . . . {l-n-l/p) = l,
p=oo
we have
.(-')'
l+.r+-+. . .+f^ + 7.'„ (5),
2! w!
Bn being still subject to (3).
We may now, if wo choose, consider the effect of increasing
n. When this is done, x"+y{7i+iy.{l-x/{n + 2)] (see chap.
XXV., § 15) continually diminishes, having zero for its limit when
n = 00 ; wc may therefore write
l+x + %+. . . + —. + . . . ad CO (6).
2 ! 11]
Thus the value of Z(l + l/s)'^ is obtained in the form of an
infinite series, which converges for all values of .2-. For most
purposes the form (5) is, however, more convenient, since it gives
an upper limit for the residue of the series.
IL
224 CALCULATION OF C CU. XXVIH
§ 3.] Tlie comlitions of the demonstration of last paragraph
will not be violated if wo put x=\. Heuce, using e, as in chap.
XXV., to denote L {1 + lis)', we have
where IL < (« + 2)/(« + 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 ou. Proceeding as far
as n = 1 2, we have
1 + 1 - '.
'000,000000
1/2! -
■500000000
1/3! =
166666067
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 10 x 5,
that i.s 5. Also the residue 7i'„<-j.^ (l/13!)<i J -0000000002
< 0000000003, so that the neglect of 11^ would certainly not
affect the eiglith place. Hence we have as the nearest T-placo
approximation for e
e=-271828I«.
It is usual to give a demonstration that the numerical constant <■
is incommensurable. The ordinary demonstration is as follows : —
Jjot iiR suppose that e is comiiicnstirahle, Bay =pl<l, whnrc f nnd q are
finite positive iiitogurs. Then we bave by (7)
f/(j = 2 + l/21+. . . +I/9! + i?„
where ^«<(« + 2)/(8 + l}'8l.
§§ o, 4. INCUMMENSUllAUIHTY OF e 225
Hence, multiplying by g!, we get
where p{q- 1)1 and I are obviously integral numbers. Hence q\Iiq must be
integral. ,
Now 3liJ,<(g + 2)/(j +!)••',
<(3 + 2)/{'/(? + 2) + l},
that is, q\Eq is a positive proper fraction.
The assumijtion 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 : —
1{ Ti, T„, . . ., r„, . . . be an infinite series of integers given in magnitude
and in order, then it can be shown {see chap, ix., § 2) that any commen-
surable number p/q (where p and q are prime to each other, and p< s) can
be expanded, and that in one way only, in the form
P^Pl+P^ + _P^ + ... + ^t^ +... (9),
q Ti rir„ r^r.,r3 r^r„ . . . r„
where Pi<rj, P2<r.2, . . ., iJ„<:r„, . . .; and that the series wUl always
terminate when either g or all its factors occur among the factors of the
integers rj, rj, . . ., r„, . . . Hence no infinite series of the form (9) can
represent any vulgar fraction whose denominator consists of factors which
occur among r^, r^, . . ■, r„, . . .
In particular, if r,, Jo, . . ., r„, . . . contain all the natural primes,
and, a fortiori, if they be the succession of natural numbers {excepting 1),
namely, 2, 3, 4, 5, . . .,» + !,. . ., then the series in (9) caimot represent
any commensurable number at all'.
The ineommensurabilitj' of e is a mere pai'ticular case of the last con-
clusion ; for we have in the series representing « - 2
ri = 2, r^=Z r„ = u + l, . . .;
j), = l, i>2=l, . . ., p„=l
Hence e - 2 is incommensurable, and therefore e also.
§ 4.] Returning to equation (5) of § 2, since L{1 + l/z)' has
a finite value e, we have i (1 -i- l/sY"" = {L{1 + l/z)'f ^ <f, 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 rj , r„ , . . . , r„ , . . . , for example,
112 3 4
2 = 3 + 375 + 37177 + 3:5:7:9+ ••• '^ " •
This point seems to have been overlooked by some mathematical writers.
c. a. 15
226 cau(;uy's summation cu. xxviii
«'=l + f, + ^ + - • ■ + f^ + ^'. (10).
where If^ is subject to the inequality (3).
Finally, since a'=e^, where X = log,rt, we liave
^...(^).<M%...,!M-.„. ,„,
where /^,<(Xa:)»+V(« + 1)!{1 - kt/(ti + 2)} (12).
Since LR„ = Q when n=oo, the series (10) and (11) may of
course each be continued to infinity.
This completes our second demonstration of the exjwnential
theorem.
§ 5.] Summation of the Exponential Series for real values ofx.
A third demonstration was given by Cauchy in his Analyse
Algebrique. It follows closely the lines of the second demonstra-
tion of the binomial theorem ; and no doubt it was sugge.sted
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 ch;i]). .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 iuhuite scries
a? a*
1+0: + .,, + . . , + -:+. . .
2! n\
by the symbol f{x). Tlicn, 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
scries, we have, by the rule for the multiplication of series
(chap. XXVI., § 14),
/(-)/(i/) = l + (.r+y)+(^+j^ + ^)^...
■*■ \h! ^ (i, -1)!T! ■*" ("^^2)121 ■*■••• "^ Hly/ "^ •• • •
§§ i, 0 EXPONENTIAL ADDITION THEOREM 227
Now
til '*'(w-l)!I!'^(?»-2)!2!"'"' ■ ''^ III
= (a; + #/«!,
by the binomial theorem for positive integral exponents.
Hence /{■c)f{y) = 1 + S (x + yflnl,
-f{x+y) (1).
Hence f{x)f{y)f{z) -^fix + y)f{z),
=f{x+y + z);
and, in general, x,y, z, . . . being any real quantities positive or
negative,
f('^)f{y)fi^). . .=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 n, we deduce
{/(!)}"=/(«) (3).
Also, taking tlic number of the letters to be q, and each to
be p/q, we deduce
where p and q are any positive integers. From (4), by means of
(3), we deduce
{/(W?)}' = {/(!)}" (5).
Finally, from (1), putting i/ = — a-, we deduce
f{x)/{-x)=/{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 qth root of [/(Ojp Now,
since p/q is positive, the value of f(j)/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 {/'(OH", or «^>
is real and positive. Hence /(pjq) must be the real positive
2'th root of e'', that is, ti''"'. Hence
15—2
22» caucuy's summation cu. xxvm
p and q being any jwsitive integers.
Finally, siiioe/(0)= 1, we see from (G) tliat
/(-P/<J) = 1//{PI'1).
= «-"«.
Hence
. = e-'f*
(8).
1! 2!
where pjq is any positive cumniensurablc number.
By combining (7) and (8) we complete the demonstration of
the theorem, that
..•ex' a*
for all commensurable values of x, e being given by
,11 1
.= 1^-, + ^ + ...-.^.... .
The student will not fail to observe that e is introduced and
defined in the course of the demonstration.
The exteujiioii of the theorem to the case where the ba.sc is
any positive quantity a is at once effected by the transformation
(j' = fl**, as in la.st demonstration.
§ 6.] From the Exponential Series we may derive a largo
number of others ; and, conversely, by means of it a variety of
series can be summe<i.
Bernoulli's Numbeni. — One of the most important among the
series wliicii can be deduced from the e.x|>inential theorem is
the expansion of j-/(1 -«"*), the coefficients in the even tenn-
of which are closely connectc<l witli tlie famous numbers of
Bernoulli.
We shall first give Cauchy's demonstration, which shows, a
priori, tliat j-/(1 - «"') fan be exjxtiided in an a-stcending series qf
puWiTS of X, provided X lit within certain limits.
§§ 5, 6 EXPANSIBILITY OF x/(l - fi"^) 229
We have
' ' (1),
l-e-' {\-e-=')lx l-.v
where 2'=l-(l-0/« (2)-
Now, from (1), we have
.r/(l -«-^) = 1 + 2/ + 2/' + • • • arl » (3) ;
and this series will be absolutely convergent provided - 1< ?/ < + 1.
Also, from (2), using the exponential theorem, we have
y = xl2\ - a?li\ + a-V4! - ... ad » (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! + pV4! + . . . ad =c < 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). Tliis double series may therefore be
arranged according to powers of x, and the result will be a
convergent expansion for xj(\ — e~^).
It is easy to show that a value of p can be found to satisfy
the condition (A) ; for we have
p/2\+p'jS\ + . . . = {e''-l)lp-l.
We have, therefore, merely to choose p so that
e''-l<2p (5).
If the graphs of e^ — 1 and of •2x 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, tliat is to say, at a gi-eater angle. There-
fore, since e'—l increa,ses as x increases, and that ultimately
much faster than 2x, the graph of e^ — 1 will cross the graph of
2a; just once. Therefore the inequality (5) will be satisfied pro-
vided p be less than the unique positive root of the equation
e'—l = 2x. Since e'-l<2 x 1, and e^— 1>2 x 2, this root lies
230 COEFFICIENTS IN EXPANSION OF .t/(1 - CT*) CM. XXVIII
betweeu 1 and 2.* It will, therefore, certainly be possible to
expand z/{l-e~') in a convergent series of powers of a; if
-l<a:<+l.
If we make the substitution for y, and calculate the co-
efficients of the first few terms, we find that
«_ 1 l^_i.^ ±^_ /e\
»-•" ^■'■'j^'^Aoi anil ■''ii-^ fir ••• W-
1-fl-" 2 62! 304! 4261
Knowinj;, a priori, that the ejtpansion exists, we can easily
find a recurrence formula for calculating the successive co-
efficients. Let
xl{l-e-') = At + A,x + Afi^ + AtT' + ... (7).
Then, putting - x in place of z, wo must have, since
-xj(l-e') = e-'x/{ !-«-'),
e-'3:/(l-e-') = A,-A,x + A3r'-At3:* + . . . (8).
Since both the series are convergent, we have, by sub-
tracting,
x = iA,x + 2A^ + . . . (9).
Hence j4, = J; and all the other coefficients of odd order
must vanish.
Therefore, from (7), we have
x = {A, + ^x + Atx'+At3* + . . .)(!-<;-'),
= (At + JiX+A,x'+AiJ^ + . . . + Jj,x*' + . . .)
x' X* a-"- a*-*'
\U 2! "^3! ■ • • (2n)!'^(2» + !)!"• ' T
The product of these two convergent series will be another
convergent series, all of whose coefficients, except the coefficient
of X, must vauisli. Hence, equating coefficients of odd powers of
r, we deduce A»= \, and
1! 3! (2n-l)l 2(2»)!(2n + !)!""■
• Mor<- noArlv, thr> root ir l-'i^O . . . ; bat tbo aotiul value, at will I..'
Men iircHiiUy, U uot u( luucli luipuiunoo.
1
§ n Bernoulli's numbers 231
Iq like manner, if we equate the coefficients of even powers
of X, we deduce
A^ A^n-2 , , -^1 _ ^^ (\\\
2! 4! (2")! 2(2« + 2)! ^ ''
If, as is usual, we put A.n = {-f''^BJ{2n)\, our expansion
becomes
and the equations (10) and (11) may be written
S„+l C>n -5„ — 2„+l C2B-2 -Oii-l + . • • ( ~ )"~ 2n+lC2-"l = ( - ) (""-)
and ^^ ^ ^_^ (^*»;)
2„+2 Con-^n — 2n+2 Can-s /^>i-l +• • • ( " )" 2n+2C'2 i>i - ( — )" « (H)
respectively.
If we put n = \, n = 2, m = 3, . . ., successively, either in
(10') or in (11'), we can calculate, one after the other, the
numbers Bi, B^, . . ., Bn, . • ., which are called Bernoulli's
numbers*. Since we know, a priori, that tlie 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' ^''^SO' ^'^42' ^'"30'
5 „_ 691 „ 7 „_3617
^5-gg, ^«-2730> -^'-6' ^*"510'
43867 „ fl222277^' — = il^h^lJ
* There is considerable divergence among mathematical writers as to the
notation for Bernoulli's numbers. What we have denoted by £„ is often
denoted by iJ.,„, or by fijn-i. For 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, Jlrit. Assoc. Rep., 1877, p. 8,
also Cambridge Observations, 1890, App. i. ; Staudt, Crelle's .Tour., Bd. xxi. ;
Boole's Finite Dijfferences (cd. by Moulton) ; and, for a useful bibliography
of the relative literature, Ely, Am. Jour. Math. (1882).
L
232 KXPANSiONSOFa:(c»+e"')/(e*-e^)ANDa;/(l+c-*) ch.xxviii
We sliall rettirn to the proi)erties of these imiiibers iu
cluip. XXX.
Rrmiirk Ttgarding the limitt tcithin tchieh the expaniion of i/(l -e~*) u
ralid. — IT we dcnoto tbe serieg
by 0(z), we may state the problem we have just solved as follows: — To find
a convergent teriei <p (x) nich that(X- e'*) 0 (x) = x, that it, $uch that (x - x'/21
+ I>/3I- . . .)4,{z) = i.
Now, since x-x-/2! + x'/3l -is absolutely convcrt-ent for all values of x,
and tbe cocflicicnts of <t>{x) satisfy (10') and (IT), <t>{i) will satisfy the con-
dition (x-x'/2!+x'/3! - . . .) ^(x) = x BO long as ^(x) is convergent. Hence,
so long as <f> (x) is convergent, it will be tbe expansion of x/(l - e-'). As a
matter of fact, it follows from an expression for Bernoulli's nnmbers given io
diap. iix. tbat ^(x) is convergent so long as -2)r<x< +2t. The actual
limits of the validity of tbe expansion arc tbcrcfore much vridcr than those
originally assigned in the a priori proof of its existence.
Cor. 1. Since a: (^ + e-')/{e' - «"") sx/(l - «"") - x/(l - e^),
we deduce from (12)
Cor. 2. Since .r/(l + e-') = 2x/(l - «"»') - j-/(I - «-),
j-^, = |(2'-l)x + ||(2^-l)^-§(2'-l)y+. . . (14).
§ 7.] Bernoulli's Theorem. — We have alrcafly seen that the
sum of the rth powers of the first n integers (,5r) is an integral
function of n of the r + 1th degree (see chn]). xx., S 9).
We shall now show that the coefficients of this function can
be expressed by means of Bernoulli's numbers.
From tlie identity
(«"-!)/(«'- l)Hl+<^ + t^ + . . .+el-'l',
that is,
(e« - 1)/(1 -«-') = «■' + *>*' + c** 4- . . . + ^,
we deduce at once
[nx nV nV W, 1 B, . li,
{-11 "^^•••■^H ^•■ll'^'-^ 21^-47
r' -t".
I
§§ 6-8 Bernoulli's expression for In'' 233
wherein all the series are ahsolutely convergent, so long as n
is finite, provided a; do not exceed the limits within wliich
l + ^x + Biar/2l-B2X*/4:l + . . . is convergent. The coefficient »
of a:'*' on the right of (1) must therefore he equal to the co-
efficient of a^"*"' in the convergent series which is the product of
the factors on the left. Hence
^Sr_jr^ j^ B^if-^ B.jf-' BsW-'
r! "(r+l)!'^2.rl"^2!(r-l)! 4!(r-3)! "^ 6! (r-5)! ' * * •
Therefore
""^^ = 7^ ^ 2 " "" 2^: ^- " ~ ^ 4! ^^"
6!
the last term being ( - )i''"-'' Bi^n, or |( - )il''-'lr .Bl(r-l)7^^ accord-
ing as r is even or odd.
This formula was first given by James Bemonlli {Ars Gonjectandi, 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 horce ! the sum of
the 10th powers of the first thousand integers, and found it to be
91,409,924,241,424,213,424,241,924,242,500.
It will be a good exercise for the reader to cheek Bernoulli's result.
SUMMATION OF SERIES BY MEANS OF THE EXPONENTIAL
THEOREM.
§ 8.] Among the series which can be summed b}' means of
the Exponential Series, two, related to it in the same way as the
series of chap, xxvn., § 5, are related to the Binomial Series,
deserve special mention.
We can always sum the series 2<^r ("») .?■"/«!, w?ie7-e <^r («) ^'s ««
integral function of n of the rth degree. {Integro-Exponential
Series.)
2^4 l(f,r(n)ln\, f <^,(n)/;i!(n+a)(n + 6) . . . (n +/.) C». xxviii
For, as in chap, xxvii., § 5, we can always establish an identity
of the form
<^r(n) = -fl<> + ^in + -4,n(n-l) + . . . + Arn{n-1) . . . (»-r + l).
Tlion wo liave, tikiiig, for simplicity of illustration, the lower
limit of summation to be 0,
• nl on! 1 (»- 1)! 1 (n-2)l
= (At + AiX+ A,a^ + . . . +Ar!t^)e'.
Cor. We can in general sum the series 'S,<t>^(n)3^/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
cliap. XXVII., § 5, only the details are a little simpler. (See
Kxample 5, below.)
Example L To deduce the formnliB (3), (4), (6) of chap, ixvn., § 9, by
means of the exponential theorem.
(x + n)'-,C,(x + n-l)'+. . . (-)',C,(x + n-r)'+. . . ( - )*x'
ia evidently the cocQicient of 2' in
The lowest power of t in the product last written is t', and the ooelGcicnta
of«", I*", £»+' are il, «!(i + Jii), Jt!{x' + njr + ^ n(3n + l)} rcspoctiveJy.
Hence
(i + n)'-,C,{x + n-l)' + . . . (-)'.C,{x + n-r)' + . . . (-j-x-
= 0, if i<n;
=nl, if (=ri;
= (n + l)!(x + Jn), if i = n + l;
= J(fi + 2)!{x' + iLr+,>5n(3n + l)}, if(=n + 2.
Example 2. If n and r be positive integers, show that
^ii+ " XI .n(n-l)...{n-. + !)_., n(n-l)...l 1
1 n + T+\ {n + r+l)(n + r+2) . . . (n-hr + i)
§8
EXAMPLES
235
The right-hand side is the coefficient of z"+'' in
n+r
(z+l)"+l (2 + j)
(Z + X)''+''-^ + . . .+i '-
(z+j;)"+'-+'
+ . . . + ^ , ' „ +. . .
= (^ + ^)"(■»+^
.+»C,
.x"}x |l + ^,
+ 21 + -
J"
.+ -. + .
til
Now the coefficient of c""*^ in this product is
n(n- 1)
Vl ll(r4
+ 1)1
x + . . .+
nl(;
1) ■ ■ . 1 J
Hence the theorem.
If we put 7=0, and x=l, we have
n + 1 (n+l)(n + 2)
^+(lir^ + ' (2!)=
- + . . . ad 00
J n n(n-l) ,"(«-!) • • • Ij
Example 3. Sum the series
IS is + 23 „
.+
13 + 23+. ■ . + ll3
2"+.
ad 00.
We have (by chap, xx., § 7)
l' + 2' + . . . + n'=()i'' + 2)j3 + ,i2)/4,
=4{4<, + /Ii7i + .l2?!(n-l) + J3n(7i-l)(«-2) + ^4m(n-l)(n-2)(n-3)},
where ylj,^,, . . ., ^4 may be calculated as follows : —
A,= 0,
A,= 4,
A,= 8, A, = l.
+1
1+ 2+ 1+ 0 + |0
0+ 1+ 3+ 4
+ 2
1+ 3+ 4 + |4
0+ 2+ 10
+ 3
1+ 5 + |U
0+ 3
1 + 18
Hence
^13+23+ +n»^,^^^^ 7^,^ *'
ml
+ 2x3S
ajo-s
(n-l)r2 (n-2)l
= (x + 4xH2i3+Jx-')<;*.
If we put :r = 1, we have
2(1' + 2' + . . . + n3)/«l=27<>/4.
Example 4. Show that S n'/n! = 5e.
Since jj' = ji + 3h (n -!) + ?; (n-1) ("-2),
Sh=/hI = £!/(» - 1)1 + 3Sl/{;j - 2)1 + 21/(ii - 3)1,
=5e.
+ tX*2
(n-3)l 4 (n-4)l'
L
236 EXERCISES XII Cll. XXVIII
Example 5. Evalaato ^ (n - 1) x'>/(n + 'i) nl.
(n-l)x'_l (n'-l)x-^
(n+2)nl z' (n + 2)l "
Now n'-l = 3-3(»i + 2) + (ii + 2)(n+l).
Therefore
■;(n + 2)Hl i')t(n + 2)I , (n + l)l^ , n!( '
= {(i»-3x+3)<r*+(Jx>-3)}/i'.
ESEBCISGS XII.
(1.) Evaluate 1/^ to f^ii places of dccimalH.
(2.) Calculate x to a second approximation from the equation
501op,(l + x) = 49x.
(3.) If e* = \ + xe'^, and x* be negligible, show that
ft = l/2!+x/4l-x*/4!5l.
(4.) Show that, if n bo any positive integer,
(l-l/n)-«>l + 1/11 + 1/2! + . . . + l/n!>(l + l/<i)«.
(5.) Sum from 0 to cc S (1 - 3n + «=) i»/»J.
Sum to infinity
(fi.) l»/2! + 2»/3! + 3'/4! + . . . .
(7.) l»/2! + 2'/3! + 3'/4l + . . . .
(8.) l-2>/ll + 3»/2!-4V3! + . . . .
(9.) l* + 2«/21 + 3V3! + . . . .
Show that
(10.) l/(2n)l-l/l!(2ii-l)l + l/2!(2it-2)l- 1/I!(2ii- l)I + l/(2n)I = 0.
(11.) If n>3, n> + .C,(n-2)» + .C,(n-4)' + . . . = n» (n + 3) 2»-«.
(12.) n"-.C, (n-2)- + ,C,(n-4)«-. . .=2"n!.
(13.) By expanding fW-*!, or otherwise, show that, if
Ar=''i (n + r-l)!/nl(n-l)!, thenil^,-(9r + l).4,+r(r-l)^^, = 0.
*-' (Math. Trip., 1882.)
(14.) Prove that
(x-x»/31 + x»/51- . . .)(l-x'/21 + x</41-. . .) = 2(-)'2»x«^'/(2r + I)l.
(16.) Solve the equation x»-x- l/n=0; and »how that the nth power of
its greater root has e for its limit when n = oc .
(IG.) For all positive integral values of n
--'m'wr ■ ■ ■ (.!-,)<--•
(17.) If
'"-'<. + =5',('-l) + ^|('-l)(x-2) + . . . + ^"(x-l)(x-2) . . . (x-n).
•how that /<, = (» + l)"-,C,f»+,C,(»-l)"-. , . (-)"/'.>•.
^ 9 EXERCISES XII 237
(18.) Show that l(n' + -2n' + n-l)ln\ = 'Je + l.
1
(19.) Sum :S,(n + a)(n + b)(n + c)x"ln\ from 7i=0 to k=qo.
(20.) Show that e cannot be a root of a quadratic equation having finite
rational coefficients.
(21.) Sum the series 2i"/(n + 3) n\ from n = 0 to n = QO.
(22.) Sum to infinity the series l»/3. 11 + 33/4. 2! + 5'/5. 31 + . . . .
If £, , iJj, . . ., i3„ denote Bernoulli's numbers, show that
(23.) ^+iCj„-iBn-:m+iC=»-sB»-i + - ■ • (-)''-'»»«CiBi = (- 1)""'-
los\ r n in+i^an-a -^n-i . / \„-i a.+i <^2 -^i , _ >„_! ^n
l^*') 2n+1^2n-"ii 22 • • ( ; 2-" ^ ' 2*"
(25.) 4,.Ci-Bi-i»CsB2+}„C5B3-. . . = (n-l)/2(n + l), the last term on
the left being (-)""-=• £^2. "^ 4( -)»("-=) »lB(„_i)/2, according as n is even or
odd.
(26.) By comparing Bernoulli's expression for 1'' + 2'' + . . .+n'' with the
expressions deducible from Lagrange's Interpolation Formula, show that
"rV)'-s.«c,'^-(-)p-i.v>
1 '
(-2p+a ,s.
Also that
fr=.27) 9
t ^ ' »w-l^'t(«+l)-"-
(Kronecker, Crelle's Jour., Bd. lxxxiv.; 1887.)
(27.) x(e- - e-)l(c- + e-) = § (2^ - 1) 2=x2 + ff (2^ - 1) i^x' + § ('i^ - 1) 2«x« + . . .
LOGARITHMIC SERIES.
§ 9.] Expansion of log {I + x). — It is obvious that no function
of X which becomes infinite in value when a; = 0 can be expanded
in a convergent series of ascending powers of x. For, if we
suppose
f(x) = Aa-¥AiX+A.x~ + . . .,
then on putting ^ = 0 we have 00 = ^1„ ; and the attempt to
determine even the first coefficient fails.
There can therefore be no expansion of log a: of the kind
mentioned.
238 EXPANSION UK U)U(l+x) Cll. XXVllI
]Ve ain, howenr, expand lng{\ +x) in a series of ascending
powers of X, prtrvided x be uumericaUii less than unity.
Tlio bafie iu the first instauce is understood to be « as usual
By § 4, we have
(l + a;)'=l+£{log(l+a')}+£»{log(l+3-)}V2! + . . . (1);
and this series is couvergeut for all values of z.
Again, by the binomial theorem, we have, provided the
numerical value of z be less than 1,
{\ + xY =1 + :x + z{z-\)3>l-2\ + z{z-\){z-2)x'IV. + . . .,
= l + zx-z{l-zll)x'!2 + z(l-z/ini-z/2)j'l3 + ... (2).
If we arrange this as a double series, we have
(l + xy = l+zx- {ca.-»/2 - z'x'/2] + {zuf/S - (1 + l)^x'/3 + i s'x'/S] +
( - )-' {c^/n - ...P, :r'jf/n + ,.,P, ^jfjn -. . .
(-)— .->P.-,s"^/»}
(3),
where ,_iiV stands for the sum of all the r-jjroducts of 1/1,
1/2, . . . , l/(n - 1), without repetition.
In order that Cauchy's crittTiou for the absolute convergency
of the double series (3) may be satisfied, it will be sufficient if
the series
zjf/n + .-, A z'x'jn + . . . + ,.,/',., z'^ln (4)
and
1 + r.r + ; (1 + z/l)x'j2 + z(l + s/l)(l + z/2) af/S + . . . ( :,)
be both convergent when z and x are positive.
Now the sum of (4) is always s (c + 1) . . . {z + n- 1) j-'/n! ;
and this has 0 for it« limit when n=ao, provided x<l. Also,
the series (5) is absolutely convergent when x< 1.
Hence, by chap, xxvi., S 34, we may rearrange tlie scries (3)
according to pnwcrs of z, and it will still converge to (1 1- x)'.
Confining onr attention to the first power of z, for the
present^ we thus find
(l+;r)' = l + {a:/l-«'/2+a:'/3-. . .\z + . . . (5).
Now, since there can only be one convergent ejcpausiou of
^§9,10 EXPANSION OF (lOG (!+«)}" 239
(1 + xf in powers of z, the scries in (1) and (5) must be
identical. Therefore
hg(l+x)'-^x/l-af/2 + .v'/-S-. . . ( - )"-^ a-"/?* + . . . (6).
The series on the right of (6) is usually called the logaritiimic
series. It is absolutely convergent so long as - l<a;< 1, and it
is precisely under this restriction that the above demonstration
is vaUd.
If we put x = l on the right of (6), we get the series
l/l - 1/2 + 1/3-. . . {-l)'^~yn + . . ., which is semi-conver-
gent. Hence, by Abel's Theorem (chap, xxvi., § 20), equation
(6) will still hold in tliis case ; and we have
log 2 = 1/1-1/2 + 1/3-. . . + ( - 1)''-Vm + . . . (7),
provided the order of the terms as written be adhered to.
If we put x = -l in (6), the series becomes divergent. It
diverges, however, to - oo ; so that, since log 0 = - oo , the
theorem still holds in a certain sense.
Cor. 1/ we arrange the coefficients of the remaining powers
of z in (5), and compare with (I), we find
{log (1 + x)Y = 2! {,Pi arl-2 - .Pi 3?IZ + ^Pi a.-*/4 - . . . },
{log (1 +0;)}"= h! {n-iP,.-!^-"/" -„Pn-. *-"+V(« + 1)
+ „.iPn-ia;''+V(« + 2)-. . .} (8).
These formula3 and the above demonstration are given by
Cauchy in his Analyse Algehrique.
§ 10.] A variety of expansions can be deduced from the
logaritiimic theorem. The following are some of those that
are most commonly met with : —
We have
log(l+a:) = ;r/l-.r=/2 + ar'/3-. . . ( - )"-'.z'"/« + . . .;
also
\og{\- x) = -xl\-irl-2-a?IZ-. . .-x"/n-. . . .
Hence, by subtraction, since log(l +a;) - log (1 - .r)2log
{{l+x)j{\-x)}, we deduce
log{(l+a:)/(l-.r)}-2{x/l + .i'V3 + . . . +.r--"-V(2«-l)+. . .} (0).
240 VARIOUS LOGAIUTUMIC EXPANSIONS CU. XXVIII
Tutting in (9) y = (l + j-)/(1 - j), and therefore x--(y-\)l
(y+1), we get
(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 +j- = x(l + 1/j-), and log ( 1 + J-) = log a- + log
(1+1/J-), putting in (10) y=l + l/ar, so tliat (y-l)/ty+l) =
l/(2jr+l), we have
log(l+*) = logx + 2{l/l(2x+l) + l/3{2x + l)' + . . .1 (11).
Finally, since x + 1 = x" (1 - l/x»)/(x - 1 ).
log(x+l) = 2logx-log(x-l)
-2{l/l(2x*-l)+l/3(2a!»-l)' + . . .} (12).
I^ in any of the above formidrc, we wish to use a base a
different from e, we have simply to multiply by the " modnlus "
1/log.o (see chap, xxi., § 9). Thus, for e.xample, from (10) we
derive
ON THE CALCULATION OF HXJARITHJtS.
§ 11.] The early calculators of logaritlims largely usicd
methods depending on tlio repteated e.xtraction of the square
root. Thii process was comhiued with the Metliod r.-,
which seems to have arisen out of tlie practical nee — thi-
Logarithmic Calculator*.
• See GlaiKhcr, Art. "^ -,' Fnqiclopjdia BriUuuiica, 9th e<L,
(rom wludi luuch ol what ..kuu.
§§ 10, 11
CALCULATION OF L0Gj2
241
Thus, Briggs used the approximate formula
logio 2 = (2">"" - 1) 2'710 lege 10,
depending on the accurate formula
L{af-l)/z = logea;,
which we liave already established in the chapter on Limits,
and w^hich might readily be deduced from the exponential
theorem. The calculation of logio2 in tliis way, therefore, in-
volved the raising of 2 to the tenth power and the subsequent
extraction of the square root 47 times !
Calculations of tliis 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 y = 2 in (10) we have
Iog2 = 2{l/l. 3 + 1/3. 3^ + 1/5.3= + . . . }.
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
1/3'
4,115,226
1/5 .3=
8-23,045
1/3'
457,247
1/7 .3'
65,321
l/3»
50,805
1/9 .3"
5,645
1/3"
5,645
1/11.3"
513
l/3>'
627
1/13.3'^
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/19.3"}/(l-i).
c. 11. 16
242 NAPIKUIAN LOGARITHMS OF 1, 2,
10 cii. xxvin
tliat is, less tli;in -OOO.OOO.OOO.OG ; anil the utmost error from
tlie Ciirri;ij,'e to tlie hust line is + 4. Tlio utmost error in our
calculation is + 8. Hence, subject to an error of 1 at tlie utmost
in the la.st place, we have
log 2 = -693,147,18.
Having thus calculated log 2, we can obtain log 3 more
rapidly by putting a: = 2 in (11). Tims
Iog3 = log2 + 2{1/1.5 + l/3.5'+l/5.5' + . . . }.
Knowing log 2 and log 3, we can deduce log4-2log2, and
log 6 -= log 3 + log 2. Then, putting ^ = 4 in (12), we have
log5 = 2log4-log3-2{l/3H- 1/3.31' + . . .1.
Also, putting x = & in (12), wc have
log 7 = 2 log 6 - log 5 - 2 {1/71 + 1/3 . 71' + . . . }.
It will be a good e.xercise in computation for the student to
calculate by meun.s of these fonnulie the Napierian Loj.'iirithms
of the first lU integers. The following table of the rc^iults to
ten places will serve for verification : —
No.
liOgaritlim.
1
0000,000,000,0
2
0 693,147, 1S(>,G»
3
l()98,(;i2,2S8,7
4
r;!H(;,-_>;i4.3Gi,i
5
r6(l'.t,437,91'>,4
r,
r79i.7.vj.ir,'.t,2
7
1-94.5,9 10,1 49.1
8
2079,441,541,7
9
2-197,--'24,577,3
10
2-302,585,093,0
From the value of log, 10 we deduce the value of it« re-
ciprocal, namely, J/= -434,294,481,903,2.")1; and, by multiplying,'
by this number, we can convert the Napierian Logaritlim of
* <! mr.ins thai tlic lOlb digit luu been incrcancJ by a unit, bccmuae lh«
lltb exoooUa 4.
§§11,12 FACTOR METHOD OF CALCULATING LOGARITHMS 243
auy 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 xm., "2, below) Professor
J. C. Adams has calculated these numbers to 260 places of
decimals.
§ 12.] The Factor Method of calculating Logarithms* is one
of the most powerful, and at the same time one of the mo.st
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>„/(i-Wio)(i-/',/io=)(i-^Vio^) . . , (1).
where p^, Pi, p^, ■ . . each denote one of the 10 digits, 0, l,
2, . . ., 9, jt?„ being of course not 0.
Take, for example, 314159 as the given numljcr. First
divide by 10\ 3, and we have
314159 = 10\ 3. 1-047,196,660,006 ....
Next multiply r047,196,666,666 by 1-4/10=, that is, cut
off two digits from the end of the number, then multiply by 4
aud subtract the result from the number itself The effect of
this will be to destroy the first siguiticant figure after the
decimal point. We have in fact
1-047,196,666,606 x (1 -4/10==)= 1-005,-308,800,000.
Next multiply r005,308,800,000 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 Glaishcr's article above quoted ;
or the Intioduclion to Gray's Tubles for the Formation of Logarithms and
Auti-LoijaiiDims to Twiiity-four Places {1S7G).
16—2
24+ KACTOUMETHOD OF CALCULATING LOOAKITIIMS CU. XXVIII
10 17,i96,(;GG,6|66
41,887,866,666
5,308,800,1000
5,026,544,000
282,25|6,000
200,056,451
8 2, 1|9 9, 5 4 9
80,006,57 6
2,119 2,9 7 3
2,000,004
|1 9 2, 9 6 9
100,000
9 2, 9 6 9
4/10'
5/10'
2/10*
8/10*
2/10*"
1/10'
9/10*, 2/10*, 9/10", G/10", 9/10".
Tlie remaining factors being obvious without farther calcula-
tion. Hence we have
. (1-9/10")
= 10».3(l + x/10"), 3'>9.
314159 x(l-4/10')(l- 5/10")
Therefore
314159 =10*. 3 (l+a-/10")/(l-4/10')(l- 5/10') . . . (1-9/10")
(2).
Since log(l + j-/10")<j-/IO", it follows from (2) th.it, as far
as the twelfth place of decimals,
log 314159 = 5 log 10 + log 3 -log (1-4/10') -log (1-5/10*)
- log (1 - 2/10*) - log (1 - 8/10*) - log (1 - 2/10')
- log (1 - 1/10') - log (1 - 9/10') - log (1 - 2/10")
-log(l- 9/10") -log(l-6/10")-log(l- 9/10").
All, th(>rcforc, that is required to enable us to i-alculato
log 314159 to twelve places is an au.xili.ary tible containing the
logarithms of the first 10 integers, and the logarithms of l-p/W
for all integral v.ilucs of p from 1 to 9, and for all integral values
of r from 1 to 12. To make quit« sure of the last figure this
au.xiliary table should go to at lea.<t 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 wore required, the Metliod uf Differences would I*
employed to find the grcil majority of the numbers to be entcrod.
§^ 12-14 FIRST DIFFERENCE OF LOG ;B 245
A full discussiuu 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 {w + h) - logio X = logio (1 + hlx)
= M{hlco-lAhl.'cr + l{l,l.vf-. . .} (I).
Hence, iih<x, we have approximately
logio {x + li) - logio X = Mhjx (2),
the error being less than \M{hlx)-.
The equation (2) shows that, if ^31{k/.r)- do not affect the
nth place of decimals, then, so long as h:!f>k, the differences of
the values of the function are proportional to the differences of
the values of the argument, provided we do not tabulate beyond
the ?!th place of decimals.
Take, for example, the table sampled in chap, xxi., where the numbers
arc entered to five and the logarithms to seven places. Suppose a = 30000;
and let us inquire within what limits it would certainly be safe to apply the
rule of proportional parts. We must have
ix-4343(/!/30000)2<5/108,
if the interpolated logarithm is to be correct to the last figure, that is,
ft<:3V23'04,
<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.
SUM5IAT10N OF SERIES BY MEANS OF THE LOGARITHMIC
SERIES.
§ 14.] A great variety of series may, of course, he 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, l.c.
i
2W l<^(»).r"/(n + a)(n + ?>).. .(n+i-) cil. xxvm
to the lo^jaritliinic tlicorem as do the tlieorems of cha]). xxvii., §5,
and chap, xxviii., §8, to tlie binomial and exponential thcoreuia: —
T/ie set-ies whose general term is <^(n)x"/(n + a)(»» + 6) . . .
(n + k), where <f> (») is an integral function of n, and a, h
k are positive or negative* uneqtuil integers, can alwai/s be
summed to infinity prnvided tlw series is omvergent.
It can easily be shown that the series is convergent provided
X be numerically less than unity, and divergent if 2: be
uunierically greater than unity.
If the degree of <^ (n) be greater than the degree f>f (n + o)
(n + l>) . . . (n + k), the general term can be split into
\l>{n)af + x{fi)^Kn+a){n + l>) . . . (n + k) (1),
where i/'(h) and x{ti) are integral functions of », the degree of
the latter being less than the degree of (h +a)(n + b) . . . (n + k).
Now Si/f (n) x" is an iutegro-gcometric scries, and can be
sumnied by the method of chap, xx., § 13.
By the method of I'artial Fractions (chap, vui.) we can
express x(n)/('» + «)(» + i) . . . {n + k) in the form
Af(n+a) + Bl{n + b) + . . . + Kl(n + k),
where A, B Jf are independent of n. Hence the second
part of (1) can be split up into
Aafjin + a) + Bjr/{n + b) + . . . + Ax"/(h + k) (2) ;
and we have merely to sum the series
A 2a:»/(n + a), B 2x»/(n +b) K^j^/(n + k) (3).
Now, supposing, for simplicity of illustration, that the sum-
mation extends from n=l ton=ao, we have
A ir"/(» + a) "Ax-'l-af^'/in + a),
--^ar-{a-/l + ar"/2 + +jf/a+log(l-x)\ (4).
Each of the other series (3) may be sumnied in like manner.
Hence the summation can be completely eflected.
* Wlicn any of the int«i;erii a, b, . . ., k are negative, Ibe method
requires the evalunlion of limita in certain case*.
§14
^(j> (n) x^lin + a){n + h) .. .{n + k)
247
If ,r= 1, the series under consideration will not be convergent
unless the degree of <^ {}>) be less than the degree of (w + a)
(n + b) . . . {n + it). It will be absolutely convergent if the
degree of <^ (n) be less than that of (w + a) (w + 6) . . . (» + fc) by
two units. If the degree of <^ (n) be less than that of (ii + a)
{n + b) . . . {n + A-) 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 ca.ses, however, where the series is convergent we can,
by Abel's Theorem, find the sum for ;?; = 1 by first summing for
.r<l, and then taking the limit of this sum when x = \.
In the special case where 4> {») is lower in degree by two
units than {n + a){7i + l>) . . . (ii+k), and a, b, . . ., Jc are all
positive, an elegant general form can be given for 5<^ («)/('* + '*)
(» +b) . . . {n + k).
From the identity
4>{n)Hji + a){ii + b) .
we have
+ .
{n + k)
iA/{n + a) + Bl(n + b) + . . . + K, {71 + k),
. . {n + k) + B{n + a)(?i + c) . . . (n+k)
. + K{n + a){n + b) . . . {ii+j) (5),
and, bearing in mind the degree of 4> (n), we have
A + B + . . .+K=Q (6).
b, . . ., n = -k, \IQ
Also, putting in succession n = - a, n-
have
.4 = </> ( - a)/(6 - a) (c - rt) .
B = <t>{-b)l{a-b){c-b) .
(k - a) \
{k-b)
K=<j>i-k)/{a-k){b-k) . . . (j-k).
Reverting to the general result, we see from (4) that
2<^ (m) af/(n +a){7i + b) . . . {11 + k)
(7).
(8),
= -%Ax-''{xl\+arl-2 + . . .+afla)~\og{l-x).:S.Ax-
where the 2 on the right hand indicates summation with respect
iQ a, b, . . ., k.
248 KXAMPixs r\\. XXVIII
Now, since yl+/y + . . .+A' = 0, iyLr"" is an algebraical
function of .f wiiicli vanishes wlitii .r-1. ^Vlso l-x is aa
algebraical function of u: iiaving the same property. Therefore,
by chap. XX v., § 17, we Imve
L log(l-.r).2^ij;--= L log{(l -;r)*'"-},
= logl.
= 0.
Hence, taking the limit on both sides of (8), we have, by Abel's
Theorem,
i<f.(n)/{n + a){n + 0) . . . (n + k) = -:S.A (l/l +1/2 + . . . + 1/a),
^.^(-a)(l/l-i-l/2 + . . .-t-l/g) .
~ " {b-a){c-a) . . . (c-k) ^ '•
the i on the right denoting sumiuutiou with respect to
a, b, c, . . ., k.
Eiample 1. Evaluate Sn'i"/(n - 1) (n + 2).
Wchave f.'j"/(n- l)(« + 2) = (n- l)i- + ii"/("- l) + !a^/(" + 2).
Now £(n-l)x»=lx» + 2x' + 3i* + . . .,
s
(1-x)-^(k-1)x»=1x' + 2x' + 3z« + . . .
-2.1x»-2.2x«-. . .
+ lx^ + . . .,
= i'.
Hence l(n-l)*»=x»/(l-x)'.
1
AlBO j£x»/{n-l) = ixLx-i/(n-l).
t 1
= -ixiog(i-x):
I Lvi" + 2) = |x-« £x"+'/(n + 2).
= - 1 r-> {*/l +at'/2 + x»/3 + log (1 - x)}.
Bcnco the whole sum is
x»/(l - x)' - 5x-' -\-lz-\(x + 8x-«) log (1 - x).
Example 2. Evaluate £ l/(n - 1) (n + 2).
*
TSy the same process as before, we find
L"/("-l)(n + 2)-J'"' + i + i^ + l('"'-')'»t5U--)-
§ 14 EXAMPLES 249
Now, since L {l-a;)»"'-»=l (chap, xxv., § 17), L (x-2-x)log(l-i) = 0.
X=l 1=1
Therefore 21/(7.-1) (H + 2) = J + J + i = Ji.
This result miRlit be obtained in quite another way.
It happens that -l/(n - 1) (n + 2) can be summed to n terms. In fact,
wo have
l/(n-l)(n + 2) = 5{l/(»-l)-l/(» + 2)}.
Hence, since the series ia now finite and commutation of terms therefore
permissible,
„5,,, ,,, „ 1 1 1 1 1 1 1
s '12 3 n-4 n-3 n-2 n-1
1 J. 1 1 1_
"!"■■■ n-4 re-3 n-2~n-l
_ 1 _ _1 1_
n n+1 n+2'
1 1 1_ 1_^^ 1_
""l''"2"'"3 n n + 1 n + 2'
Hence, taking the limit for n=» , we have
-l/l 1 1\ 11
T~3 Vl'*'2"'"3y~18"
Example 3. To sum the series
(Lionnet, Nouv. Ann., ser. n.,t. 18.)
Let the (n + lUh term be «„, then, since «„=0, association is permitted
(see chapter xxvi., § 7), and we may write
111
■4n + 1^4n + 3 2;i + 2'
11111
, + TITT^ - TTT-r + :
4n+l 4h + 2 47J + 3 4rt + 4 4n + 2 4h + 4'
^ / 1 1 1 _ 1 \ 1 /_^ 1_\
~ V4n + 1~ 471 + 2"*" 4h + 3 4/1 + 4/ ■*" 2 1.271 + 1 271 + 2 j'
= v„ + U!„, say.
Now, as may be easily verified, u„ 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. xx\x, § 6) 2r„ and 2i(i„ are absolutely couveigent
series. Therefore (chap. xxvi. , § 13)
S«„=S (»„ + «>„),
0 0
=2u„+2ic„.
2riO INEQnAIJTY THEOREMS CII. XXVIII
Uence, again dissociating i', and ir, (ni* is evidently iKTmisaible) wo liave
- ,1111111
,1/, 1.111111 \
+ 2i^-2 + 3-4 + 5-G + 7-8+---j
= lop, 2 + i loR, 2, by § 9 above,
= 5 log, 2.
Tliiit example is an iiittircsting Bpecimon of Die somewhat dvlicatc opera-
tion uf evaluating a scnii-convergcnt series. Tlic process may be described
B!) consisting iu the conversion of the semi-convergent into one or more
absolutely convergent scries, whose terms can be commutatcd with safety.
It sliould be observed tlint the terms in the yiven series are merely those of
the series 1-1/2+1/3-1/4 + 1/5- . . . written in a different order. Wo
have thus a striking' instance of the truth of Abel's remark that the sum of
a Bcmiconvergent series may be altered by commutaling its terms.
APPLICATIONS TO INEQrALITY AND LIMIT THEOREMS.
g 15.] Tlie Expiiiientiiil and Loijarithiiiic Series may be
applied with eflcct in establishing theorems regarding inequality.
Thus, for example, the reader will find it a good exercise to
deduce from the logarithmic expansion the theureiu, already
proved in chapter xxv., that, if a: be positive, then
.r-l>logx>l-l/a; (1).
It will also be found that the use of the three fiiuda-
mcntiU .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
log -!■>-+ =•+ -,+ ... + ->log
m-1 m m+1 m+2 n ° m
If we put l-l/x=l/ni, that is, z = m/(m-l), in the second pnrl of (1) abova,
and then replace m by m + 1, m + 2, . . .,n successively, we get
log m - log (m - 1) > 1/m,
log (m + 1) - log m > l/(m + 1),
log n - log (n - 1) > 1/n.
Ilcnce, by adlition,
logn-log(m-l)>l/m + l/(m + l)+. . . + 1/n (2).
J
§ 15 LIMIT THEOREMS, EXERCISES XIII 251
Next, if we put x - 1 = l/m in the first part of (1), and proceed as before,
we get
log (m + 1) - loH J« < l/m,
log (hi + 2) - log (m + 1) < l/(m + 1),
log (n + 1) - log n < 1/n.
Hence
log(H + l)-Iogm<l/m+lAm + l)+ . . .+l/n (3).
From (2) and (3),
log{H/(ni-l)}>l/m + l/(m + l)+ . . . +l/n>log{(« + l)/»i}.
Example 2. If p and q be constant integers, show that
L {l/m + l/(ni + l)+ . . . +ll{pm + q)} = \og2>-
(Catalan, Traite Elcmentaire dcs Scries, p. 58.)
Put n=pm + q in last example, and we find that
log{(pm + 3)/(m-l)}>l/m + l/{m + l) + ... + l/(7)m + g)>log{(iim + (; + l)/m}.
Now L log{(i)7n + 3)/(m-l)}=logi),
and L \os{{pm+q + l)lm}=logp.
Hence the theorem.
Example 3. Evaluate L (c==- l)=/{.T-log(l + x)} when x = 0.
Since {e'-l)^={x + hx-+ . . .)••'= x2(l + i.i;+ . . .f;
x-log(l + j;) = ix2-ix3+. . .=i.r=(l-|x+. . .).
Therefore
{<!»-l)=/{x-log(l + x)}=2(l + ix+. . .)-l{l-lx+. . .).
Since the series with the brackets are both convergent, it follows at once
that i(t^-l)-/{x-log(l + x)} = 2.
Exercises XIII.
(1.) If P=1/31 + 1/3.3P + 1/5.31»+. . .,
g = 1/49 + 1/3. 49^ + 1/5. 495+. . .,
B = 1/161 + 1/3.1613 + 1/5.1G1°+ . . .,
then log2 = 2(7P + 5Q + 3i?),
log3 = 2(llP + 8Q + 5fl),
log5 = 2(ir,2^ + 12Q + 7fl).
(See Glaislier, Art. "Logarithms," F.ncy. Brit., 9th cd.)
(2.) If a= -log (1-1/10), 6= -log (1-4/100), c = log (1 + 1/80), d =
-log (1-2/100), c = log (1 + 8/1000), then lo-2 = 7a-26 + 3c, log3 = llrt-36
+ 6c, log 6= 16a -46 + 7c, log7=4(39a-106 + 17c-d) = 19a-46 + 8c + e.
(Prof. J. C. Adams, Proc. E.S.L. ; 1878.)
(.'!.) Calculate the logarithms of 2, 3, 5, 7 to ten places, by means of the
foimulffi of Example 1, or of Example 2.
(4.) Find the smallest integral valne of z foi which (1-01)';> lOx.
252 EXERCISES XUl Ctl. XWIll
Sum tbo series : —
(5.) 2'/l(r>-3j:)' + 2:'/3(x»-3j-)'+ . . .
(7.) x>/1.2-x»/2.3 + x'/3.4- . . . ( - )"-'i»/;i(h + 1) . . ,
(8.) i'/3 + z</15+. . . +i»"/('in«-])+ . . .
(9.) i/l» + x'/(l' + 2») + r'/(l> + 2' + 3-) + --- + x"/(l' + 2' + ... + ,!») + ...;
also l/i= + l/(l' + 2') + l/(l' + 2' + 3») + . . . + l/(l> + 2« + . . . + «')+ . . .
(10.) 4/1.2.8 + G/2.3.4 + 8/3.4.6+. . .
(11.) If x>100, thon, to seven places of decimals at knst, log(x-t.8) =
2 lo^ (x + 7) - log (x + 5) - log (x + 3) + 2 log X - log (x - 3) - log (x - 6) + 3 log
(x-7)-log(x-8).
(12.) Expand log(l + x + x') in ascending powers of x.
(13.) From log (x'+l)s log (x+l) + log(j'-z + l), show that, if m be a
positive integer, then
6m -2 (Gm - 3) (Gwt - 4) (Cm -4) (Cm -5) (6m -6) , _
^~~2r'*' 31 " 41 ■•" •
(Math. Trip., 1882.)
(14.) {Iog.(l+x)}'=2x»/2-2(l/l + l/2)x>/3 + . . . (-)"2{l/l + l/2 + . . .
l/(n- l)}x"/n . . . Does this formula hold wlicn x= 17
(15.) log(l+x)'°«('-'l=-(?,x>/l-Q,x*/2-. . .-(?^.,x~/n-. . .;
where $,,-, = 1/1-1/2 + 1/3 . + l/(2n - 1).
(16.) Ifx<l, show that
x + Jx' + ix' + ,',x'«... = log{l/(l-x)}-JP,-tP,+ JP,-|P,-|P, + AP„...:
whore P,=i"+x'" + x*» + x'" + x""+ . . ., and the general term ia i-)'PJn,
unless n is a power of 2, in which case there is no term.
(Trin. Coll., Camb., 1878.)
(17.) Ite-'xc^'xe"'' ... = A^ + AtX + .. ., thou ^,,.=^^, = 1.3.6 . . .
(2r-l)/2.4.6. . .2r.
(18.) Ifx + a,x» + a,x» + . . . + y + <t,!/' + <i,y» + . . .= {rx + y)/(l -X!/)}' +
a,{(x + y)/(l-r)/)}' + (i,{(x + !/)/(l-xi/)}'+ . . ., for all values of x and y
which render the various aeries convergent, find a^, a„ , , .
Show that
(19.) log(4/«) = l/l. 2-1/2. 3 + 1/3. 4-1/4. 6+. . .
(20.) log2 = 4(l/l. 2. 3 + 1/5. 6. 7 + 1/0. 10. 11 + 1/13. 14. 15 + ...) (Eulor.)
(21.) (l-l/2-l/4) + (l/3-l/6-l/8) + (l/8-l/10-l/12) + ... = Jlog2.
(See Liounet, lYour. Ann., scr. ii., t. 18.)
(22.) <T,/ll-ncrJ2! + n(n-l)(r^,1l- . . . ton+1 terms =l/{n + l)', where
«r, = 1/1 + 1/2 + 1/3+. . .+l/r. (Math. Trip., 18S8.)
(23.) «~(l + l/m)"' lies between ^/(2m + l) and e/(2m + 2), whatever m
may be. {Souv. Ann., sur. ii. , t. 11.)
(24.) L{x/(i-l)-l/logx}=4.whenx = l. [E\i\eT, Iiut. CaU. Diff.)
(25.) I, { f* - 1 - log ( 1 + x) 1 /x« = 1 , when X = 0. (Euler, J.c. )
(28.) L(x'-x)/^l-x + logx)=-2, whcnx=I. (Eulcr, *.«.)
§ 15 EXERCISES XIII 253
(27.) I,(l + l/;!)""(l + 2/;i)""- • ■ (1 +«/n)'''"='l/<'. when n = oo.
(28.) I,{(2h-1)!/)i'-"-i}''»=4/c2, whenn = oo.
(29.) c^> 1 +1, for all real values of x.
(30.) a;-l>logx=-l-l/x, for all positive values of x ; to be deduced
from the logarithmic expansion.
(31.) e";> (1 + 7i)"/;il, n being any integer.
(82.) If n be an integer >-e, then Ji"+' > (n + 1)".
(33.) If A, B, a, b be all positive, then {a-l)l(A- L) + {A^ - Bh)
\os{BIA)l{A-B)^ is negative. (Tait.)
(.34.) Ilx>y>a, then {(x + a)/(i -«)}=:< {(y + a)/(»/-a)}i'.
(35.) L{ll{n + l) + ll{n + 2) + . . . + l/2H}=log2, when 71 = 00 . (Catalan.)
(36.) log{{7i + i)/(ni-i)}>l/m + l/(m + l) + . . . + l/n>log{(K + l)/m}.
(Bourgnet, Nouv. Ann., ser. 11., t. 18.)
(37.) log3 = 5/1. 2. 3 + 14/4. 5.6 + . . . + ('J7( -4)/(3n-2) (3n-l) 3k +. . .
(38.) If i(-)''->0(n)/(n + a) (n + h) . . . (n+k), where a, b ft are
1
all positive integers and </>(«) is an integral function of n, be absolutely
convergent, its sum is
S= S .^(-a){l/a-l/{a-l). . . (-)''-il/l}/(&-a) (c -«) . . .{k-a);
a,h it
and, if it be semi-convergent, its sum is
S + log2 S (-)«0(-a)/(6-a)(c-a). . .(i-a).
a,b k
(30.) Show that the residue in the expansion of log {1/(1 -«)} lies
between
x''+i{l + (»i + l)x/(n + 2)}/(;i + l)
and x''+i{l + (n + l).r/(l-x)(K + 2)}/(n + l).
(40.) In a table of Briggian Logarithms the numbers are entered to
5 significant figures, and the mantissie 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/x)^+» continually decreases as x increases.
(42.) Show that 5l/)i (4)i=-l)-= J- 21og2.
L
CHAPTER XXTX.
Summation of the Fvmdamental Power Series for
Complex Values of the Variable.
GENERALISATION UF THE ELEMENTARY TRANSCENDENTAL
FUNCTIONS.
§ 1.] One of the objects of the present chapter is to generalise
certain cxpan.sion theorems establislied in the two chapters which
precede. In doing this, we are led to extend the definitions of
certain functions such as a', log„;r, cos;r, &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 tlie circular functions play an iinj>ortant part in
what follows, it will be convenient here to rec.-ipitulate their
loailing properties. Thi.s is the more nece&sary, because it is
not uncommon in Engli.sh elementary courses so to define and
di.scuss the.se functions that their general functional character is
lost or greatly obscured.
§2.] Dt'finitionandPropi-rticsof the Direct Circular Functions.
Taking, as in cliap. 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 0 radians, count<'r-clockwise if <* bo a p<jsitive, clockwise
if it be a negative quantity. If (x, y) be the algebraical values of
the coonlinatcs of P, any point on the radius vector of 0, then
xjr, yjr, yjx, xjy, r/x, r/y are obviously all functions of 0, and
of 0 alone. The functions thus geometrically defined are called
§§ 1, 2 EVENNESS, ODDNESS, PERIODICITY 255
COS 6, siu 9, tau 6, cot 6, sec 6, cosec 6 respectively, and are spoken
of collectively as tlie circular functions.
All the circular functions of one and the same argument, 6,
arc algebraically expressible in terms of one another, for their
definition leads immediately to the equations
tan 6 = siu 6/cos 6, cot 6 = cos 6/sin S ; \
sec 6 = 1/cos Q, cosec 6 = 1/sin 6 ; \ (1) ;
COS" B + sin- 6 = \, sec" 6 - tan^ ^ = 1 ; )
from which it is easy to deduce an expression for any one of the
six, cos 6, sin 6, tan 6, cot 0, sec 6, cosec 0, in terms of any other.
When F{6) is such a function of 6 that F{- 6) = i^(6i), it is
said to be an even function of & ; and, when it is such that
F{-0) = -F(0), it is said to be an odd function of 6. For
example, 1 + 6^ is an even, and 6 - ^6^ is an odd function of 0.
It is easily seen from the definition of the circular functions
that cos 6 and sec 0 are even, and sin 6, tan 9, cot 6, and cosec 6
odd functions of 0.
When F{e) is such that for all values of 0, F(0 + nX) = F(e),
where X is constant, and 7i any integer positive or negative, then
F{6) is said to be a periodic function of 0 having the period X.
It is obvious that the graph of such a function would consist
of a number of parallel strips identical mth 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 repetitious of the same.
Since the addition of + 27r to d corresponds to the addition
or subtraction of a whole revolution to or from the rotation of
the radiiis vector, it is obvious that all the circular functions are
periodic and have the period 2^. Tliis is the smallest period,
that is, the period par excellence, in the case of cos 0, sin 6, sec 0,
cosec t/. It is easily seen, by studying the defining diagram, tbiit
tau 6 and cot 6 have the smaller period ir. Thus we have
256
ZERO AND TUUNINO VALUES
CH. XXIX
COS (6 + 2nT) = cos 0, sin {6 + 2«r) = gin 0,
sec (0 + 2nn) = sec 0, cosec (0 + 2nir) = cosec 6, j- (2).
tan {6 + nw) = tan ^, cot (0 + nir) = cot 0.
Besides these relations for whole periods, we have also the
following for half and quarter periods : —
cos(?r+e) =-cos^, sin(7r + e) = + sintf;
cos{W±6) = + 9in0, !im(hTr±0)=+cos0; \ /^\
tan(j7r + e) = + cote, cot(h-!r + 0) = + Uu0;
&c.,
all easily deducible from the definition.
We have the following table of zero, infinite, and turning
values : —
(•»).
which might of course bo continued forwards and backwards
by adding and subtracting whole periods
Hence cos 6 has an infinite number of zero values correspond-
ing to 0 = ^{2n+ 1)t, where n is any positive or negative integer ;
no infinite values; an infinite number of nuLxima and of minima
values corresponding to 0 = 2nir and 0 = {2h + 1)t respectively;
and is susceptible of all real algebraical values lying between
-1 and + 1.
Sin 0 is of like character.
But Uin 6 is of quite a diflerent character. It has an infinite
number of zero values corre.'.ponding to 0=ffr ; an infinite
number of infinite values corresponding to 0= h{'2n + l)v ; no
tuniing values ; and is susceptible of all real algebraical values
between - «■ and + oc .
Cot 0 is of like character.
e
0
J'
T
3'
2w
*c. \
cos^
+ 1
0
-1
0
+ 1
sin^
0
+ 1
0
-1
0
tantf
0
00
0
00
0
&c.
cote
oc
0
00
0
00
sec^
+ 1
oc
-1
00
+ 1
cosec 0
00
+ 1
00
-1
X
J
_l
X
_I
,
.'"^
^^-^""'^
f^^^Zl^__
„/"
'^^ 1
III>"
CO,/
i
^''
s,
...... y \^
y
k
v
"«.,
^>7<^IIZ
II>
,„-""'"
>|><cil_
~^~^^
/^
,^
W"^"'
<
o
' ^v
'''' \
;<d"
II^>
<
"~V Vr'- '"
/^ V ■ —
?
:isC;__
^-— """''^^-^^^^'S^^
•:^
^
C. IL
17
258
ADDITION FORMULiE
CII. XXIX
Sec 6 and coscc 6 have again a distinct diameter. Kiwli of
tbeiu lias infinite and turning values, and is susceptible of all
real algebraical values not l3^ng between - 1 and + 1. The
graphs of the funetions y = sin ar, y = cos a-, <S: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 dott<;d. The
curves that lie partly between and partly outside the parallels
KL, K'L', belong either to tana; or to cot a;, the graph of the
latter being dotted.
Agiiin, from the geometrical definition combined with
elementary considerations regarding orthogonal projection are
deduced the following Addition Formulw : —
cos (0±<l)) = cos ^ cos <^ + sin 5 sin ^ ;
sin (6 ±<t>) = sin 6 cos <^ ± cos 6 sin <t> ;
tan (^ ± </>) = (tan 6 ± tan </>)/(! + tan ^ tan <(>).
As consequences of these, we have the following : —
cos ^ + cos <^ = 2 cos h{6 + <}>) cos h(,6-<p);
cos <^ -cos 0=2 sin i(6 + <^) sin i{d - </.) ;
sin 6 ± sin <^ = 2 sin A(fl ± ^) cos i(^ + <^).
cos 0 cos <!> = ^cos (6 + <f>) + Jcos (6-<l>);\
sin 6 sin <^ = A cos {B-<f>)~i cos {0 + <t>); t
sin 6 cos <^ - Asin (6 + <^) + isin {0 - <f>). J
C082e = cos'e-siu'e = 2cos'tf-l = l-2sin'tf '
= (l-tAi\'6)f(l+t&u-e).
sin 20 - 2 sin 6 cos 0^2 tan 0/(1 + Un» 0).
tan 20 = 2 tan 6/(1 - tan' 0).
(.'■.).
(C)
(7)
(8).
§ 3.] Liri'r.i<' drrnhir Functions. When, for a continuum
(continuous stretch) of values of y, denoted by (y), we have a
relation
x-r(^) (1),
§§ 2, 3 INVEKSE CIRCULAR FUNCTIONS 259
wliich enables us to calculate a single value of x for each value
of y, and the resulting values of x form a continuum (*•), theu
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 x as a continuous function of y, but also
y 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-'{x) (2)
is identically equivalent to (1).
It must be noticed, however, that, although F'^ {x') is con-
tinuous, it mil 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 -oc <y< + oo,
the continuum {x) is given by 0;:)>a;<+ oo ; and each value of x
occurs twice over. We have, in fact, y = ±a^ \ 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, i^"' {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 -a)<y<+oo. The continuum (x) is given by
0^x< + <x) ; hence y is defined, by the above relation, as a
function of x for values of x between 0 and + -x> and i'ur 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 siny that the equation
x = smy (3)
completely defines x us n single-valued continuous function of
y, for — CO < ^ < + CO . Hence, we may write
^ = sin-' X (i),
n—2
260
JI LILTI PI,E-VALU EDNESS
Cll. XXIX
where tlie inverse function, sin"'ar*, is continuous, but neither
siugle-vahied, nor completely defined for all real values of t.
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"' :r 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"' a: is multiple-valued to an
infinite extent.
If, however, we confine ourselves to
values of sin"' x lying between - \-r and
+ \ TT, 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"'x. If we draw
parallels to the axis of x through the
points A, B, C, , . ., A', B", ....
whose ordinates are + irr, + § -, + f t, . . . , - i t, - f t, . . . , then
between every pair of consecutive parallels we find, for a given
value of a: (- \1cx1f'+ 1), one, and only one, value of y = sin"'ar.
'n»e values of y corresponding to points between the parallels
A' and A constitute what we may call the PriiicijHil Branch of
the function. Similarly, the part of the graph between A and B
represents the 1st positive branch ; the part between B and C
the 2nd positive branch ; the part between A' and B" the Ist
negative branch ; and so on.
If, as is usual, wc understand the symbol sin"' a; to give the
value of y corresponding to x, for the principal branch only, and
use y„ or „ sin"' x for the wth branch, then it is easy to see that
Fig. 2.
y» = „8iu"' a; = HTT + ( - 1 )" sin"' a-
(5).
• ThJB may 1)C road "angle whose sine i« x" or "aro-sincx." In
Coiitiuont.ll works the latter name is coutracted into uro-iiiax; and lhi> u
used iustcoU of biu~ ' z.
§ 3 BRANCHES DEFINED 261
where ii is a positive or negative integer according as the brancli
in question is positive or negative.
It is obviously to some extent arbitrary wliat portion of tlie
graph shall be marked oif as coiTesponding to the principal
branch of the function ; in other words, what part of the function
shall be called the principal branch. But it is clear!}- necessary,
if we are to avoid ambiguity — and this is the sole object of the
present procedure — that no value of 1/ should recur within the
part selected ; and, to secure completeness, all the ditierent values
of 1/ should, if possible, be represented. Attending to these con-
siderations, and drawing the coiTesponding figures, the reader
will easily understand the reasons for the following conventions
regarding cos~^ar, tan~'^, cot"'^ a;, sec~'a^, cosec~'a;, wherein y
and the inverse functional symbols cos""'ir, &c., relate to the
principal branch only, and ?/„ to the ?ith branch, positive or
negative.
y = COS"' a-, y between 0 and +77; i
3'n = (n + | + (-)"-'i)T+(-)"cos-'.r. | ^^^
y = tan~'.r, y between -\Tr and + ^t; l
y„ = WTT + tan"' X. J
y = cot"' X, y between 0 and tt ; ■»
y„ = riTT + cot"' X. j
y„ = niT + cot"' X. )
y = sec"' X, y between 0 and ir ; 1
y« = (» + ^ + (-)''-'|)T + (-)"sec-'.r. J
y = cosec"'a;, y between —\-i: and +1-; \
«^„ = WTT + ( - )" cosec"' X. J
(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 formnlte 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
2G2 INVERSION OF w ^ :" rii. xxix
If X be jiositive, the fonuula cos 0 -±J(l - sin' 6) becomes
sin"' X = COS"' J(l-^);
but, if z be negative, it becomes
sin"' x= — COS"' ^(1 - J-').
If 0<r<\IJ2, 0<y<l/^2, we deduce from the addition
formula) fin" the direct functions
siu-'a- + 8in"'y = cos-'[v/{(l-a^)(l-y')| -ry] ;
if 0<a:<l, 0<y<l,
tan"' X + tan"' y = tan"' [(x + y)/(l - xy)].
\{ X and y be both positive, but such that ry>\, then
tan"' X + tan"'y = it + tan"' [(x + y)/(l - ay)] •;
and, in general, it is esisy to show that
«tan-' X + „tan-' y = {m■¥n^■p)v■^■ tan"' {{x + .v)/(l - xj/)},
= .+.+ptan-'{(a: + y)/(l-ay)} (U),
where p= 1, 0, or - 1, according as tan"' a: + tan"'y is greater
than \-ir, lies between \v and -^t, or is less than -\-!t.
ON Tnr. INVERSION OF w = r".
§ 4.] When the argiunent, 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
ir = s» (1),
where n is a positive integer, s is a complex number, say
2 = 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
* Id En^'lioh Text-nonka cqantionB of this kind aro »rtcn loonlj
■taU'J ; and tlic resall Ini.'i bccu Bomc confusion in tbo blKlicr liranchat
of ninthematica, anch as tho integral calculus, wlioro tbcae invcnc fuuctiona
pla> au importaut port.
J
§§ 3, 4 INVERSION OF w = c" 263
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 y is continuous for all
finite values of x and y, that, if we form two Argand Diagrams,
one for x-^yi (the s-plane), and one for « + vi (the w-plane), then,
whenever the graphic point of s* describes a continuous curve, the
grapliic 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 c;-plane.
We shall also represent z by the standard form r (cos B-^i sin G),
and w by the standard form s (cos <^ + •( sin </>) ; but we shall, con-
trary to the practice followed in chap, xii., allow the ampHtudes
6 and <^ to assume negative values. Thus, for example, if we
wish to give s all values corresponding to a given modulus r,
without repetition of the same value, we shall, in general, cause
B to vary continuously from - ir to + ir, 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 + co , we shall get
every possible complex value of z, once over ; and thus effect a
complete exploration of any one-valued function of z.
Substituting in (1) the standard forms for w and z, and
taking, for simplicity, n = 3, we have
8 (cos ^ + i sin <^) = r^ (cos 0 + i sin Of
= r'{coa3e + ism36) ('->)
by Demoivre's Theorem. Hence we deduce
* For shortness, in future, instead of "graphic point of z" we sliall say
"z" simply.
2CA
CIRCUI-O-SPIRAL, GRAPHS
r-II. XXIX
or, if (as will be Buflicieiit for uur purpose) we confine ourselves
to a single complete revolution of the graphic radius of z,
s^r', * = 3fl (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 tlio w-plano (Fig. 4). Also, if we suppose z to describe
its circle (Fig. 3) with uuiform 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
Fio. 3.
Fia. 4.
corre.spond three complete revolutions of ir. In other words, the
values in the (MO-continuum which corre-ijwnd to those in the
(c)-continuuni (ire each rejmifi'd three times ox\r*.
The actual cour.se of «• is the circle of radius r* 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, r, 1, 0', which re-enters into it.^elf at O' and 0'.
The actual course may then be imagined to be what this spiral
becomes when it is .shrunk tight upon the circle.
• To indionto this poculiiiiitv "f ic wo gliall occasionally urw thi" term
"RcpoatinK Fonotion." A rc|>onling function need not, howi'vor, Ih' jx-iioilia
an »=:' u.
§§ 4, 5 riemann's surface 265
If we now letter the corresponding points on the s-circle with
the same symbols we have the circle O'll' in the w-plane, cor-
responding to the circular arc O'lI' in the £-pLine, 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 gi-aphical discussion that the 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 ofrv.
In other words, since, in accordance with a notation already
defined, (1) may be written
z = yw (1'),
we have sho\vn 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 i<;-plane, and a trisected circumference in the z-plane.
Hence, if we consider the equation
w = .5" (4),
aaid its equivalent inverse form
z=^w (4'),
all the alteration necessary is to replace the triple by an m-ple
spiral, returning into itself on the negative or positive part of
the M-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 to
for all values of w.
§ 5.] Riemann't Surface. It may be useful for tliose who are to pursue
their mathematical studios beyond the elements, to illustrate, by means of
the simple ease w = 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 brancli
of modern algebra whose fundamental conceptions we are now explaining.
2G6 BRANCHES OF ^W PH. XXIX
luatcad of Bupposing all the spircB of the le-path in Fig. 4 to lie in oue
plaue, we may conceive each complete spire to lie in a Boparate plane snper-
posed on the tc-plane. Instead of the sinBle ir-plane, we have thus three
separate planes, P,, /*„, P, . superposed upon each other. To Becnre continuity
between the planes, each of them is supposed to bo slit along the u-axis from
0 to - 00 ; and the three joined toRothcr, so that the upper edge of the slit in
P, is joined to the lower edge of the slit in P, ; the lower edge of the slit in
P(, to the upper edge of the slit in P, ; the lower edge of the slit in P, to the
npper edi^e of the slit in P, , this last junction taking place across the two
intervening, now continuous, leaves. Wc have tlius clothed the whole of the
irplane with a three-leaved continuous flat belicoidal* surface, any continn-
ou<i 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 Ritvmnn'i
Surface. The origin, about which the surface winds three times before
returning into itself, is called a JVinding Point, or Branch I'oint, of the
Third Order. Upon this three-leaved surface w will describe a continuotu
single path corresponding to any continuous single path of t, provided we
suppose that there is no continuity between the leaves except at the junctions
above described.
§ G.] If we confine 0 to tliat part T'Ol' of its circle which
io bi.sected by OA', and <^ to the corresponding .spire T'Ol' of its
path, so that <t> lies between — ir and + t, and 0 between - v/n
and -1- n/n, then s becomes a one-valued function of w for all
values of tc. We call this the princip.il brancli of the n-valued
function !!/w; and, as we have the distinct notation tr"" at our
disposal, we may restrict it to denote this particular branch of
the function z. In other words, if
w = « (cos <^ + / sin <^), — ir<<^<-t- jr,
we define ir"" by the equation
w^ = s"" (cos . <i>ln + i sin . <f>/n) ;
and we also restrict (cos <^ + 1 sin <^)''" to mean cos . <j>/n + 1 sin . <t>/n.
Just as iu § 4, we take the next spire after T'Ol' in the
positive direction (counter-clock) to represent the first positive
branch of yw; the ne.xt in the negative direction to represent the
first negative branch of ^w; and so on, the last positive and the
la.st negative being full spires, or only half spires, according as n
is odd or even.
If, as is usual, we repre.sent the actual analytical value of w
* Like a spiral BUurcaM.
§§ 5, 6 PRTNCIPAL VALUES 267
by the form s (cos <^ + i siu </>), where <^ is always taken between
- TT and + TT, then it is easy to find expressions for the values of z,
belonging to the m - 1 positive and negative branches of ^w and
corresponding to any given value of w, in terms of the value
belonging to the principal brancli. We have, obviously, merely
to add or subtract iimUiples of Stt to represent the successive
positive and negative whole revolutions of the graphic radius of
w. Thus, if z, z,, Z-, relate to the principal, tth positive, and
<th negative branches oi z= 'ijw respectively, we have
z = s"" {cos . ^In + i sin . </>/«} ;
2;t = s""{cos . (</> + 2<7r)/rt + / sin . (<^ + 2tTT)ln] ;
z-t = s""{cos . (<^ - 2tir)/n + i sin . (<^ - 2t7r)/n]. ,
(5).
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, if z denote the principal value of the nih root
of w, and u)n — cos. ^irfn + i sin . iirjii, then
Zt=e'»n, s-( = ««)„"'; ) ,„.
that is, c, = w'"W, S-« = w'"'o)„-'i
Cor. 2. The principal value of the nth root of a positive real
numhe)' r is tlte real positive nth root, that is, what has already
been denoted by r"" (sec chap, x., § 2).
For, in this case, we have w = r (cos Q + i sin 0), that is, "^ = 0.
Hence 'ijw = r^".
Cor. 3. The/re is continuity between the last values of any
branch of IJiv and the first values of tJie next in succession, and
between the last values of the last positive branch and the first
values of the last negative branch; but elsewhere tivo values of
^w belonging to different branches, and cm-responding to the
same value of w, differ by a finite amount.
It should be noticed as a conseixueace of the above that the principal
value of the jith root of a real negative number, such as - 1, is not definite
until its amplitude is nssif^ned. For we may write -l = eos7r + isin tt or
= cos( - jt) + isin (- tt) ; and the principal value in the former case is
cos.Tr/M + isin.Tr/ji, in the latter cos( - 7r/H) + i sin (-x/n). This amhiguity
doeR not exist for complex numbers differing; from - 1, even when they differ
inQuitely little, as will be at once seen by referring to Figs. 3 and 4.
268 DISCUSSION OF wP = i^ CH. XXIX
%!.] It should be observed tliat if, instead of restrictJDg ^
in the expression a = s""{co8. <^/» + f sin.<^/n} to lie between
— :r and +7r, we cause it to vary continuously from -nw to
+ MTT, then «■■* {cos .</)//» + t sin. <^/n} viiries continiiously and
passes once through every possible value of ijw, where | tc | is
given =s.
It follows also tliat, if to describe any continuous path
starting from P and returning; thereto, the value of ^'w will
vary continuously ; and will return to its original value, if w
have circulated round tiic origin of the rr-plane pn times, where
p is 0 or any integer ; and, in general, will return to its original
value multiplied by <i)„', where t is the algebraical value of
+ /1— V, fi and V being the number of times tliat 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 ^w.
§ 8.] Let us now consider briefly the equation
M'P = £« (1),
where ;) and q are positive integers. We shall suppose p and q
to be prime to each other, because that is the only ca.se with
which we shall hereafter be concerned*.
Our symbols having the same meanings as before, we
derive from (1)
s" (cos/x^ + 1 sin p<t>) = f^ (cos qO + 1 sin qO) (2).
Hence, taking the simplest correspondence that will give a
complete view of the variation of both sides of the etjuation
last written, we have
.,p = ,w, j,^ = qe (3).
If, then, we fix r, and therefore s, the p.nths of z and w will
be circles abo>it the origins of the z- and w-planes resjiectively ;
and, since p is prime to ^, if s and w start from the positive part
•Up and q hail the Q.C.M. k, no that p = hp', q — kq', whoro p' nnd q' are
mutually prime, then the equation (I) could \k written («>'>')* = (r«')», which
in equivalent to the k cqnationH, «*"=!<', ic<>' = <i»jj«', m>»' = «j'x*', . . ., k**
z:u^*z^', where w^ Ik n primitive k\.\\ root of 4- 1. Kach of thme k equation!
falU under the case above dincnniiod.
7,8
DISCUSSION OF WP = 2l
269
of tlie X- and «-;ixes slmultaueously, they will not again be
simultaneously at the starting place before z has made p, and
M) has made q revolutions.
To get a complete representation of the variation we must
therefore cause 6 to vary from -j»7r to + jott, and i^ from — qir to
+ qTr. The graphs of z and w will therefore be spirals having
p and q spires respectively. To each whole spire of the (/-spiral
will correspond the p/qth part of the j[?-spiral. The case where
p = 3 and (/ = 4 is illustrated by Figs. 5 and 6.
Fig. 5.
Fig. 6.
It follows, therefore, that the equation (1) determines w as a
cuntinumis p-vaiued function of z, and z as a continuous q-valucd
function of w. Taking the latter view, and writing (1) in the
form
z^'Jw" (1').
and (3) in the form
r = «"/«, 6=p4>lq (3'),
wc see that, if we cause <^ to vary continuously from -qir to
+ qir, then s*"' (cos -<t> + i sin - <^ j will vary continuously through
all the values which :^w^ can assume so long as | w | = s, and
will return to the same value from which it started. In fact, we
270 BIUNCHES OF yw' CH. XXIX
sec in general tliat, if w start from any point and return to the
same point again after circulating ft times round tlie origin in
the iwsitive direction, and v times in the negative direction,
then ijw'' returns to its original value multiplied by cos . iptvjq +
t sin . iptnjq where < = + /x - »• ; that is, by «,", where «, denotes
a primitive gth 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 ijuf corresponding to these,
then each of these branches is a single-valued function of 6.
The spire corresponding to -jr<<^< + 5r is taken as the
j)rincipal spire, and corresponding thereto we have the principal
branch of the function z = ^w', namely,
s = s'^|cos^<^ + tsin^<^|, --<4><+T.
For the (+ t)th. and (-r t)t\i branches respectively, we have
z, = «»"«{cos .p{<t> + 2tir)lq + » sin . jt> (<^ + 'it-^Vq],
z., = «*"« {cos . J? (<^ - 2«7r)/«/ + i sin . p (^ - 2«ir)/<7l>
As before, we may use w'"' to stand for the principal branch
of ilw", and we observe, as before, that the principal value
of llw' when w is a real positive (juautity is the real positive
value of the gth root, that is, what we have, in chap, x.,
denoted by if'''.
§ 9.] It mast be observed that, when p is not prime to q, the erprcasioni
fn/«{co8.p(^±2(T)/g + ipin.p(^±2(T)/<;} no loDRcr furnish &1I the q Taluos
of i!\C, but (na may be easily vi rifiL-d) only qfk of them, where * is the
O.O.M. of p and q. The appropriate expression in this case would bo
»p/»{co8.(p0i2«T)/g + iBin.(p^±2(T)/9}.
This last expression Rives in all cases the q different value* of !j^ ; but
it has this great inconvcnicnoe, that, if we arrunge the branches by taking
succcssivfly t = 0, « = 1, f = 2 the end value of each branch is equal,
not to the initial value of the sacceeding branch, but to the initial value of
a hrnnch several orders farther on. There will therefore l>e more than one
cruMinis in the graphic epiraL The invesUgution from this point of view will
§§ S-10 EXERCISES XIV 271
be a good exercise for the student. Wheu p is prime to 7, the two expres-
sions for ^wP are equivalent ; and WG have preferred to use the one which
leads to the simpler grajjhic spiral.
If we adopt Eiemann's method for the graphical representation of the
equation w''=z'>, then we shall have to cover the z-plaue with a p-leaved
Eiemann's surface, having at the origin a winding point of the j)th order ;
and the w-plaue with a j-leaved surface, having at the origin a winding
point of the qt'a order.
Exercises XIV.
(1.) Solve the equation
tan-i{(a; + l)/(j:-l)}+tan-i{(2 + 2)/(x-2)}=jF,
and examine whether the solutions obtained really satisfy the equation wheu
tan"' denotes the principal branch of the inverse function.
(2.) If 27r-<4q^, show that the roots of the equation x'- qx-r=0 are
2 (9/3)1/2 cos a, 2 (4/3)'/= cos (J 7r + a), 2 (7/3)1/2 cos (^tt- a), ^jj^-g „ jg jgtgr-
mined by the equation cos 3a = J r (S/i/)-'/^.
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 x^-5x + 3 = 0 to six
places of decimals. (See Lock's Higher Trigonometry, § 135, or Todhunter's
Trigonometry, 7th ed., § 200.)
Trace the graphs of the following, x being a real argument : —
(3.)
y = sin X + sin 2x.
(4-)
j/ = sinx + cos2x.
(5.)
1/ = sin X sin 2x.
(6.)
^ = taux + tan2x
(7.)
j/=xsiux.
(8.)
y = sin x/x.
(9.)
i/ = sin3j/cosx.
(10.)
^ = sin~ix^.
(U.)
?/-=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 w-paths
when the s-paths are circles about the origin of the z-plane, or parallels to
the real and to the imaginary axis.
(13.) «>2=zS.
(14.) w = llz.
(15.) WJ=l/23.
(16.) w^=llz\
(17.) to"-={z-a){z-b).
(18.) w- = {z-a)'{z-b).
(19.) w^={z-a)-.
(20.) w"-={z-a)<.
(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 OEN'EKAUSATIONOFINTEliKO-nKOMETRICSEUIES (11. XXIX
part of the subject dn not arise where we have to lU'iil merely
witli a finite summation ; that is to say, tlie summation of a
series to « terms. For any sudi siinnnatinn involves merely a
statement of the identity of two ciiains of oj)erations, eacli con-
taining a finite number of links, and any such identity rests
directly on tlic 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 detennine which
branch of the function represents the series ; for the series, by
its nature, is always one-valued.
We commence with some caaes where the lai^t-mentioued
point does not arise.
GEOMETRIC AND INTEGRO-GEOMETRIC SERIES.
§11.] The summation
l+c + =^+. . .+z' = (l-z'*')/(l-:) (1),
since it depends merely on a finite identity, holds for all values
of z. We may therefore .suppose that z = x + ifi = r (cos 0 + i sin 6),
and the equation (1) will still hold.
Also, since L s"+' = Zj-^' (cos n + ifl + i sin » + \$) = 0,
when r<\, we have, provided |s|<l, the infinite summation
l+c + s^+ . . . adoo = l/(l-c) (2)
for complex as well as for real values of z.
In like manner, the finite summation of the integro-geoinetric
series 2<^(H)i", 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<^(h)^ is converfjent provided |s|<l, the
infinite summation deducilile from the finite one will hold good
for all complex values of z such that 1 2 1 < 1.
§§10,11 EXAMPLES 273
By substituting in (1) or (2), aud in the corresponding
equations for 2<^ (h) s", the value r (cos 0 + i sin 6) for s, and then
equating the real and imaginary parts on both sides, we can
deduce a large number of summations of series involving circular
functions of multiples of 6.
Example 1. To sum the series
S„=l + rcose + r-cos2e + . . . + r»cos>i9,
r„ = rsinfl + r=sin2fl + . . . + r"smne,
U„=cosa + rcos{a + 0)+r-cos{a + 2e) + . . . + r''cos{a + n0),
F„ = sina + rsin(a + 9) + r=siii(o + 2«) + . . . +r'" sin {a + nO),
to n terms ; and to x when r<l.
Starting with equation (1), let us put 2 = r (cos (? + f sin ff), and equate real
and imaginary parts on both sides. We find
l + r(cosS + i sinfl) + )-^(cos2^ + isin29) + . . . + r"(cos?!9 + i sin jiS)
= { 1 - r»+' (cos (n + l)e + i siu {n + 1) 0)}/{l - r {cose + i sin 8)} (3) ;
whence*
S„={l-rcos9-r»+icos(n + l)e+j«+»cosn9}/{l-2rcose + rS} (4);
T„={r am 6- r"+' sin (n + 1) fl + r"-" sin )ifl}/{l - 2r cos fl + r=} (5).
Again, since U„ = cos aS„ - sin a7'„ ,
F„ = sin oS„ + cos oT„ ,
we deduce from (4) and (5) the following: —
i7,= {coso - rcos (a - 0) - r''+' cos (n + 19 + a) + r"*' cos (nff + a)}/
{l-2rcose + r2} (G),
r,= {sin a - r sin {a -6)- r"+i sin {n + lO + a)+r''+- sin (n« + a)}/
)l-2rcos9 + r=) (7).
From these results, by putting r=+l, or r=-l, we deduce several
important particular cases. For example, (6) aud (7) give
coso + coa (a + fl) + cos(a + 29) + . . . + oos(a + Ji9)
=cosJ{a + (o + ne)}sin J(n+l)fl/sinie (G');
sino + sin (o + S) + sin (a + 29) + . . . + sin(o + ;i0)
= sin J{o+(a + ne)}sinJ(n + l)9/sinJ0 (7').
Finally, if r< 1, we may make n infinite in (4), (5), (G), (7) ; aud we thus
find
S„ = (l-rcose)/(l-2rcos9+rS) (4");
T„ = r8ine/(l-2rcose + r=) (5");
i;„ = {cos a - r cos (a - e) }/{l - 2r cos e + r=} (6") ;
V„ = {sin o -r siu (a - »)}/{! - 2r cos fl + 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. II. 18
274 EXAMPLES Cll. XXIX
Exuinplo 2. Sum to iutiuity tbu aerios
S=l-2rcoBff + 3r*coB2»-4r»co93tf + . . . ('•<1)-
If 2 = r (cos $ + i siu 0), then S in tiic real part of tlie niim of the series
r=l-2i + 3»«-U' + . . . .
Now, by chap, xz., § 14, Example 2,
r=l/(l + «)«.
Ileiieo S = iJ (1/(1 + rcoBe + ri sin «)'},•
= J? {(1 + r COB e - n sin ej'/(iTrco80» + ri sin' e)»),
= (1 + 2r COB 9 + r* cos 2fi)/( 1 + 2r cos e + 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 = C0Ba + i sina, 2> = cob /3 4- i sin /3, e = cos 7 +
t sin y, and observe that
cos ^ + i Bin /3 - cos 7 - 1 Bin 7 = 2i Bin i (^ - 7) {cos i (/S + 7) + 1 sin J (/J + 7)} .
We thus get
4UsinJ(/3-7){coBj(/S + 7) + iBinJ(/S + 7)}
=S8inJ(^-7){coBp + <Binp}|oo87 + t8in7}{cosJ(/J + 7)
+ t8ini(/} + 7)}.
whence
4 COB (a + /S + 7) n Bin J (/S-7) = S sin i (/9-7) COS I (/9+7) ;
4 sin (a +/3 + 7) n Bin i 03- 7) = 2 sin J (/3-7) sin I (^ + 7).
formula: connected with demoivke's theorem and
tllk binomial tueohem for an integral index.
§ 12.] By chap, xn., § 15 (3), we have
cos(fl, + e,+ . . .+0,) + iBm{0i + 6t+. . .+e,)
= (cos 0, + i .sin 6,) (cos 0, + 1 sin 0,) . . . (cos 6, + 1 sin 0,).
If we expand the right-hand side, and use P, to denote
Sees 6, cost/, ... cos 6,m\ d^+i . . . sin^,, that is, the sum of all
the partial i)roducts tliat can be formed by taking the cosines
of r of the angles ^i, ^,, . . ., P» and the sines of the rest, then
we tiud that
co8(e, + tf,+ . . . 4-e,) + «8in(e, + 6*,+ . . . +0n)
* Wc aac R/{z + yi) and //(z + yi) to denote Uie teal and imaginary parts
of / (x + yi) respectively.
§12 EXPANSIONS OF cos (01 + 0, + ... + ^,.), &c. 275
Heuce
cos{0, + o,+ . . . +0„) = P„-iV« + P„-4-^„-6+. . . (1);
sin (d, + e,+ . . .+e,)= p„_, - p„_3 + i^,-5 - P„-7 + . . . (2).
LVom these, or, more directly, from
cos {0i + 6^+ . . . + 6„) + i sill (^1 + 63+. . . + 6n) = cos 0i cos 0,
... cos 6„ (1 + i tail ^^i) (1 + i tan 0.,) . . . (l+i tan On),
we derive
tan(e.+ 0, + . .. + e„)^{T,-T, + T,-. . .)I{\-T,+ T,-. . .) (3),
where TV = 2 tan ^i tan 6^ . . . tan 6^.
The formula) (1), (2), (3) are generalisations of the familiar
addition formulae for the cosine, sine, and tangent.
From tlie usual form of Demoivre's Theorem, namely,
cos nO + i sin nO = (cos 6 + i sin 6)'\
we derive, by expansion of the right-hand side,
cos nO + i sin nO = cos" 0 + i„Ci cos""^ 0sm6-„C2 cos"~^ 0 sin^ $
Hence
cosme=cos"fl-„(7aCos"-'6'sin=e + „C4Cos"-'esiu«6i-. . . (4)*;
sin w0 = „C, cos"-' e sin 61 - „C, cos"-» e sin' 0
+ ,.C;cos"-'esiii=e-. . . (5);
. . „Citan6l-„C3tan'(9 + „C5tan''0-. . . .„,
**''"^= l-„(7.tan-'^ + „(7.tan^e-.. . <^)-
These are generalisations of the formula; (8) of § 2.
The formulae (4) and (5) above at once suggest that cos nO
can always be expanded in a series of descending powers of cos^;
that, when n is even, cos 7i9 can be expanded in a series of even
powers of sin 9 or of cos 9; sin ?(6/sin 0 in a series of odd powers
of coa 9 ; and sin nO/cos 0 in a series of odd powers of sin 9 :
and, when n is odd, cos n9 in a series of odd powers of cos 9 ;
cos n9/cos 9 in a series of even powers of sin 9 ; sin nO in a series
of odd powers of sin 9 ; sin n9/sin ^ in a series of even powers
of cos 9.
* The formula; (1), (5), (G), (8) were first giveu by John Bernoulli in 1701
(seeOii., t. L.p. 3a7).
18—2
27f) EXPANSIONS IN POWERS OF STN <? AND ens ^ TH. XXIX
Knowing', a priori, that these series exist, we could in variniis
ways iletermiue their coefficients ; or we could obtaiu certain
of them from (1) and (2) by direct transformation, and then
deduce the rest by writing hir-6 'm place of 6. (See Todluiuter's
Trigonamitry, §g 2S6-28S.)
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 o = cos6 + i8in ©, /3 = co8 6-
t sin 6, and therefore p = 2 cos 6, q= 1, we deduce
2 cos ne = (2 cos 6)' - ", (2 cos «)""' + "^""^^ (2 cos «)-' - . . .
^_^,»(»->-l)(>'-r-2)...(»-2r^li^.^^^^^^,.,^ ^.^,.
sin ne/siu 6 = (2 cos 6)-' - — j^, ' (2 cos «)"-• + ('iZL^Hwjii)
(2cose)--. _^_)M-r-l){n-r-2). . .{n-2r)
(2 cos 6)"-*'-'+ ... (8);
cos «g = (-)"' jl - ^' cos' 6 + "'^"^7 ^'^ cos' e - . . .
, ,.«'(n»-2'). . . (n'-27^») „„ 1, , ,„,
( - )• — ^ (ily. e+. . j(n even) (9) ;
cos n6 = (~ y-'y i - COS 0 ^^ — ' cos* 6 + — ^ ^p ^
5/1 / X. » (»' - 1') («' - 3') . . . (»' - 27-"l') ^ . , . )
cos'O-. ..{-y^ ^2^1 1)! ^C08**'fl + . . .j
(«odd) (10);
sin «g/sin e^{- )'^-' |^ cos g - " ^"g~ ^"^ cos* g -t- . . .
(-)' (27^)1 'co8**'e+. . .J(«even) (11);
• Tlic Bcrica (7). (9'), (10') were first given by James Bcmoalli in 1703
(ace Op., t. II., p. 92C). He deduced them from the formula
2.in»n0=g(2,in .).-"'<";- ^•)(2,in.)..H"'<"'-^;)i!!l:igl>(a.in.)«-...,
wliidi lie ciitabliahed by an induction baaed ou the preTioui r«iulla of Vieta
rcgarJjug ilie mulliaection of au angle
§§12,13 EXPANSIONS IN POWERS OF SIN 0 AND cos ^ 277
sin ne/sin 6 = (-)('-»^ |i - !^ cov 6 + ("'-I'K^'-S") ^^^4^ _ _ _
(-)" ^ ^^-^ 'cos-^e+ . . .| («odd) (12).
If in the above six formulse we put hir-O'in place of 6, we
derive six more in which all the series contain sines instead of
cosines. In this way we get, inter alia, the following : —
cos ?ig = 1 - |j sin° 6 + ^ ^^' ~ ' sin* B - . . . (n even) (9');
sm ne = - sm e — ' sm^6+ ^ '- sin' 6- . . .
1! ol 5!
(modd) (10');
m nO I cos 9 = -j sm 0 - -~^ — -' sin' 6 + -^ -^ ' sin' d-. ..
(w even) (11');
sm
cosw
., . , n'-l» . ,„ («^-l=)(»»-3") . ,„
-fl/cos 0=1 -7— s\v? 6 + ^ ^-^ ' sm* 0 -
^! 4!
(n odd) (12').
The formulfe of this paragraph are generalisations of the
familiar expressions for cos 26, sin 29, cos 30, and sin 39, in terms
of cos 9 and sin 6.
§ 13.] The converse problem to express cos" 9, sin" 0, and,
generally, sin™ 6 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 + { sin 6 by cc, then we have,
by Demoivre's Theorem, the following results : —
x = cos9 + ism9, l/x = cos9 — ishi9; ^
af = cos n9 + i sin n9, IjaP = cos n9 - i sin nO :
i
cos9 = --(x+l/x), sin9 = —:(x-l/a;):
2 2z
cos ne = 5 («" + l/x"), sin nO = ^. {x" - l/x").
y (•)•
278 EXPANSIONS IN fOSLNES AND SINES OF MLLTII'LES OK 0
Hence
+ ^C, {a^^ + 1/.!*-^) + . . . + ,„.r.},
= -5^, {cos 2m0 + tt.C, cos {2m - 2)0 + ^.C, cos (2m - 4)5 +
. . .+UC7.} (2).
Similarly,
CDS'"*' tf = „L {cos (2m + 1)0 + a„+,C, cos (2m - 1)0
+ a«+iCjC08(2m-3)6+ . . . +»+,C«co8fl} (3);
8in'^0 = ^^^{cos2me-t„CiCos{2m-2)6
+ ^C,CM(2m-4)e+. . .(-)"4«C«} (4);
sin"-*' 6 = ^^ {sin (2m + 1)0 - ^+,C, sin (2m - 1)6
+ ,„+,C;sin(2m-3)0+. . .(-)"«+, C. sin 0} (5).
These formulae are generalisations of the ordinary trigonometrical
formula; sin' B = -\ (cos 20-1), cos' Q=\ (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' 0 cos* 0 in a aeries of sines of multiples of B.
Bin' 9 cos> tf = 2^, (X - l/x)» (i + l/z)>.
Starting with the coefficients of the highest power which happens to bo
remembered, sny the 4th, we proceed thus —
CocUidonU of MulUpllcr.
CoclBdcnU of Product
1-1
1-4+ 6- 4 + 1
1-6 + 10-10 + 5-1
1 + 1
1 + 1
1 + 1
1-4+ 6+ 0-5 + 4-1
1-8+ 1+ 6-6-1+8-1
1-2- 2+ 6+0-6+2+2-1
Thccopflicicntain the laat lino are thow in the expansion of (x- 1/j)'(t + 1/t)'.
Honec, arranging together the terms at the beginning and end, and replacing
§ 13 EXERCISES XV 279
i (j' - 1/x') by sin S9, .j-. (.c" - 1/x") by sin 69, aucl so on, wc find
8in»ecos'9 = s7{sin8e-2sin(i9-2sin4tf + Gsin2ff + 4.0},
= _ {sin 8ff - 2 sin 69 - 2 sin 49 + 6 sin 29}.
128
The student will see that sin'" ^ cos" 6 can be expanded in a
series of sines or of cosines of multiples of 6, according as m is
odd or even. The highest multiple occurring will be (m + n) 6.
Example 2. If 9 = 27r/n, and a any angle whatever, and
,„P„=coB'"a + cos'»(a + 9)+. . . +cos"'(a + 7^- 19),
„;"„=sin'»a + sin'"(a + 9)+. . . + sin™ {a + ?»-!«),
where m is any positive integer which is not of the form r + snj2, then
«nP»=am^«="-l'3- • • (2m-l)/2.4. . .2m;
Sm+l ^n = 2ni-f 1 '^n — ^*
This will be found to follow from a combination of the formula) of the
prcBent paragraph with the summation formula of § 11.
Exercises XV.
Sum the following series to n terms, and also, where admissible, to
infinity :—
(1.) cos o- cos (a + 9) + cos (a + 29)-. . .
(2.) sin a -sin (a + 9) + sin (a + 29)-. . .
(3.) Ssin'nS. (4.) ncos9 + (n- 1) cos29 + (?i-2) cos39 + . . . .
(5.) 2 sin n9 cos {?i + 1)9. (6.) S ain h9 sin 2k9 sin 3n0.
(7.) sin a - cos a sin (a + e) + cos= a sin (a + 29) - . . . .
(8.) l + co3 9/cos9 + co3 29/cos=9 + cos39/cos''9 + . . . to n terms, whore
e='mr.
(9.) l-2rco3 9 + 3r=cos29-4r'cos39 + . . . .
(10.) sin 9 + 3 sin 29 + 5 sin 39 + 7 sin 49 + . . . .
(11.) Sh= cos (n9 + a). (12.) S« (u + l) sin (2;( + l) 9.
(13.) 6in2H9-j„CiSin(2n-2)9 + j„C2sin(2n-4)9-. . . (n a. positive
integer).
(14.) sin(2« + l)9 + o„+,CiRiu(2n-l)9 + 2„+iC2sin(2n-3)9 + . . . (n a.
positive integer).
(15.) 2m(m + l). . . (m + ii- 1) r" cos (o + n9)/Hl to infinity, m being a
positive integer.
(16.) Does the function
(sin= 9 + sin' 29 + . . . + Bin'n9)/(co8'9 + co8=29 + . . .+cob»7i9)
approach a definite limit when n = co ?
(17.) Expand l/(l-2cos9.i + x=) in a series of ascending powers of x.
280 FUNOAMKNTAL SEIUKS KOU COS0 AND SIN(^ Cll. XXIX
(18.) Exi)and 1/(1 - 2co3 9,x + x')' in a Bcrics of ascending powers of x.
(19.) Expand (l + 2x)/(l -x*) in a scriee of ascending powers of z ; and
show that
„_, (3»-l)(3n-2) (3H-2)(3n-3)(3n-4). _
l-3n+ -, 5i + . . . = (-!)•.
(20.) Show that l/(l+i+x') = l-x + x»-x« + x«-x' + x'-x"+ . . .;
and that, if the sum of the even terms of tliis expunuion be ^(x), and tlio
sum of the odd terms ^ (x), then {0(x)}'- {f (x)}»=0(x') + ^(x').
Prove the following identities by means of Demoivre's Theorem, or
othenvise. S and IT refer to the letters o, /9, 7: —
(21.) 2sino/(l + 2coso)= -11 tan } a, where a + ^ + y=0.
(22.) S sin (9 - /9) sin (9 - 7)/sin (o - /3) sin (o - 7) = 1 .
(23.) Ssini(a + /3)sin J(o + 7)coso/sin4(a-/3)sinJ(a-7) = co8(o+^+7).
(24.) cos a cos (a - 2o) cos (<r - 2/3) cos (<r - 27) + sin a sin (cr - 2o) sin (a- - 2/3)
sin (<r - 27) = cos 2a cos 2/3 cos 27, where <r = a + ^ + 7.
Expand in series of cosines or sines of maltiples of $ : —
(25.) cos"fl. (26.) sin'ff. (27.) siu'tf.
(28.) cos*0Bin'«. (29.) cos* 0 sin' 0.
Expand in series of powers of sines or cosines : —
(SO.) cos 109. (31.) sin 79.
(32.) sin 30 cos Cd. (33.) cosn>9cosn0.
EXPANSION OF COS 6 AND SIN 6 IN POWERS OF 6.
§ 14.] We propose next to show that, for all finite real
values of 6,
cos« = l-^/2! + ^/4!-^/6! + . . . adoo (1);
Bine = e - e'/3< + e'/5l-6'/ll + . . . ad« (2).
These expansions* are of fundamental importance in the
]iart of algebraical analy.si.s with which we are now concerned.
Tiiey may be derived by the method of limits cither from the
formula! of § 12, or from two or more of the equivalent formula)
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 Atialytit per aguationei ttumero
terminorum rnfnilat, which was shown to Barrow in 1GC9. The lending idea
of the above demonstration was given by Euler (7n(rod. in Anal. Inf., t. I.,
§ i:!2), but bis demonstration was not rigorons in its details.
§ 14 FUNDAMENTAL SERIES FOR COS 0 AND SIN 6 281
From (4) and (5) of § 12, writing, as is obviously permissible,
6/?« in place of 9, and taking n = m, we deduce, after a little
rearrangement,
cos^ = cos'"ifl-^}/^^"-(tan^/iy
771 1. 2! V m/ m)
^(1-1M(1-2M(1-3A»),.A i/^Y_ I (,)
4! \ ml mj )
a
= cos"' - {1 - Ms + «4 - . . . }, say, (3') ;
and
sin Q = cos"
I \ ml ml
_(i-iM)(i-2M^A j/^V^^ 1
3! V ffj/ »!/ J
= cos" - {Mi
■!(, + .
say,
(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 mthout limit. On the other hand, we take 0 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
lUi
u,.
_ (l-2»/m)(l-2» + lM / e_ leV'
(2»+l)(2?« + 2) V^^m/mJ-
Hence, so long as n is finite,
L
W2»
e^
(2w + 1) {2n + 2) •
If, therefore, we take 2»+l>^*, we can always, by taking
m large enough, secure that, on and after the term Ui„, the
numerical value of the convergency-ratio of the series (3) shall
be less than unity.
* Strictly speaking, it is sufficient if e<J{{2n + l) (2n-|-2)}.
282 FUNDAMENTAL SEUIES VOH CCIS 0 AND S\S 0 (11. XXIX
Frnni tliis it follows that, if 2h + \>0, and m lie only taken
large enough, cos 6 will ho intermediate in value hetween
a
cos"- {!-«., + «,-. . .(-)"«„} (5),
and
COS-^fl-Kj + M^-. . .(-)"«« + (-)"^'»*«„} (C).
Therefore cosfl will always lie between the limits of (5) and
(6) for m= cc.
Now (see chap, xxv., § 23)
L cos" (0/m) = 1 , itt, = ^/2! , Lti, = 0'H\, . . .
Hence cos 0 lies between
l-6y2\ + eyil-. . .(-)"^/(2n)!
and
1 - ^/2! + ^/4! - . . . ( - )"^/(2n)! + ( - )"+' <?»"+V(2« + 2)1.
In other words, j^rovidM 2» + 1 >d,
cme=\ - 0^121 + e*l^\-. . .(-)''r'/(2M)! + (-)"+' //a. (7),
where R^ < <?»+V(2n + 2)! .
Here 2/» may be made as large as we please, tlicrefore since
L e»+V(2n + 2)! = 0 (cliap. xxv., § 15, Example 2), we may
write
cosfl = 1 - ^/2! + ^/4! - . . . ad 00 (7').
By an identical process of reasoning, wc may show that,
provided 2n + 2 > 6*, tlu-n
8iad = tf-^/3! + . . .(-)"^+V(2» + l)! + (-)"*'yA».+. (8),
wlure j?a,+, < e^*'H:2n + 3)! ,
and tlter^ore
sme = e- 0>/3\ + 0"/5! - . . . a<l oo (8').
It has already been shown, in chap, xxvi., that the series (7*)
and (8') are convergent for all real finite values of 0 ; they are
• More closely , if « < ^ { (2n + 2) (2n + 3) } .
§ 14 EXAMPLES 283
therefore legitimately equivalent to the one-valued functions
cos 0 and sin 6 for all real values of 6, 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
9 having the period 27r. This conclusion may at first sight
startle the reader ; but he can readily verify it by arithmetical
calculation tlirough a couple of periods at least.
When 6 is not very large, say :}>|t, which is the utmost
value of the argument we need 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 7th decimal
place.
We sliall 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
COBl = l-l/2! + l/4l-l/6!-Hl/8!-l/101 + iJio.
Bio<l/12!,
= 1 - -500,000,0 + -041,606,7 - -001 ,388,9 + -000,024,8 - -000,000,3 + iJ,„ ,
iJj„< -000,000,003.
= -540,302,3.
Similarly,
sin 1 = 1 -1/3! + 1/51 -1/71 + 1/9! -/Ja,
i;g< 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 fl>sin9>9-49'; l-ie'<cosd<l-ie- + ^0'.
These inequaUties follow at ouce from (7) and (8) above. They are
extensions of those previously deduced, in chap, xxv., §21, from geometrical
considerations.
Example 3. Expand cos {a + 0) in powers of $.
Besult. cos (o + S) = cos o 003 e - sin a sin 9,
= cosa-6ino9-co8ae-/2! + sinoe'/31 + co3a9741-. . .
• Seeing that the cosine or sine of every angle between Jt and i«- is
the sine or cosine of an angle between 0 and itr.
284 EXERCISES XVI CU. XXIX
Example 4. Find tho limit of
e (1 - COS <?)/(tan 0-0) wlien 9 = 0.
L9 (1 - cos 0)/(tan 9 - «) = I, see 9 />9 (1 - cos ff)/(sin ff - fl cos 0),
= lxL0(0-j2-0*lil + . . .)l{0-9'l:i\ + . . .-9 + (Pyj-. . .),
= L{e'l2-0>H\ + . . .)/(ff'/3 + . . .).
= L{ll2 + PeP + . . 0/(1/3 + <?»» + . . .).
= 3/2.
EXKRCISES XVI.
(1.) Expand sin (o + 9)sin 0 + d) in powers ot S.
(2.) Calculate sin 45^ 32' 30" to five places of decimals.
(3.) Given tan 9/9= 1001/1000, calculate 0.
(4.) Expand co!;-d, sin- 9, and sin* 9 cos 9 in powers of $; and find the
general term in each case.
(5.) Show that cos™ 9 (m a positive integer) can be expanded in a con-
vergent series of even powers of 0 j and that the coefficient of 0" in thia
expansion is
(-)"{m^+„C,(>»-2p+„C,(m-4)* + . . .}/2"-MaFi)l.
(6.) Show that, if m and n be posiitive inte^ters, and l<n<in, then
'n"-mC,(">-2)- + >C,(m-4)»-. . . = 0.
Examine how thia result is modified when n = l, or n = in.
Evaluate the following limits:—
(7.) (sin'm«-sin'n9)/(cosp9-cos59), 9=0.
(8.) {sinp(a + 9) -8inpo}/9, 9=0.
(9.) |sin»;)(o + f)-sin».Do}/9, 9 = 0.
(10.) {8in"p(o + 9)cos(o + ())-sin";)acosa|/9, 9=0.
(11.) (a*sina9-6''sin^-9)/(6*tan<I9-a''tan^9), 9 = 0.
(12.) l/2x» -t/2x tan TX- 1/(1-1^, x = l (Euler).
(13.) {sinx/x}"/*", x=0.
(14.) {(x/a)sin(a/i)}^, x = <b, (m=>2).
(15.) Show, by employing tho process used in chap, xxvii., § 2, that the
scries for sin n9/co3 9 in powers of sin 9 can be derived from tho scries for
C08n9 in powers of sin 9; and so on.
(16.) Show, b; using the process of chap, zxvii., § 2, twice over, that, if
0O8n0=l + i4,Bin»9 + /f,8in*9 + . . . + il,6in*'9+. . .,
then
-n»0O8n«=2.i, + (3.44,-2'^,)8in'9 + . . .
+ {(2r+l)(2r + 2)^^,-(2r)'.4r}»in*9 + . . . .
Hence determine the cocQicicnts.^,, A„ Ae.; and, by combining Exercise
15 with Exercise 16, deduce all the soricg (7) . . . (12') of § 12.
(17 ) Show (from § 18) that cos0 9 and sin* 9 can each be expanded in a
convergent series of powers «f 9 ; and lind an expreMiion for the coefficient of
the general term in each cnoe.
In particular, show that
8in»x/31 = i'/3!-(l + 3')x'/5! + (l + 3' + 3«)x'/7l-(l + 3' + 8« + 8«)*»/'Jl + ... .
S 15 BINOMIAL THEOREM 285
BINOMIAL THEOREM FOR ANY COMMENSURABLE INDEX.
§ 15.] If, as iu chap, xxvii., § 3, we write
/(m) = l+2„(7„«" (10),
wliere m is any commensurable number as before, but z is now
a complex variable, then, so long as |2|<1, 2,„C„c:" will (chap.
XXVI., § 3) be au absolutely convergent series ; and /(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 /(»i,)/(w?2) =/(?«i + m>) will still hold good; and all the
immediate consequences of this theorem — for example, the
equations (4), (5), (G), (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
= (l+c)" (11).
It follows tha,t f{p/q) represents part of the g'-valued function
;^(1 +z)'' ; and it remains to determine what part.
Let z = r (cos 6 + isin 6), then, since we have merely to ex-
plore the variation of the one-valued function /{p/q), it will be
suthcient to cause 6 to vary between — -n- and + -a:
Also, let
w = 1 + r = 1 + x + yi,
= \+rco%6 + ir sin 0, \ («),
= P (cos 4' + i sin <^),
so that
p = {(1 + xf + y-y^ = (1 + 2r cos 6 + ry- ■ '
tan 4> = y/{l+x) = r sin 6/{l + r cos 6),
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<l, this circle falls short of the
origin. Hence <^, the inclination to the a;-axis of the vector
drawn from the origin to the point m, is never greater than
2S6 EXPANSION OF (I +X + yir CH. XXIX
tan-' {rAl - r")"). anil "ever lesa than - tan"' {r/(l - r*)"^}.
Hence <^ lies in all cases between - Jir and + ^x. Therefore,
since /{piq) is continuous, only one branch of the function
^(1 + 2)'' is in question. Now, if we denote the principal
brancli by (l+s)*"', so that
( 1 + z)'" = p"" (cos . p<\>lq + i sin . p<l>lq),
we have, by § 8,
^(l + s)'' = (l + c)''«< (12),
where f = 0, ±1, ±2 according to the branch of the
function which is in question. Hence we have
f<j>lq) = 0-^:y<,
where t has to be determined.
Now, when s = 0, we have/{p/q) = 1, hence we must have
1 = -,".
Hence * = 0, and we have
/(j>lq) = (1 + =)"« = P"'" (cos . p<l>/q + I sin . p<t>/q),
where -Jfl-<<^<Jjr.
Next consider any nej^tive commensurable quantity, say
-p/q. Then (by cliap. xxvii., § 3 (9)),
A-p/q)=AO)l/{p/q).
= i//(j>lq)-
If, therefore, wo define (l + s)-*"* to mean the reciprocal of
the principal value of (1 +c)'"', we have
= p-"'' {cos ( -p^lq) + % sin ( -p4>lq)\ (13).
To sum up : We have now eftahlisheJ tlie fullowing exjxiii.tion
/or the principal value 0/ {\ + :)", in all cases inhere m is any
commensurable number, and | s | < 1 : —
(l + s)-=l + 2«C.5- (14).
The theorem may also be written in the following forms : —
1 + :i„r.(j- + yi)* = {(1 + xf + yr' [cos . m tan-' {y/(l + x)\
+ I sin . VI tau"' ly/(l + x))] ^li; ;
§§15-17 GENERAL STATEMENT OF BINOMIAL THEOREM 287
1 + 2,„C>,r" (cos iiO + i siu nO)
= (1 + 2/- cos 6 + r")""^ (cos ffi<)!> + i sin m<j>),
where -iTr<<^ = tan~' {rsin^/(l+rcos^)}<+ Jt (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 + 2„.+i.iC„(.r + FT
= [(1 + xf + y-Y''- [cos {m tan"' {yl{l + x)} + Ik log ((1 + xf + f-\\
+ i sin {m tan"" {y/(l + x)] + \lc log {(1 + xf + fW]
Exp[-^-tan->{i//(l + ^)}]-
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 |2| = 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 + 2„,C„ (cos nQ + i sin nO) will be convergent
if, and will not be convergent unless, each of the series
S=l + %n,G„ cos n6,
r=2„.0„sin?j«
be convergent.
In the first place, we remark that, if 7W<— 1, LmCn = ±<^
when TO = CO , so that neither of the series S, T can be convergent.
If TO = - 1, then „.a, = ( - 1)", >S' = 1 + 2 ( - 1)" cos n6,
T= 2 ( - 1)" sin nO, neither of which is convergent (see chap.
XXVI., § 9).
If -l<m<0, then L,nC„ = 0; and the coefficients ulti-
mately alternate in sign. Hence, by chap, xxvi., § 9, both the
series >S' and T are convergent, provided 6 + + 7r. When 6 has
one or other of these excepted values, then S=l +2(-l)"mC>,,
which is divergent when m lies between -1 and 0 (see chap.
XXVI., § 6, Example 3).
* Seehis Analyse Algibrique
t (Luvret Comjaletes (ud. by Sylow & Lie), 1. 1., p. 233.
L
288 GENERAL DEFINITION OF ExP C CM. XXIX
If m>0, then, as wo have already proved (see cliap. xxvi.,
§ 6, Example 4), 'S.^Cn is absolutely convergent, and, a /urtiori,
1 +■ 2.(7. cos nO and SnC. sin nO are both absolutely convergent.
It follows, therefore, that the equation
(l+rr=l + 2„^,c-
will hold u-hcii 1 5 1 = 1 , ill all ca^s where ni > 0 ; and also when m
lies between -1 and 0, provided that in this last case the imaginary
part ofz do not vanish, that is, provided the amplitude o/z is not±it.
In other cases where | s [ = 1, the theorem is not in question,
owing to the non-con vergency of 2«<7,s".
In all cases where |c|>l, the series 2„C,c" 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 FUNCTIONa
§ 18.] The series
l+s + c'/2! + s'/3! + . . .
is absolutely convergent for all complex values of z having a
finite modulus (see chap, xxvi., § 10). Hence it defines a single-
valued continuous function of z for all values of z. We may
call this function the E.xponential of c, or shortly E.\pc*; bo
that Exp z is defined by the equation
E.\ps=l+; + c'/2!+a'/3! + . . . (1).
The reasoning of chap, xxvin., § 5, presupi)o.ses nothing but the
absolute convergence of tlie Exponential Series, and is therefore
api)licable when the variable is complex. We have therefore
the following addition theorem for the function Exp z : —
* When it is ncco^snry to distinguiah botwccn the gcnpTol fanction of t
complex variable x and the ordinary exj>uncntinl function of a real variable z,
we shall ubc Exp (with a capital letter) for the former, and cither r* or cxp z
for the latter. After the student fully understands the theory, bo may of
courxc drop this distinction. It socms to be (orijottcn by some writers that
the r in «** is a mere nomxnii umbra — n contraction for the name of a function,
and not 2'71S'2s . . . Oblivion of this fact has led to some Btrmogo pieces of
luatlicmalical luifio.
^17, 18 ADDITION THEOREM FOR ExP ^ 289
Exp^iExp^;., . . . Exp z^ = Exp (zi +:., + . . . + z,„) (2),
where Sj, z.^, . . ., s,„ are any values of z wliatever.
In particular, we have, if m be any positive integer,
(Exp»)"' = Exp(»w) (3).
Also
Exp z Exp {-z) = Exp 0,
= 1;
and therefore
Exp(-c) = l/Exp^ (4).
We have, further,
Exp 1 = 1 + 1 + 1/2! + 1/3! + . . ,,
= « (5);
and, if x be any real commensurable number,
Expir=l + .r + .-r/2! + a:73! + . . ,,
= ^ (G),
by chap, xxviil., where e' denotes, of course, the principal vahie
of any root involved if x be not integral.
It appears, therefore, that Exp x coincides in meaning with
(f, so far as (f is yet defined.
We may, therefore, for real values of x and for the corre-
sponding values of y, take the graph of y = Exp x to be identical
with the graph of y = (f, already discussed in chap. xxi. Hence
the equation
y = Exp X (7)
defines a; 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
x^Exp-^y (8);
and it is obvious that Exp^^y may, for real values of y greater
than 0, he taken to be identical with logy as previously defined.
If we consider the purely imaginary arguments + iy and - iy,
we have, by the definition of Exp s,
c. II. 19
290 MODULUS AND AMPLITUDE OF ExP(x + ty) CII. XXIX
Exp ( + 11/) = 1 + i> -fl2\ - if/31 + i//i\ + iy/5! - . . .,
= (l-y/2I + t/*/4!-. . .)
+ e(y-y'/3!+y'/5!-. . . ).
= cosy + «siiiy (9);
Exi)(-/» = (l-y/2! + y/4!-. . .)
-.•(y-y'/3!+//5!-. . . ).
= C08y-t8iuy (9'),
by § U.
Fiually, by the addition theorem,
Exp (x + yi) = Exp (x) Exp (yi),
= e* (cosy + «' sin y) (10).
The Greneral Exponential Function is therefore always expressible
by means of the Elementary Transcendental Functions «*, cosy,
siny, already defined.
Inasmuch as the function Expc possesses all the character-
istics which «* has when z is real, and is identical with «* in all
cases where ^ is already defined, it i.<! usual to employ the nota-
tion e" for Exp;; in all cases. This simply amounts to defining
0* in all cases by means of the equation
0"=l+2 + ~'/2! + s'/3! + . . .,
wliich, a.s we now see, will lead to no contradiction.
§ 19.] Graphic Discuitsion qfthe General Exj)oiieiitial Fiinctitm
— Definition of the General Logarithmic Function. Let w be
defined as a function of z by tiie equation
w = Exp« (1);
and let z=x+ yi, and «> = u + r* = s (cos <^ + « sin <t>). Then, sine
Exp {x + yi) = e' (cos y ■¥ i sin y), we have
s (cos <^ + » siii<^) = ^ (cosy + 1 sin y) (2).
Hence
3 = e', 4> = y (3),
where we take the simplest relation between the amplitudes that
will suit our purpose.
Suppose now that in the c-plnne (Fin. 7) wo draw a 8trai),'lit
lino 'i'l'l'^' parallel to the y-uxis, and at a distance x from it.
§§ 18, 19
GKAPH OF Exp (x + yi)
291
■Y
K
i
IK
D
B
r
C
A
X
o"
0
0 K
c
A
B
B
1
fK
5'
Fig. 7.
FiQ. 8.
19—2
292 c.UAi'ii OK Exi*(fl; + yt) cii. xxix
Tlieii, if wc t-iiisc z to describe this line, o" will remain coii.staiit, and
therefore tf will remain constant; that is to say, the point w will
describe a circle (A') (Fi^'. 8) whoso radius is tf about the origin
in the M.'-pliine. If we draw parallels to the ^-axis in the z-plane,
at distances O'l' = ir, 0'2' = 3:7 above, and O'l' = ir, 0'2' - 3ir,
. . ., below, then, as y varies from -tt to +7r, s travels from 1'
to r ; as y varies from + tt to + 3ir, z travels from 1' to 2', and
80 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 clejirer, we may, as heretofore, replace the
repeated circle iT by a spiral sujjposed ultimately to coincide
with it. Then to the infinite nuuibor of pieces, e:ich equal to
2ir, on the line K corresponds an iutinite number of spires of the
spiral A'.
In like manner, to every parallel to the y-axis in tlie r-plane
corresponds a spiral circle in the it'-plane concentric with the
circle A'. To the axis of y itself corresponds the spiral circle
BAOAB of radius unity ;_to the parallel DO'D to the left of
the y-axis the spiral circle DO'D ; and so on.
To the whole strip between the infinite parallels Dli and
DB corresponds the whole of the w-'-plane taken once over ;
namely, to the right half of the infinite strip corresponds the
part of the M.--])lane outside the circle BAOAB; to the left
half of the strip the part of the u?-plane inside the circle
BAOAB.
To each such parallel strip of the c-piane correspondB the
whole of the u'-plane taken once over.
Hence the vahies of ic are repeated infinitely often, and we
see that the e<iwition (1) dtfnus w as a continuous periodic
function of z having the jicriod 'Iwi,
Converse/y, the above graphic discmsion shows that the equation
(1) dijirifs z IIS a continuous <x>-jile vahwd fnnrtiim (fw.
Taking the latter view, we might write the equation in the
form
£ = Exp w (1).
§ 19 Log w = log \w\ + i amp (w) 293
Instead of E.\p"' w we shall, for the most iiart, employ the
more usual notation Logw, tising, 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 arc
both restricted to be real.
In accordance with the view we are now taking, we may
write (3) in the form
«=logS, 7/=<^.
Hence z = Log w
gives x+yi = Log {s (cos 4> + i sin 4>)],
where x = log s, and y = 4>-
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 - oo and + oo , then the left-hand side of the
equation (2') will vary continuously through all values which
Logw can assume for a given value of \w\.
If we confine <^ to lie between -ir and +ir, then Logw
becomes one-valued ; and we have
Log w = log s + i<i> (4),
wheres = |w|=,y(«- + ir), andcos<^=M/V('*"+e'), sm(f>=v/J{u'^+v'),
— ■ir'^cj>^+ IT.
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 z, or tLogw
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 ir (that
is, a value of w whose amplitude differs by an integral multiple
ofiir), then
jLog W = Zt = Z + 2tTri,
= \ogs + i{4> + 2tTr) (5),
where <}> is the amplitude (confined between tlie limits — tt and + ir)
ofw, and t is any integer positive or negative.
If V) be a real positive quantity, =u say, then s = \w\ = u,
<l> ^ amp w = Q ; and we have, for the principal value of Log u,
Log u = log u.
294 DEFINITION OK EXI' „; CH. XXIX
Iltnice, for real p<mt'tve vulws of tlie nnjument, log u is the
princi'iial value of Lixj u. The other values are of course given
b}i ,Logu = logu + 2tni, t being the order of the branch.
We have also the following jmrticular jiriiicipal values : —
Lo'^ ( + i) = Ui,
Ij'Jg(-t) = -iTt,
Log(-l) = + Tri:
the principal value in the la-st case is not rlotenninato until wo
know the amplitude ; and the same applies to all purely real
negative arguments.
§ 20.] Definition if Exp aZ. The meaning of a', or, as it ia
sometimes wTitten, Exp aC, has not as yet been defined for values
of 2 which are not real and commensurable.
We now define it to mean E.vp (z . ,Log a), where ,Loga 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 fiiet, since ,Log a = log s + « (<^ + 2<ir), where » = |a|,
and </> = amp a ( - tt < <^ < + tt), we have, \i z = x + yi,
o^*^" = E.xp [{x + yi) {log s + t (</. + 2tir)\],
= Exp [{x log « - (<^ + 2/7r) y] + i\y log s + (■/> + 2/rr) x\],
= exp {.C log S - (<^ + 2t-n) y\. [cos {y logs + (<^+ 2tTr)x\
+ is\n\y\ogs + {i> + 2tir)x\] (1).
If we put t = 0, that is, take the principal branch of Log a,
in the defining equation, then we get what may be cjilled the
principal branch of o^^"^, namely,
o*^»' = Exp(sLoga),
= cxp{xlogs-<^y}.[cos{ylog«+<jl>j-}+t.Mn{.vloK»+<^j'}] (2).
The value given in (1) would then bo called the <-th branch,
anil might for distinction he denoted by fi'*"* or by ,Exp ^(j- +y«).
It is important to notice that the ahore definition of a' agrees
vith that already girenftr real commenjturalde ralues ofs provided
we take the corresjMnding branches. In fact, when y = 0, (1) gives
a' = exp (x log s) . [cos (<^ *■ 2tir) x + i sin (<^ + 2/>r) j-] ;
§§19-21 ADDITION THEOREM FOR LOG 2 295
that is, if X =plq,
[s (cos <^ + t sin <^)]'"«
= s*" [cos . (<^ + '2H)plq + i sin . {<i> + 2tTr)p/q] (3) ;
the riglit-hand side of whicli is the ^th branch of the left as
ordinarily defined.
Cor. It /iillows from the above that when x is an incommen-
surable number the function (f has an infinite number of values
even when both a and w 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 coiunieusurable
quantity differing Infinitely little from x.
§ 21.] The Addition Theorem for Logz.
By the result of § 19 we have
„Log w, + „Log Wi
= log I w'l I + log I ifo I + i amp Wj + i amp w^ + 2 (to + n) nri.
Now (chap. XII., % 15) \wi\\Wi\ = \wiWi\, and, if amp (wi w^)
were not restricted in any way, we should have ampwj + amp Wo
= amp (m,'i W2). Since, however, amp ( Wj w^) is restricted in the
definition of Log ( Wi w^) to lie between - ir and ir, we have
amp Wi + amp w^ = amp {wi w,) + 2pT,
where p = + 1 , 0, or - 1 according as amp Wi + amp w.,> + ir, lies
between +ir and -ir, or <-7r. Hence we have
JjOg Wi + „Log Wj = m+„+pLog (Wi Wa) (1),
where p is as defined.
In like manner, it may be shown that
„Log Wi - Jjog Wi = „_„+pLog (WiM) (2 ),
where p= + l, 0, or -1 according as amp Wj - amp W3>+ ir,
between +ir and — ir, or <— ir.
1] Taking the definition of a'^'^ given in § 20, and making use
of equation (1) of that paragraph, we have
296 EXPANSio>f OF ,Lnn(] + z) en. xxix
tLog K'"^*^ = log I fi'*'^ I + (amp k*'*'^ + 'JX-tt) /.
= x\ogs-{<f> + 2tir) y + ly log » + (<^ + 2tv) a-l «■ + 2 (i + /) «t,
where / is an integer, positive or negative, chosen so that
-Tr<7/I(>gs + (<^ + 2tTr) X + 2h < + TT.
Hence
iLog ,a'*'^ = (x + yt) {log « + (<^ + 2tw) i\ + 2 {k + 1) ^ri,
= (x + yi)tLoga + 2{k + l)-iri (3).
Tlie equations (1), (2), (3) are generalisations of formulae for
log;r with which the reader is already familiar.
If we confine each of the multiple-valued functions |Log and
(E.xpa to its principal branch, we have
Loga'+'* = (x + yt')Loga + 2W (3'),
where / is so chosen that
- Tr<i/\ogs + <t>x + 2lir<+v.
§ 22.] Ej'jKtii.^inii of ,Loij (1 + c) in poirers of z.
Con.sider first the principal branch of the function Log(l + z).
By the definition and di.scus.sion of § 20, we see that, when x is
any real quantity, the princijMil branch of (1 + £f has for its
value Exp {x Log (1 + a)}. Hence we have
(l + cr=l + {j:Log(l + r)} + {xLog(l + c)lV2! + . . .;
and, since the series 1 + 2,C,c" represents the principal branch
of (1 + zY, we have
l + 5,C,;- = l+{./-Log(l + c)} + . . . .
Now all the conditions involved in the reasoning of chap,
xxvm., § 9, will be fulfilled here, provided the complex variable
z be so restricted that | c |< 1.
Hence, if |i:|<l, we must have, as before,
Log(l + s) = £-r'/2 + s'/3-2'/4 + . . . (1).
In other words, so long as\z\<\, the series z - z'Ji + c'/S - , . .
reprtseuts the principal branch of Exp''^ (1 +s)-
Cor, aince ,Log (! + £) = Log (1 + r) •♦■ 2<>ri, \re havs
.Log (1 + ;) = 2tni + s - s=/2 + z'13 - ;'/•» + • • • (-').
§§21-23 GENERALISED CIRCHLAU FUNCTIONS 297
wliicli gives us an expausiou lor tlio t-th branch of Exp~' (1 +2}
within the region of the «-plane for which | s 1 < 1.
It follows readily, from the principles of chap, xxvi., § 9, that
when I » I = 1 the series z - z-/2 + s^/3 - ... is convergent, pro-
vided amps=t=±7r (other odd multiples of tt are not in question
here). Hence, Ijy the theorem of Abel so often quoted already,
the expanssion-formuUe (1) and (2) will still hold when |s| = l,
provided amp s =t= + tt.
GENERALISATION OF THE CIRCULAR FUNCTIONS — INTRO-
DUCTION OF THE HYPERBOLIC FUNCTIONS.
§ 23.] General definition of Cosz, Sinz, Tcmz, Cotz, Secz,
Cosecz. Since the series l-2?/2! + 2^/4! -. . ., z-s^/3\+s^/5\
— . . . are convergent for all values of z having a finite modulus,
however large, they are each single-valued continuous functions
of z throughout the s-plane. Let us call the functions thus
defined Cosz and Sins, using capital initial letters, for the pre-
sent, to distinguish from the geometrically defined real functions
cos X and sin x. We thus have
Coss=l-«=/2!+£V4!-. . . (1),
Sin c = 5-2^/3! +;j=/5!-. . . (2).
We also define Tans, Cot 2, Secz, Cosec« by the following
equations : —
Tans = Sinc;/Coss; Cot s = Cos c/Sin c ;1
Secs=l/Coss; Cosec s = 1/Sin s. J ^■''•
In tlie first place, we observe that when z is real, =x say,
we have, by § 14,
Cos .2; = 1 - a-/2! + .r*/4! - . . . = cos a*,
Sin a; = a; — .r73! + a;^/5! — . . .=sinir;
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
EULERS FORMULA
cn. XXIX
(■«)*
Since
l-s'/2! + ;V4!-. . . = ^iExp(u) + Exp(-ic)},
s-c'/3! + z'/5!-. . . = i i Exp (li)- Exp (-»•--)},
it follows from (1) and (2) that we have for all valiu\'< of x
Co3s = ^lExp(/s) + Exp(-tz)},
Sin^ = i {Exp (L-)- Exp (-/--)};
with corrpsponcling expressions fur T;inc, Cots, Sees, and
Cosec z.
By (4) we have
Cos'a + Sin'a
= i [{Exp (/r)}' + {Ex-p ( - ic)}» + 2 Exp (t.-) Exp ( - is)
- {Exp (/c);' - {Exp ( - u-)}' + 2 Exp (/.-) Exp ( - iz)\.
Hence, beariiij,' iu mind that \n\ have, by the exponential
addition theorem,
Exp {%£) Exp ( - \z) = I]xp (ts - tc) = Exp 0=1,
we see that
Cos'« + Siu'2=l (5).
from which we deduce at once, for the generalised functions, all
the algebraical relations which were formerly est^iblishcd for the
circuliir functions properly so called.
We also see, from (4), that Cos (-c) = Cose and Sin(-e)
= -Sin2:; that is to say, Co8£ is an even, and Sin 2 an odd
function of z.
Since, by (4), we have
Cos z + i Sin z = Exji (/;),
Cos z-i Sin z = Exp ( — iz),
' Tbeso fiirmulm were first RiTcn by Kuler. Sec Int. in Anal. Inf., t. L,
% 13S. Ho Kiivi'. hnwi'ViT, no fnilTicipnt jiiHtifloalion for lliuir iiiia);p, ri'uliiig
uiorcly on a buM aimlu).'y, as Ucrnoiilli ni»l Dfinoivrc IiikI iluau biTure liim.
§ 23 PROPERTIES OF Cos Z, &c. 299
it follows from the exponential addition theorem, namely,
Exp (?S] + iSj) = Exp {Iz^ Exp (jc,),
that
Cos («i + 2^) + i Sin (si + So) = (Cos Sj + i Sin 2i) (Cos ^j + i Sin zi)
= (Cos 2, Cos «2 - Sin 5j Sin Sj) + i (Sin ^i Cos Cj + Cos z^ Sin c,)*.
Hence, changing the signs of «i and z,, and remembering that
Cos. is even and Sin. odd, we have
Cos {zi + Cj) - i Sin (zi + j-,) = (Cos s, Cos «2 - Sin Ci Sin z.^
— i (Sin Si Cos S3 + Cos Zi Sin s^a)-
Therefore, by addition and subtraction, we deduce
Cos («i + S2) = Cos Si Cos z^i - Sin «i Sin Co O , ,
Sin(si+2:2) = SiusiCoss2 + Cos~iSins2.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 + 2?i7r) = Cos z Cos 2mr - Sin z Sin 2ii-!r,
that is, if n be any positive or negative integer, so that
Cos 2mr = cos 2iiir = 1, and Sin 2mr = sin 2mr = 0, then
Cos (z + 2mr) = Cos z.
In iilie manner, Sin (s + 2mr) = Sin s ; Tan {z + mr) = Tan z ; &c.
That is to say, the Generalised Circular Functions have tlie 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 equatiojis (3) of § 2. Thus, for
example,
Cos {ir + z) = Cos TT Cos z - Sin tt Sin z,
= cos TT Cos z — sin ir Sin z,
= - Cos z.
Also all the equations (5), (6), (7) of ^ 2 will hold for the
generalised functions ; for they are merely deductions from the
addition theorem.
* We cannot here equate the coefBcient of i, Ac, on both sides, because
Siu(i, + j.j), Ac, are no longer necessarily real.
800 DKFIMTION UK UYI'KUU(JLIC KUNCTIONS CU. XXIX
§ '-'1.] We proct'ed next to discuss brielly tlie variation of
the generalised circular functions.
Consider first the case where the argunient is wholly
iniaginiiry, say z = li/. In this case we have
<^'"s ('» = 2 !''^P (''!/) + ^^<^- »"'»}.
^lie-'-^-e*) (1):
= I («»-«-») (2).
We are thus naturally led to introduce and discuss two new
functions, namely, hie' + «"') and J (e* - e""), which are called
the Hyi)erbolic Cosine and the Ilj'porboiic Sine. The.se functions
are usu.ally denoted by cosh// and siiiliy ; so that, for real values
of y, coshy and sinhy arc delined by the equations
cosh y = J (e* + e""), sinhy = J(e»-e"'') (3).
In general, when y is complex, we define the more general
functions Cosh z and Sinh z by the equations
Coshc=i{E.xp(5) + Exp(-c)(,
Sinh 2 = i{E.xp(c)-I-:.xp (-.-)), (3').
We also introduce tanhy, cothy, sechy, and co.sechy by the
definitions
tauh y = sinhy/cosh y, coth y = cosh y/sinh y ;
scch y = 1/cosh y, coscch y = 1/sinh y ;
and the more general fmutions Tanh z, Coth z, &c, in precisely
the same way.
From the equations (1) and (2) we have
Co8(iy) = co.sh y. Sin (»y) = i sinh y ; 1
Tan (iy) = « tanhy. Cot (M/) = -«cotliy ; > (1),
Sec (/y) = scch y, Cosec (ly) = - « coscch y ;J
and, of course, in general, Cos iz = Cosh z, &c.
J
\
^/c
y/r ^^^^5:1
!></
0 X
/s \
I
i
Fio. 3.
302 GRAPHS OF HYrEHIlOLIC FUNCTIONS CH. XXIX
The discussion of tlie variatioti of the circular functions for
purely imagiuary argumeuts reduces, therefore, to the discussion
of the hyi)erbolic functions for purely resd arguments.
§ 25.] Variation of the IIi/j>erMic Functions /or real argu-
ments. The graphs of y = coshx, y = 8inha:, &c., axe given in
Fig. 9 as follows : —
co.shar, CO; siuhar, SOS;
cothar, T'TTT; tanhar, TTOTT;
scchar, G'C; cosecha-, S'S'S'S'.
By studying these curv-cs the reader will at once see the tnith
of the following remarks regarding the direct and inverse hyper-
bolic functions of a real argument.
(1) cosh a; is an even function of x, having two positive
infinite values corre.sponding to x = ±<a, no zero value, and a
minimum value 1 corresponding to a: = 0.
cosh"'^ is a two-valued function of y, defined for the con-
tinuum 11i>i/^oo, having a zero value corresponding to y=l,
and infinite values corresponding to j/ = oc , but no turning value.
(2) sinh a; is an odd function of x, having a zero value when
x = 0, and positive and negative infinite values when ar= + oo aud
x = - (x> respectively.
8inh~'y is one-valued, aud defined for all values of y ; it has
a zero value for y = 0, and positive and negative infinite values
when y = + 00 and y = - oo respectively.
(3) tanha; is an odd fiinction, has a zero value for x = 0,
positive maximum + 1, and negative minimum - 1, corresponding
to ar= + « and x = - as respectively.
tanh"'y is a one-valued odd function, defined for -1 ;^y5» + 1 ;
has zero value for y = 0, positive and negative infinite values
corre.sponding to y = +l aud y = -l.
(4) cotha: is an odd function, having no zero value, but an
infinite value for x = 0, and minimum + 1, and maximum - 1, for
a;= + 00 and a: = - oo respectively.
coth~'y is a one-valued odd function, defined, except for the
continuum -\^;/^-^\, havini; ))nRitivo and negative infinite
values corresponding to y=+l uud y--l respectively, and
a zero value fur y - oo .
§§ 2-i-27 INVERSE HYPERBOLIC FUNCTIONS 303
(5) sech X is an even function, having a maximum + 1 for
a; = 0, and a zero value for x ^± x.
sech'^}/ is a two-valued function, defined for 04>J'^1, having
a zero value for y=l, and infinite values for y = 0.
(6) cosech a: is an odd function, having zero values for
x = ±co , and an infinite value for x = 0.
cosech"'y is one-valued and defined for all values of jr, haviug
zero values for 3/ = + qo , and infinite values for y = 0.
§ 26.] Logarithmic expressions for cosh'^y, sinh~'y, ttc.
If X = cosh~'y, we have
^ = cosh a; = J («*-!- 6"^) (1).
Therefore
±v/(y-l) = H'''^-e-^) (2).
From (1) and (2),
e'=y±J{ir-i)-
Hence
x = \oz{y±^(y--\)];
that is, cosh-'3^ = '^og\y±J{f-\)} (3),
the upper sign corresponding to the positive or principal branch
of cosh"'?/, the lower sign to the negative brauch.
In like manner we can show that
siDhr^y = log {y + J{f + 1)} (4) ;
tanh-'y = ilog{(l + 7/)/(l-3/)} (5);
coth-'3/ = ilog{(y+l)/(y-l)} (6);
sech-'y = log[{l±V(l-r)}/2/] (7);
cosech-'^^ = log [{1 -I- V(l + f)\jy] (.s).
§ 27.] Properties of the General Hyperbolic Functions ana-
logmts to those 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 2=2 ^^•''P ( + *^) + E^p (-'*)};
Sin;: = l{Exp(-^^^)-Exp(-L-)}
(A).
In like manner, the properties of the general hyperbolic
functions spring from the defining equations
304
I'UOI'KKTIES OK IIVI'EKIIOLIC KUNCTIONS CH. XXIX
(B).
Coshc^JiE.xi.(+c) + Exp(-c)}n
Sinh s = A {Exp ( + c) - Exp {-z)\ )
Wo should tliercforo exi)ect 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 projwrtics of the circular functions tabulated in § 2. The
demonstrations are for the most part omitteil ; they all depend
on the use of the equatifms (B), combined with the properties of
the general cxjwnential function, already fiilly discussed.
The demonstrations might also be made to dejiend on the
relations connecting the general circular functions with the
general hj'perbolic functions given in § 24*, namely,
Cosh z = Cos iz, I Sinh z = Sin iz ;
+ 1 Tauh z = Tan iz, - i Coth z = Cot iz ;
Sech c = Sec iz, - i Cosech z = Cosec iz ;
(C).
(1).
Algebraic Relations.
Cosh' z - Sinh' z = \, Scch' z + Tanh' z = 1
&c.
Pcriodicifi/. — All the hjqjerbolic functions have the period
25r» ; and Tanh z and Coth z have the smaller period jti.
Thus
Cosh (s + 2iivi) - Cosh z; &c.\
Tanh {z + mri) = Tanh z ; &.c.)
Also,
Cosh (tt/ ±z) = - Co.sh z, Sinh (W ±z) = + Sinh z ;
Cosh (4ir/ ± s) ^ ± »■ Sinh z, Sinh {Wi ±z) = i Cosh z ;
Tanh (^w ± s) = ± Coth z, Coth ( JW ± c) = ± Tanh z ;
Addition FormuUr.
Cosh (;, + ;,) - Cosh c, Cosh Cj ± Sinh c, Sinh r, ;
Sinh (s, + c,) = Sinh c, Coshc,+ C<)shi, Sinh c, ;
Tanh (j, ± .-,) = (Tauh z, ± Tanh i,)/(l ± Tanh c, Tanh r,).
('-').
(3).
(5).
* This connection furiiUboa the umplost memoria technica for the bjpor-
bolic foriuuln.
(
I
§§ 27, 28 GENERAL HYPERBOLIC FORMUL.E
Cosh z, + Cosh 23 = 2 Cosh I (z, + z,) Cosh J (z, - c,) ;
Cosli z, - Cosh s. = 2 Sinh i («r. + -,) Siuh J (^^ - z.) ;
Sinh «, ± Sinh ^^ = 2 Sinh ^ (s, ± z^) Cosh J (z, + c,)-
305
(G).
Cosli z, Cosh ^2 = i Cosh (z, + Z2) + h Cosh (r, - ;r„) ;"
Siiili S-, Sinh z,= i Cosh (cj + «o) - i Cosh (cr, - c.) ;
Siuh ;. Cosh 2„ = 1 Sinli (s, + So) + J Siiih (s, - c,). .
Cosh 2z = Coslr z + SiiJr a = 2 Cosh' z-1, ^^
= 1+2 Siuh- « = (1 + Tanh= c)/(l - Tanli" c).
Siuh 2- = 2 Siuh z Cosh z = 2 Tanh ^/(l - T;iuh' z).
Tanh 2^ = 2 Tanh «/(l + Tanh=«).
(T).
(S).
Inverse Functions. — Regarding the inverse functions Cosh-^
Sinh~\ &c., it is sufiicient to remark that we can always express
them by means of the functions Cos"', Sin"', &c. Thus, for
example, if we have Cosh~^a = iv, say, then
z = Cosh w = Cos iw.
Hence iw = Cos~'^z;
that is, w = -« Cost's.
So that Cosh-'s = - i Cos-'c ;
and so on.
In the practical use of such formula;, 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->s = 27nTri ± i Qos-'^z,
where m is some positive or negative integer, whose value and
the choice of sign in the ambiguity ± both depend on circum-
stances.
§ 28.] FormulcB for the Ihjperholic Functions analogous to
Denutivre's Them-em and its consequences.
We have at once, from the definition of Cosli s and Sinh 2,
c. 11. 20
306 ANAT.OOUE TO DEMOIVRE's THEOREM CH. XXIX
Cosli (c, + r, + . . . + Zn)± Sinli (;, + c, + . . . + r,)
= I'-xp ±(z,+z,+ . . . + :,),
= ILxp + z, Exp + 2i . . Exp + r„
= (Cosh Zi ± Sinh z,) (Cosli s, + Sinh i-j)
. . . (Cosh r, ± Sinh c„) (A);
and, in particular, if n be any positive integer,
Cosh fiz ± Sinh nz = (Cosh z ± Sinh z)' (B).
Tliese correspond to tlie Denioivre-formulae, with which the
reader is already i'aniiiiar*.
We can deihice from (A) and (B) a series of formnl.T fur the
hyperbolic functions analo^'ous to those established in § 12 for
the circular functions.
Thus, in particular, we have
Cosh (c, + c, + ...+-.) = /'. + P,_. + /^.. + . . . (1'),
where Pr = 2 Cosh Si Cosh r, . . , Cosh «r Sinh Jr+, . . . Sinhc,.
Tanh(c, + «, + . . . + c„)
= (7',+ 7',+ 7', + . . .)/(! + 7", + 7'. + . . .) (.r).
where 7; = 5 Tanh c, Tanh :, . . . Tanli c^.
Cosh nz = Cosh"s + ,Cj Cosh"-";: Sinli^c
+ ,C4Cosh"-«cSinh*r + . . . (»').
Sinh nz = ,(7, Cosh-'* Sinh z + ,C, Cosh"-'c Sinh' z
+ ,C.Cosh"-'sSinh''r + . . . (5').
Cosh nz = {- )-« {l - 2' cosh' z + "'("'-^') cosli* :-. . .
(-)•— i (2«)! ^'co8h-z + . . .j (9),
(n even) ;
• As ft mutter of liistory, Dcmolvre first fonnd (B) in the form
V = i[l/\'{v'(l + «")-''}-'J'{\/(l+0-«'}]. "''"''■o y '» ">» ordinate of l> in
I'ig. 10 bcluw, aiiil v the orclinnto of y, y corrvBjwndinK to a vector OQ nuch
tliat the nrca AOQ in n times AOV, and OX is taken to be 1. He then
deduced the corienjiondinK formula for the circle l>y an imagiiury traoi-
formaliuu. (Sue il'ucMaxuiX Aitiili/tica, Lib. 11., cup. 1.)
^ 28, 29 HYPERBOLIC INEQUALITIES AND LIMITS 307
Smbn5/siuhc = (-)i''-=)^|jicosli;;-'-?-^:'^f^cosli^i; + . . .
(« even) ;
and so on.
We may also deduce formulie analogous to those of § 13,
such as
Smh''"+'z = ^{sinh(2;« + l);;-^+,(7,sinh(2;«-l)a + . . .
{-)'"5™+iC'„sinhs}.
§ 29.] Fundamental Inequality and Limit Thmrems for the
Hyperbolic Functions of a real argument.
Ifube any positive real quantity, then
tanhM<M<siiihtt<cosha (1).
By the definitions of § 24 we have
sinh u = \ {exp (m) - exp ( - m)} ;
= m + mV3! + mV5! + . . . (2);
cosh«=l+MV2! + MV4! + , . , (3);
whence it appears at once that sinha>2<.
Again, cosh?t = +;^(l+sinh=M), so that cosh m> sinh w.
Finally, since
tanh u = sinh m/cos1i m
= «(l+M-/3! + ?*V5! + . . .)/(1+m72! + mV4!. . .),
and mV3!<m72!, mV5!<mV4!, &c.,
we see that tanhw<M.
Cor. When u = 0, L sinh u/u = l, and L tanh n/u = 1. 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 eke has a finite limit when n = oo , t/ien, when « = oo ,
20—2
808 GEOMETRICAL ANALOGIES CU. XXIX
We have
Hence, if wc put 1 +6-'""=^ 2 -2c, so that 2 = 0 coirpsiinmU
to n = oo, then we have
L ('cosh-y = f- L {(1 -;)-'••} -WorP-*!
n=oo\ «/ t-0
Now, L {I -£)-"• = €, and i2;/log (1 - 2j) = - 1. Hence, by
chap. XXV., § 13,
,(co.sh;;)'
= <^e"
We leave the demonstration of the second limit as an exer-
cise for the reader. The third is obviously dcducible from the
other two.
A very simple proof of these theorems may also be obtained
by using the convergent series for cosh . a/n and sinh . a/n.
§ 30.] Geometriail A nalogies between t/w Circular and I/ifjter-
bolic Functions.
If 6 be continiiously varied from —ir to +ff, and we connect
X and y with 0 by tlie equations
a; = acos^, y = osinfl (1),
then we have
a^ + if = a^{cm*6 + sWe)=a' (2).
Hence, if (r, y) bo the co-ordinates of a point P, as 6 varies con-
tinuously from — IT to + TT, P will describe continuously the
circle A'AA" (of radius a) in tlie direction indieated 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. Wo
can determine the form of this function as follows : —
Divide 6 into n equal parts, and let /•, , f,, . . ., Pr, . /'
be the points corresponding to 6/n, 20/)i, . . ., rO/n, . . . nO/n
respectively. Then we have, by the lemmas of Newton,
Area ^10/^" L ^'i'^ PrOPr^i.
§§ 29, 30
AUliA OF CIRCULAR SECTOR
Y
309
Fio. 10.
Now
PrOPr^,
= OMr^, P,+, + i)/.« P,« PrMr - OMrPr.
= h {^r+l^r+l + (]/r+l + I/r) {^r - ^r+i) - ^r^^r},
= ^tt' {cos . r9/n siu . (r + l)0/>i, - sin . rO/n cos . (r + l)9/u],
= ia''sin. d/n.
Hence
0 = |a' Ln sin . 6/«,
= ^''9i(siii.e/n)/(0/n),
= W^- (3).
Hence, i{ 0 = 2@/a^, we have cos6 = x/a, s\n6 = j/la, t3,n0=i//x,
cot 6 = ic/y, &c.
let
Thou
Next, let u be continuously varied from - « to + oo ; and
x = a cosh a, y = o sinh ?/ (1').
x'-if =0' (cosh^ u - sinh" u) = a' (2').
i
310
AitrA OK riYPEniiouc sector
CM. XXIX
Hciico, if (x, y) be the co-ordinates of 1\ jis «• varies con-
tinuously from — CO to +Q0, P will describe continuously the
right-baud brauch A'AA" of the rectangular hyperbola, whose
Fio. IL
Bemi-ftxis-major is OA-a, in the direction indicated by the
arrow-heads in Fig. 11.
If f be the point corresponding to u, 7',, /%+, the points
corresponding to ru/n and {r-*-\)u/?i, and f^ the area AOP
agreeing in sign with u, tlien, exactly as before,
* Adopting an axtrononiioal term, ire may oall u the hTperbolio exoentrio
anomaly of P. The quitiitity u lOayg in the theory of iho hyp<>rl>ola, in
general, tlio saiiiu |iart an Uio cxccutrio angle in the theory of the ellipae.
§§ .30, 31 GUDERMANNIAN oil
r=n— 1
U=^r L 2 (.rr^r+l-irr+iy,);
n = «j r=0
= a" {cosli . ru/n siuh . (r + 1) «/» - sinh . ru/n cosh . (r + 1 ) »/"}.
= a' sinli . u/n.
Therefore ^ = I "^^-^^ si"h • «/".
= \a-uL (sinh . u/n)/{ti/n),
= ia-u, by §29, (3').
Hence, if the area AOP=U, and u = 2U/a-, then, a- and
^ being the co-ordiuates of P, we might give the following
geometric definitions of cosh u, sinh u, &c. : —
coshM = a;/a, smh.u = 2//a,
tanh M = Tf/x, c oth M = x/^, & 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 6 is
the product of Ja- into the number of radians in the angle AOP. It there-
fore follows from (3) that the variable $ introduced above is simply the
number of radians in the angle AUP. Our demonstration did not, however,
rest upon this fact, but merely on the functional equation cos- fl + sin- 9 = 1.
This is an interesting point, because it shows us that we might have intro-
duced the functions cos 9 and sin 9 by the definitions co» fl = J {Exp (i9)
•hExp(-i9)}, Bine = ^. {Exp(ie)-Exp(-ie)}; 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 A I', 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.] E.rpression of Meal Hyperbolic Punctioim in terms oj
Real Circular Functions.
312 GUDERMANNIAN CH. XXIX
Since the rnnjje nf the variiition of cosh » when « varies from
- 00 to +00 is tl>e saiiio as the range of sec 6 when 0 varies
from - JjT to + jTT, 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 - « and + oo and of 0 between -\v and + ja- such that
cosh u = sec tf (1).
If wo determine 0 in this way, wo have
sinh « = ± ^(cosh' « - I),
= ±^/(sec'^-l);
hence, bearing in mind the understanding as to sign, we have
sinh u = tan 6 (2).
From these we deduce
e' = cosh tt + sinh u,
= sec ^ + tan 6 ;
u = log (sec 0 + tan 6),
= logtan(j7r + i(9) (3).
Also, as may be easily verified,
tanhitt = tanlfl (4).
When 6 is connected with « by any of the fotir equivaii^nt
equations just given, it is called the GuJn-mannian* of u, and we
write ^ = gd M.
* This name was invented by Cnyley in honour of the Ocrman mathe-
matician Gnderniann (179^-1852), to whom the introduction of the hyperbolic
functions into modern analytical practice ia largely due. The origin of the
functions goes back to Mercator's discovery of the logarithmic quadrature of
the hyperbola, and Dcmoivre'e deduction therefrom (sec p. 30G). According
to Houel, F. C. Mayer, a contemporary of Demoivre's, was the first to give
shape to the analogy between the hyperbolic and the circular functions. The
Dotation co.^h. 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
fiirmuliD were indopondcntly deduced by William Wallace (Professor of
Mathematics in Edinburgh from IHIO to 1838) from the geomotriral pro-
perties of the rectangular hyperbola, in a little-known memoir entitled S'eit
Serif for the Quadrature of Conic Section* and the Computation of Lojarithmt
(Trnnj. li.S.E., vol. vi., 1812). For further historical information, iie«
Oiintlier, Die Lehre i^on den j/cirii'iri/iV/K-n und verallgrmcinerlen Hyprrbel-
funktioncn (Halle, 1881) ; also, Heitrfigetur GetchichU der S'eueren Malltematik
{Programnuchri/t, Anabach, 1881).
§ 31 EXERCISES XVII 313
It is easy to give a geometrical form to the relation between $ 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 3IQ-=0^P- 0Q- = 3i^-a"=y-, we have
Beoshu=a;=asec QOlf. Therefore QOAf=S, and we have j/ = J[/Q = atan9.
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
{unctions to the following authorities: — Greenhill, Differential and Integral
Calculus (Macmillan, 1886), and also an important tract entitled A Chapter
in the Integral Calculus (Hodgson, Loudon, I88S); Laisant, "Essai sur leg
Fonctions hyperboUques," il^m. de la Soc. Pltys. et Nat. de Bordeaux, 1875 ;
Heis, Die Uyperbolischen Functionen (Halle, 1875). Tables of the functions
have been calculated by Gudermann, Theorie der Potential- oder Cyclisch-
hyperbolischen Functionen (Berlin, 1833); and by Gronau (Dautzig, 1863).
See also Cayley, Quarterly Journal of Muthematics, vol. xx. ; aud Glaisher,
Art. Tables, Encyclopcedia Britannica, 9th Ed.
Exercises XVII.
(1.) Write down the values of the six hyperbolic functions corresponding
to the arguments Atti, vi, ^ri.
Draw the graphs of the following, x and y being real : —
(2.) y = sinhxlx. (3.) y = xcothx.
(4.) t/ = gdi. (5.) !/ = 6inh-i{l/(.r-l)}.
(6.) Express Sinh~'z, Tanh-^z, Sech"'2, Cosech~'z, by means of Sin-'z,
Cos~'z, &c.
(7.) Show that cosh'u-sinh'u=l + 3sinh'uoo8h'u.
(8.) Show that
4 cosh'u - 3 cosh u — cosh 3u = 0 ;
4 sinh'u+ 3 sinh u- sinh 3u=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 ;^(7/3) cosh u, 2J{ql'S)(cos^Tr
cosh II ±i sin jTTSinh u), u being determined by cosh 3u = 3r,^S/2^/(;'.
(10.) Solve by the method of last exercise the equation a^ + 6a + 7 = 0.
Express
(11.) tanh"'x + tanh-'!/ in the form tanh~'2.
(12.) cosh-' X + cosh"' ?/ in the form cosh^'z.
(13.) sinh-'i-Binh-'?/ in the form cosh-'z.
Expand in a serios of hyperbolic sines or cosines of multiples of u : —
(U.) Cosh'i'u. (15.) sinh'u. (16.) cosh»uBinh»u.
314 EXERCISES XVII CH. XXIX
Expnnd in a serica of powers of bjrpcrbolio sines or ooaincs of u:—
(17.) CoshlOu. (18.) siiihTu.
(19.) cosh Cu siiih 3u. (20.) siub mu cosli nu.
Establish the following identities : —
(21. ) tanh J (u + 1') - tanh J (u - r) = 2 sinh t7/(co6h u + cosh v).
,-- . 8inh(u-r) + sinhu + sinh(« + r) , .
(22.) , , ' , .; ' = tanhu.
' ' cosh (u - r)+ cosh « + cosh (u + r)
(23.) tanh u + tanh (J»-i + u) + tanh(3ri + u) = 3tanh 3«,
cosh 2u + cosh 2i' + cosh 2ui + cosh 2 (u + r + ir) = 411 cosh (p + »).
(24.) Tan Hu + iv) = (sin u -{- i siuh p)/(cos u + cosh r).
(25. ) Express Cosh* (u + ii') + Sinb* (» + iv) in terms of functions of u and p.
Eliminate u and v from the following eqauliuns:—
(20.) x = aeosh (u + \), y = b anh (u + ft).
(27.) y cosh u-XBiiihu:=a cosh 2u,
y sinh u -t- z cosh u = a sitih 2u.
(28.) X = tanh u + tanh r, y = coth u 4- coth r , u-t-r^e.
(29.) Expand sinh(u + A) in powers o( h.
(30.) Expand tanh-'i in powers of x; and deduce the expansions of
cosh-'x and Binh"'x. Discuss the limits within which your expansions ar«
▼alid.
(31.) Given 8inhu/u = 1001/1000, calculate u.
" 1 /x'''*~'-l\ ^
(32.) Show that the series S ^j ( — j is conrergcnt, and that ita
sum is (xi+l)/(x>-l)-l/logx (Wallace, I.e.).
(33.) Prove that the infinite product cosh ,r| cosh ^ cosh ,tj • • . >■ oon-
vergent, and tliat its value is sinh u/u.
(34.) Show that
«-x-» 3 3 _
(Wallace, Le.)
I » — * - • « * ,
from 1/log X (in defect) by less than
{1 + 1 (x'/'"*' + x-"'*^')}/3. 4»+'P,.
Evaluate the following limits: —
(30.) (sinhx-Einx)/x>, x=0.
(87.) (sinh' mx- sinh' nz)/(eosb;>T- cosh 9x), x=0,
(88.) (tan' x - Unh' x)/(cos x - oush x) , x = 0.
d
§31 EXERCISES XVII ^15
Show that, when /i=0,
(39.) L {cosh a (x + /') - cosh n.T}/A = a sinh ax,
(40.) L {smYi a (x + h) - sinh rjx}/ft = acosh<(j;.
(41.) L {tanh a(x + h)- tanli ttx\jh=a sech- ax.
(42.) L {cotho (x + ft) - coth ax)lh= - acoBech'o*.
(43.) Show that
1 -
2-« '=°'''' 2^ = """^ " - - .jl. '""'^ 2^' •
1 " 1 «
- = coth u-S^i tanh— ,
u 1 2" 2" '
and state the corresponding formuls for the circular functions (Wallace,
Trans. R.S.E., vol. ti.).
(44.) From the formulie of last exercise, derive, by the process of chap.
XXVII., § 2, the following : —
2S coth'-^ |„=coth»u- 2 .-jj-„ tanh^-|i ,
i,=coth«u-sist''nt'|.
(Wallace, I.e.)
In the following, 0 is the centre of the hyperbola x-/n^ - 2/-/ft" = 1 ; A one
of its vertices ; F the corresponding focus ; F and F' any two points on the
curve, whose excentric anomalies are u and ii, and whose co-ordinates are
(x, y){^, y), so that s = acosh«, y = b siahu, &a. ; and iV is the projection
of P on the axis a. Show that
(■15.) Area JWP=Ja6(sinh2u-2u).
(46.) Area of the right segment out off by the double ordinate of 1'
= -xJ(x-- a-) -ab cosh"' - ,
a a
= -xJix^-a^)-ab\oB — 2L! '. ,
a a
(47.) Area of the segment cut off by PP'=Ja6{sinh(u'-ii) -(u'-u)}.
Express this in terms of x, y, x', y'.
(48.) If 7i be the middle point of PF', and Oil meet the hyperbola in S,
the co-ordinates of S are {a cosh J (ii + u'), 6 sinh J (u + u')}.
(49.) OS bisects the hyperbolic area POP'.
(50.) If PP" move parallel to itself, the locus of ji is a straight line passing
through 0.
(51.) If PP" cut ul! a segmuut of couslaut area, the locus of ii is a
hyperbola.
316
GRAPH OF Cos(a; + yt)
CH. XXIX
GRAPHICAL DISCUasiON OF TlIK GKNKHALISED CIRCULAR
FUNCTION'S.
§ 32.] Let U8 now consider the gciienil functional (Miuation
w = Cos z, or, as wo may write it,
u + it> = Cos (a; + yi) (1),
wliore M, V, X, y arc till real.
Since (,'os {x + yi) = Cos x Cos yi - Sin x Sin »/«' = cos x cosh y -
i sin X sinh y, we have
« = cos .T cosh y, t) = -sina;8iuhy (2);
and therefore , ,
M7cos'a;-«'/8in'a:=l (3),
u'/cosh'y + cV.siuh''y = 1 (4).
V
U 0
TJ-^
M
L
T
•< 1
L
M
N UjU
N
M
L K
K
TH.
-0
BR.
PRIN.
BR.
[
DR.
CC
=5
D
R
g B
R
D
sec
S
D
R S
B
G
r.
0
A
P
F
F
p
A
Q G
a
Q
A
P F
GX
r.
0
s
D
R
B
ii
R
5
S C
c
s
D
R B
B_
D
U
N
M
L
K
R
L
fa
N 0
u
N
M
L K
R
Fio. 12.
In ortlor to avoid repotition of the v.nliios « and v, ari.'sinp
from the periodicity of co.sj- and sin J-, wo confine z, in the first
instance, to lie between the axis of y and a parallel UCGCU U^
this axis at a distance from it eqnal to ir (Fig. 12).
If we draw a .series of paraliols to the y-axis within this strip,
we see, from equation (3), that to each of these will belong half
§
32
GRAPH OF Cos (x + yi)
317
of a hi-perbola in the w-plaue {Vig. 13), having its foci at the
fixed points i<'and G, which are such that 0F= 0G= 1. Thus,
for example, if in the s-plane FP = \Tr and FQ = f tt, then to the
parallels LFL, iVQi\' coiTespoud the two halves LFL, N(^N oi
a hyperbola whose transverse axis is PQ = J2.
\
V
M
5
/
/
/
^
7^
TT
/
c
\
\
1
/
/
\
-^
u
c
>
G
r
A
5
M
\
■JB
K
Fig. 13.
K
B
K
B
L
TH
(-n
R
M
BR.
D
N U
S C
a ■
c
PRIN.
S
Y
M
BR.
D
C 1<
R B
K
B
L
R
M
BR.
D
N U
s c
0
C
F
F
P
A
Q G
G
Q 1
A
P F
F
P
A
Q G
F X
B
K
B
(3
SI
s c
N U
C
U
s
N
D
M
J
R B
L k
B
K
R
L
D
M
S C
N U,
C
u
FiQ. 14.
318 QRAPU OF Cos(x + yt) CH. XXIX
To the parallel MAM, wliich bisects the strip, correspoiKk
the axis of v (which may be regarded as that hyperbola of the
confocal system wliich has its transverse axis equal to 0) ; and
to the parallels KFK -AmX UGU, which bound the strip, corre-
spond the parts KFK and UGtJ oi the u-axis, each regarded as
a double line (flat hyperbola).
Aj,'ain, if we draw parallels to the a"-axis across the strip, to
each of these will correspond one of the halves of an ellipse
belonging to a confocal system having /"""and G for common foci.
Thus to BllDHC and BIIDHC equidistant from the x-axis corre-
spond the two halves BKDSG and BIIDSC of the same ellipse
whose semi-axes are coshj/ and sinhy. In particular, to FPAQG
on the X-axis itself corresponds the double line (Hat ellipse)
FPAQG. _
Thus, to the whole of the first parallel strip between KOK
and UU corresponds uniquely the whole of the ir-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 s,
having the real period 2n- ; and s is a multiple-valued function
of w of infinite multiplicity, having two branches for each period
of w.
The value of s corresponding to the first strip on the right
of the ji/-axis is called the principal branch of Cos"' w, and the
others are numbered as usual. We therefore have for the /-th
branch
,C08-'fC = S, = (t + i+(-)'-'i)ir + (-)'(;03-=M> (5),
where Cos"' w is the principal value aa heretofore ; and Cos"' w
= x + i/i, X and y being determined by (3) and (4), when u and ti
are given.
It should be noticed tli.it for the .same branch of : there is
continuity from H to li not directly across the M-axis, but only
by the route BFB; whereas there is continuity from li to Ti
§§32-34 GRAPH OF Sin (a; + yO 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 + oo and
from G to -co, as indicated in Fig. 13. If we were to con-
struct a Kiemann's surface for the w-plane, so as to secure unique
correspondence between every tt'-poLnt and its z-point, then the
junctions of the leaves of this surface would be along these slits.
The reader will find no dilhculty in constructing the model.
Since to the line KFPAQGU (the whole of the w-axis) corre-
sponds in the 3;-plane the three lines KF, FPA QG, G U taken
in succession, we see that as w varies first from + oo to 1, then
from 1 to — 1, and finally from — 1 to - <» , Cos~^ w varies first
from CO i to 0, then from 0 to jt, and finally from tt to tt + oo « ;
so that an angle whose cosine is greater than 1 is either wholly
or partly imaginary.
§33.] If w = Sin 5, say
u + iv = Sin {x + yi) (1),
then, as in last paragraph,
u = siu X cosh y, « = cos x sinh y (2) ;
wYsin" X - 'wYcos^ x=l (3) ;
u^/co!i\i^y + v^/sm\i'y = l (4).
The graphical representation is, as the student may easily
verify, obtained by taking Fig. 13 for the w-plane and Fig. 14
for the a-plane.
We have also, for the t-th branch of the inverse function,
eSin~' u' = 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 = Tans;, say
u + ic = Ta,n(x + yi) (1),
then (u + iv) Cos (x + yi) = Sin {x + yi),
that is,
(m cos X cosh y + vsmxsinhy) + i {-u sin « sinh y + « cos x cosh y)
= sin X cosh y + i cos x sinh y.
320
Tlierefore
UKAPII OK TAN(x + y»)
CU. XXIX
tt COS X cosh y + » sin X siiih y=«va.x cosli y,
— u sill X sinli y + » cos a; cosh y = cos a- sinh y.
From the last pair of equations it is easy, if we hear in mind
the formulae of § 27, to deduce the following : —
H = sill 2j-/(co8 2a; + cosh ly), v = sinh 2y/(cos 2x + cosh ly) (2) ;
tt» + «• + 2m cot 2z - 1 = 0 (3) ;
«• + «=- 2p coth 2// + 1 = 0 (4).
Tlie graphical representation of these results is given by
Figs. 15 and 16.
Y
|*w
1,
»«■
l.*»
I-
--
r
R.
PR
N.
f
Ft.
(•
"t
6
1
'
-i
,i
'
i
>.
1
1/ .
J
'
7
J
■
■'■ ■
J
7
" <
t
1
K
V
**
**
^
H
"'
■•
L
w
r
-
H
"■
^
s
'
*' ■
X
*
J
J
e
-
■'■
•
J
-
■'
-
'
*
"
"
M ,
4
'
'
W
it
'^
*'
■■
■
■'
■•
M
s
'
-
•'
• T
K
'I
r
J
C
^>
7
V
,)
f .
k
f,-
•CO
i
u«
1,
-»
Fio. IS.
When ar is kept constant, the equation to the path of w is
given by (3), which evidently rcpresentii a series of circles passing
thiuuKh the points (0, +1) and (0, - 1).
When y is constant, the equation to the path of ic is (4),
which represents a circle having its centre on the tva.xis ; and it
is easy to verify that the sriuare 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
orthiitomie.
If we consider a parallel strip of the c-plane boundwl by
« = - JjT, «-- + i^-, wo find that to this correspouda the whole
§34
GRAPH OF Tan {x + yi)
321
to-plane taken ouce 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 ^-plane correspond the circles
passing through / and / in the w-plane, and to the horizontal
parallels correspond the circles in the «<'-plane which cut the
former orthogonally.
It should be noticed that / and I in the w-plane correspond
to + 00 and - oo in the direction of the y-axis in the s-plaue, and
Atco
U
J
K
T
/
-t
'iS
L^
V
/
.)
5v^
m
\
J
/
Tp
"v
\
^
A
A -
B
V
cl
D E
' F
)g
Hi
J V
\
i^
\Ji
V
y
L\
K
A-»
r
J
Is
Fia. 16.
that to A and / iu the c-plane correspond the points at oo on
the u- and v-axes in the w-plane ; also that there is no continuity
directly across /Z'-JD or /A'co in the if-plaue, except in passing
from one branch of Tan~' w to the next.
For the t-th branch of the inverse function we have
tTa,n-^w = z, = tir + Ta.n-^v; (5),
where the principal value Tan"' w is given hy Tan'' w = a; + yi,
X and y being determined, under the restrictions proper to the
principal branch, by means of (o) and (1).
c. II. 21
322 GiiAPHS OF /(x + yi) AND il/i<c + yi) cu. xxix
§ 35.] It will bo a useful exercise for the student to discuss
directly the graphical represeutiition of «; = Sec2;, w = Co8ec«,
and M; = Cots. The iij^ures iu the w-jilane for these functions
may, however, be derived from those already given, by means of
the following interesting general principle.
Jf Z be any z-path, W and U" the corrc.yxmding w-p<tths/or
w =J\x + yi) and w = l/Ji-t+yi), tlien W is the image withresi>fct
to the u-ajcis of the invrrse of \V, t/ie centre 0/ inversion being the
origin of the w-plane and the radius of inversion being unity.
This is Ciusily proved ; for, if (p, <^), (p, <^') be the polar
co-ordinates of points on I^aud W corresponding to the point
{x, y) on Z, then we have
P (cos </> + »' sin <^) =f{x + yi),
p (cos <^' + 1 sin <^') = l/f{x+ yi).
Hence p (cos <^ + 1 sin <^) = l/p'(cos <^' + » sin <j>'),
= ( 1/p') (cos (-.^') + .- sin (-«')).
Therefore p = \/i>', <^ = - <^' + 2Xir, which is the analytical ex-
pression of the principle just stated.
From this it appears at onco tbat, if we choose for our Btandard t-psths
a doable sjBtem of orthotoiuic parallels to tlio .r- and y-axes, then the w-p«tha
for ic=Cot2 will be a double ry intern of orthotomic circles, and tbe ir- paths
for u;=Secz aDdu> = C!osccr a double system of orthotomic Bicircul&rQuartica.
Example 1. If u + vi = See [x + yi), show that
u = 2 cos z cosh y/(co8 'Jx + cosh '2y) ;
ti='2sinx8uih j//(cos2x + coBh 2y\;
(u' + c')' = u'/cos' X - I'/sin' x ;
(u» + 1')' = u'/cosh' y + t)»/siiih' y .
Piscuss the graphical representation of the functional equation, and show
bow to deduce the (-th branch from the principal branch of the function.
The curves represented by the lost two eqoations are most CMily traoad
from their polar c<iuations, which arc
p»=2(cos2^-co8 2x)/sin' 2i,
p' = 2 (cosh 2y - coa 2^)/6inh* 2y,
respoctivcly.
Example 3. The same problem for u + t'i = Cosoo (x + yi).
Kiauplu 3. The same problem for u + ii = Col (x i- y 1).
§§ 35, 36 ORTHOMORPHIC TRANSFORMATION 323
§ 36.] Before leaving the present part of our subject, it will
be well to point out the general theorem wliich underlies the
fact that to the ortliogoual parallels in the s-plane in the six
cases just discussed correspond a system of orthogonal paths in
the it'-plaue.
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 ;i-plane f{z) can
always be expanded in a convergent series of powers of z-a,
so that we have
/(-) =/(«) +A,{z-a)^ A, {z -af + . . . (1),
where ^I,, A.^, . . . are functions of a and not of 3.
Then we have the following general theorem, which is funda-
mental in the present subject.
If Ai^O, the angle between any two z-ixtths emanatinfj from
a is the same as the angle between the corresponding w-paths
emanating from the point in the w-j)laiw which corresponds
to a.
Proof. — Let z be any point on any path emanating from a,
{r, 0) the polar co-ordinates of z Avith respect to a as origin, the
prime radius being parallel to the a;-axis. Let w and b be the
w-poiuts con-esponding to z and a, (p, <^) the polar co-ordinates
of w with respect to b. Then we have
P (cos <f> + i sin <^)
= w-b=f(z)-f(a),
= A,iz-a) + A3(z-a)' + . . ., by (I),
= A^r {cos6 + isin 0) + A.^i-{cosO + i sin 6)-+ . . . (2).
Let now J , = r, (cos uj + i sin a,), A. = r^ (cos a., + i sin a,), . . . ,
then (2) may be written
P (cos </) + J sin <^) = r^r (cos (a, + 6) + i sin (uj + 6)]
+ r^r" {cos (oj -I- 25) + i sin (a, + 2(9)} + . . . (3).
Whence
P cos ^ =r^r cos (o, ^■6)^■ r^r' cos (a, + 20) + . . . (4) ;
p sill <^ - r^r sin (a^ + 6) + r-ji' sin (a.^ + 25) + . . . (5).
21—2
3"24 OKTIKiMOllPIIIC TllAN.SF<»KMATI(i\ CH. XXIX
III tlie limit, when r mid consoiticiitly p are inacle iiiHiiitcly
small, (4) ami (5) reduce to
(p/r) cos <)!) = r, cos (a, + tf), (p/r) sin <^ = r, sin (a, + <?) (G).
Since p and r arc both positive, these equatious lead to
p/r = r,, and ft> = 2kir + ai + 0 (7).
Hence, if we take any two paths eniauatiug from a iu directions
determined by 0 and 6", we should have <f>- <t>' = 0-0', which
proves our theorem.
We see also, from the first of the equations in (7), that if we
construct any intinitely small triaiij;le in the r-plane, having its
vertex at a, to it will correspond an infinitely small similar
triangle in the u?-plane having its vertex at b.
Hence, if we establish a unique correspondence between points
(m, v) and {x, y) in any two planes by means of the relation
u + vi =/(x + yi) = X (x, y) + t> (x, y),
then to any diagram D in the one plane rorres/tonds a diagram
D' in the other which is similar to D in its infinittsimal detail.
The propositions just stited show that, if we hate in the
z-plane any two families of curves A and B such that each curve
of A cuts each curre of B at a constant angle a, thiti to these
correspond respectively in the w-plane families A' and B' such
that each curve of A' cuts e<ich curve of B' at an angle a.
Since the six circular functions sjitisfy the preliminary condition
reganliug the function f{x + yi), the theorem regarding the
M-c-curves for these functions which correspond to j: = const.,
y = const, follows at once.
If .1, = 0, ^la = 0, . . ., vl,_, = 0, .'l, + 0, then the above con-
clusions fail. In fact, the equations (7) then become
plf'-r,, <l> = 2U + a, + n0 (7');
and we have </> - <^' = w (0 - 0').
In this ia.sc, us the point ; circulates once round a, the point
w circulates ;» times round b. That is to say, b is a winding
point of the «th order for z ; and the Kiemann's surface for the
tp-plane ha.s an w-fuhl winding ])oint at b. We have a simple
example of this in the ca^u of u'-.^, already discussed, fur which
§ 36 EXERCISES xvm 325
w' = 0 is a wiudiug point of the third order. The points w = ± 1
and z = ±0 are coiTOsponding points of a similar character for
tv = cos z.
The tlieorem 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 0 ( j, y ) + 7x (j^, 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 Gnindlagen fiir eine allgcmeine Theorie der Functionen einer
verdnderlichen compleien Grbsse, in which Kiemann 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. JNIany of the ordinary geometrical
transformations are particular cases of this ; for example, the student will
readily verify that the equation \D = 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 IG may be taken to represent lines of force and lines of equal
potential; so that every particular case of theequation u-l-fi=/(a; + yi) gives
the solution of one or more physical problems.
Exercises XVIII.
(1.) Discuss the variation of 6in~'u and sin~'n', where « and v are real,
and vary from — oo to -h x .
Draw the Argand diagrams for the following, giving in each case, where
they have not been given above, the to-paths when the z-paths are circles
about the origin and parallels to the real and imaginary axes: —
(2.) ui = log«. (3.) jc = exp2.
(1.) w = cosh2. (5.) «; = tauh2.
(0.) Show that co-s"' (u-H't)) = C03-' {/-icosh"' V ;
sin"' («-)-ir) = sin~i P+icosh~i V,
where iV=^{(n + lf + v-}-J{(ii~\Y-^v''},
the principal branch of each function being alone in question.
326 EXEHCISES .Win Cll. XXIX
(7.) Sbow that the prineipnl branch of tau"' (u + iv) u given by x + yi,
where y = J tanh"' {2u/(u' + t!'+ 1)};
and «=itsn-'{2u/(l-u»-i')>. if u» + t''<l;
= ± Jt + i tan-' {2ii/(l-u« -«•')}, ifu» + w»:.l,
the upper or lower sign being takin according aa u ii positive or negative.
(8.) If u + fi = cot(x + j/i), show tliat
u = sin 2x/(cosh 2i/ - cos 2x), ti = - Riiih 2y/(cosh 2y - cos 2x) ;
u' + »'-2ucot2jc-l=0, u' + 1>' + 2t' colli 2i/ + 1=0.
(0.) If u + ri = C08ec(x + yi), show that
u = 2 ?in X cosh y/(cosh iy - cos 2x), v = -2 cos x siiih y/(coBh 2y - cos 2x) ;
(u' + tJ*)' = u'/oos'x - c'/sin'y, (u' + r')'= u»/cosh»y + p'/sinh'y.
Express the following in the form u + ri, giving both the principal branch
and the general branch when the function is mnltiplc-vnlucd: —
(10.) Cosh->{*+yi). (11.) Tanh-'(x + yi).
(12.) iLog{(x + yi)/(i-y.)}. (13.) Log Sin (x + yi).
(1*.) (cosff + isin*)'. (15.) Loga+tf (x + yi).
(16.) Show that the general value of Sin"' (coscc «) is ((-(-i) r + ilog
cotJ((ir + e), where t is any integer.
(17.) Show that the real part of Exp^ {Log (l + i)} i»«-^'cog (It log 2).
(18.) Prove, by means of the series for Cos $ and Sin 0, that Sin '20 = 2 Sin 0
Cos0.
(19.) Deduce Abel's generalised form of the binomial theorem from
§§ 20, 22.
(20.) Show that
l + «+,iCi' + m+,(C,x'+. . . adeo
= (1 + x)"" [cos {n log (1 +z)} + 1 sin {n log (1 + j))].
(21.) Show that the families of curves represented by
sin X cosh y = X, cosxsinhy=/<
are orthotomic.
(22.) Find the equation to the family of curves orthogonal to r*
oosn0 = X.
(2.'i.) Find the condition tliat the two families
/(x5 + 2/.xy + Cy' = \, il'x» + 2/('xy + Cy«=^
be orthotomic.
(24.) If tan (z + iy)=siD(u-l-ii7), prove that coth r siuh 2y = cnt u sin 2x.
SPECIAL APPLICATIONS OF THE FOREGOING THEORY TO
THE C.IRCULAK FUNCTIONS.
§ .37.] In order to nvoiii breaking; our o.\po.'»ition of tlic
geiicrul theory of the eluiiicuUiry trauscendeulA, wu did not stop
I
§§ n7, S8 APPLICATIONS TO CIROTTLAR 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 + 2m(7„s" is
convergent, we have seen that it represents the principal value
of (1 + s)'". Hence, if z = r{cos6 + is\u6), where r is positive,
and -•«■:}> 6 :|> +7r, we have
1 + 2 „,C„r" (cos 110 + i sin nO)
= (1 + 2r cos 6 + »-)""' (cos m<l) + i sin «i0),
where - ^7r::|> 0 = tan"' {r sin 6/(1 + r cos 6)}> + i^.
Hence, equating real and imaginary parts, we must have
1 + 2,„C„r'' cos 110 = (1 + 2r cos 0 + rY'' cos m4> (1) ;
2„^„r'' sin n0 = (1 + 2r cos ^ + rT" sin m4> (2).
These formuhc will hold for aU real commensurable values of
m, provided r<l.
When r = l, we have
</> = tan-' {sin 6/(1 + cos 0)}=10,
and (1) and (2) become
l + 2,„(7„cosHe = 2""cos"'^J6'cos^?w0 (1'),
2„.a„ sin 710 = 2" cos"'i6' sin lm0 (2').
These formulte hold for all values of 0 between — ir and + tt*,
when w>— 1; and also for the limiting values — tt and + t
themselves, when m>0.
§ 38.] Series for cos m(l> and sin m<l>, wheti m is not integral.
If in (1) and (2) of last paragraph we put 6 = |n-, and
r = tan 4>, so that <^ must lie between - \ir and + ^tt, tlien
(l + 2rcos6 + r')'"''=sec'"<^; and we find
cosm<^ = cos"'<^(l-„Cstan''^ + „,C4tan^<^-. . .) (3),
sin 7n<l> = cos'" <j>UCi tan <p-,nCs tali-' <i> + . . .) (4).
* Since the left-hand sides of (1') and (2') are periodic, it is easy to
Bee that, for 2fm - n > $ > '2i-Tr + ir, the right-hand aides will be •2'"cos"'A0
eosim(e-2t>Tr) and '-""cos'njSsinim (fl-2pj-) respectively, where 2'"cos'"itf,
being the value of a muduluii, must he made real and positive.
32S SERIES FOR COam^ AND SIN m(^ en. XXIX
WLence
,C,taii«^-„C,tan'0-t-. . . , .
These foniiuliB are the generalisations of forimilaB (4), (5), (6)
of § 12. They will hold even when <^ lias either of the limiting
values + Jn-, j)rovided m>- 1 ; so that we have
2""'cosim7r=l-„C, + „C4-. . .;
2"^8inJw7r = „C, -„(7, + . . . .
Since
cos"-*-./. = (1 - sin'.A)'"-*'^'' = 1 + 2 ( - )\„.^nC. sin*-.^,
and the t«rnis of this series are ultiniatoly all positive, it follows
that the double scries deducililc from (3), that is to say, from
2(-)''m^jrC0s'""''<^sin*<^ by substituting cxpan-sions for the
cosines, satisfies Cauchy's conditions (chap, xxvi., § 34), for
there is obviously absolute convergency everywhere under our
present restriction that — \-ir^<l>^ +\-n:
Hence we may arrange this double series according to powers
of sin <f>.
The coefficient of ( - )"■ sin*"/!) is
»— r
r-o
OT(m-2) . . ■ (m-2r + 2) x r r
= 1.3 .. . Cir^ 5(«-.)«t'.(*-il,C,-,.
Now, by chap, xxiii., § 8, Cor. 5,
Hence the coefficient of ( - )' sin*".^ is
wi(m-2) . ■ . (m-2r + 2)(m + 2r-2) . . . {m + 2)m
1.3... (Jr- 1) 2 . : . (2r- 2)2r
^ fH'(w'-2') • • j^ (m' - 2r - 2*)
(2r)!
Hence
cos m<l> -1--^ 8in'<^ + — ^-, siu*<> - . . . (C).
2! 4:
§ 38 (/) IN POWERS OF SIN (}> 329
In like manner, we can sliow that
m . m{m/-\-) . .
sin »;</> = Y", sin 0 —— sin <^
+ — ^ Tp 'sin'<^-. . . (7).
Also
cos m<p = cos ^ j 1 — j— sin-0
^<"^'-y-^'>sinV-...} (8);
sin m<p = cos <^ -^ —, sin <p ^—-^ s\v?<j>
5!
sin^t^-
.} (9)-
Tlie demonstration above given establislies these formulae
under the restriction - ^7r:}>(/):j>^jr. It can, however, be shown
that they hold so long as -^7r:}>^:}>i7r ; 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
, . ^ Isin'.^ 1.3sin=<^ 1 . 3 . 5 sin'^
p = sin <i H + + + . . . .
^ ^2 3 2.4 5 2.4. G 7
If we put a; = sin 4', this gives
. , 1^ 1.3 a.-' 1.3.5.r' .,„,
sm-^ = ^ + -3+--+2-^^^^... (10).
In particular, if uo put a; = \, we obtain
'^=^{^2i:2-»-^2-:ji:¥^---} ^''^'
from whicli the value of t might be calculated with tolerable
rapidity to a moderate number of places. The result to 10
places is 7r = 3-U1.5926r)3G ....
330 EXAMPLKS Cri. XXIX
The important eerics (10) for expanding bid'' x ia here demoogtratod for
values of I lying between - 1/^2 and +1/^/2. It can be shown that it ia
valid between the limits i = - 1 and x= + 1.
The series was discovered by Newton, who gives it along with the series
for sinz and cosz in powers of t in a small tract entitled AnaUjiii per
yEquationet Numero Tfrminonim Iiifinitait. Since this trnct wafl shown by
Newton to Barrow in iri'iD, the series (10) is one of the oldest oxaD)|>lea of an
infinite series applicable to the quadrature of the circle.
Example 1, If m>0, and
C = 2-" S „C,cos(m-2;i)x,
S= 2-" 2 „C, Bin (m - 2n) x,
C=2-«'2 (-)->„C.C08(Fn-2;i)x,
n-O
.9'= 2-" 2 (-)-'„C.8in(m-2n)x.
then, p being any integer,
1°. C=(co3i)'"co8 2nipx, S=(coBx)"8in2mpT,
from x = (2p-4) r to x = (2/>-i-i) r.
V. C = (-cosx)'»cosm(2p+l)ir, .S = ( -cos x)'"8in m(2p + l) «•,
from x=('_'p + i) T to x = (2p + ?)T.
8°. C'^(8ini)'»oo8m(2p+4)T, S' = (sin xj'-sin m(2p + J) t,
from x = 2pir to x = (2p + l) r.
4°. C' = (-8inx)'»co8m(2p + J)ir, S' = (- sin i)" Bin m (2p +$)»•,
from x = (2p + l)r tox = (2p + 2)»-.
These formula] will also hold when m lies between -1 and 0, only that
the extreme values of x in the varioos stretches most be excluded. (Abel,
(Eiivrfi, t. I., p. 249.)
If we multiply {!') and (2') above by cos a and sin a respectively, and add,
we obtain the formula]
COS a + 2„C, cos (o - nS) = 2" cos"* JS cob {a-im8 + mpr),
wherein it mast bo observed that cos^JO is the modulus of (1 +2reo8ff + r')'^
when r = 1, and must thoroforo bo always ko adjusted as to have a real positive
value.
From the equation jof t written, Abel's formulas can at once be deduced
by a series of substitntions.
Example 2. Show, by taking the limit when ni = 0 on both sidea of
(1) and (2) above, that the series (1) and (2) of § 40 can be deduced from tlie
generalised form ol the binomial theorem.
Rxaiuple 3. Sum to infinity the aertes 2n'„t', sin")? cos n9. This seriea
is the real part of 2n'„C,8in*0(co8fl0'f '«inn9). Uenco
S = «(2ii'„C.Bin»e(o08* + »Wn»)»),
= /il{m'Bin'e(oostf+i sinP)»+m {Sm - 1) sin'O (cosfl + i sin 0)'
+ m din e (cos 0 + it)iud)\{lr »in (((cos e + i sin C}J""'J,
J
FORMULA FROM EXPONENTIAL & LQGARITHMIC SERIES 331
by Example 5 of chap, xxvii., § 5,
= [m»sin»ffoos{39 + (m-3)^} + m(:ini-l)sm«9cos{29 + {m-3)^}
+ m sin e cos {9 + (m - 3) 0}] (1 + 2 sin e cos e + siu= e)(^-^ift,
wliere 0 = tan-' {sin-S/{l + sin 6 cos 6)}.
^ 39.] Formulw deduced from the Exponential Series.
From the equation
^ (cos y + 1 sin 3^) = 1 + S (a; + yiTln\ ,
putting x = rco&6,y = r sin 6, we deduce
grcosfl jgQs^^gjn g) +jsin(rsiuO)}= 1 + 2r"(cos?i0 + «sin«O)/)i!.
Hence
grcos»cos(r sin (') = ! + 2r" cos wS/«! (1) ;
gr cos 8 gin (,. gin e) = -S, r'^sm n6/n ! (2) ;
wliich liold for all values of r and 0.
In like manner, many summations of series involving cosines
and sines of multiples of 6 may bo deduced from series related
to the exponential series in the way explained in chap, xxviii.,
§8.
Thna, for instance, from the result of Example 3, in the paragraph jnst
qaoted, we dedace
2(1»+2H. . .+n')x"/n! = c'"<*^*{rco3(« + rsinfl) + tr=cos(2S + rsine)
' +2r'cos{3e+rsine) + Jcos(49 + rsine)}.
§ 40.] FormulcB deuced from the Logarithmic Series. Since
the principal value of Log(l + 2:) is given by Log (!+«) = log
1 1 + s| + 2amp(l +s), and since the series s - s^/2 + 2^/3 — . . .
represents the principal value of Log(l +z), if we put « = r(cos6
+ 1 sin 6), we have
log (1 + 2r cos e + r')" + / tun"' {r sin 6/(1 + r cos 6)]
= 2 ( - )"-' r^ (cos n6 + / sin n6)lv,
where -|7r:}>tan-' {r sin 6/(1 +r cos 6)}:t>|:r, that is, the prin-
cipal value of tlie function tan"' is to be taken.
Hence we have the following : —
i log ( 1 + 2»- cos 61 + r*) = 2 ( - )" -• r" cos nOln ( 1 ) ;
tan-' [r sin (/y\l + r cos e)} = 2 ( - )"-' r" sin nejn (2).
332 SIN^-isIN20 + iSlN3<?-. . .=J0 CH. XXIX
Altliougli, strictly speaking, wc Lave cstablislied tlicso results
for values of 6 betwceu -ir and +ir both inclusive, yet, since
both siiies are periodic functions of 6, they will obviously hold
for all values of 6, provided r<\.
If r=l, (1) and (2) will still hold, provided ff + ±ir; for the
serie.a in (1) and (2) are l>oth convergent, and wc have, by
Abel's Thcoroiu,
cos^-Jcos2fl + ^cos35-. . .= Z |log(l + 2rcostf + r*),
r-l
= log(2co8je) (3);
s\n0-l sin 25 + J sin 3^ -. . . = tan"' {sin 0/{l + cos 6)),
= tan-'{tan|(fl + 2/l-7r)},
= ie + kn (4),
wlicre k must be so chosen that ^6 + kir lies between - J*
and + Jtt. Thus, if 6 lie between -n- and +jr, k = 0, and we
have simply
sin 5 - § sin 26 + lsm30- . . . = JO (1).
In particular, if we put 0 = ^ir, we get
which is Gregory's quadrature ; see § 41.
When 0= ±(2p + l)T, the scries id (3) diverges to - to , and tlio right-
band side becomes log 0, that in - x , so that (3) still holds in a oertain
Bense.
The behavioar of the scries in (4) vhen 9= ±(2^ + 1) r is very onrioiu.
Let us take, for eimplicitr, Uic caKe 0= i^r. With this value of 0 we bav*
for values of r as near unity as wc please tan~> (rain 0/(1 + r cos 0)} = O.
Tlence, by Abel's Theorem, when 0=^w, sin 0-}sin2tf -f . . .=0, as il
otherwise suflicicntly obvious.
On the other hand, for any value of 0 didering from ±t by however little,
we have X. tan"' {rsin 0/(l + rco»(?)} = 49. IIcDCa, again, by Abel's Theorem,
r— 1
toT $= ^T'rp, where 0 is infinitely small, wc have
sin0-^sin '20+. . . = ±JttJ*.
The porics y = sin 0 — J sin 2tf + . . .is therefore discontinnons in the neigh-
bonrhood of (>=*»; for, when 0= ±t, j/ = 0, and when 0 dilTcrs infinitely
little from <tr, y dilTcni infinitely little from *-wji. This discontinuity ii
accompanied by the phcnomeuon of infinitely slow ooDTergencc in the
neighlxinrhood of r = l, 9= *»; and the sudden alteration of the value of
the sum ia ai>.->ociatud with the fact that the values of the double Uniita
i-;^ iO, 41 Gregory's series 383
L L tan-i {rsinfl/(l+rcose)} and L L tan-' {r sin (?/(! +rcos 0)}
r = l fl = ±ir e=±ir r=l
are not uliUe.
When 0 lies between tt and 3ir, we may put 9 = 2jr + 6l', where 9' lies
between -ir and +7r, then, for such vaUics of 9, we have
j/ = sin^-j8in20'+ . . .,
= W, as we have already shown.
Hence, however small <j> may be, we have, for 9 = n + (l), ?/ = i0- Jr. But,
as we have just seen, for 9 = 7r- 0 we have )/= - J0 + Jir. Hence, as 5 varies
from TT-.^ to 7r + 0, y varies abruptly from -^tp + ^ir to ^^-^ir. In other
words, as 9 passes through the value jr, 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 paragrapli
we put 6= lir, we deduce the espausion
tau-V = r-l^-^+i/-^-. . . (6),
where tau-'r represents, as usual, the principal value of the
iuver.se function, and — i:j>r:)>l.
In particular, if r = 1, we have
x = 4(l-i + i-. . .)•
The series (G), 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 Shai-p and John Machin (Professor of Astronomy
at Gresham CoUege), used the series to calculate t to a large number of places.
Sharp, using the formula; iir = tan-'l/,y3 = (l/^3){l- 1/3.3 + 1/5.33-. . .},
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, Jr = 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-' J + tan-' J. Gauss {Werke, Bd. ii., p. 501) found, by means
of the theory of numbers, two remarkable formulte of this kind, namely : —
Jr = 12 tan-' 1/18 + 8 tan-' 1/57-5 tan"' 1/239,
= 12 tan-' 1/38 + 20 tan-i 1/57 + 7 tan-' 1/239 + 24 tan"' 1/268,
* The reader should now draw the graph of the function y, for all real
values of 9.
3:54 EXERCISES XIX CH. XXIX
by means of which w could be calculated with great rapidity aboold its value
over be required beyond the 707th place, which was reached b; Mr Shanks
in 18731*
Exercises XIX.
Sum the following series to inljuity, pointing out io each caae the limits
within which the summation is valid : —
« X , 1 - 1-8 o^ 1-8.5
(1.) l-^cosg+^^co82a-^ ^ gC0B3g+. . . .
cos tf 1 , COS 3* 1 . 3 , cos 68
(2.) . -j- + 2.«'-3-+27^x«-^+. .. .
,_ , cos tf 1 COS 39 1.3 cos 59
<3) — +2^-+2-4-5-+-- =
result }cos-> (1 - 'i sin 9).
(4.) 2(2n-l)(2ri-3)cosn9/iiI (5.) 2 sin ne/(n + 2) lU
(6.) «-'6in9- J<>"''Bin38 + i<-*'sin50-. . . .
(7.) sinff-^— -8in29 + -— jsinS*-. . . .
(8.) Bin*9-^8in>29 + isin'39-. . .;
result \ log SCO 0.
(9.) Sco8 2n0/n(n-l). (10.) Ssinn9/(ii'- 1).
(11.) |sin0sin0-^ sin29sin'6 + lsin38Bin'9- . . . .
(12.) co8(o + /9)-cos(o + 3/3)/31 + cos(o + 50)/5l-. . . .
(13.) cos « - 1 cos 29 + J cos 39 - . . . ;
result \ log (2 i- 2 cos 9), except when 9 - (2p + 1) r.
(14.) oos9 + Jcos29 + }cos39+. . .;
result - i log (2 - 2 cos 9), except when 9 = 2j>r.
(15.) 8in94'i si°-^ + l ">°89-i- . . .;
result =0, if 9=0; =\(t -a),UO<.e>ii , Ac.
(16.) 6in9-}Ein 39-t-t BinS9- . ■ • •
(17.) j:co8 9-}i'co8 39+lr'co<i59- . . .;
result J tan-' {2xco8 9/(1 - jr)).
(18.) oos9co8 0- i cos29cus2^'f icoii39cos3^- . . .;
rcKult ^log {4 cos) (e + ip)iM»\ (9-^)).
(19.) xcos9cos0- Jx'oos39coBS^-f(x*oos59oos 5^- . . .;
iMult itan-'(4x(l -«»)co89cos0/{(l+x»)«-ix»(co»'»-ooe»0)}l.
(20.) Show that log (l+x + x») = 22; (-)•-' co« inri"/n, provided |x|<l.
and examine whether the result holds when |x| = l.
* For the liistory of this subject see Ency. BnU, art. "Squaxinj{ the
Circle," by Muir.
I
^ 41 EXERCISES XX oo5
(21.) Show that, under certain restriotiona upon e,
log (1 + 2 cos ff) = - 22; COS \mr cos nfl/ii ;
d= — S cos |H]r sin nS//i.
(22.) Show that
TT ,11111 1 1
2;7^=^"*"3"5"7 + y+n-i2~i3+-" •
(Newton, Second Letter to Oldenburg, 1676.)
Exercises XX.
(1.) Calculate ir to 10 places by means of Maohin's formula.
(2.) Show that, if j: < 1,
(tan-' 1)2
=x'-{l + ll3)x*l2 + . . .(-)"-'{! + 1/3 + . . . + l/(2n-l)}x-'"/«. . . .
Does the formula hold when x = 17
(3.) Expand tan"' {x + cot o) in powers of x.
(4.) Deduce the series for sin"' x from Gregory's series by means of the
addition theorem for the binomial coeflicients.
(5.) If X lie between 1/^2 and 1, show that
^(I-:r=)J, 11 -x\ l(I-x-)^ )
,,„ 'x=x-^^^ |l-3 ^5-+5 ^i . . j .
(6.) Show that § 38 (10) is merely a particular case of (7).
(7.) Show that
8 2 2 4 2 4 6
- = sin9 + -8inSe + 5-^sin5e + .5^„sm'fl + . . . .
cos 0 i 3.5 i.O. I
(Pfaff.)
1 „ sin^e 2 sin* 9 2.4 sin" 9 ,„, . .,, ,
{8.) 2»^=-^ + 3^- + 3-:5-6 +••• • (btamville.)
(9.) e3^sin30 + ^.|(l + pjsin5e + . . .
3.5...(2«-l)^_/ 1+ + J^ '\sin'^»Hff + . .. .
+ "~4.6...2n 2n + lV 3=^^ ^{2«-l)V
(10.) e<=sin»e + |.|^l + .i)siu«fl + . . .
4^(2„:2) 2 /I .^ 1 Vi„..,^. . . .
^5.7...(2ii-l)nV •^- («-l)V
(11.) Deduce from § 38 (6) and (7) an expression for e"'/sin"'9 in powers
of sin d.
(12.) If 6ine = i8in(9 + o), show that e + r7r = 2x"sin ;m/«.
(13.) If c^=a^- 2ab cos C + b"; then
log c = log a - (6/a) cos C - i {bja)^ cos 2C - ^ {bja)" cos 3C - . . . .
(14.) Show that
- n-3 (n-4)(n-5) _ («-5) (n- 6) (n-7) ^ 1 + (-)»-" 2 cos jht
2 "*" 2.3 2.3.4 +• • • „ ^
33G EXEUCISES XX CIL WIX
8I10W that
(15.)* e>=8in«e + 2'Bin«' + 2'8in<|; + 2«Bin«.'+. . . .
(IG.)* u'=8inb=u-2'Binh*"-2«Binh*^-2*8inh«^j-. . . .
(17.)* ?e = 6me + 3Bin»*+3'Bin>^ + . . . .
(18.)* ?BinO=^ 3^„..Bin3'"tf + 2 3^.sm»3-'(;.
Q • / _ 1 \ni— I
(19.)* 2coB»=2 5_^__eoB»3'»-'*.
* See Laisant, " Essai siir les Fuuclioua bjpeiboliques," Mim. de la Soc.
de Bordeaux, 1875.
I
CHAPTER XXX.
General Theorems regarding the Expansion of
Functions in Infinite Forms.
EXPANSION IN INFINITE SERIES.
§ 1.] Cauchys Tlieorem regarding the Expansion of a Function
of a Function.
If
y = a„ + 'Sanaf' (1),
the series being convergent so long as |a;|<^, atid if
z = h + lKy'' (2),
this series being convergent so long as\y\<S, t/ien from (1) and
(2) we can derive the expansion
s = C, + ^C„x\
provided x be such that \x\<R, and also
|a„| + 2|a„||a;i"<-Sf.
This theorem follows readily from cli.np. xxvi., §§ 14 and 34.
We have already used particular cases of it in previous chapters.
§ 2.] Expansion of cm Infinite Product in the form of an
Infinite Series.
If 'S.ii,, be an absolutely convergent series, and „2i<i, „2«, m^,
. . ., „2«, Uq. . . Ur, . . . denote the sums of the pi-oducts of its
first n terms taken one, two, . . . , r, . . ., at a time, then
L „tu,= T„ Ln~UlUi=Ti, ..., L n'S.lHUi. . .Ur=Tr, . . .
where Ti, T^, . . ., Tr, ■ ■ . are all finite.
Also the infinite series 1 +S7'„ is convergent ; and converges
to the same limit as the infinite product n (1 + «„).
c. II. 2:2
338 INFINITE PRODUCT REDUCED TO A SERIES CH. XXX
After wluit has been laid down in cimp. xxvi., it will
obviously be sufficient if we prove the above theorem on the
assumption that all the symbols «i, m, «,, . . . represent
positive quantities. In the more general case where these are
complex numbers the moduli alone would be involved in the
stiitements of inequality, and the statements of equality would
be true as under.
Since «i, «,,...,« are all positive, we see, by the
Multinomial Theorem (chap, xxiii., g 12), that
0<«2»/iU,. . .«r <(«, + «,+ . . . + «,)7rl
<(«, + w,+ . . . + «„+. . .adx)7rl
<^/H, (I),
where S is the finite limit of the convergent series Su, ; and the
inequality (1) obviously holds for all values of r up to r = n,
however great « may be.
Tlicrcforo „2«i «, . . . «r has always a finite limit, T, say,
such that
0>7;>.S7r! (2).
By (2), we have
0<l+7',+ Z', + . . . adoo<l+«/l! + /S72! + . . , adx,
that is,
0<l + 27',<«» (3).
Hence 1+27', is a convergent series, whose limit cannot
exceed e*.
Again, since /i,2i/, »/, . . . Mr = TV when » = oo , we may write
,1U,U,. . .Ur = (l+rAn)Tr (4),
where LrA, = 0 when n= «.
Hence, A, being a mean amiuig ,^4., ,A , ..4., and
therefore such that LAn = ^> "lien h = oo, we have
11(1 + II J _ 1 + ,2m, + ,2m, m, + ,
a + (l + -l.)2/'. (:,).
I
§ 2 INFINITE PRODUCT AND SERIES 339
If iu (5) we put H = 00 , we get
n(l + «„) = l + Z{(l + .l„)2 7;},
=i + 2r,. ' (6),
1
since LA„ = 0, and 27'„ is finite.
1
Tills completes the proof of our proposition.
Cor. 1. 1/ 2«„ be ahiJutcli/ convergent, then, 7",, having the
above meaning, \ + So:"?',, will be convergent for all finite values
of x; a/nd we shall have
\l{l + xu,) = l + ix"Tu (7).
This follows at once by the above, and by chap, xxvi., § 27.
Cor. 2. Let
nn = n% + n'Vl-'>: + nV-zX- + . . . (8),
where n'Wo, n^'i) <&c-, <*'■« independent of x, and the series on the
right of (8) may either terminate or not ; and let
Mn'=|nlV|+U'Wl|k|+|ni'2lk|"+. •• (9).
Then, if 2i(„' be convergent for all values of x such that
\x\<p, it follows that for all such values n (1 + n„) is convergent,
and can be expanded in a convergent series of ascending powers ofx.
For, if Tn have the meaning above assigned to it, then it will
obviously be possible to arrange 7',, as an ascending series of
powers of x. Moreover, if we consider the double series that
thus arises from 1 + STn, 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+27'„ as
a convergent series of ascending powers of x.
Example 1. Toexpand (l + x)(l + x-')(l + r')(l + x8) . . . in an ascending
series of powers of i. (Euler, Introd. in Anal. Inf., § 328.)
The series 2;|xp" is obviously convergent so long as |x|<l. Hence, so
long as |x|<;l, we may write
(l+i)(l + x=)(I + x«)(l + a«). . . = 1 + Cix + Cji2 + . . . + C„x"+. . . (10).
To determine the coefficients Cj, C„, C„, we observe that, if we multiply
both sides of (10) by 1-x, the left-hand side becomes L (1-x-"), that is,
1, since | x | < 1 . We must therefore have
l/(l-x)--.l + C,x + C„x-+. . . + C„.i''+. . .,
22—2
a4-0 I'UODUCTS OF EUI.Klt AM) CArCllY CIJ. XXX
llmt is,
therefore C, = C,= . . .=C,= . . .=1.
Another way is to put x^ fur x on both aides of (10), and then multiply by
(l+i). Wo thus got
l + 2C„i»=l + x + C,i'+. . .+C,T*' + C, !»"+'+. . .;
whence Cj„=Cj,+, = C,, C, = l,
from which it is cosy to prove that all the oooHJcicnts are unity.
Example 2. To show that
(l+j-.:)(l + x=r). ..(l + x-z)
-'\r, (i-xj(i-x^)...(i-x-) "
(Caucby, CompUt lieiuiut, 1810.)
Let
(l+xr)(l + x'r)...(l+x'"2)
= l + Ait + A„z'+. . .+A,z'+. . .+A„t'' (2),
where j4,, /I,, . . . are functions of x which have to be dotormincd.
Put Tz in place of z on both eidcs of (2), then multiply on both sidea by
(l + xz)/(l+x'"+'i), and we get
(H-xz)(l + xa;) ...(l + xmj)
= {l + (l + ^,)xr + (^, + ^5)x3r' + ... + (^„_, + ilJx":- + ... + .l„x'"+>i"+'},
x{l-x»+»2 + x=("'+')c3 + . . .(-)»x»<"^'lr"+. . .J (3).
Hence, arranging the right-hand side of (3) according to powers of »,
replacing the left-hand side by its equi\-alcnt according to (2), and then
c<jualiug the cucllieicnts of z" on the two sides, we get
^«={-',. + J,-i)a:"-'"*'(^.-i + ^.-J»-*
• • • ■ •
(_)»-llO>-I)(i»+l)(^,-hl)z
(-jl-X-^^ll);
whence
Putting n - 1 in place of n in (4), we have
,=4^-^,_,x"-l-^..,x*-- . . . (-).->x<"-»'- (S).
«"->(l-x'") "-•■
U we multiply (5) by x" and odd (4), we derive, after an obvioot
reduction,
(l-x-)^, = (x--x»+').<,., (6J.
In like manner,
(l-x-')^.., = (x-->-x-^')^,-, ((y,
(1 -x--') .<,.,= (x-'-x^')^^, (OJ.
(l-x).J, = (x -*«•+') (CJ.
§§ 2, 3 EXPANSION OF SeCH X AND SEG X 3-H
Multiplying (6,), {65), . . . , (G„) together, we derive
_ (j - x'»+') (j- - 1°'+') ■ ■ ■ (g;" - x'"+')
"" ("l-a:)(r-i=).'. .(1-j:") ^'''
(l-a;)(l-x-). . .(1-x") ^ '
which establishes our result.
If |a;|<l, the product (l + xz){l+x'z) . . . will be convergent when
continued to infinity, and will, by the theorem of the present paragraph, bo
expansible in a series of powers of z. The series in question will be obtained
by putting j«=oo in (1). We thus get
(l + x.)(l + x».)...ad«=l+^Sj^-^j^j-^^,y— ^5-^^ (1.).
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 a; + Exp - <r) = 1/(1 + 2.;-""/(2w)!) (1)-
Heuce, if y = 'S.x"'l{2n)\ (2),
2/(Exp X + Ex-p -x) = 1/(1 +y),
= l + S(-r2/" (3).
The expansion (3) will be valid provided ]2/]<l ; and the
series (2) is absolutely convergent for all finite values of x.
Hence, if i=|a-|, it follows from § 1 that the series (3) can
be converted into a series of ascending powers of x provided
i P/(2«)!<1 (4).
n=.l
This last condition involves that
that is, that ^<log (2 + 73).
This condition can obviously be satisfied ; and wo conclude
that 2/(Exp X + Exp - x) can be expanded in a scries 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 a; + Exp - a;) = 1 + 2 ( - TE„x^l{2n)\ (5).
To determine E^, E^, . . ., we multiply one side of (5) by
3+'2 EULEIl's NUMnEltS CH. XXX
i (Expx + Exp - x), and the otiicr by its cqiiivnlont 1 + 2«*/(2n)!;
wu thus have
1= {1+2 (-)"/;„ .r*/(2H)!)!l+2.J^/C.'H)!} (6).
El, Ei, ... must be so detennincd that (6) becomes au
identity. Wc must therefore liave
(2H):0!~(2«^)!2!'^(2n-4)!4!~* ' '^"^ oTcJ^"" ^'^ •
or,
a; = ^c./i",-, - ^c, /i-,-, +...(- r-'^c^.,E, + ( - 1 )-' (8).
The last eiiuation enables us to calculate E,, E~, Et, . . ,
successively. We have, in fact,
Ei = l; E.. = GE,-\; E, = \5E,-\ 5/i', + 1 ;
Ei = ^inE, - lOEj + 2,s£'. - 1 ; &c
whence
E,= 2702765,
E=- 199360981,
E,= 193915121-15,
£■, = 2404879075441,
JS,= 1,
E3= 5,
i> 61,
JE'«= 1385,
£•. = 50521,
Tiiese numbers were first introduced into analysis by Ruler* ,
and the above table contains their values so far as he calculated
them.
Since the constants Ei, E„ . . . are determined so as to make
(G) an identity, (6), and therefore also (5), will be valid for all
values of x, real or complex, which render all the series involved
convergent. Hence, since 1 + 2ij*'/(2H)! is convergent for all
values of or, (5) will be valid for all values of x which render the
series 1 + 2(-)''£',a*'/(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 X = 1 + 5 ( - )" A', j*/(2«)l (9) ;
and, if we put ix in place of x, it gives
Sec X = 1 + 2A\a*'/(2n)! (10).
• Sco Intt. Calc. Diff., $ 2'.' I : the last five diffits of K, are JncorrocUy
given by Kiiler an fil671.
Fnr aiiumliiT of curioas pro|Mrticaof tlio Kulcrinn numl>«r*i)oe Sjrlvoalei,
Comptei Rtndiu, t. 52 ; anil Stcru, CrtlU'i Jour., hi. Lxxix.
§§ 3, 4 EXPANSION OF Tanh X, &c. 343
Cor. Sech"x and Sec"x can each be 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 + 2^„.r-7(2?))!)
cos"ar; expand cos" a; first as a series of cosines of multiples of x\
finally in powers of x ; and thus obtain a recurrence formula for
calculating A^, ^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,
xGoix, Coseca;*.
We have already shown, in chap, xxviii., § 6, for real values
of X, that
xl{\ - e-') = 1 + ^a: + 2 ( - )"-' i?,..r="/(2«)!,
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
a;/(l -Exp-a-) = 1 + ^^r + S( -)"-' Z?„.r™/(2«)! (I),
where Exp — a; is defined as in chap, xxix., and x is such that
|ir| is less than the radius of convergency of the series in (1).
From (1) we derive the following, ail of which will be valid so
long as the series involved are convergent :
X (Exp X - Exp - ;r)/(Exp x + Exp - x)
= 4.r/(l - Exp - 4.^:) - 2.r/(l - Exp - 2.r) - x,
= 2 ( - )"-' 2-" (2=" - 1) B^3?^l{2n)\ (2) ;
X (Exp X + Exp - ir)/(Exp x - Exp - x)
= a-/(l - Exp - 2x) - xl(l - Exp 2x),
= 1 + 2 ( - )"-' 2^" B„a?"j{2n)\ (3) ;
2.r/(Exp X - Exp -x) = 2.r/(l - Exp -x)- 2x1 {I - Exp - 2x),
= 1 + 22 ( - )" (2-"-'- 1) /;„ar"/(2«)! (4).
From these equations, we have at once
Tanh a: = 2 ( - )"-> 2-" (2"' - 1) BnX-"''/{2ny. (5) ;
X Coth a: = 1 + 2 ( - )"-' 2=" Bnx"'/(2n)\ (G) ;
X Cosech a: = 1 + 22 ( - )" (2*-' - 1) B^ar"/{2n)\ (7).
• Euler, l.c.
344' EXERCISES XXI CM. XXX
If in (2), (3), and (1), we replivce x l)y tjr, we deduce
Tun X = Sa*- (2" - 1) BnX^-'l(2n)\ (S) ;
;rCotx = l-22»iB,ar"/(2n)! (9);
arCosecar=l + 22(2»-'-l)Zf.a*'/(2n)! (10).
Cor. Eiirh of the functions ( Tank x)', (x Coth x)', (x Cosech x)',
(Tan x)', {x Cot x)", {x Cvsic x)" can be expanded in an ascendimj
series of powers of x.
EXERCISKS XXI.
(1.) If 0=gdu (sec chap, xiix., § 31), show that
0 = a,ii -<;,»' + <;,«'- . . .,
u = a,tf+a,^ + a,0» + . . ..
where a^+, = ^J(2fi + 1)!.
(2.) Find expressions for the coeflicicnts in the expansions of Sin'z and
COB»X.
(3.) Find recarrcncc-fonnolie for calcalating the coefficients in the
expansions of (xcosccj')* and (seci)*.
In particular, show that
Sec'i*+'x= " •^r^»+^p-i^iH->-»-- • • + ^i^>n>-i + ^»»p ^_
^ (2p)I •(2n)l'
where 5, denotes the sum of the products r at a time of 1*, 8*, 5', . . . , (2p - 1)*.
(Ely, American Jour. Math., 1882.)
(4.) ir|z|<l, showthat
{l+x'){i. + x*)(l+z*) . . .ad cr =l + ±r"**'/(l-x')(l-x«) .. .(l-x'-).
(5.) If {x|>l, and p be a positive integer, show that
*,r,(x-l)(x'-l)...(x«-l)
(G.) Show that the Binomial Theorem for positive integral exponents U
a particular case of § 2, Example 3.
(7.) Show that
(l + iz)(l+x>r) . . . (l + i»»-»*)
_ m (l-x*")(l-r'--')...(l-x^-»^») ,
~ «::i (i-x»)(i-x')...(i-x*') '^'^•
(Cauchy, CompUt lUndiu, 1840.)
(8.) Show that
(l-x»)(l-x'*)...(l-*"x)~ (l-x)(l-x»)...(l-«")
also that, if |x|<l, |ix|<l,
l/(l-xz)(l-x«j). ..ad 00 =l + Zx*««/(l-i){l--r*). . .(I-x*).
(Euler, Inl. in Anal. Inf., § 813.)
§ -1 EXERCISES XXI 345
(9.) If m be a positive integer (1 - a"*) (1 - x'"-') ... (1 - x"*-"*') is exactly
divisible by (1 - 1) (1 - x=) . . . (I - x").
{Gauss, Summatio quarumdam sericrum singulariuvi,
Werke, Bd. ii., p. 16.)
(10.) If/(..,,0 = l + ^(-)"'^-(f.'g-;:;;;V.';;_y"'.where |x|
>1, show that
/(x, m)=/(x, m-2X)(l-x'"-i)(l-x"'-S) . . . (1 - x"'--''+>)
1 - x'"-! 1 - x">-- 1 - x">-5
1-X-l ■ 1-1-3 ■ 1-X-S
Hence show that, if |x|<:l, then
1-x- 1-r* 1-3-5
l + 2x»(»+')/'=i-^.'-^.i— ^ . . . ad «>.
1-x 1-x' 1-x*
ad 00 .
(Gauss, lb.)
(11.) Show that, if in be a positive integer,
(l + x)(l + x2) . . . (1 + a"') = 1 + 2a" ^^ " "''"^ ^\~,'!°""°1 " ' ' ^}, ~ "'
(12.) Show that
1
(1 - l2j (1 - X*) ... (1 - X=")
(Gauss, lb.)
(1 - X2) (1 - i^j) . . . (l-x=^-iz)
„ Jl-a'^Xl-a""-") ■ ■ ■ (l-x=>"+^-»)
"^ ^ (l-x-)(l-x^) . . . (l-a»»)
Also that, if |a|<l, and |2x|<l,
l/(l-x^)(l-x32) ... ad » =l + 2x"2»/(l-x=){l-x<} . . . (l-x=").
(13.) Show that, if |x|<l,
l/(l-i)(l-a3)(l-x5) ... ad QO =(l + x)(l+x=)(l+x3) ... ad x.
(Euler, I.e., § 325.)
(14.) If lx|<l,
+«
(1 - x) (1 - x=) (1 - x=) . . . ad 00 = 2 ( - )nxi'n'+'')l^.
(Euler, Nov. Comm. Pet., 1760.)
(15.) If |x|<l,
log)(l-x)(l-x=)(l-x') . . . nd oo}=-2j(H).c"/n,
where ^{n) denotes the sum of all the divisors of the positive integer n ; for
example, J(4) = l + 2 + 4.
Hence show that
(Enler, lb.)
(16.) If d(n) denote the number of the different divisors of the positive
integer »(, and |xj<:l, show that
2d(n)x"=S= -.
1 1 l-a»
(Lambert, Essai d'Architectonique, p. 507.)
346 EXPANSION IN INFINITE PRODUCT CH. XXX
Also that
m m /I + r<»\
(Clausen, Crelle'g Jour., 1827.)
(17.) I(|x|<l, show that
I i> _^ _jc_ «» J*
1-x l-x'"^l-x»" ■ ■ ■~l + x»'^l+x*'^l + *«'^" ■ • •
(18.) Sj«"+'/(l-**'*')'=2nx"/(l-x*').
2(-)»-'nx»/(l + x") = 2(-)"-'x»/(l+i")'.
(19.) The sum of the products r at a time of x, x', . . . , x" U
x'tr*l««(xr+l_l)(jrM_x) . . . {x»-l)/(z-l)(x«-l) . . . (x"-'-l).
(20.) If Sf be the sum of the products r at a time of 1, *, . . ., x^', then
Sr='S,_,X-<»-'l(»-*T/».
(21.) Show that, if x lie between certain limits, andtherootsof ox' + fcx + c
be real, then {px + q)l{ax' + bx + c) can be expanded in the form u,+
S (i/,x* + t',z~*) ; and that, if the roots be imaijinary, no expansion of this
kind is possible for an; value of x.
ON THE EXl'KESSION OF CERTAIN FUNCTIONS IN THE FORM
OF FINITE AND INFINITE PRODUCTS.
g 5.] The following (Jeneral Theorem covers a variety of
cases in which it is pos.*ible to express a given function in the
form of an intinite 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 misunderstomi.
Let /(n, p) be a function {with real or inuujinary coefficifnti)
of the integrcU variables n and p, such that L f(n, p) is finite for
all finite values of n, suiy L f{n, p) =f{n); and let us suppose
that for all vnln<'s of n and p {n<p), hmn'rer great, tfhich exceed
a certain finite value, [f(n, p) \/\f(n) \ is not infinite.
Then L n {1 +/(",;')l = n {1 +/(»)} (1).
provided 2|/(n)| be convergent {that is, providid IT {1 +f(»)\ he
absolutely convergent).
Let us denote fi {1 +/(«, p)\ by P,; L n {1 +/(n, p)\ by
P ; l/(n, P) i by y. («. p) ; and \f{n) \ hyf^n).
§ ■") GENERAL THEOREM 347
We may write
P,= n{l+/{n,p)\ 11 {!+/(",/')},
= i^m§m, say, (2).
Just as ill chap, xxvi., § 26, we have
|ft„-l|>> n {l+A(n,2y)}-l.
n=ni+l
Now, by one of our conditions, if m, and therefore p, exceed
a certain finite vahie, we may put /i (n, p)/J\ (n) = A„, where An
is not infinite. If, therefore, A be an upper limit to An, and
therefore finite and positive, we have/, (», /))4>vl/, (?j). Hence
|ft.-l|> n {l+AMn)]~l.
> n {l + AA{n)}-l, (3).
m+l
Let US now put p= x> in (2). Since m is finite, and
^ /(«. i') =/{>')' ^TO tave
p=» 1
m
Therefore P= n {1 +/(;,)} Q,,. (4),
where Qm is subject to the restriction (3).
Let us, finally, consider the effect of increasing ot.
Since n (1 +/i (n)} is absolutely convergent, 11 {1 + A/j {n)\ is
absolutely convergent. It therefore follows that, by sufficiently
increasing m, we can make II {I + A/i(n)} -1, and, a fm-timi,
m+l
|Q„-1| as small as we plea.se. Hence, by taking m sufficiently
great, we can cau.se Qm to approach 1 as nearly as we please.
lu other words, it follows from (4) that
P = n{i +/(»)} (5).
In applying this theorem it is necessary to be very careful to see that both
the conditions in the fir.st part of the enunciation rej^ardiug the Talne of
f(n,p) are satisfied. Thus, for example, it is not sufficient that L f{n, p)
p=a5
have a finite definite value f(n) for all finite values of n, and that -/,(«) be
348 INFINITE PKODUCTS FOR SINU pu, SINH U CH. XXX
absolutely convergent. This seems to be taken (or granted by many mathe-
matical writers ; bnt, as will be scon from a striking example given below,
sach an ossauiption may easily lead to fallacioas results.
§ 6.] Factorisation of sink pu, sink u, sinpd, and sin 6*.
From the result of chap, xu., § 20, we have, p beiug any
positive integer,
x^-l = {x'- 1) ll' (x' - 2areos - + l) (1).
From this we have
—3 — - = n ( j:" - 2.r COS — + 1 ;
a^-1 -A P y
whence, putting « = !, and remembering that Z<(.r*-l)/(j:»-l)=;>,
we have
p = 2'-' n (1 - cos . htt/p) (2) ;
= 4'-' n sin'.«>r/2/> (3);
1
anil, since sin . ir/2/?, siu.^2v/2p, . . ., sin . (/>- 1) jr/2/> are
obviously all positive,
v/p = 2"-' n sin . nir/ip (4).
If wc divide both sides of (1) by a*, we deduce
3f-x-'' = {x- X-') n (x + x-' - 2 cos . nv/p) (5),
where for brevity we omit the limits for the product, which are
as before.
If in (5) we put ar = «*, we get at once
siuh pu = 2''"' sinh u U (cosh u - cos . nwfp) (6),
= 4'-' sinh u n {sin*.fi-/2p + sinh'. m/2) (7).
Using (3), we can throw (7) into the following form : —
8inh/>M =p sinh m IT {1 + sinh'. H/2/8in'.tnr/2/>} (8).
Finally, since (8) holds for all values of u, wo may replace u
by u/p, and thus derive
• The resnlts in §,5 f>-9 were nil (rivpn in one form or another by Enler in
bis IntToductio in Analyhn Infinilonim. His demonstrntionH of the funda-
mental theorems were not satinfactory, altlioui;h they are still to b« (uuod
unaltered in many of oar elementary text-books.
§§ 5, G INFINITE PRODUCTS FOR SINH pu, SINH U 349
smh «« = « smli - 11 -^ 1 + - .-- '.-f-^ (0).
i' a=i l snr. mr/'2p) ^ '
Wo pliall next apply to (9) the general theorem of § 5.
pK'fiirc lUiiiig so, we must, however, satisfy ourselves that the
rc(iuisite conditions are fulillled.
lu the first place, so long as n is a finite integer, we have
J sinh" ■ w/2j.) ti^
p=ocSm". M3r/2^J n'tr' ^
This can be deduced at once, for complex values of ?<, from
the series for sinh . ul'lp and sin . mrj^p. When u is real it
follows readily from chap, xxv., § 22.
The product n {l + u-jn-Tr) is obviously absolutely convergent.
We have, therefore, merely to show that, for all values of n and 2)
exceeding a certain finite limit,
siuh° . m/2/? / ii
sin'' . tnr/2p,
7—1
<A
(11),
where ^ is a finite positive constant. That is to say, wo have
to show that
remains finite.
Now
u/2p
A sin . 7i-n-j2p\
nnl2p ~)
sinh . uj^p //sin . n-n-j'2p\
= 1 + .
sinh . nj^p
uj2p
1 (:iL\
>\ +
3\\2p
2J"
(12).
Since the series within the bracket is absolutely convergent,
its modulus can be made as small as we please by taking p
sufticiently great.
Again we know, from chap, xxix., § 14, that, if 6:}>^(G x 7)
:^6'4S, and, a fortiori, if e:t>2jr, then
that is, if 6 be positive,
sin e/(9<t;l -!(/-.
860 INFINITE I'RODUCTS FOR tilV pd, Hlti 0 CU. XXX
Now, since ti^p - 1, mTJ'iplf \-r. Tliereforo
sin. nn/2p ^, i /'"^V
*!-*£
fMr/2/> "^ *V2;i/
<tl-g<-53 (13).
From (12) and (13) it is abundantly evident that the con-
dition (11) will be satisfied if only p 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 ajjplica-
bility of the General Limit Theorem being fulfilled, we may make
p infinite in (9). Remembering that Lp siuh . u/p = «, we thus get
sinh « = u n (1 + u'/u-'ir') (14).
To get the corresponding fonnuhc for RinpO and sintf, we
have simply to put in (5) x = exp iO. The steps of the reasoning
are, with a few trilling modifications, the same as before. It will
therefore be sullieient to write down the main results with a
corresponding numbering for the equations.
p-i
8in;;6i = 2'-' sin 6 Xl (cos 0 - cos . nx/p) (C') ;
ii-i
= 4'-' sin eil (8in'.Mir/2;> - sin'. 6/2) (7').
sin|>e=j»sin 6a (1 -siu'.e/2/sin'.«jr/2;>) (8').
Bintf=»sin- II \l- . ,--- t (9)-
ji> „.i I sin'.HJr/2y>J
s\ne=e ini -eyn^T'i (u').
H-I
It should be noticed that, inasmuch as (f>), (7), (8), (9), and
(14) were proved for all values of «, re.-U and complex, we might
have derived (6'), (7'), (8'), (9), and (14') at once, by putting
u = i6.
Cor. 1. T/if foUowiuij Jiiiite products for siiijtO and sinJipu
should bo noticed : —
J
§§ 6, 7 WALLIS'S THEOREM 351
smpd = 2''-^sm6sin{9 + Tr/p)sm{d + 2Tr/p) . . ,
sm{0 + p-l-n-/p) (15);
5mhpu = (-2i)''~^sm'hus\Tih(u + iir/p)smh.(u + '2iTr/p) . . .
sinh (u+p- Utt/p) (16).
The first of tliese may be deduced from (6'), as follows : —
smp$ = 2^"' sin ^11 (cos 6 — cos. nv/p),
= 2P-' sin en {2 sin {mr/2p + 6/2) sin {mr/2p - 6/2)},
= 2"-' sin 6n {2 sin {mr/2p + 6/2) cos {p-n-rr/2p + 61/2)}.
Hence, rearranging the factors, we get
smp6 = 2^-' sin ^n (2 sin {n7r/2p + 6/2) cos {mT/2p + 6/2)],
= 2"-^ sin 6 n sin (5 + nir/p).
We may deduce (16) from (15) by putting 6 = -iu.
Cor. 2. Wallis's Theorem.
If in (14') we put 6 = \t, we deduce
1 = irfl (1 - 1/2V) (17);
, TT 2^^ 4= (2?«)'
Wlience - = ,— - . r— ^ . . . r: h-j- r-, . . . ad 00 ,
2 1.3 3.5 (2m-1)(2«+1) '
2 2 4 4 2n 2» , ,,„,
= r3-3-5---2;^^-2^rri-- •''^=° ^^^^-
This formula was given by "Wallis in his Aritlimetica 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, coshpu, cosku. Following
the method of chap, xu., § 20, and using the roots of -1, we
can readily establish the following identity : —
arP+lH n(ar'-2^cos^^""^^'^+l) (1).
Putting herein «= 1, we get
2 = 2''n(l-cos.(2n-l)ff/2ij) (2);
= 4''Usin^(2H-l)7r/4^ (3).
352 INFINITE I'UODUCTS FOR COS p0, COH 0 CU. XXX
Ueace, since all the sine:) are positive,
^2 = a' n sin . (2n - 1 ) njip (4).
From (1),
jJ + a;-" = n (a: + a;-' - 2 cos . (2n - 1 ) jr/2;;) (5) ;
whence, putting a: = Expt5, we deduce
cospe = i . 2''n (cos e - cos . (2» - l) 7r/2^>) (C) ;
= i . .fn (siu\(2H - 1) 7r/4;, - siii'.e/2) (7).
From (7), by means of (3), we derive
Ltispe=U. (1 - sin'.6'/2/8in'.(2M - 1) ir/ip)
From (8), putting 6/p in place of 6, we get
sin'.tf/2p
cosfl= n n- -. -j-T^ — -,\-^7-\
,-1 I sill'. (2« - 1) njAp]
(8).
(9).
For any finite value of n we have
. sin'.g/2;>
40*
p.. sin'. (2h - 1) 7r/4;; (2» - Xf-n'
Also the product U (1 + 46''Y(2h - l)'7r') is absolutely con-
vergent.
Moreover,
I sm.Ojip
ej2p
(12);
80 that I sin . 6/2pl0l2p \ can be brought as near to 1 as we plea.se
by sulliciently iiicreiising/>.
Also, since (2h - l)5r/ly):^j7r, we have, exactly as in !a'<t
paragraph,
(13).
6m^(2n-l)j/4p .
(2n-l)^/4p^^^^
We mayi therefore, put p=co in (9) ; and we thus get
coaO= I'l {l-4</V(2«-l)V'} (14).
§§7,8 INFINITE PRODUCTS FOR COHU pu, COSU a :353
In like mauuer, putting ar = g" in (5), we get
cosh^M = i . 2'' n (cosh « - cos . (2« - 1) Tr/2p) (6') ;
n=I
= i . 4" II (sin' . (2« - 1) n/ij) + sinh' . «/2) (7').
coshpu = n (1 + sinh' . M/2/sin . (2» - 1) 7r/4/i) (8').
, p f, sinh\w/2/> ) ,„,,
coshM= n -^1 + . , — frVr r (9)-
cosh » = n {1 + 4«7(2h - 1)= ir=} (14').
We might, of course, derive tlie hyperbolic from the circuhir
formulse by putting 6 = 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
^'^'=^"{rT/(2«)=}'
TT t(2»-l)7r'(2«+l)ffj ■
Hence, putting lir-6'm place of 0, we deduce
cosf^-- — U|~-^2„_i)^ • (2« + 1)^ J'
= (1 - 2e/7r) n {(1 + 2e/(2» - 1) tt) (1 - 26i/(2?« + 1) tt)},
= (1 - 26/77) (1 + 2^/7r) (1 - 2^/377) (1 + 26/377) ....
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 clianging its value ; we may, liowever, associate
them as they stand in groups of any finite number. Taking
them in pairs, we have
cos (9 = (1- 40^/77=) (1- 46^/3 V=) . . .,
= n{l-46-/(2»-l)V}.
§ 8.] From the above results we can deduce several others
which wiU be useful presently.
c. II. 23
354 VAIUOUS INKINITE PRODUCTS CII. XXX
We Imve, since all the iirodiu-ts involved arc absolutely
couvergeut,
sin (0 + «^) ^ e + ^ U{l-(g-t- <i>fl>'''f^
sinfl ~ e UII-^/m-V} •
provided fl + 7i7r.
Hence, provided O^tfr,
cos <^ + sin <^ cot e = (l + ^) II {l - ^^43 ^' )•
In like manner, starting with cos (0 + <>)/cos tf , we deduce
cos * - sin <^ tan <> = n {1 - 4 ^.~^tff_^^) (-').
provided tf + H2» - 1) T.
Also, from the identity
sin «^ + sin 6 _ sin |(</» + 0) cos A («^- 6)
sin 6 ~ sin JO cos i^ *
we derive
1 + cosec 6 sin <^
-(^^±\ n r<l - ('/' ->■ g)V4>»^^l {1 -(</.- g)V(2» - D^ir'n
=(-i)"{'-'4?f;*'} <*
provided 64= nn-.
A great variety of other results of a similar character could
be de<lucc(l ; but these will sulhce for our purpose.
§ 9.] Before leaving the present subject, it will be instructive
to discuss an example which brings into prominence the ueccjs-
sity for one of the least obvious of the conditions for the applica-
bility of the General Theorem of § 5.
We have, 0 being neitlicr 0 nor a multiple of r,
a* - 2j* cos e + 1 = {j* - (cos (? + 1 sin 0)\ { j* - (cos 6 - 1 sin tf)}.
The pi\\ roota of cos d + 1 sin fl are given by
COB . (2n7r + tf)/;; + I sin . (2Mn- + <>)//>, n = 0, 1, . . ., p-\ (1).
The j)i\\ roots of cos 0-i sin 6, that is, of cos (-<?) + i
8in(-e), by
cos. (2Hir-0)//^+« sin. (2«n^ -*)//>, M = 0, 1, . . ., ])-\ ('.').
§§ 8, 9 PKODUCT FOR COS (j> - COS 6 S55
Since cos . {'2inr - 6)/p = cos . {2 (p -n)-n- + 0\lp,
sin . (27jjr - 0)Ip = — sin . {2 {p -n)Tr + 6\lp,
(2) may be replaced by
COS. {'imr + 6)lp-i^n\.{2,mr + 6)lp, « = 0, 1, . . .,p-l (2').
We have, therefore,
arP-2a*cose + l
= {x" - 2x cos .6lp+\)n {or - 2x cos . (2«:r + e)lp + 1 } (3).
n=l
Since cos . {2mr + 6)lp = cos . 12 {p -n)ir- 6\lp, wo may, if ^ be
odd, arrange all the factors of the product ou the right of (3)
in pairs. Thus, if ^ = 2q + 1, we have
a;*2+a_2a;28+icos6i + le
(a?-ixcos-^ +l\h{ (•^^-2^cos.(2«^ + e)/(2.y+l)+l)|
\ar ix cos 5^ + ^ + ^)liA>^ {or - 2x cos . (2«7r - 0)li2q + 1) + 1)1
(4).
If we now put x=l, we get
2nir + 0 . „ 2«ir-
4g' + 2„'^ir'"' ' iq + 2 ' iq-
If we divide both sides of (4) by x^*^, and put x = Ex^i<l>,
we deduce
2 (cos (29- +!)</) -cos 6)
= 2'^+'{cos <^ - cos . e/{2q + 1)} n {cos .^ - cos . (2?«7r + e)/{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 0
- 2 ^mne h - !H'LMii±2)l ^ f, sm\<i>/{4q 4-2) I
- ^ siu JP |i g.^, _ ^^^^^ ^ 2)J„ii l^ sin= . (2«7r ± 6»)/(4<? + 2)J
(7).
Since n:!f>q, (2n-!r±6)/(iq + 2):!^{2qTr±6)/{4q+2); and the
limit of this last when 5-= 00 is ^tt. Heuce, by taking q large
enough we can secure that {2mr±6)/{4:q + 2) shall have for its
23—2
4.sm-- = 4-«+'sm- -n^sur. -sin-.— -} (5).
2 4g + 2„=il 4(7 + 2 4g + 2j ^ '
356 I'KODUCT Full COSH U- COS ^ C'll. X\X
upper limit a tpiautity wliicli diO'cni from Jr by as little as
we please; aud therefore (see § 6) that sin. (2nir + e)/(4g + 2)/
(2nir±d)/{4q + 2) shall have for its lower limit a quautity uot
less than "58.
We may, therefore, put g - « , &c., in (7). We then get
cos<^-cosd = 2siu'Jtf(l-<^76f)n{l-0»/(2«7r + e)'} (8),
n-l
that is,
cos 4> - cos 0
Putting <p = iu in (8), we deduce
cosh U-C08 tf = 2 sin' A« (1 + u'/9') n {1 + «V(2n» + df, (9).
The fonnula (8) might have been readily derived from those
of previous paragraphs by using the identity cos <^ - cos tf
= 2 sin J (^ + '^) s'" i (^ ~ ^) and proceeding as in the latter part
of § 7.
Itenuirk. — At first sight, it seems as if we might have dis-
pensed with the transformation (4) aud reasoned directly from
(3), thus—
From (3) we deduce
p-i
2 (cos;^0 - cos 6) = 2' (cos <^ - cos . 0/j>) 11 {cos ^ - cos . (I'/iir + e)/p\.
Hence
cos ^ - cos d
= 2 sin' Je fl - «!";t/-n u7l - _!i5l^*/?P_l
Put now p ^ <x>, &c, aud we get
cos ^ - cos e - 2 sin' i e ( 1 - ^'/^') fl j 1 - .j!,'/(2»ir + 6)').
This result is manifestly in contnuiiction with (8), although
the reasoning by which it is established is t\w s;iniu as that often
considered sutlicieut in such coses.
§ 9 INSTANCE OF FALLACY 357
In point of fact, however, the condition of § 5, tliat
M=fi{n, p)/fi(n) must remain finite wlicn n and p exceed certain
limits, is not satisfied.
lu the present case the upper limit of {2)i7r + 6)/2p, namely,
{2 (p — 1) 5r + 6}/2p, can be made to approach as near to ir as we
please. Hence iu this case 31 may become infinite. We have,
in fact,
^^1 ^m.{4>l2p)l{^lip) \
I sin . (2??7r + e)l'2pj{2n-7T + 6)/2p
hence, if we give n its extreme value p — 1, and put p= cc, 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 + irja) (1 + ixlh) (1 + ixjc) ... = A+iB, then
2 tan-' (.r/«) = tan-' (BjA).
Hence show that S tan->(2/n2) = 3ir/-l.
1
(Glaisber, Quart. Jour. Math., 1878.)
(2.) Find the n roots of
"("-3).„-4_
x''-nx''-»+ 2j
( _ j,«(«-r-l)(n-r-2)...(n-2r+l)^„_,, ^ _
(,S.) If n be an odd integer, find the n roots of the equation
x+— gpar' + 5 — — 5! '^'^ y, ' 'x'+- ■ .=a.
(1.) Solve completely
x'' + „CiCOsax''~' + „C„cos2ax"--+ . . .+cosna = 0.
(Math. Trip., 1882.)
(5.) The roots of
s»sinn9-„C,s»-isin(H(7 + c^)+„C„x»-2sin(?ie + 20)- . . .=0
Rra given by x = sin [d + (p ~ k-!rln)cosec (0- liirjn), where /i = 0, 1, . . ., or
I! a = vj'2p, prove the following relations:—
(6.) J) = 21^' sin 2a sin 4a. . .sin (2p - 2)o;
l = 2i>-'siuo sin 3a. . .sin(2j)- l)o.
358 EXERCISES XXII CH. XXX
(7.) VP=''^'^'"'""''08 2o. . .cos (p- l)a.
(8.) l = 2P-'siD.a/2 6in.3o/2. . .8in.(2p-l)a/2;
= 21^' coa . o/2 COS . 3o/2 ... cos . (ip - 1) o/2.
(0.) Binpe=2P-'(!in<»8in(2a + 0)sin(4o + tf). . . sin {"ip - 2a + e);
COBp9 = 2'^'Bin(a + (?)8in(3a + e)Biii(5a + tf). . .sin (2;> - la + fl).
(10.) tan p(? = Ian 0 tan (ft + 2a) . . . tan(0 + (2p- 2)o), wlierep in odd.
( U .) ten « tan (fl + 2o) . . . tan (9 + (2/» - 2) o) = ( - 1)''/», where p is even.
(12.) Show that tbo modulus of
C08(ff + t»C08(e + i> + r/p). . .coB(9 + i> + {p-l)»/p)
is {co8h/)#-c03(pir + 2pe)}/2»*-i.
(13.) If n he even, show that
• »" / ,„flo.-. * * + 2»- e + ir ff + (2n-2)»-
Biu' -: = ( - )»/>2"-' COB - COB COS . . . COB ^ ,
2 ^ n n n n
(14.) Show that ri(l + 6ec2"(?) = ten2»fl/lan9;
and evaluate " fl jll^/''^' j .
(15.) Show that
\l (l-4sin' - j = coBO;
and write down the corresponding formoUB for the hyperbolic fanctions.
(Laisant.)
Prove the following resnltB (Ealer, Int. in Anal, Inf., chap, a.):—
no '^+'^'-n fi, 4(fc-c)x+4x' 1 .
''"-'^ = /l + ifL^ „ |l + *ibzAf±±'2 \
*»-«• V 6-cr r (2")'ir> + (6-c)'r
C08hy + coBhc_„ j, , J=2^;/ + y« )
' ' l + coshc ~ t ■^(2h-1)»ip»+c-4 •
coshy-coahc_ / y'\ „ /i *2q/ + y'l
1 - cosh c ~ V c'/ I (2n)* T»+ c»[ '
Writ«>down thooorroBpondinK rorniiilio lot Iho circular function*, and dnliiM
them by trannroriuulion fruni § U.
§10
EXERCISES XX 11
359
C08^+C03g L 0^ )
'^^•' 1 + C03 9 -"■} {(2n-l)ir±e)H-
(19.) cos,^ + tanpsm« = n j(l + ^2ir=lt^ ('l-^'J-^^)}.
Bin (e* - 1
(21.) Show that
cosh 2v - cos 2u = 2 («2 + 1-') H j'"'^ ^g*^, f 5
„ (((27i-l)7r±2a)2 + 4r=)
cosh2o + cos2M=2n y^ (2n-lpT^ ('
( 4u*)
C03h2u-00B2u=4u'II a+^4~4> ;
( 2*u* I
cosh 2u + cos 2« = 2n <1+.., _^^4y4( •
(Schlomilch, Handb. d. Alg. Anal., chap, xi.)
(22.) Evaluate n(,-^^j35^J.
(2.S.) U tn-=log (l + x/2), show that
EXPANSION OF THE CIRCULAR AND HYPERBOLIC FUNCTIONS
IN AN INFINITE SERIES OF PARTIAL FRACTIONS.
§ 10.] By § 8 we have, provided 6 + ^{2n-l) ir,
200 + <^=
cos
0 - sin <!> tan 6 = H {l - 4 (,,_,).^_,gj (D-
Now, referring to § 2, Cor. 2, we have here
«„' = 8
(2h - 1)V - 46
;Jl'^l + 4
1
,<^' + ,
(2» - l)'w= - 46
4
1*1'.
<^".
I (2» - 1)V - 46'= r * |(2» - 1)V - 46'= I
where 6' = 1 6 1, <^' = | c^ |. It follows, therefore, that the product
in (1) may be expanded as an asceudmg series of powers of </>.
360 INFINITE SERIES OK PARTIAL KUACTIONS Cll. XXX
Expaiuling also on tlie left of (1), we have
+ 1 6 (26* + <^')' :S |(2«-l)'«»-4e'}{(2«-l)'7r»-4^1
(2)-
Since the two scries in (2) must bo identical, we have, by
comiiavin^ the coefficients of <^,
*""^ = «^!(2;.-l)'^-4^ (^)-
This series, which is analogous to the expansion of a rational
function in partial fractions obtained in chap, vni., is absolutely
convergent for all values of 0 except Jir, ^n, Jn-, . . . It should
be observed, however, that when 0 lies between J (2n - 1) t and
i (2n + 1) IT, the most importAiit terms of the scries are those in
the neiKhbourhood of the wth term, so that the convergence
diminishes as 0 increases.
We may, if we please, decompose 8^/{(2n- l)V-4^} into
2/{(2« - 1) TT - 26) - 2/{(2rt - 1) IT + 261, and write the aeries (3)
in the semi-convergent form
2 2 2 2
taa6 = -
20 IT + 26 3ir - 26 3ir + 26
2 2
5n- - 26 r>n + 26
,^ + ... (3').
In exactly Ihe same way, we deduce from (1) and (3) of § 8
the following ; —
6cot6=l-26'S ,-^-5 (4),
or
6 6 6 0
6 cot 6 = 1 .+ -. - 2 + ,r a
+ r, — . . . (4 ).
3r-6 37r+6
§ 10 INFINITE SERIES OF PARTIAL FRACTIONS 361
provided ^4=t, Stt, Sir, . . . ;
and
0 cosec 0 = 1 + 2^- S-^rP"^l (■'"').
or
.3 /) 1 ^ ^ ^ ^
6 cosec t' = 1 + j; 2, ~ "^ a + S a
provided 5 =f= -, 2t, 3^, .
_« 6__
We might derive (4) from (3) by writiBg (Itt-B) for 9 on
both sides, multiplj'ing by 6, decomposing into a semi-convergent
form hke (3'), and then reassociating the terms in pairs ; also
(5) miglit be deduced from (3) and (4) by using the identity
2 cosec 0 = tan ^6 + cot ^0.
When we attempt to get a corresponding result for sec 0,
the method employed above ceases to work so easil}' ; 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 p = :5 + 7. n ~ r: o +
6 TT-O TT + e 27r-6l -iTr + d
1
iir-e Srr + e ' " "
which we may do, provided 5 + 0, and then put Jtt-^ in place
of 6, we get
2 2 2 2
sec Q = -7; + -
■_2e 7r+2t> 3^-20 37r+25
2 2_ _ . ,.
■^5ir-2e'^5^ + 2^ • • • ^^'
or, if we combine the terms in pairs,
sec e = 42 C - ^"-' (gj - 1) ^ ... /gN
secp 4-.( ; (27»-l)"-7r'-4e» ^*'''
where 6 =t= \Tr, fir, \-k
The series (G), unlike its congeners (3), (4), and (5), is only
362 INFIVTTR SERIKS OF PARTIAL FRAmOXS CH. XXX
simii-convergent ; for, when n is very large, its nth term is com-
parable with the Hth term of the series 21 /(2n - 1).
We might, by pairing tlie terms differently, obtain an abso-
hitely convergent series for sec 0, namely,
but this is essentially clifTorent in form from (3), (4), and (5).
Cor. 1. The sum o/all the products two and two of the terms
of the s<'ries 21/!(2«-l)'7r^-46'} is {tan 0 - e)jVi8e*; and the
like sum for tite series il/lH':!^ - ^^} is (3 - ^ - 3fl co< e)/8^.
This may be readily established by comparing the coefficients
of <^' in (2) above, and in the corresponding formula derived from
S 8 (1).
Cor. 2. The series 21/{(2»- l)=5r'-4^}' converges to the
value (6 tan' e- tan 6 + 6)/ 640'; and 2l/(H'ir*-ff')' to the value
{&' cose<? 6 + 6 cot 6- 2)/i0*.
Since the above series have been established for all values of
6, real and imaginary, subject merely to the restriction that 0
shall not have a value which makes the function to be expanded
intinite, we may, if we choose, put 6 = ui. We thus get, inter aiia,
tanh M = 8«2l/{(2n - lY^r' + 4t<'i (8) ;
ttcoth«=l+2M''2l/{n'ir' + M'} (9);
u cosech M = 1 - 2h'2 ( - l)"-'/{n'>r' + «'} (10) ;
sech « = 42 (-)"-■ (2n - 1) T/((2n - l)'7r' + 4H'} (11).
EXPRESSIONS FOR THE NUMBERS OF BERNOULLI AND EULER.
RADIUS OF CONVEUfiENCY FOR THE EXPANSIONS OF
TAN ^, COT^, COS EC d, AND SEC^.
§ 11.] If |0|<»r, then every term of the infinite scries
20'/(n'ir' - e*) can be expamlcd in an ab.solutely convergent scries
of ascending powers of 6. Also, when all the powers of 0 are
replaced by their moduli, the series arising from l/(»''r' - ^)
will simply become l/{n^i!'~\0\'l, whicli is positive, since \0\<'-
The double scries
^10-12 EXPRESSION FOR BERNOULLI S NUMBERS 3G3
« r 6= (/•• e-'" 1
thorofiire satisfies Caiich)''s criterion, and may be arranged
according to powers of 6. Hence, if
<r,„=l/P"'+l/:i'^"' + l/3="'+. . . (1),
we have, by § 10 (4),
6'cot6l = l-2261V(MV"--e'),
= l-2So-,„e="'/^"™ (2).
Since o-2m(<o'2) is certainly finite*, the series (2) will be
convergent so long as, and no longer than, 6<-ir.
Now, by § 4 (9), we have
6 cote = 1- 22-"'/?„.e^"'/(2»i)! (3),
provided 6 be small enough.
The two series (2) and (3) must be identical. Hence we
have
^2(2m)!<r,„.^2(2»^)! fill 1
"■ (27r)='" (27r)-"' Ip™ "^ 2="" 3="' ■ •/ ^ ''
§ 12.] If, instead of using the expansion for OcotO, we had
used in a similar way the expansion for tan 6, we should have
arrived at the formula
Bm =
2(2?w)!
(1 - 1/2="') (2:r)"
ll^"" 3="" 5="* ' ' ")
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-m due to Eulert. We observe that the result of multiplying
the convergent series a-^m by 1 - 1/2"'" is to deprive the series of
all terms whose denominators are multiples of 2. Thus
(1 - 1/2='") <r2„ = 1 + 1/3='" + 1/5='" + . . . .
• It may, in fact, be easily shown tliat La;^=l when m=a); for, by
chap. XXV., § 25, we have the inequality l/('2m- l)>l/2^' + l/:!-"' + l/4^
+ . . .>l/(2m-l)2™-', which Bhowe that /-(l/2-'" + l/:!2'"+l/4-'"'+. . .)=0,
when m = « .
t See Inlrod. in Anal. Inj., ^ 2S<i.
364 PKOPERTIES OK BERNOULLI'S NUMIIEUS CH. XXX
If we take the next prime, namely 3, and multiply
(l-l/2'")<r„ by 1-1/3™, we shall deprive the scries of all
terms invohnng multiples of 3 ; and so on. Thus we shall at
lost arrive at the equation
(1 - I/a'") (1 - 1/3»") (1 - 1/5=") ... (1 - !/;>*") <r«
= 1 + l/g«- + . . . (G),
where 2, 3, 5 p are the succession of natural primes np to
p, and q is the next prime to p. We may, of course, make q
as large a.s we please, and therefore l/q^+. . . (which is less
than the residue after the y— 1th term of the convergent series
o-j„) as small as we please. Hence
a-„= 1/(1 - 1/2"") (1 - 1/3'") (1 - 1/5*) . . . (7),
where the succession of primes continues to infinity. Hence
5„ = 2 (2»»)!/(2>r)'" (1 - 1/2'") (1 - 1/3*") (1 - 1/5'") . . . (S).
§ 13.] Bernnulltn Numhcrs are all j>f>.<i!(if^ ; th/y increase
after B^; and have oo for an tipper limit.
That the numbers are all positive is at once apparent from
§ 11 (I)- The latter part of the corollary may also be deduced
from (4) by means of the inequality of chap, xxv., g 2.'). For
we have
l/(2m-l)>l/2'"+l/3*" + l/4*"+. . . >l/(2w-l)2'" ' (9).
Hence
^„+, ^ (2OT + 2)(2OT+l)crM..n
B, (2:r)V„ ^ •
(2/» + 2)(2OT-H){l-t-l/(2w-t- 1 ) 2*"^'}
^ ' (2»-)''{l + l/(2»»-l)} "
^(2w)'-l
^ W ■
Hence 77„+,/5„>l, provided ♦n>^/(T*+^), that is, if
w>316. Now lh>B,, hence B,<Bt<Bt< . . ..
Again, it follows from (9) that A<t,„ = 1 when m = oe , and
Z(2OT)!/(2n-)'" is obviously inrmitc ; hence LB^ is infinite.
Cor. //„/(2m)! ullimntely dtrreasfs in a (fi'omffriral }tro-
gressitin hunngj'ur its common ratio l/4ir'. From which it/olU>ir$
^ 12-1-i CONVERGENCE OF SERIES FOR TAN 6, &C. 3(j5
tiat the series for tan 9, 6 cot 6, and OcosecO, given in § 4, have
for their radii of convergence 6 = iir, tt and t- respectively.
§ 14.] Turning now to the secant series, we observe that
42 ( - )"-' (274 - 1) 7r/i('2« - Ifrr' - 461^} does not, if treated in the
above way as it stands, give a double series satisfying Cauchy's
criterion, for, although when | ^ | < |ir the horizontal series are
absolutely convergent after we replace 6 by |6|,yet the sum
of the sums of the horizontal series, namely, 42 (-)""' (2« - 1) t/
{(2h- l)"?r--4|^|"J, is only semi-convergent. We can, however,
pair the positive and negative terms together, and deal with the
series in the form
f (In-S)^ (in~l)7r 1
* l(4«-3)V-4e» (An-lfTT'-m ^ ^'
o ,. (4» - 3) (4»- 1)77^ + 46' ,,,,
that IS, ^""-{(^i^^rff^i^^WHIM^l^f^i^^^W} ^^^^-
Since (11) remains convergent when for 6 we substitute
\6\, it is clear tliat we may expand each term of (10) in as-
cending powers of 9, and rearrange the resulting double series
according to powers of 0. In this way we get
/I _ . V r ;- f 1 1 ]-i2-'"ff^
^^^ ~ %i« L..-1 1(4« - 3)='"+' ~ {in - ir+'j] 1^^ '
= 2 2="'+=T^+.e^"/'^"'+' (12),
where t2,„+i=1/1="'+i-1/3™+'+1/5="'+'-. , . (13).
Comparing (12) with the series
sec6»=l + 2£'„^"/(2m)!,
obtained in § 3, we see that
2''""(2m)!r,„^.
(2\2m+I /J 2 I ■•
-) li^^TH-giSiiri+SSi^x-- • •} (14),
which may be transformed into
E^ = 2(2,„)! gPy(l . ^i^.) (l - ^i^.) (l . ^.i) . . .
in the same way as before. (15)*.
• See agaiu Euler, IiUrvd. in Anal. Inf., § 284.
366 I'UOrERTIES of EULEH's NUMBEltS CU. XXX
Cor. 1. Etder's numlurs are all positive; thry omlinually
increase in vutijiiitude, and liave injinity for their upper limit.
For we liave
l>T3^+,>l-l/3*^> (IG).
Heuca
E„-n _ (2m + 2)(2«+l)4T»+,
H- ir^T,
m-fi
{-2m + 2){2m + 1)4(1- X/S-"*')
> :^ .
But this last constantly increases with m, and is already
.■.'leater tlian 1, wlieu »»=1. Hence E,<L\<E,<. . . Also,
from (16), we see that Ltm+i = 1 when m = «, and
Z,(2OT)!(2/7r)'-^' = ao, hence LE„=x.
Cor. 2. Em/{2m)\ ultimatelif decreases in a geometrical
progression whose common ratio is 4/^*. Hence the rrnlius of
convergence of the secant scries is 6= Sir.
§15.] Wehave, by§ll (4),
1 1 1 021M-I D
and hence
' - JL JL JL
/ l\2"'-'/?«
■ T^ (2) ;
and
2 (2m)
111 ^, 2\2™-'//„^,
1 - -^1— "ir-
2»>' (2w)! '
» j« 2'" ■ 3"- • ■ • \* 2"y (2w)
^(^-'-1)5.
(2».)!
Again, from (14) of la.st paragrajjh
(3).
__i L_+_i__ - _^ V-.+I (u
'"+'~l«'«+i 3»"+' 5"+' ■ ■ • ~2'"+'(2ni)! ^''
* Tho rcinnrkablc suiuiniitinns involved in t)io formulm (I), (2), (3) wen
discovcroJ iiiilopcndcutly by John liuruoalU (occ Op., t. iv., p. ID), and by
Euler {Cumin. Ac. Ptlrop., 1740).
§§ 14-lG
SUMS OF CERTAIN SERIES
367
Inasmuch as we have iiidepeudent means of calculating the
numbers Bm and £"„,, the above formula; enable us to sum the
various series involved. It does not appear that the series (^^,,,+1
can be expressed by means of Bm or £',„; but Euler has cal-
culated (to 16 decimal places) the numerical values of a-^m-vi 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 formulce involving ir for o-^
are accurate when m is even ; but only approximations when
m is odd.
<r2= 1-6449340668 .
. =7^76.
0-3= 1-2020569031 .
. = -71725 -79436 .
. ,
<T4= 10823232337 .
. =-n-V90.
a6=l'0369277551 .
. =7r7295-1215 .
, ,
0-6=1 -01 73430620 .
. =7r7945.
o-,= 1-0083492774 .
. =71-72995-286 .
, .
0-8=1-0040773062 .
. =7r8/9450.
<rs= 1-0020083928 .
. =ir729749-35 .
. .
EXPANSIONS OF THE LOGARITHMS OF THE
CIRCULAR FUNCTIONS.
§ 16.] From the formulae of g§ 6 and 7, we get, by taking
logarithms,
log sin e = log e + 2 log (1 - 6l=/?jV),
= loge- 2 (T„^(r-'"lnnT-"'
(1),
since the double series arising from the expansions of the
logarithms is obviously convergent, provided | ^ | < tt.
If we express a-^ by means of Bernoulli's numbers, (1) may
be written
log sin e = log ^ - 2 2-'"-'B^6^'"lm (2m)! (1').
* Imt. Calc. Diff., ohaji. vi.
308 STIULINO'S THEOREM CU. XXX
The corresponding fonniiltc for cos 0 aro
log cos fl = - 2 (2=» - 1 ) <r.„. (P^lvi^' {'>);
— 2'.>""-' (2'" - 1) D„e'-'lm (2m)! (2').
Till- like fonniilu; for log t:in 0, log cotO, log siiili u, log cosh «,
&c., can be derived at once from the above.
If a table of the values of o-j„ or of //,„ be not at hand, the
lirst few may be obtained by expanding log (sin OjO), that is,
log(l -673! + tf*/5! — . . .), and comparing with the series
-'^<T^ff""lmir^, For example, we thus find at once that
stikijng's theorem.
§ 17.] Before leaving this part of the subject, we Fhall give
an elementary proof of a theorem of great practical importance
which was originally given by Stirling in his Methodus Differen-
tialis(ll-M).
]l'/icii n is very great, n\ approaches equality with J(2mr)(n/e)*;
or, more accurately, when n is a lurge number, we have
«! = ^/(2IrH) («/c)''exp {1/12« + e\ (1),
where - l/2i«' <e< l/2i« (« - 1 ).
Since log {«/('* ~ 1)1 = ~ ^"o (1 ~ !/'')< we have
, 7) 1 1 1 1 I
^ M - 1 M 2/r 3«' 4«' mw "
We can deprive tliis expansion of its second term by multi-
]ilying by n - i. We thus get
,,,,», 1 1 m - 1
(tt - A) log — I = 1 + r.i J + , .."1 + • • • + a — 7 rr~m + • • • •
- "h-I 12h' 12«' 2wi(»»+l)»"
Hence, taking the exponential of both sides, and writing suc-
cessively H, M — 1, H-2, . . .,2 in the resulting cijuation, wo
deduce
/ n V'-* /, 1 1
\7«- 1/ ' \ 12n' 12;i
w - 1 \
■*■ 2m (m+1) «■"■*"■ • 7'
§516, n Stirling's THEOREM 369
/«-l\''-i-4 /, 1 1
m-1 \
m-\ \
* 2}n{m+l){7i-2)"''^' ' ')'
/3\3-J /, 1 1
33 + . . .
m-1 \
"^ '2)» (ot + 1) 3"" "*" ■ ■ ')'
/2\2-i / 1 1
"^ 2»i(7« + l)2"''''' ■ 7"
By multiplying all these together, we get
nz — 1 „, 1
where S',^ = 1/2"' + l/S"" + 1/4"" + . . . + l/w".
Now
-»'„ = /S„ - l/(« + ir - l/(« 4- 2)-" - . . . (3),
where S,„ = 1/2"" + 1/3'" + . . . + l/»i"' + ... ad 00 .
By the inequality (6) of chap, xxv., § 25, we have
1/(ot - 1) «"-' > l/(;i + 1)'" + l/(« + 2)-" +...>!/{?« - 1) („, + i)>"->.
Heuce
-S'„- l/{m - 1) {u + !}"'-'> S-^>S„- lj{m-l) «"-».
Therefore
i2*'»^l2^''"'-' •^2J(^+1)'^'"'+---
•^2W2(?«+1) "2 m (to + !)«*""' ^''
-2 7n(/n + l) ^2 ?«(/« + 1)(«+ 1)"'-' ^'•
c. 11. 24
370 STIRUNO'S THEORKM CU. XXX
Since ('^„<l/(m- 1), the scries 2(m - l)iS'„/m (»» + 1) con-
verges to .1 finite limit which is independent both of m and of n.
Again,
S 1
(c);
f »» (w« + 1 ) n
m-l
1
■^3
1
.4«'
n
1
r„i' '
1
1
12;j
.{■
1
+ -
II
1
<h^
12»
1
-1)
Also,
by (6),
1
(7).
Tw»(m+1)(»+1)"-'
T\m OT + !/(» + IT""
=(-i)(-;rTi-'''«0-;rli)}
= + -+n-(n' + «)log(l + -);
1 J_
(8).
Combining (2), (4), (5), (7), nnd (8). we hftvo
-^>«'''l'{"-^^i^«,(,irrD-12« -247(7^1)1 <^)'
'I - J m (m + 1) 12« 24m'J
1
-»M
1
2
1
3h
1
^4«»
1
1
"I
1
2.3;i
3
4h^
.5«'
1
1
12n'
§ 17 STIRLING'S THEOREM 371
Hence, putting
(7=exi,{l-ji(^^4 (11),
so that C is a finite numerical constant, we have
n\ > Ce-n^i exp (i - ^,) (12),
< Cfe-''w"H exp (~ + 1 -^ (13) :
or, since the exponential function is continuous,
«! = (7.-»«"+iexp(jl-^+e) (14),
where -l/24«-<e<l/24n(re-l).
Hence, putting »i = qo on both sides of (14), we have
Z»i! = CZe-"m"+5 (15).
Tiie constant C may be calculated numerically by means of
the equation (11). Its value is, in fact, ^/(27r), as may be easily
shown by using Wallis's Theorem, § 6 (18).
Thus we have, when « = 00 ,
IT 2"'(w!)'(2?t+l) ^ J 2^''(«!)*(2»+l)
2 P3V . . (2w+l)^ {(2«+l)!p •
Hence, using (15), we get
'^ = (^T 2"'^~"»'"'^' (2» + 1)
2 e-'"-''(2a+l)«+» '
^ c;^ , e»
4 {(l + l/2?i)-T{l + l/2n}-'
~ 4V'
Therefore, since C is obviously positive,
e=V(2-) (16).
Using this value of Cin (14), we get finally
w! = 7(27r») {njef exp {1/12» + 0]* (17),
where - l/24n'^<0< 1/24h («- 1).
* An elementary proof that Ln\ = LJ(2im)(nje)'^ was given by Glaisher
(Quart. Jour. Math., 1878). In an addition by Cayley a demonstration of
the approximation (17) is also given ; but iuasmucb as it aasames that series
24—2
372 EXERCISES XXIII CH. XXX
Cor. By combining (11) and (16), tw dedure that
where 5,= 1/2"+ 1/3- +1/4" + . . . ad ».
Exercises XXIII.
(1.) Show that, when |i|>t, f cotz can bo cipnndcd in the form
il,+ 2(7?,x-"+C,x"); and determine the coeOicients in tlie particnUr esM
where ir<i<2r.
(2.) Show that the snm of the prodacts r at a time of the tiqnares of the
reciprocals of all the integral nambers is w'/ISr + l)!; and find the like sam
when the odd integers alone are considered.
(3.) Sam to n terms
tan9 + tan (9 + r/n) ■«• tan (0 + 2T/n)-t-. . .;
tan»tf + tan»(e + r/n) + tan'((? + 2T/n) + . . . .
Snm the following: —
(4.) 1/(1« + j^ + 1/(2' + x») + 1/(3' + i») ....
(5.) l/x'-l/(x>-T^) + l/(i>-2'x')-- ■ • •
(6.) l/x + l/(z-l)+l/(x + l) + l/(x-2) + l/(r + 2) + . . . .
(7.) 1/(1 -«) + 1/(1-') + 1/(9-') + - • . + !/(«' -f) + . ..
(8.) 1/1. 2 + 1/2. 4+1/3. 6 + 1/4. 8 + . . . .
Show that
(9.) (ir'-6)/6 = l/l».2 + l/2'.3 + l/3>.4 + . . . .
(10.) T/8-l/3 = l/1.3.6-l/3.S.7 + l/5.7.9-. . . .
(11.) If /r (n) be an integral function of n whose degree i* r, Rhow that
— /r ('■)/(^'> - 1)*** can be expressed in terms of BcmouUi's nombera, proridod
r > 2m - 2 ; and 2 ( - )"-'/r ('')/(2'' - l)*"'*'' in terms of Ealer's nnmbcrs, pro-
vided r J»2m - 1.
In particniar, show that
1 1 + 2 1 + 2 + 3 »»/, ir>\
(13.) Show that
Sl/(«i-+«)«=oo8ec>#;
2 l/(n» + «)*=oosec*9 - } eoitt?0,
11=0 being incladcd among the valaes to bo given to n. (Wolslenholme.)
of the form of 1/2" + 1/3"+ . . . can be expamled in powers of 1/m, it cannol
be said to be elementary. The proofs nsuallr given by means of the Mio>
laurin-iom-fonnola are nnsatisfactoty, for they depend on the oae of a seriee
which does not in grnenil convergn nhen cnntinned to infinity, and which can
only b« used in cuujanctiou with ita rcnidue. Sui lUabc, CrtlU'i Juur., xxr.
^^ 17, 18 EXERCISES XXIII 373
nn " 1 _ ■^'J'i sinh . irx^2 + sin . -rxji 1
T "H x' ~ 40-" cosh . 7rj-^/2 - cos . irx^/2 ~ 'ic" ■
(Math. Trip., 1888.)
(14.) Sliowtbat
5 1 ir^ 1^
,^i{(2n)»-(2m-l)«}2 I6(2m-1)3 2(2m-l)»'
Si 5r»
„=i {(2n - 1)2 - (2m)2}« ~ Urrfi '
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
jr«/384.
(15.) If|)<7i, show that
cos "9 _ 1 "-^^ _ w sin . (2r-H) 7r/2ra . cos ■ P(2r+ 1) jr/2n
cosHtf nr=o cos « - cos . (2r + 1) wftii *
(IG.) Show that
*''""' ^" ^ itau-i— " tan-i— ^i=tau->(tanhj)cot«):
„i. r°"'(2«-l).-2«-^^°"(2„-l). + 2,.[ =t-"-Mtanh.tan«).
(Schlomilch, Ilandb. d. Alg. Anal., cap. xi.)
(17.) If X(x)srfI{l-(x/Ha)2}, /i(x) = n{l-(2x/2^r^a)=}, express
X (a; + a/2) in terms of /i (i) , and also |it (x + a/2 ) in term s of X (x).
Hence evaluate I, 1 . 3 . 5 . . . (2m-l)^(2m + l)/2"7n!.
(Math. Trip., 1882.)
(18. ) Show that, if r be a positive integer,
.i('-r('-r--('-'-^)'"--»-
(19.) Show that
(20.) If 71, p, X be all integers, prove
(7t + x)(7t+a + l) . . . (71+y + x-l)
^. (l + x)(2 + x) . . . (j + x)
EEVERSION 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 Analyst per jEquationes Numero Terminorum Injinitas.
374- STATEMKNT OF EXPANSION PROBLEM CU. XXX
Let US consider the function
2(;«, u)x''ir={h 0)ar+(0, l)y + (2, O.r'+d, l)j-^+(0, 2)^ + . . .,
where the indices m and n arc positive integers, and wo use tlio
symbol (m, n) to denote the coefficient of x"y", so that (m, n) is
a coustiint. We suppose tlie absolute term (0, 0) to l>e zero ;
but the coefficients (1, 0) (0, 1) are to be different from icro.
The rest of the coefficients may or may not be zero ; but, if the
number of terais be intinit<?, we suppose tiie double series to be
absolutely convergent when |a:| = |y | = 1*. From this it follows
that the coefficient (;«, n) must become iufuiitely small when m
and n become infinitely great ; so that a jinsitive (piantity X c^in
in all Ciises be assigned such that |(/n, h)|;^X whatever values we
assign to m and n. It also follows (see chap, xxvi., § 37) tlrnt
2(ni, n)afif^ is absolutely convergent for all values of x and y
such that \x\1f>\,\y\1c\.
We propose to show tluit one value, and only one value, of yat
a function of x can be found which has the following projKrties: —
1*. y is expausihle in a connrgcnt series of integral jnniYrx of
X for all values of x lying within limits which are not itijinitcly
narrow.
2°. y has the initial value 0 when x ^0.
3*. y makes the equation
2(m, u)x"y'-0 (l)
an intelligible identity.
Let us assume for a nioincnt that a cnnvrrpent sorios for y
of the kinil deniandcil can bo fouml. It« absolute tenn must
vanish by condition 2°. Hence the series will be of the form
y = btX + b^x' + b,3^ + . . . (2).
In order that this value of y may make (1) an intelligible
identity, it nnist bo possible to find a value of x<l such tliat
(2) gives a value of y<l. The series (1), when transfomietl by
means of (2), will then satisfy Caucliy's criterion, and may be
arranged according to powers of x. All that is further necessai .
* The more goniTal coiio, when the scriei U eonvargont no long an |<| >•
and ly I >/3, can eimily ba bronght uuder Uia above by a Dimple traDifonniu
tiou.
§18
GENERAL EXPANSION THEOREM
S75
to satisfy condition 3° is simply that the coefficients of all the
powers of x shall vanish.
It will be convenient for wliat follows to assume that
(0, 1) = - 1 (which we may obviously do without loss of
generality), and then put (1) into the form —
y = ((l, 0).c + (2, 0)a;" + (3, 0)a,-' + . . . }
+ {(1, l)a;+(2, l)«= + (3, l).i^ + . . . ]y
+ {(0, 2) + (1, 2) .V + (2, 2) .1' + (3, 2) ,r' + . . .}f
+ {(0, n) + (1, n).t + (2, n) a? + (3, n) o,-^ + . . . } ?/"
(3).
Using (2), we get
biX+ LaP+ b3a^ + . . .
= {(l,0)a; + (2,0)ar' + (3,0)«' + . . .}
+ {(l,l)a; + (2, l).r' + (3, l)x-= + . . .}\b, + b.x+b,,T'+. . .}x
+ {(0,2)+ (1, 2).r + (2, 2)^ + (3, 2).r' + . . .\{b, + b.x+b,.v'+. . .^x^
+ {(0, h) + (1, fi)x + (2, n)x~ + (3, n) J-' + . . . } {b^ + b,.c+b,x'+. . .}"«"
(4).
Hence, equating coefficients, we have
J. =(1,0),
6,=(2, 0) + (l, 1)6, +{0.2)b^
b,={3, 0) + (l, 1)6. +(2, 1)6, +(l,2)6.'+2(0, 2)6.63 + (0, 3)6^
6» = (», 0) + (1, 1)6„-. + (2, 1) 6„_, + . . . + (0, «) 6."
(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 6,, all the coefficients are uniquely determined.
There is therefore only one value of y, if any.
In order to show that (5) really affords 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 6„ 60, . . . have the values
assigned by (5).
376 GENERAL EXPANSION THEOREM CM. XXX
This, following a method invented by Cauchy, we may show
by considering a jjarticular case. The moduli of the coefficients
of the series (3) have, as we have seen, a finite upjjcr limit A,
Suppose that in (3) all the coefficients are replaced by A, and
tliat X has a positive real value <1. Then we have
y=X{a; + ar' + jr' + . . . [
+ \{ar + a;^ + «* + . . . \i]
+ X{1 +a:+ar' + a!* + . . . [y*
(fi).
This scries is convergent so long as j^<l and |y|<l. It
can, in fact, be summed ; for, adding X + Xy to both sides, we get
(l + X)y + X = X/(l-x)(l-y),
that is, (1 + X)y--y + Xj/(l-ar) = 0.
Hence, remembering that the value of y with which we are
concerned vanishes when x = Q, we have
y = [1 - 7(1 - 4X (1 + X) x/(l - a-)}]/2 (X + 1) (7).
Now, provided 4X (1 + X) j-/(l - a:) < 1, that is, x< 1/(2X + 1)',
the right-hand side of (7) can be expanded in an absolutely con-
vergent series of integral powers of x, the absolute term in which
vanishes. Also, when x<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 = C,.r+(7jJ^+C,a:^ + . . . (8).
then the equations for determining C|, C%, Ci, . . . will be
found by putting (1, 0) = (2, 0) = (1, 1) = . , , = X in (5), namely,
e, =X,
C,=X(1 + C. + C,»),
C, = X (1 + C, + C, + C + 2(^,(7, + C),
C,= X(1+C,., + C,., + . . . + C',"),
(9);
from which it is seen that C'„ t'„ C'„ . . . are all n-al and
positive.
§ 18 GEtfERAL EXPANSION THEOREM 877
Returning now to the system (5), and denoting modnli by
attaching dashes, we have, since (1, 0)', (2, 0)', &c., are all less
than X,
6/ = (l,0)'<\<^„
^-oX2, oy + (1, 1)'^' + (0, 2)7V^<\(1 + c, + Ci=)<c;,
^'3>(3, 0)'+(l, l)V+(2, l)'6i'+(l, 2)V+2(0, 2)Vi; + (0, 3)V,
< X (1 + (7j + (7, + (7.=' + 2Cx(7a + Ci') < C„
.• .■ ■ • (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 \x\< 1/(2X + 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
y' <hix' + bix^ + h3x'^+ . . .,
<C,x + C^x"'+C,x'^ + . . .,
<[l-V{l-4X{l+X)^7(l-^')}]/2(X + l) (11).
Now the right-hand side of (11) is less than 1, provided
a;'<l/(2X + l)''. If, therefore, |a:|<l/(2X + 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 x can be found to satisfy the equation (1) ;
and the like follows for x as regards y. The functions of x and y
thus determined, being representahle 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 tliat 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 KEVERSION or SERIES CH. XXX
be found to satisfy (I). We sli.ill settle these qiie-stions presently
in the ca,«e where tlie series 2i (m, ti)jfi/' terminates.
§ 19.] The problem of the Ileivrsion of Series pmiwrly so
called is as follows : —
Given the equation
a; = rto + o-.v'" + <»-+iy"+' + . . . (i;.
tv/iere a^=^0, but a, may or may not be zero, and the series
Om y" + <»ii.+i y""*"' + . . . is absolutely convergent so bmg as
\y\^a fixed poMtive quantity p, to find a convergent exiHuifion,
or convergent ejrpansions, /or y in ascending powers qf x-a,.
Let ^ denote {(jr-aoVa.}""', that is, tlie principal \'alue of
the with root of (x-ao)/(J„, and w„ a primitive nith ri>ot of
unity, then (1) is efpiivaleut to m cnuations of which the
following is a type : —
Now, the series inside the bracket in (2) being absolutely
converj;ent for all values of y such that lyl^^p, it follows from
the binomial theorem combine*! 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, siiy,
-<"»\* + i/+C,y+(7,y' + . . .=0 (3).
It follows, therefore, from the general theorem of last para-
graph that we have, provided |(| does not exceed a certain
limit,
y=*.«--'^ + V-'f' + 6.«^.^''^-. . . (4).
We have, of course, m such results, in which the coefficients
bi, bt, b,, . . . will be the same, but r will liave the different
values 0, 1, 2, . . ., (ot-1).
Each of these solutions is, by chap, xxti., § 19, a continuous
function of x. If wc cause x to circulate about a, in Ar^'and'a
Diagram, the m branches of y will piss c*>ntinuou.<<ly into each
other; and after m revolutions the branches will recur. The
point a, is therefore a liranch Point of the f/ith order for the
function y, just as the puinl U ia for the iuuctiuu w''* in
chap. XXIX., g 5, 6.
§§ 18-20 EXAMPLES OF REVERSION 379
Cor. In the particular case where aa = Q, m = \, we get tlw
single solution
y=hiX + biX^ + bs^ + . . . (5).
Example. To reverse the series
r = l + ;//ll+2/='/2l + 2/3/3! +.. . (G).
Let ; = 1 + X, Iheu we have
Hence, provided | x | lie withiu certain limits, we must have by the
general theorem
y = b^T.^-b„x'' + b,x' + . . . (S).
Knowing the existence of the convergent expansion (8), we may determine
the coeflicicnts as follows.
Give 2/ a small increment i-, and let the corresponding increment of a; be /(;
then, from (7), we have
A- jj + 2, + 3j +. . . .
Hence, since i{(2/ + fc)"- !/"}/'>" = «!/""' when k=.0, and since, owing to
the continuity of the iseries as a function of y, /i = 0 when /i; = 0, we have
^fc-^+ii + 2"i+---
= \ + x
(9).
Again, from (8), we have, in like manner.
ft
Lr = 6i + 26aX + 3i3a:- + . . .
(10).
Combining (9) and (10), we have
6i + 26ji+3&3i»+. . . = l/(l + x),
= l-x + x»-. . .
We must therefore have
.
fc, = l, 6,= -1/2, 63 = 1/3
Therefore
X x' x»
S' = i- 2" + ^"- • •■
_z-\ (z-lf (z-lf
~ 1 2 ' 3
(11).
It must be remembered that (11) gives only that branch of the function y
which is expansible in powers of z-1 and which vanishes when 2 = 1. We
have, in fact, merely given another investigation of the expansion of the
principal value of logz.
§ 20.] Expansions of the various bianches of an Algebraic
Function.
The equation
2(;«, »)a-V + (0, 0) = 0 (1),
380 DEFINITION OK ALOEISKAICAL KUNCTION CO. XXX
w/wTo the series on the left terminates, gives for any aiwigned value
of a: a finite number of values of i/. If the highest power of y
involved be the Hth, we might, in fact, write the equation in the
form
Anif + A„.,y'-' + . . .+A^y^■A, = 0 (2),
wliere Ao, Ai An are all integral functions of x. If, then,
we give to x any pjirtioular value, a, real or complex, it follows
from cliaj). xil., §23, that we get fmni (2) n corresponding values
of y, say b„ b-^ b^- If we change x into a value o + /<
difl'ering slightly from a, then i,, 6, 6, will chango into
ft, + A,, hi + /■;, . . ., bn + /"„ ; that is to say, we shall get n values
of y which will in general be different from the former set. We
may therefore say that (2) defines y as an n-valued function of
x; and we call y when so determined an nhjibraic function of a:.
Since every equation of the form y=F{x), where Fix) is an
ordinary synthetic irrational algebraic function (as defined in
chap. XIV., § 1), can be rationali.sed, it follows that every ordinary
irrational algebraic function is a branch of an algebraic function
as now defined. Since, however, integral eqtiations whose degree
is above the 4th cannot in general be formally solved by means
of radicals, it does not folK)w, 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 aUo supi)osc 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 root.'* of the equation formed by equating
this factor to 0 would be the common root or roots in question.
Therefore the eqtiation (2) could be broken up into two integral
equations in y whose cot'iTicipnts wo\ild be integral functions of x;
and each of these would dofiuf a separate algebraic function of j*.
The condition of irreducibility involves that (2) cannot have
§ 20 SINGULAR POINTS 3S1
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
Vu Vi, ■ ■ ■, yn, the equation
^(^-yi)(y-2/-2) ■ ■ . {y-y<,-i){y-ys+i) ■ ■ ■ {y-yn) = 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 ^o, Ai, . . ., A„, and there-
fore integral functions of x*. 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 speciahsed) 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 commont. In
other words, it may hajipen that the n branches of y have points
in common; hut it cannot hajipen that any two of ttw n branches
wholly coincide.
When, for x = a, the n values b,, b^, . . ., 6„ are all different,
a (or its representative point in an Argand-diagram) is called an
ordinary point of the function y, and 61,62, . . ., 6„ single values.
If 61 = 62 = . . . = br, each = b, say, then a is called an r-ple paint
of the function, and b an r-j)le value.
For every value of x (zero point) which makes Ao = 0, one
branch of y has a zero value ; for every value of x (double zero
point) which makes ^o = 0 and Ai = 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 A„ = 0, one branch
of y has an infinite value ; for every value of x (doubk pole)
which makes -4„ = 0 and ^„_i = 0, two branches have an infinite
* See chap. xvin. , § i.
t These are the values of x for which
and n^„;/"-' + (n-l) J„_ii/»-2 + , . .+J, = 0
have a root iu common.
382 EXPANSION AT AN OKDINARY POINT CH. XXX
value; and so on. These may be calloil ainoU, (ImliU-
in/inities of the function.
Tlie 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, ex]>ansible in an ai:rrnding or descending
power series of a jtarticular kind ; and thus to shmc that every
branch is, except at a pole, continuous fur all finite values of x.
§ 21.] If, at the point x = a, the abjibraic function y has a
single value y = b, then y-b is, within certain litnits, expansible
in an absolutely convergent series of the form
y-b=C,{x-a) + Ct{x-a)''+ Ct(x-ay + . . . (4).
Let x = a + (, y = b + r), then the equation (1) becomes, after
rearrangemeiit,
(0, 0) + (l,0)f + (0, l)i7 + (2, 0)i' + &c. = 0 (5).
Since y = b is a single root of (1) correspoutling to x=a, it
follows tliat 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 + C,i' + . . ..
and no other of the kind will satisfy the equation (5) ; that is,
y = b + Ct (x - a) + Ct(x - a)* + C,{x - ay + . . . (6)
will satisfy (1).
The function y determined by (6) is continuous so long as
|ar-a| is less than the radius of convcrgeucy of the series
involved; and it has the value y = b when x = a.
If we suppose all the values of y, say b,, b,, . . ., b„ corre-
sponding to a: = 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 nf ihi' branches of an algebraic function
have a point in common, each branch ui a continuous J unction qf x;
and the increment of y at any point of a particular branch it ex-
^ 20-22 EXPANSION AT A MULTIPLE POINT 383
pansible in an ascending series of positive integral poicers of the
increment of x 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 x = a is a multiple
point of the function y.
We shall prove that for every multiple point of the qth order, to
vhlch corresponds a q-ple value y = b, we can find q different con-
vergent expansions for y of the form y = b + 'S,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 analj-sis that follows if we premise the follow-
inj5 remarks regarding expansibility in ascending power-series
in general : —
If 17 be expansible in an absolutely convergent ascending
series of positive powers of i, of the form
, = C, i-. + d'".+'^ + (7s£».+-^+''^ + . . . (A),
where a,, oo, . . . are all positive, then we can establish a series
of transformations of the following kind: —
';„-. = l'«(C„ + '?„) (B),
where i/i, v-i, • • ■, Vn all vanish when ^ = 0; Ci, C,, . . ., C,
are all independent of ^, and all diflerent from zero ; and
C, = Lnli''', G^ = Lnili''' C, = Lr,n-,/i'^ when f=0.
(Conversely, if we can establish a series of transformations of
the form (B), and if we can show that rin is expansible in a series
of ascending positive powers of $, it will obviously follow that 17
is expansible in the form (A).
Let now y = bhe a y-ple value of y corresponding to x = a,
and put as before x = a + i, y = b + r], tiien the equation (1)
becomes
2K «)?"»;" = 0 (7).
384 EFFECTIVE nnOUP OF TERMS CIl. XXX
Since q values of y become b when x = a, q values of >7 must
become 0 when ^ = 0. Hence the lowest power of rj in (7)
whidi is not muitiiilied by a power of ^ must be if. There
must also be a power of t' which is not multiplied by a jwwer of
T), otherwise (7) would be divisible in general by some power of
ij, which is impossible since (1) is irreducible. Let the lowest
such power of i be i''.
Put now
, = ^((7, + 7j.) = ^» (8),
and let us seek to determine a positive value of X such that
Ci = i/T = Zij/i* is finite both ways* when f=0.
The equation (7) gives
2(m, «)f^*"e" = 0 (9).
Now (9) will furnish values of r which arc finite both ways when
^ = 0, provided we can so determine A that at least two tonus of
(9) are of the same positive degree in i, and lower in degree
than all the other terms.
Assume for the present that we can find a value of X for
which a group of r terms has the character in question, so that
8=»n, + Xh, = OT, + Xn, = . . . = njr + Xwr (10),
where »», :^ w, :J> . . . :^ «r ;
and ^ = {f»i - mr)/{nr - th) = g/fi, sixy, (11).
where g is prime to h,
S = {mi/i + n,g)/h.
Then, putting d = f'*,t so that f, = 0 when ^ = 0, and dividing
out ^i"*!**"!', we deduce an equation of the form
<t>{i„ «)^, + (mr, «r)i'"' + (;nr-i, nr-i) «"'-' + . . .+("1,, n,)«"i = 0
(12),
where <^(^i, v) is an integral function of ^, and v.
For our present purpose we are concerned only with those
* That in, noithcr zero nor infinite — n Dscful phrniio of Do MorRan'i.
t It i» Hiifl'icicnt for our iiurjioiio to tukv tku |>riDci]i»l value iuurcl>' of tho
/ilk root of {.
§ 22 EFFECTIVE GROUP OF TERMS 885
roots of (12) whose initial values are finite both ways. There are
evidently rir-ih such roots, and their initial values are given by
(nir, llr) »"'-"! + {nir-i, «r-l) V^r-l-"! + . . . + (tBi, «,) = 0
(13).
If the roots of (13) are aU different, then we get % — jti trans-
formations of the form (8) ; and the coiTcspouding values of v,
that is, of Ci + r]i, 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
Vx = diii + d.2$i- + . . .,
= dj"^ + d,$"'^ + . . . (14);
and each of these will give for rj a corresponding expansion of
the form
7, = Cr&'^ + d,$'9+^V^ + d^^+'^"' + . . . (14').
If a group of the roots of (13) be equal, then we must
proceed by means of a second transformation,
v. = i.''{C, + v.) (15),
to separate those roots of (12) which have equal values. If the
next step succeeds in finally separating all 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 «r - »i 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, otiierwise 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
1) is single- valued in the neighbourhood of the point i = 0 ; that
is to say, if we cause i to circulate about the point | = 0 and
c. II. 25
386 NEWTON'S PARALLELOGRAM CI!. 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 scries will Uiko
the form
where one at least of the fractions afq, pjq, . . . , is at its lowest
terms. The function >; is then g'-valued and the series (14")
will as in g 19 lead to a cycle, as it is called, of q branches
which pass continuously into each other when i is made to
circulate q times round f = 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 5th 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)
Euilicient to give us q exi)ansions corresponding to the q branches
which meet at the g'-ple point x = a.
Tiie 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 0 Y (Fig. 1) be a pair of rectangular axes,
the first quiulrant of which is ruled into squares (or rectangles)
for convenience in plotting points whose co-ordinates are positive
integers. For each term (m, n)f"i>" in equation (7) we plot a
point K {dt'ijree-point) whose co-ordinates are 0M~ m, iIK= n.
We obser\e that^ if KP be drawn so that cotA7-'0 = A, then
OP = OM + MP = m + nK Hence OP is the degree in f of the
term in (9) which corresponds to (ni, w)i'">;''- If. tiiercfore, we
select any group of terms whoso degree-points lie on a stniight
line A, these will all have the same degree in ^, namely, the
intercept of A on OX.
Tlie necessary and suflicient conditions, therefore, that a
group of two or more terms furnish the initial values of a group
of e.\]>ausions, let us say be an ejffectlve group, are : —
1°. Tliat the lino A containing the degree-points shall cut
OX to the riglit of 0, and 0 Y aliove 0, Tliis secures that X be
positive.
§22
NEWTON S PARALLELOGRAM
887
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
than those of the selected group.
(9) be of higher degree in $ ■
Y
V
1
-
x.
'r
^
---
-*
i.\
x,l
/
F
•v,
/
H
\
i
J
K
\,
\
\
N
y
\
N
*
\
s
s
1
V
D
s
■s
L
V
s
s
\
c
1
\
-
■-
~t^
B
\
t
>S
-^A
( \
\
J
'3
t.
*
^h
P
X 1
I
V,
V
Fio. 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 t) alone, so that OA =p, 0E= q. Let a radius vector,
coinciding originally with AX3, 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 o\
them taken in order from A be C. 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
tukeu in order from G be D. Then let the radius turn about D ;
25—2
388 EXAMPLE OF EXPANSION CH. XXX
anrl so on, until at last it passes through E, or through a group
of which E is the last.
We thus form a brokeu 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 diflerouce between the degrees of ■>} in the first
and hist tonus C and D ; but this diflerence is the projection of
CD on OY. The sum of all the projections of AC, CD, &c., on
Oy is OE, that is to .say, q. Ilcuce we shall get, by taking all
tiie groups AC, CD, &c., q difl'erent expansions for y correspond-
ing to the q different branches that meet at the multiple point
x^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 x-a.
Example 1. To separate the branches of the function i) at the point { = 0,
») being dutermined by
+ /J|'»,,"=0. (ir,).
The lowest term in i\ alone is i)'", eg that { = 0 is a multiple point of the
10th order. Plotting the decrees of the terms in Newton's diagram, and
naming the points by affixing the cocfiioients, we find (see Fig. 1) that the
effective groups are AliC, CD, DK. Taking, (or simplicity of illustration,
A = +-i, B=-3, C=+l, D=-l, E=+l,
we get from the group AUG
\=C/2^3/l, so that h = l, aud ti°-3t> + 2 = 0 gives the initial values of d,
that is, v = l, or 2, the corresponding expansions being
From the group CD, wo get
X=l/3, t;'-l = 0 gives the initial values of r,
that is, i> = l, u, u', where u is a primitive imaginary cube root of 1, the
corresponding expansions being
,={'/>(! +d,{"' + rf,fw+. . .),
§ 22 EXAMPLE OF EXPANSION 389
In like manner, DE gives five expansions of tlie 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 ij at the point {=0, ij being
determined by
ie-Si*- 4^- h -i) + i(r,- f )- = 0 (17).
The effective group for (17) at the point f =0 corresponding to branches
which have the initial value ij=0 is 4(ij-t)-; as will be readily seen from
Newton's diagram.
X = l, /i = l and, if i;=|(Cj + i;,)=|t), we have
4|'-3«-4|(v-l) + 4(t>-l)= = 0 (18).
Hence two branches have the same initial value for v, viz. v = l. For
each of these ri = i{i + Vi)', and we have for tj^ the equation
'i4^-3e-iiVi + H-=0 (IS').
If we draw Newton's diagram for (18"), we find that the effective group is
*Vi^-iiVi~^?'; and that X=l. Put now i;i = {(C„ + i;o)=|i;i; and we get
4| + (2f,-3)(2r, + l) = 0 (19).
The initial values of v^ are given by (2t)i - 3) {2i\ + 1) = 0, which give the
«i7i<;te values i', = 3/2, «i= - 1/2. Hence for the two branches we have
,,=|(3/2+,j); r;i'=l(-4+%');
and the farther procedure will lead to integral power series for ijj and ijj .
We have therefore for the two branches
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 au
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 we have
Ae^ + Bi^rj + CiW^O.
This gives, when we drop out the irrelevant factor ^,
390 ALOKRUAIC FUNCTION ALWAYS EXPANSinLE CH. XXX
which breaks up into two equations,
and thus determines two functions, each of which has a l>ranch
coincident to a first approximation with a branch of >; (as deter-
mined by (16)) which has zero initial vahie.
In lilvc manner, CD gives Ci* + />>;' = 0 ; and DE gives
We tluis get a number of binomial equations, each of which
gives an approximation for a group of branches of the function
ij 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), convergimt
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 a; or y or both become infinite.
Ist. Let us suppose that the value of the function y tends
towards a finite limit b when x tends towards oc. Then, if we
put >; = y - 6, X = f , we shall get an equation of the form
2(ffi, n)r",," = 0 (17),
which gives »? = 0 when i=^.
Jjdt us suppose that Fig. 1, as originally constnicted, is the
Newton-diagram for (17), and let i* bo the highest power of (
that occurs in (17) so that 00, = k. Now in (17) put i=l/^,
and multiply the equation by ^'* ; we then get the equation
2(m, «)f*-"v" = 0 (18),
which is obviously equivalent to (17).
But the Newton-diagram for (18) is obviously still Fig. 1,
provided 0,X, and 0,1', be taken, instead of OA' and OF, as
the positive parts of the axes.
Hence, if we make a boundary convex towards 0, in the
same way as we did for 0. we sliall obtain a series of branches
of r) all uf wliich are exp)ln^>ible in a-scending powers uf i', tiuit
§§ 22, 23 EXPANSION AT POLES, &C. o91
is, in descending powers of $, and all of which give jj = 0 when
^=00. For each such branch we have
that is,
(ij-b)a^ = c + d/a^ + e/x^ + . . . (19),
where A, a, /?, . . . are all positive, and c is finite both ■wa}-s.
2nd. Suppose that a: = a is a pole of y, so that i/ = <x> when
a; = a ; and put i; = y> ^ = x — (t, so that we derive an equation
2 {m, n) f'Tj" = 0 (20),
for which Fig. 1 is the Newton-diagram with OX and 0 F as
axes. Then, putting t; = l/V, we get an equation of the form
2 {m, n) e'v''-" = 0 (21),
I being the highest exponent of rj in (20).
The Newton-diagram for (21) is then Fig. 1 with 0,A'i
and Oil^i as axes; and we construct, as before, a boundar)',
EFG say, convex towards Oi, every part of which gives a series
of branches of 77', that is, of l/»;, expansible in ascending powers
of i. For every such branch we shall have
7y.^=l/(c-Hrf|'' + ^^3 + . . .),
where X, a, /8, . . . are all positive, and c is finite both ways.
Hence also, by the binomial theorem combined with § 1,
7?^^ = l/c + rf'4" + e'^'+. . .,
that is,
ij(.v-a)'' = l/c + d'{x-a)'+e'(x-ay+. . . (22),
where X, a, ji', . . . are all positive, and c is finite both ways.
3rd. Suppose that y has an infinite value corresponding to
a: = 00 (pole at infinity). Then, if we put a; = f = 1/f , 2/ = •^ = l/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
7)' = i'>'{c + d^"' + ei'^ + . . .}.
Whence, as before, for every such branch
yla^ = \j{c + dl3f + elafi + . . .),
= l/c + d'/af + e'/a^+. . . (23),
where X, a, P', . . . are all positive, and c is finite both ways.
392 AU3EnHAIC ZEROS AND INFINITIES CH. XXX
If we ciiiiibine the results uf the present with those of the
foregoing paragraphs, we arrive at the following important
general theorem regarding any algebraic function y: —
lfy = 0 when x = a (a4=oo), then L yl(x-aY is finite both
ways.
Ify = 0 when ar= oo , then L yjx'*- is finite both ways.
Ify=<x> irhen a: - a (a 4= oo ), then L y/{x - o)"* is finite both
ways.
Ify='X) w/ien ar= co , then L y/x^ is finite both ways.
X is in all cfise.i a finite positive commensurable number
which may be called tlie order of the particular zero or ii\finite
value qfy.
This tlicorcm leads ns naturally to speak of algrhrnical trro- or infinity-
rallies of functions in t;<^neral, moanini; such as bave the property just
stated. Thus 8in:r = 0 when x = 0; but Lsin jr/x = l when z = 0 ; therefore
we suy that sinx has au algebraic zero of the tirsc order when x = 0. Attain,
tanx = ao when x = hr; but Lt8nx/(x- Jt)-' is finite when x=Jf; the
inlinity of tan x is therefore alKebraieal of the first urdor. On the other
hand, r'=x) when x=x ; but this is not an algebraical infinity, since no
finite value of \ can be found such that Le'lx^ is finite when x = cd. (See
chap, ixv., § 15.)
§ 24.] Application of the method of successive approxima-
tion to the crpansion of functions. This method, when applied
in coiijnnctiou with Newton's diagram, greatly increases the
practical usefulness of the general theorems which have just
been cstiibli.shed. Tiie method is, moreover, of great historical
ititcrest, because it apiiears from the scanty records left to us
that it was in this form that the general theorems which we have
been discn.ssing originated in the mind of Newton.
Let us suppose that the terms of an ecpiation (which may be
au infinite series) have been plotted in Newton's dijigram, and
that an effective group of terms hits been found lying on a line
A; and let ij"-f" (the coeflieients are taken to be unity for
8iini)licity of illustration) be a factor in the group thus selected,
repeated, say, a times, so that the whole groui) is ijl>,(f, j?) (>?'"— f")".
Let A bo moved parallel to itself, until it meetj) a term or group
§§ 23, 24 SUCCESSIVE APPROXIMATION 393
of terms </)j {$, '/) ; tlieu again iintil it meets a grcmp <^3 {$, ?/) ;
and so on.
The complete equation may now be arranged thus —
<^i {^, v) ('?"' - r)" + 4>. (f . v) + i>. (f, v) + . ■ . = 0,
or tlms —
say, (V"-t")° + '-2 + '-3 + - - -=0.
Now, by the properties of the diagram, when ■>] = f '"",
<^a (^, '?), "^3 (^, v). • • • are in ascending or descending order as
regards degree in ^, and the same is true of tj, tj, . . . Let
us suppose that f and r] are small, so that tj, tj, . . . are in
ascending order.
As we have seen, rj'" = ^, that is, »/ = ^""", gives a first
approximation. To obtain a second, we may neglect t,, tj, . . .,
and substitute in t._, the value of rj as determined by the first
approximation. To get a third approximation, neglect tj, . . .,
substitute in tj the value of »? as given by the second approxima-
tion, and in t, the value of ■>/ 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, 17 = 1^'", in t^
does not cause r^ to vanish, which will happen, for example,
when <^o(^, rj) contains i?™-^" as a factor. In such a case the
beginner might be tempted to put Ta = 0 and go on to substitute
the first approximation in tj. This would probably lead to error.
For, if we were to substitute the complete value of 17 in tj, 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 t,. The best course to follow in such cases may be
learned from Example 5 below, which deals with a case in point.
394 EXAMPLES Cn. XXX
Exiimple 1. Taking Uio equation (10), to flud a third approximation to
one of the brandies of the group CD.
Next in order to C and D a parallel to CD meets Bucccssivcly li and A.
Hence, putting, for simplicity, D=+l, C=D = A=-1, the equation (16)
may bo vrritten
r'l'('7'-f*)-f"ij-{"+. . - = 0.
Whence »;'-£*- r/-! - {'7')' + • • - = 0 (Z.-^.
The first approximation is 17 = 1*'' ; hence, neglecting {"/i?* in (25), wo j,'il
for the second
Whence i) = f«'''(l + f')">=f*»(l + J{''^) (26).
If we nee this second approximation in {^/i), and the first approximation
in {'"/i)' now to be retained, (25) gives for the third approximation
1)' - f* - r/f ''' (1 + If") - f "/i"* = 0.
Whence, if all terms higher than the last retained bo neglected,
which gives
,= {«.■> (l+^/»+3£10,»)t.
= {*'(! + if" + ii'") (27),
which is the required third approximation.
This might of course bo obtained by successive applications of the method
of transformation empIo}-cd in the demonstration of § 22, or by the method
of indeterminate coeOicicnts, but the calculations would be laborious. It
will be observed on comparing (27) with the theoretical result in § 22 that
dj=d, = df = di = df=dj=df=df = 0; a fact which, in itself, shows the advan-
tages of the present method.
The other branches of the cycle to which (27) belongs are given by
, = (a,{> ')« {1 + i (a,£''')» + J («£'-»)■•} .
where u is any imaginary cube root of unity.
Example 2. To find a second approximation for the branches corre-
sponding to AUC in equation (16), in tlic special case where A=i +1, B= -2,
C= + l, D=-l.
The terms concerned in this approximation aro {ABC) and {D). We
therefore write
r(')-w-e'v=o.
or (.(-a'-Wf^o.
The first approximation is i) = ('; hence the second is given b^
(v-i')'-V'lt=o,
that is, (')-f')'-{"=0.
Whence ,-{»i|ii/"=0,
which givrs the two second approximations oorrosponding to the group.
I'here arc two, because to a firm approximation the hranc-lics are coincident
This, therefore, is a cose where a second approxiuuliuu is necessary to
distinguish the brauchug.
§ 24 EXAMPLES S95
Example 3. To fiutl a second approximation, for large values both of
f and 7), to the branch corresponding to HI in equation (16).
Referring 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
(H|'V' + ^l'V) + GW = 0-
For simplicity, let us put H = l, 7=G= -1, and write the above equation
in the form
Confining ourselves to one of the five first approximations, say ))=t^'°, we
get for the second approximation
which gives 1?=!*'^ (1 + ir^'"')-
The other branches of the cycle are given by
where u is any imaginary fifth root of unity.
Example 4. Given
x=y + y-li\ + y'ji\ + y*li\+ .. .,
to find 2/ to a fourth approximation. We have
j/=x-J/=/21-!/3/3!-2/-'/4!- . . . .
Hence 1st approx. y = x.
2nd ,, y=x-ix\
3rd „ y=x-)i{x-ix-)--i^^
= x-ix'' + i:i?.
4th „ t/ = x-4(i-ix= + ix=)2-i(x-Jx-)'-,\xS
Example 5. To separate the branches of ij at f =0, where
4|5_3|4_4|2(,_|) + 4(,-f)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 i;={ is common to the two branches.
Since ij-J is a factor in { },, we cannot obtain a second approximation
by negleotmg { }s and putting 7;=| in { }.,. In obtaining the second
approximation we therefore retain { },, treating jj-J as a variable to be
found. We thus get
4(n-tT-4s"(')-«) = 3|*;
whence {2 (t) - I) - t''}'=4i'',
which gives v=^ + '^i'l^i
or v'=i-m-
The branches are thus separated.
396 HISTORICAL NOTE CH. XXX
If a third approxiiuatioa wcro required, we should now retain { }^, and
write
i2{.;-0-P}'=4{*-4i'.
Henoe 2(,-f)-{»= ±2{'(l-f)»,
= ±2{'(l-{/2).
We thus get
i,={ + 3f-/2-f'/2;
V=f-r/2 + f/2.
Historical Note. — As has alrcoily been remarkwl, the fandnmrntal idea of the
reversion of series, and of the expaiiHinn of the roots of algebraical or other equa-
tions in power-scries. on(rin:ilcd with Niwton. Uis f.-'nions " P:iralUlo(;min " is
first mentioned in the second letter to Oldeiihur);; but is more fully ciplainod
in the Geometria Anahjiicn (sec HorsUj's c<lilioii of Newton's Workt, t. I.,
p. S98). The mctho<l was well nnderstoml by Newton's followers, Stirling and
Taylor; bat sceius to have been lost sight of in Englaml after their time. It was
much used (in a nuidilicd fonn of Dc Gua's) by Cramer in his well-known Analy$»
des Ligne.1 Courhes Algdhriijiut (1750). Lngritnge gave a complete analytical form
to Newton's method in his "Mi'Uioiro snr I'Usngu dcs Fractions Continues," yi>uv.
Mint. d. I'Ac. roil. <i' Sciences d. Berlin (177G). (See (Kitrres de Larrramje, t. rv.)
Notwithstanding its great utility, the method wns everywhere all but forgotten
in the early part of this century, as has been pouited out by Do Morgan in an
interesting account of it given in the Cambridge Philosophical Traiuaetions,
vol. IX. (\»:>5).
The idea of demonstrating, a priori, the i«>ssiliility of expansions such as the
rcTcrsion-formnhc of § is originated with Cauchy ; and to him, in effect, arc duo
the methods employed in §5 IS and I'J. See his memoirs on the Integration of
Partial Differential Kqimti'iis. on the Calculus of Limits, and on the Nature and
Properties of the Itonts of an Equ:itl«n which contains a Variable Parameter,
Exercices d* Analyse et de rhysiquc Afath^matique, t. 1. (IS-iO), p. 327; t. ii.
(1841), pp. 41, 10!l. The form of the demonstrations given in §§ 18, ly baa
been borrowed partly from Thonme, A7. Theorie der Anali/tischen Functiontn
einer Comjilexen Ver/tnderlichcn (llallo, ISSO), p. 107 ; partly from StoU, Alhje-
meine Arillimelik, I. Th. (Leipzig, 18S5), p. -I'M.
The Parallelogram of Newton was used for the tlieoretical pnrpose of cstablisli-
ing the expansibility of the brnnches of an algebraic function by Puiaeux in
his Classical Memoir on the Algebraic I'unctinus (Liour. Atalh. Jour., 1850).
Puiseux and Briot and Bouquet {7'hcone des functions f.'llii'liques (1S75), 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 denumstra-
tion given in § *J1 depends U|Kin the iiroof, a priori, of tlie possibility of an
expansion in a {wwer-serios; and in this respect follows the original idoa of
Newton.
The reader who desires to pursue the subject farther may consult Dnr^go,
Elrmentc der Theorit dir Functionrn einrr Ct'mpUren Veranderlichen OrGste, for
a go<Hl inlrwluction to this great branrh of modern function-theory.
The English student has now at his di.sposal the two treatises of Harknew and
Miirhy, and the work of Forsyth, which deal with function-theory from Tarious
points of view.
The applications are very numerous, for example, to the dnding of corvatores
and curves of closest contact, and to curve-tracing generally. A number of
iM-autifnl cxamiiles will be fuuud iu that much-to-bo-rocouuueuded text-book,
Frost'a Curve Tracing,
§ 24 EXERCISES XXIV 397
Exercises XXIV.
Eevert the following series and find, so far as you can, expressions for
the coefBoient of the general term in the Eeverse Series : —
(1.) j, = i + ^+'i(^W!L(!iZ^_(!Ll^W ... ,
(2.) y = x-ix^ + ix''-}x'+ ....
x' ^ x"^
(3.) 2/=^- 37 + 5! "71+ •• • •
(4.) y = i+x2/2=+xS/3= + xVi=+ ....
(5. ) If !/ = sin i/sin (x + a), expand x in powers of y.
X 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 | i/ 1 , or both, become either infinitely small or
infinitely great : —
(G.) y^~2y=x*-x"-.
(7.) a^(y+x)-2a-x{y + x)+x^=0, (F. 69*).
(8.) {x-yY-{x-y)x^-ix*-iy^=0, (F. 82).
(9.) a{y"--x^){y-2x)-y*=z0, (F. 88).
(10.) ax{y-x)--y* = 0, (F. 96).
(11.) x(y-x)--a3=0, (F. 115).
(12.) x^y-^-2a^x-y + a*x-h^=0, (F. 121).
(13.) y{y-x)-{y + 2x) = 9cx^ (F. 131).
(14.) {x{y-x)-a-'ry^=a-', (F. 140).
(15.) a? -x*y^ + a^y*-axy^=0, (F. 143).
(16.) a{x^+y'')-a-x^y+3?y*=0, (F. 143).
(17.) x^y* + ax-y^ + bx*y + cx + dy'- = 0, where a, b, c, d are all positive,
(F. 155).
(18.) If e„ be any constant whatever when ti is a prime number, and
Buch that e^=epe^e^ . . . when n is composite and has for its prime factors
p, q,r, . . ., then show that
If a, i, c, . . . be a given succession of primes finite or infinite in number,
s any integer of the form a^b^cy . . ., t any integer of the forms a, ah,
abc, . . . (where none of the prime factors are powers), and if
i^(a:) = 2e./(x»),
then /(x) = 2(-)"6',F(x'),
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, xviii., p. 518 (1884).)
* F. 69 means that a discussion of the real branches of this function,
with the corresponding graph, will be found in Frost's Curve Tracing, § 69.
CHAPTER XXXI.
Summation and Transformation of Series
in General.
THE METHOD OF FI.MTE 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
cliaracter 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 a<lditional 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 heret<ifore, «, be the nth term of any series ; in other
words, let u, be any one-valued function of the integral variable
n; u,_,, «,_,, . . ., u, the same functions oi n-l, n-'Z, . , ,, I
respectively.
Farther, let Au„ Am,_,, . . ., Sii,
denote «<i.+i-«», ««-".-i, . . ., «,-«,;
also A(Au,), A(A«,.,) A (Am,),
which we may write, for shortuuas,
§ 1 DIFFEllENCE NOTATION
t.
A=«„, A'm„_i, . . .,
A"«i,
denote
Ak„+,-A«„, Ai<„-AM„_i, . . ., A»o
- A«i ;
and so ou. Thus we have the successive series,
Ml, Mj, Ms, . . •> «». •
. .
(1)
AUi, A«,, A«3, . . ., Au„, .
. .
(2)
A-tii, A^M^, A^Ms, . . ., A=M„, .
. .
(3)
A»«,, A'lu, A'a^, . . ., A^(„, .
• •
(4)
399
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 «„ = n', the series in question are
1, 4, 9,16, . . . n", . . .;
3, 5, 7, 9, . , . 2« + l, . . .;
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'-«,, A'-«.„ A'-Mj, . . ., A'-«„
then the series of 1st, 2nd, 3rd, . . . differences will be
A'+'m,, A-'+'m,, A-'+'ms, . , ., A'-+ii<„, . . .;
A'->--u^, AT+^u.^, A'-+-u^, . . ., A^+-«„, ...;
A^+»«., A-'+^K,, A-'+'ms, .... A'-+»«„, ...;
In other words, we have, in general, A''A'm„ = A'"+*«n. 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 formulas 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
^<^ (n) = «^ (n + 1) ; £"«*, = «,+, ; £*«, = «, ;
and 80 on.
In accordance with this definition we have E(Eii^), which we
contract into £Pu^, = Eii^+i = «»+, ; and, in genend, A'"tt, = «,+«.
We have also, as with A, E''L''u^ = E''*'tt„, for each of these is
obviously equal to u^+r-n-
Example 2. £'ii»= (ii + r)'.
Example 3. The nitli iliflercuce of an iutcgral fanction of n of the rtb
degree is an integral fuuction of the (r-i»)tb degree if ni<r, a cocstant if
r=ni, zero if m>r.
Let
^r (n)= an' +in'^'+ en'-' +. . .;
then
A^,(n) = a(n + l)'- + 6(n + l)'^>+c(n + X)'^>+. . .
-an""- brf-^- cn''~'+. . .,
= r<i«'-' + {ir(r-l)a + (r-l)l}n'--» + . . .,
= *'r-l(").
say, where <t>^i (n) is an integral function of n of the (r - l)th degree. Then,
in like manner, we have i^^., (n) = 0,_, (n). But A^p_,(n) = A-0,n ; hence
AVr(") = *'r-i(")- Similarly, AVr(") = 0r-j('>); wl. in general, A'"<i,(n)
= 0,._„(n). We see aUo that A''<t>r{n) will reduce to a constant, namely, r'.a;
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 - 1) 6), 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 tlie exponent, and 6 the difference of the
faculty. This notation wo shall occasionally Uf=c in the slightly modified
form a""'^ which is clearer, especially when the exponent is compound.
Since
a{a + b) . . . (a + (m-l)t) = i'"(a/6)(fl/t + l) . . . (<i/6 + m-l),
any faculty can always be reduced to a multiple of another whose differenoe
is unity, that is, to another nf the form c""", which, omitting the 1, we may
write c""'. In this notation the ordinary factorial ml would be written 1 "".
The reader should carefully verify and note the following properties of
tlie differences of Faculties and Factorials. In all cases A opiiatc^ as unual
with respect to n.
Example 4.
A(a + fcn)"»'»=mi{a + 6(n + l)}'"^'i».
Example 6.
A{l/(a + 6n)i»i»}=-m6/(o + tn)'"«-'i».
Example ti.
a-e{a-b)<**>'*
'a-b 0'"+''* *
§§ 1, 2 FUNDAMENTAL DIFFERENCE THEOREMS 401
Example 7.
Acos(o + |8n)= -2sini^sin(a + Jj3 + /3H) ;
Asin (a + |3n)= +2sin4/3co3(a + i/3 + /3H).
§ 2.] Funrlamcntal Theorems. The following pair of
theorems* form the foundation of the methods of differences,
both direct and inverse : —
I. A'"«„ = M„+„-„C, ?<„+„,-! +mC;M«+m-2 + . . . + (-)'"«„.
To prove I. we observe that
Am„ = «„+!- </„;
A M„ = tt„+2 — W„+i
- tln+i + Un,
hence
and so on.
- «„+2 + 2(/„+i-?/„,
= Mn+3 - 3K„+a + 3«„+i - Un ;
Here the numerical values of the coefficients are obviousl}'
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
^«mi «m+i = «m + ^Mm- Heuce, siuce the difference of a sum of
functions is obviously the sum of their differences, we have, in
like manner, «„+, = «„+! + Att„+j = «„ + Ai<„ + A («,„ + Ae/„,) =
«m + Au,„ + Ae/„ + A=«„. We therefore have in succession
* The second of these was given by Newton, Principia, lib. m., lemma v.
(1G87) ; and is sometimes spolten of as Newton's Interpolation Formula. See
his Itact, Methodiis Differentialis (1711) ; also Demoivie, Miscellaiiea Analytica,
p. 152 (1730), and Stirling, Methodus Differentialis, &c., p. 97 (1730).
c. 11. 26
402 SUMMATION BY DIFFERENCES Cll. XXXI
+ Am,+ A'm„
M„ + 2A«„+ A'm^;
««+.= «-+2Au.+ A'm«
+ A», + 2A'»„ + A'w„,
«„ + 3Ai/„ + 3A»«„ + A'««;
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 sj-mbol K, 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",/, -.(£-!)"'«„ (I.); «„+. = (! +A)"«« (II.).
§ 3.] The following theorem enables us to reduce the sum-
mation of any series to an inverse problem in the calculus of
finite diflerences.
If i\ be any function of n such that Ai', = ««, then
2H, = e„+,-r, (1).
This is at once obvious, if we add the equations
«,_i = Ap,.,=t>, -r,-,,
«. = At>, = v^i - v..
The difficulty of the summation of any series thus consists
entirely in finding a solution (any solution will do) of the finite
difi'ereuce equation AV, = u,, or t\+, - 1), = «,. This solution tan
be effected in finite t<,'rms in only a limited number of ca>c-,
borne of the more important of which arc exemplified below.
Ou the other baud, the above tlieurcu enables us to con-
§§ 2, .'5 EXAMPLES 403
struct an iiifiuite number of finitely summable series. All we
have to do is to take any function of n whatever and find its
first difference ; then this first diflerence is the ?ith 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.
Example 1. i: {a + nb} {a + {n + l)b} . . . {a + {n + m.-l)b}.
Using Kramp's notation, we have here to solve the equation
Av„={a + nbY"^"> (2).
Now we easily find, by direct verifloation, or by putting m + 1 for m and
r - 1 forn in § 1, Example i, that
A[{« + (n-l)6}i"'+'ii'/("< + l)&] = {a + ni}""'*.
Hence «,,= Ja + (7!- 1) ii}""+i'''/(m + l) ii is a value of v,, such as we
require.
Therefore
a ' ' {m + l)b ^' '•
Hence the wcll-knoicii rule
n
2{a + nb}{a + {n+l)b} . . . {a + {n + m-\)b}
-GJr{a + nh){a + {n + l)h\ . . . {a + {n + m-\)h} {a + (n + m)b}l(m + \)b
where C is independent of n, and may be found in practice by makinrj the two
sides of (4) aijree fur a particular value of n.
Example 2. To sum any series whose nth term is an integral function of
n, say/(ji).
By the method of oliap. v., § 22 (2nd ed.), we can express /(«) in
the form a + hn + cn{n + \) + dn(n + \)(n + 'i) + . . . Hence
if(n) = G + an + lbn{n + \) + \,:n(n + l)(n + i) + idn{n + l)(n + 2)(n + Z) + . . .
(5),
■where the constant C can be determined by giving n any particular value
in (5),
Examples. 'Zll{a + bn} '"">.
Proceeding exactly as in Example 1, and using § 1, Example 5, we deduce
_1 l/{a + fcg}""-'i''-l/{a+'^(fi + l)}i'"-'i''
", {a + lm}"»''> {m-l)b
Hence a rule for this class of series like that given in Example 1.
(G).
Example 4. To sum the series 2f {n)l {a + bn}"""', f {n) being an integral
function of n.
26—2
404. EXAMPLES Cn. XXXI
Decompose /(n), as in Example 2, into
a + p(a + bny"^ + y{a + bn)i*>i+i{a + bn)<*l> + . . . (7).
Then wc have to evaluate
a5l/{u + 6n}"»i' + /3Sl/{o + J(n + l)}"»->i» + . . . (R),
which can at once be done by the rule of Example 3*.
Example 5.
V lL'^ _ " ((0 + 6)""* (a + &)i»-'i>) . .
Thin 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 aeries whose terms are the Figurate Numbers of
the mth rank.
The Cguratc numbers of the Ist, 2nd, 3rd, . . . mnks are the numbers
in the let, 2nd, 3rd, . . . vertical columns of the table (II.) in chap, rv.,
§ 2,'5. Hence the (n+l)th figurate number of the mth rank is ,4^_,C„_,
= ii+m-i^» = "'("' + ^) • • • (m + n- l)/"!- Hence we have to sum the seriea
, , » i;i(>n + l) . . . (m + n-1)
^■^t 1.2 ... Fi •
Now if in (9), Example 5, we put a = m, b = 1, e = 1, we get
« ml"! _ (m+l)i«i m + 1
film- II. I - 1 •
Hence
ij.„ j-^'^ + ^'x m(m+l) . . . (m+n-1)
^ (m + l)(m + 2) . . . (m + l + B-l) .
1 . 2 . . . n
that is to Bay, the mm of thefirit n figurate numbert of the mth rank it the nih
figurate number of the (m + l)th rank.
Thiii theorem is, however, merely the property of the function „//,, which
wo have already established in chap, xxiii., § 10, Cor. 4. The present
demonRtration 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,
acme of which will be fuupd among the exercises below.
* The methods of Examples 1 tn 4 arc all to bo found in Stirling's iletho4u$
Differentialit. Ho applies them iu a very remarkable way to the npi<roxi-
matc evaluation of series which caunut be summed. (See Exercise*
xxvu., 17.)
§§ 3, 4 DIFFERENCE SUMMATION FORMULA 405
Example 7. To sura the series
S„=cosa + cos(a + /3) + . , . + cos(a + (n-l)/3);
r„=sina + siu (a + ^) + . . . + siii (a + {n- l)/3).
From § 1, Example 7, we have oos(a + j3«) = A {sin (o- Ji3 + j3«)/2sin J/3}.
Hence
S„={sin(a-4^ + /3n)-sin(a-i/3)}/2sm4ft
sin 4/371 , , , . , ,,,
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 Mi + «<2 + • • • + 'U ! s^utl let us add to the
beginning an arbitrary term «o- Then if we form the quantities
. . . , Sn=Uo+ lh + U2 + - • . + «n, • • • >
we have
A>S„ = Mn+i, A=^„ = A«„+„ .... A'",S?„ = A"'->«„«, ....
Hence, putting n = 0,
AS, = tH, A^S'o = A«„ . . . , A^^S^, = A"-' «,,... (1).
Now, by Newton's formula (§ 2, II.),
S^ = S, + „C, A^„ + „aA'S, + . . . + A"^„ (2).
If, therefore, we replace S^, ^So, A^aS'q, ... by their values
according to (1), we have
2!<„ = «o + nCilh + nCjAi<^ + nCA'lh + . . . + A""' (<j (3) ;
0
or, if we subtract Uo from both sides,
2m„ = „Cith + nCjAtt, + nCsA'w, + . . . + A"-'?*, (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 2a;" it gives
* This formula, which, aa 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 Ara Conjeetaiidi, p. 98 (1713).
406 hontmort's tueoreh cu. xxxi
J-' + i* + . . . + x'
n(n-l) / ,\ n(»-l)(n-2) , ,,.
= HJ+ \jj— 'dr(x-l)+ -^ ^ 'x(;r-l)' + . . .
+ x(x-ir-\
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 tiie word sum, only
when the »ith differences of the terms become lero, 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
nth 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 »»th degree must have for its nth
term an integral function of n of the (m — l)th degree- We have
thus reproduced from a more general point of view tlie results of
chap. XX., § 10.
Example. Sam the Beries
2{n + l)(n + 2)(n + 8).
1
If we tabulate the Cist few terms and the eaocessive diffeieooes, we get
1, 2, 3, 4, S
"«
A",
A'u.
A* II,
Hence, by (4),
r(n + l)(n + 2)(n + 3)
= „.24^"i"^.36^"J:^L^!l^>.84^."("-^)("j'><n-3) ^^
= i("* + 10n' + S5n»+50n).
§ 5.] Montmort's Theorem regarding the summation q/'Si/.j^.
An elegant formula for the transformation of the power-
series 2u,.r* may be obtained as follows. Let ua in the first
place consider S= 2u,a:", which we suppose to be convergent when
|xl<l; and let ua further suppose tliat |x|<|l-«|. Put
ar = y/(l +y); so that
24,
60,
120.
210,
336.
36,
60,
90,
126.
24.
30,
36.
6,
6.
0.
§§ 4, 5 montmort's theorem 407
\yl{\+y)\ = \x\<\,
and |?/| = |ir/(l-«)|<l.
Then, since
(l+7/)-"'=l-„(7i?/+„+i(722/'-m+2C'3/ + . . .,
we have
£f=i«„2/"/(i+#,
1
= u,ij-Uif+ Ujf- Mi/+ n^f-. . .
+ Ihy" - iGi^kl^ + sGitky^ - iPsU^lf + ■ . .
+ Usf - iCiUsy* + iC.ttsi/ - . . .
+ ItiV* - iCi^hf + ■ ■ ■
+ thf-- ■ ■
This double series evidently satisfies Cauchy's criterion, for
both \y\<i and \y/{l+y)\<l. Hence we may rearrange it
according to powers of y. If we bear in mind § 2, I., we find
at once
S=tiiy + ^iiiy- + ^"nif + ^'uiy* + ^*ihf + - • • •
Hence, replacing y by its value, namely, x/(l - x), we get
r'" i-a;^(i-^r-^(i-a;)=^-- ^^^ •
When the differences of a finite order 7n vanish, j\Iontmort'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 have for the finite mm
+ (A^«,-a;''A^M„+i)(j^a+. • • (2).
For, if we start with the series ?<„+ia;"+' + ?<,i+2.?;"+- + . . ., and
proceed as before, we get
From (1) and (3) we get (2) at once by subtraction.
• First given by Montmort, Phil. Trans. R.S.L. (1717). Demoivre gave
in his Miscellanea a demonstration very much like the above.
40^ EULER'S TnEOnEM CH. XXXI
The formula (2) will furnish a sum in the proper sense only
when the dilVurences vanish after a certain order. The summa-
tion of the intcgro-geometric series, already discussed in chap.
XX., ^13 and 14, may be effected in this way. It siioidd bo
observed that, inasmuch as (2) is an algebraic identity between
a finite number of terms, its truth does not depend on the con-
vergeucy of 2j/„;r", although that suppoi^ition was made in the
above demonstration.
Cor. 2. 1/ u„ be a real jmnitive quaii^itt/ which comtantly
diminishes as n increases, and if Lun = 0, then
Uj-th+ti,-. . .=-?/,- ^ Ak,+ -,A'm,-, . . (I)*.
Tliis is merely a particular case of (1) ; for, if in (1) we put
-X for x, we get
i(-)-«,;r- = i(-)-A-X.(j^)" (5).
Since the differences must ultimately remain finite, the right-
hand side of (5) will be convergent when x=l. Also, by Abel's
Theorem (chap, xxvi., § 20), since 2 (-)"«, is convergent, the
limit of the left-hand side of (5) when a; =1 is 2 (-)"«,. Hence
the theorem follows.
The transformation in formula (4) in general incrca-ses 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, tfte /olloiping formula : —
«.-«, + . . . (-)"-'«, = 2 ("■-(-)""»+.) -2»('^"'-(-)"'^"«+>)
in which the restrictions on u„ will be unncces.sary if the right-
hand side be a closed e.xpres-sion, which it will be if the differences
of tt, vanish after a certain order.
* Enler, Itut. D{ff. Cole., Tort II., cap. i. (1787).
§ 5 EXERCISES XXV 409
Example 1. We have (Gregory's Series)
T , 1 1 1
4=^-3 + 5-7 + - •• (^>-
If we apply (4), we have ii„=l/(2ra-l). Hence
A'"«„=(-)'"2.4 . . . 2)/(2«-l)(2n+l)(2» + 3) . . . (2(i + 2r-l);
A'-Ui = (-)'-2.4 . . . 2r/1.3.5 . . . (2r+l),
= (-)'-2'-.1.2 . . . r/1.3.5 . . . (2r + l).
-n f IT, 1 1.2 1.2.3
Therefore ^ = 1 H 1 h h . . . (S)
2 8 3.5 8.5.7 ^ '
Example 2. To sum the series
S„ = l=-22 + 32-. . . (-)''-'«'.
Since A«^i = 2in-3, A«-i = 3,
'i-"n+i = 2, A-iti = 2,
A=«„+, = 0, A3u, = 0,
we have, by (fi),
Sn=Hl-(-)''(« + l)'}-i{3-(-)"(2n + 3)}+J{2-(-)"2},
= (-)"-' 4" (n + 1)-
Exercises XXV.
(1.) Sum to n terms the series whose Jith term is the >ith r-gonal
number*.
Sum the following series to n terms, and, where possible, also to
infinity : —
(2.) 2»{7i + 2)(n + 4). (3.) 21/(k=-1).
(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(aH + 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.) 2(n+l)=/;i(n + 2).
(12.) 4/1.3.5.7 + 9/2.4.G.8 + 1G/3.5.7.9 + . . . .
(13.) 2?ecnescc(n + l)<'. (14.) 2 tan ((?/2'>)/2".
(15.) 2 tan-i |(na - n + 1) a"-'/(l + n (n - 1) a-"-')}.
(16.) 2tan-'{2/H=}.
(17.) ml + (m + l)I/ll + (m + 2)I/2! + . . . .
(18.) l!/m! + 2!/(m + l)! + 3I/(m + 2)I + . . . .
* The sums to n terms of arithmetical progressions whose first terms are
all unity, and whose common differences 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 Cn. XXXI
(19.) i-„c, + „r,-. ..(-)-„c,.
(20.) Show tbnt ibe CRurato Duiiibers of a gircD rank can be Bammod bjr
the fonnula of § 3, Example 1.
i 12 1.2.3
* ' '^m'*'m{m + l)'^m{m + l)(m + 2)'*'' ' ' '
a(a + l) . . . ja + r) . a(a + l) . . . (g + r+l) .
*''''•' c(c + l) . . . (c + r)*e(c+l) . . . (c + r + l)** ' ' *
(24.) 2(a + n)i"'-»7(c + n)"»'.
1.3 1.3.5 1.8.5.7
'""'■' 1.2.3.4"'"l.2.3.4.5'*'l.a.3.4.5.6^" • ' *
/Ofi X (l+jllljL^r) (1 + r) (1 + 2r)(l + 3r)
* ' 1.2.3.4.5 1.2.3.4.6.6 "
(27.) jm-j— gm(m-l) + j--g--gm(m-l)(m-2)-. . . .
(28.) Show that
^/^^ • • ("+^)-ri/M • • • ("-D+^/]| • • • {"-I)--'-
(GlaUhcr.)
(20.) Show that
l + 2(l-a) + 3(l-<i)(l-2u) + . . . + n(l-a)(l-2n) . . . (l-(»i-l)a)
= u-'{l-(l-<i)(l-2<i) . . . (1-na)).
_1__^ 21 3]
^ ' i + l~«-l"(x-l)(i-2)'*'(x-l)(«-2)(r-S)"* * •
(-)'*'nl /, n + l\
{x-l){x-2)...{x-n)V-x+-i)-
(31.) If a + 6 + 2=e + (l, then
;. ai»it»""_ ab ((g + l)'" (fe-Hl)'"' (o + l)i'-"(t + l)i»-")
7ci"'(J'»'~(a + l)(6 + l)-ed( cl«i<|i«i ~ eit-iidn-ii (•
(32.)
, 9-r 7(^-l).r(r-l)
^~(p-, + l).(l) + r-r)'^(p-? + l)(i.-j + 2).(p+r-l)(p + r-2) ' ' '
^(p-?)-(P + '-)
p.(p-9 + r) •
(Educational Timu Rrprint, toI. xu., p. 08. t
(33.) Transform the equation
log2 = l-i + l-J + . . .
by § 6, Cor. 2.
(34.) Show, by moans of § 2, I., that, if m bo a positive integer, then
1 r "^ r "<"-') r -(<»- IX^-S) ,
<^-:){^-bh)'"{^-b-^y
§ G DEFINITION OF RECURRING SERIES 411
RECURRING SERIES.
§ 6.] We have already seen that any proper rational fraction
such as {a + bx + cj^)/(1 +px + q.i- + rx^)* can always be expanded
in an ascending series of powers of x. In fact, if | a; ] be less than
the modulus of that root of ra? + qar+px+l = Q which has the
least modulus, we have (see chap, xxvu., §§ 6 and 7)
a + bx + cx' ., , n , /i\
\ + px + qx" + rx^
We propose now to study for a little the properties of the
series (1).
If we multiply both sides of the equation (1) by l+p.r
+ qx^ + rx^, we have
a + bx + cxr = {l+px + qx' + rx^){tio + u^x + u.2ar + . . .+u„af^ + . . . )
(2).
Hence, equating coefficients of powers of x, we must have
Mo = a (3i);
Vj+pUo = b (Sa);
v._+pih + qUo = c (83);
v-i+piu + qih + r 11^ = 0 (84);
«» +pUn-l + qUn-i + rUa-s = 0 (3„+,).
• •••••
Any power-scries which has the property indicated by the
equation (3„+i) is called a Recurring Power-Series] ; and the
equation (3„+i) is spoken of as its Scale of Eelation, 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, . . ., r, . . . constants, the recurring series is said to
be of the 1st, 2nd, 3rd, .... »-th, . . . order. When x=l, so
that we have simply the series «„ + «i + «2 + • • • + «» + . . ■ ,
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
generalised.
t The theory of Recurring Series was originated and largely developed
by Demoivre.
412 MANIFOLDNESS OF RECURRING SERIES CII. XXXI
tlie scries as a recurriwj scries simply* ; so that every recurring
series may be regarded as a particular ca-st- of a recurring power-
series.
It is obvious from our definition that all the coefficionts of a
recurring power-series of the ?-th order cau be calculated when
the values of the first r are given and also the constants of its scale.
Hence a recurring scries vf 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 ; a.s 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 bo
given, two conditions must be satisfied in order that it may be a
recurring series of the (r- l)th order; four in order tlrnt it may
be a recurring series of the (r - 2)th order ; and so on.
Example. Show that
is a rconrriog series of the 2ud order. Let the scale be u„ +i»u,_, + ?•',-, = 0.
Then we must have
The first two of these equations give p= -2, g= +1 ; and these values
are consistent with the remaining two equations. Hence the theorem.
§ 7.] The rational fraction {a->i-bx + cx')!{l-¥px + q3? + r3^),
of which the recurring power-series «,, + M,a; + f^ar" + . . . 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 tlie
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 tlic scale, and consider the coelhcienta
* Wo might of course regard a reonrring powor-Bcries as a particular casa
of a recurring scries in general. Thns, if wo put (/,=u,t', wo might rvgarj
the series iu (1) as a recurring series whose scale is
§§ G-8 GENERATING FUNCTION 413
as determined by the equations (3i), (3j), . . ., (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),
{3s), . . .,(3n+i), . . .of§6, that
{mo + (mi +pUo) X ->- {Ui +pih+ qUo) ar}l{l +px + qx^ + ra?)
is the generating function of the series «o + u^x + u^ir + . . . ,
whose scale is M„+j9i<„_i + 5'(/„-2 + rM„-3=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 rectirring 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 of from the several expansions.
Example. Find the general term of the recurring series whose scale is
u„ - 4u„_i + 5u„_5 - 2u„_3 = 0, and whose first three terms are 1 + 0 - 5 . Con-
sider the corresponding power-series. Here p= -4, g = 5, r= - 2; so that
a = «(| = l, 6 = u-i+pU(|= -4, c = U2+i)Ui + 2"o=0.
The generating function is therefore
l-4x _ 1-4j
l-4j; + 5i»-2i»~(l-if(l-2x)'
2 3 4
~l-x^ (l-x)- (l-2x)
Expanding, we have
= l + 2(3« + 5-2»+»)x».
The general term in question is therefore 3n-l-5-2"+-.
§ 8.] If Un bo any function of an integral variable n which
satisfies an equation of the fonn
M„ + p«„_i + qUn-i + rUn-s = 0,
or, what comes to the same thing,
«n+3 +pnn+i + Q>t«+i + rUn = 0 (1),
• We might also regard the series as deduced from the generating
function by the process of ascending continued division (see chap, v., § 20).
4M- UNEAU niKFEKENCE-EQUATION CIl. XXXI
we sec from the reasoning of last paragraph that «, is uni(iuely
determined by the equation (1), provided its three initial values
Mo, u,, J/, are given; and we have found a process for actually
determining u».
It is not difficult to see that we might assign any three
values of u„ whatever, say «., u^, Uy, and the solution would
still be determinate. We should, in fact, by the process § 7,
determine u„ as a function of n linearly involving throe arbitrary
constants «/„, u,, u.^, say/(ao, «^, «,, n) ; and u^, «,, «, would bo
uniquely determined by the three linear equations
/(««, «.,Wi,,a) = Ma, /{u„ih,th,P) = Uff,/(u„u^,u,,y) = Uy (2).
An equation such as (1) is called a Linear Difference- Equal ion
of the 3rd order with constant coefficients ; and we see generally
that a linear difference-equation oj the rth order vith constant
coefficients has a unique solution when the values of tlie function
involved are given for r different values of its integral argument.
Example. Find a function u, snoh that u„+,-4u,+j + 5u,4., -2u, = 0;
and u„ = l, t»j = 0, Uj=-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
then
pxS^ =pu^x + puiar+. . . +/?«,_, ar"+ pu^x'^^,
qar'Sn= qu^a^ +. . .+qu^_jx"+qu,.ia:"*^ + qu^af*"^,
r3?Sn= . . . + n/„-3ar"+rM,.aX"+'+»-tt,-,j-"+'+ru,j*^'
Hence adding, and remembering that «, +p«,_, + (ytt,_,
+ r«,-j = 0 for all values of n which exceed 2, we liave
( 1 + /).r + ^jr' + ra?) Sn = u, + («, + puj) ^ + (m^ + />(/, + qu^) 3*
+ (/'«» + gitn-x + rUn-3) a;"^' + (!Z«, + r«,_,) x'** + ru,x^*' (1) ;
whence -S„ can in general be at once determined by dividing by
I +px + ijx' + ra?.
The only exceptional case is tliat where for the particular
value of X in question, s^y x = a, it happens that
\+pa + qa? + ra.' = 0.
§§ 8, 9 SUMMATION OF RECURRING SERIES 415
In this case the right hand of (1) must, of course, also
vanish, and *S'„ takes the indeterminate form 0/0. S^ may in
cases of this kind be found by evahiating 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 S^t^a;" be convergent, then 7^«„a:" = 0 when
w = cc ; therefore the last three terms on the right of (1) wiU
become infinitely small when ?i = qo . We therefore have for
the sum to infinity in any case where the series is convergent
■"" 1 +px + qor + rx'
The particular cases
Mo + "i + '^ + • • • + "'• + • • • (3)'
«o-Mi + «--. • .+(-)"(<»+. . . (4),
are of course deducible from (1) and (2) by putting x=+l
and x = -\. Exceptional cases wiU arise if \+p + q + r = Q, or
if l—2) + 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,
"^.n-af and 1 + 2 ( - Y-'^2nx" being, in fact, recurring series whose
scales are «„-3«„-i+ 3i«„-2- '<n-s=0 and m„ + 2«„_i + m„_o = 0
respectively.
Exercises XXVI.
Sam 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+1-2+. . . .
(3.) 2 + 17i: + 95i= + 461xS+. . . .
(4.) 5 + r2j; + 30x2 + 78x3 + 210j;^+. . . .
(5.) l + 4x + 17j;= + 76x» + 353i-'+. . . .
(6.) 1 + 4x + 10j;2 + 22i3 + 46x<+. . . .
(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 tenn 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 r,, T„+i, r,« be consocntive terms of the reonrring aerica
whose scale is J',+j = '>r,4., - bT,, then
(T,.^,' - aT, r„+, + 6r.')/( r._^,' - or,_, r._^+, + fcr,_,») = fc'.
(9.) Form and sum to n terms the teria each term in which is half the
difference of the two preceding terms.
(10.) Show that every integral series (chap, zx., § 4) is a recarring scries;
and show how to find its scale.
(11.) If u,=u,_,+u,_,, and u, = au,, show that
«,'- ".+■«,-, = (- )"(«'-« -l)"!*.
(12.) If the series u,, u,, u,, . . ., u,, . . . be ench that in every four
consecutive terms the sum of the^cxtrcmes exceeds the sum of the means by
a constant quantity e, find the law of the series ; and show that the sum of
im terms is
Jm(m- l)(4m-5)e- m(m- 2) 11] + mU]+in(n> - 1) u,.
(13.) If u,.^=u,^i + u„, U] = l, u,= l, sum the series
(14.) By French law an illegitimate child receives one-third of the portion
of the inheritance that he would liave received had he been legitimate. If
there be / legitimate and n illegitimate children, show that the portion of
inheritance 1 due to a legitimate child is
1 n n (n - 1) n(n-l). . . 2.1
J~3/(i + l)'*'8''J(J + l)(i + 2) •••' '3«J({ + 1) . . . (Jin)*
(Catalan, Nouv. Ann., scr. ii., t. 2.)
SIMPSONS METHOD FOR SUMMING THE SERIES FORMED BY
TAKING EVERY AtII TERM FROM ANY POWER-SERIES
WHOSE SUM IS KNOWN.
§ 10.] This method depends on the theorem that the sum of
the ptb powers of the kth roots of unity is k if p be a multiple
of k, but otherwise zero.
This is easily seen to be true ; for, if w he a primitive /ttli
root of 1, tlicn the k roots are <«', <•>', <u' w'-'. If p /i/-,
then (w*)'' = (1)'''' = (o)*)"' = 1. If p bo not a multiple of k, then
we have
(<i>y -h (my + . . .+ (ui*-')" = 1 + (u^y ■*■ (a.")' + ...-•■ (o)'/-',
= {I -K)'i;(i --•').
= 0.
for (a.")* - (ui'y = 1 , oud o,' +- 1.
J
§ 10 SIMPSON'S THEOREM 417
Let us suppose now that f{x) is the sum of n terms of the
power-series «„ + 2M„a:", 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 <A-.
The coefficient of of in the equivalent series is
«,{(o.<')*-'"+'-+(<oi)*-'"+'' + (a)=)*-'"+'' + . . . + (o)*-»)*-'"+'"}/A- (2).
Now, by the above theorem regarding the ^h roots of unity,
the quantity within the crooked brackets vanishes if k-m + r
be not a multiple of k, and has the value k if k- m + r be a
multiple of k. Therefore we have
f7„ = u^af + M,„+iar"'+* + tt„+aa:"+^ + . . . (3),
where the series extends until the last power of x is just not
higher than the «th, and, in particular, to infinity if f{x) be a
sum to infinity*.
If we put 7» = 0, we get
{f{x) +f{u>'x) +f(u,'x) + . . . +/(<.»-' a:)}/^-
= Uf,+ i/tA* + tl^X^ + UjkO^ + . . . (4).
Example 1.
l + x+<c'+. . .+i''=(l-i"+i)/(l-i).
Hence, if u be a primitive cube root of 1, we have
(1 - a:"+' 1 - u"+' 1"+' 1 - u'»+=x»+'1
l + x^ + x*+. . .+x^=i\—. + —. -+ — , ^ V,
(1-X l-Ci)X 1- u-x )
where 3< is the greatest multiple of 3 which does not exceed n.
Example 2. To sum the series
i» x' x" ,
* This method was given by Thomas Simpson, Phil. Trans. R. S. L.
Nov. 16, 1758 (see De Morgan's Trigonometry and DouhU Algebra (1849),
p. 159). It was used apparently independently by Waring (see Phil. Tram,
li. S. L. 17S4).
C. II. 27
418 MISCELLANEOUS METHODS CIL XXXI
We have
e'=l+x + |-' + |J+...ad».
Ilcnco, if w bo a primitive 'lOi root of unity, say u = i, then, aiooe here
4 = 4, m = 8, k-m = l, (■)'=-!, u'=-i, we get
x* x" x"
that is, i (sinh ' - "° ') = o] + 71 + i ij + • • • •
MISCELLANEOUS METHODS.
§ 11.] When the nth term of a scries 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
((x + 1)' (x + 2) "•" (X + 2)» (X + 3) ■*■ (l+3j» (7+T) ■*" • • •[
( 1 1 1 1 _ 1
■*" |(x+l)(x + 2)«''"(x + 2)(x + 3)''''(x + 3)(x + 4)>'^' ' •|~(F+Tj'"
Denote the earns of n terms of the two given series by S, and T
respectively, and their nth terms by u, and w, respectively. Then
u,= -l/(x + n) + l/(x + n)«+l/(T + n + l);
t>,=l/(x + n)-I/(x + n + l)»-l/(x + fi + l).
Whence we get at once
S,+ r, = l/(x + l)'-l/(x + n + l)'.
Therefore S. + T. = l/(x + 1)'.
Example 2. Bosolution into partial fractions will always effect the
summation of the scries
2/(n)/(n + a)(n + 6) . . . (n + A),
whore a, b, . . ., k aro positive or ncgntivo integers, and /(n) is an integral
function of n who«« degree is less by two at least than the degree of
(n + o)(u + i<) . . . {« + *).
^10-12 euler's identity 419
For we have
f{n)l{n + a)(n + b) . . . {n + h)^ZAI{n + a),
and
/(n) = 2J(n + fc)(« + c) . . . (ii + k).
Since the desreo 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, i, . . ., 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, so that we shall have Alp + Blp+ . . . +Klp = 0.
On collecting all the fractions belonging to all the terms of the series we
shall be left with a certam number at the beginning and a certain number at
the end; so that the sum will be reduced to a closed function of 71.
§ 12.] Elders Identity. The following obvious identity*
1 - Oi + «! (1 - «2) + «ia2 (1 - «3) + . . . + ai«2 . . . a„ (1 - a^+i)
= \-a^a. . . . as„+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
flh — — , tta— t OE3 — , • • •) (ln-\-\— t
y y+Pi y+Pi y^P"
and multiply on both sides by y/(y - x), we get
X X {X +Pi) X(x+pi) . . . (x+Pn-i)
^ y+Pi ^ (y+Pi) (y +p^) ^' ' '^ (y +b) iy+pd ■ • ■ (y +p«)
^ _y ^ (x+pi){x+p.) . . . {x+p„)
y-x y-x' {y +p^) (y +2h) • • • (y+Pn)
If the quantities involved be such that
(2).
»=« iy+Pi) (y +i^2) ■ • -(y+Pn)
then
l + _^x--^.?^+...adoo = J^ (4).
y+Pi {y+Pi){y+p^) y-^
Used in the slightly different form,
Ii)(l + aj)(l + a3)(l + a,) . . .
= l + a, + a5(l + aj)+a,(l + a,)(
by Euler, Nov. Comm. Petrop. (17C0)
420 EXERCISES XXVIl CH. XXXI
If ill (2) wc put y = 0, we get
Pi PiP, ' ' ' P>P,- ■ Pn
1 + £ + £i-^ +/»>) + + x{x+p,). . . (x+p,-,)
=('v,)('v.)-('^^.) "'■
From (3) a variety of particular cases may be derived by
putting 71= x), and giving special values to pi, pt, . ■ • Thus,
for instance, if the infinite series 21/;;, diverge to + <», then (see
chap. XXVI., § 24) we have
l_£+*L(^Z£il_. . .ad« = 0 (6).
Pi PiPi
00
In general, if the continued product n(l +a-/j3,) converge to any
CO
definite limit, then the series l + 2j-(x+p,) . . . {x+pn-i)/piPi ■ ■ ■ />.
converges to the same limit.
Example. Find when the infinite Ecries
<;=l4._f- , _fj£+£)_ , x(x+p){x + 2p)
y+P {y+p){y+-2p) ly + r>){y + 2p){y + 3p) •'•
oonTcrges, and the limit to wbicb it converges.
If in (2) above we put Pi=p, Pi = 2p, *<>•, • • •> w have
„_ y X ^ (j + p)(x + 2p). . . jx + np) .
y-x~y-x ...{y + f}{y + 2p). . .iy + np)
How the limit in question may be written
I I 1 + y/np)
but this diverges to x if (x - y)/p be positive, and converges to 0 if (x - y)/p
be negative (cliap. xivi., § 24).
Hence, if p denote in all cases a positive quantity, we see Uiat
z x(x+p) . . od« = -i^.
^y+P^{y+p){y+^p) v
if y>z; and
y-p (y-pXy-ap) »-■»
if y<.x.
EXERCIBEB XXVII.
(I.) Given 1/(1 -x)' = 1 + 2x + 8x' + '1jc'+ . . .,
sum l + 4x» + 7x« + 10x»+. . . .
(2.) Sum the scries
l + i»/4+z«/7+. . .;
l + i'/3I + «»/CI+. . . .
§ 12 EXERCISES XXVII 421
(3.) If /(j-) = i(„ + )(,x + «o,r=+. . ., and a, p, y, . . . be the ;ith roots
of - 1, show that
i{a2»-'»/(<u:) + /3-"-™/(/3.r) + . . .} = ",„.c'"-«m+„.i:'"+" + w™+o„.r"'+"-»- . . .
n
where m<n. (De Morgan, Diff. Calc, p. 319 (1839).)
Sum the following series, and point out the condition for couvergeucy
when the summation extends to infinity : —
(4.) l-a:'/4 + x«/7-. . . ad oo ;
i-x*/4!+x7/7I-. . . ad 00.
(5.) l + „C, + ^Ce + „,C^+. . . adco;
i-m^'3 + m<^e-m'-»+- • ■ ad CO .
(G.) 1/1.3+1/1.2.4 + 1/1.2.3.5+ . . . to n terms.
(7.) l/1.2.3 + „,C,/2.3.4 + ,„Cj/3.4.5+. . . ad co .
(8.) 1-2j:/1 + 3x'-/2-4i»/3+. . . adoo.
(9.) cos(;/1.2.3 + cos2ff/2.3.4 + cos3e/3.4.5+. . . ad oo .
(10.) 1/12. 2= + 7/2-. 32+. . . + {2ir + iii + l)l{ii + l)-(n + 2)-.
(11.) l/l».2a-l/2-.3=+. . . (-)"-il/u2(„ + l)J+. . . adoo.
(12.) If n be a positive integer, show that
n 1 n(«-l) 1 n(7i-l)(n-2) ^
in + n'*' 2 (m + n) {m + n-1)'*' 3 {m + n) (m + n-1) (m + n-2)
_ n 1 n (n - 1) 1 7i{n-l){n-2)
~m + l~2(m + l)(m + 2)'^3(Hi + l)(m + 2)(m + 3) ••• *
(13.) Show that
„C, rfit , s9l := " .
l-xjl (l-x/l)(l-x/2)"^(l-x/l)(l-a:/2)(l-x/3) ••• n-x'
and hence show that
„Ci<ri - „C,<r3 + . . . (-)"„C„(r„=l/n,
where (Tr = 1/1 + 1/2+. . .+l/r.
(14.) Sum the series
m' , m°(m'-l') m» (m' - 1=) (m' - 2=)
F ■*■ — P:2= 13722. 3" + . . . aa CO ,
, , m\ mMm^ + l') mMm' + l") (m' + 3')
1+P+ 12.3. + iTsTsQ + • • . adco.
(15.) Show that
ai+Pi K+ftlK+Pa) {ai+Pi)K+i'2)(03+i's)
PlPa ■ • • fn-l^n _ I PlPa • ■ • Pn
(ii+PilK+Pa) ■ --K+PJ ("i+PiJK+Pa) ••• K+P«)*
(16.) Show that
,1 _l*-{V-x-)'' 3*-(3'-3:°)»
tan 2'f*- (ia_a2j2 "*■ (l3-xY(3'-xT ' " '
(Glaisher, MatU. Mess., 1873, p. 188.)
422 EXERCISES XXVII CII. XXXI
(17.) Showtliat
11.1. 1-2
n-~n(n + l) ii(n + l)(n + 2) n(n + l)(Fi + 2)(n + 3) " •'
and apply this result to the approximate calculation of w* bjr means of tho
formula
t'/C = 1/1«+1/3> + 1/3»+. . . .
(Stirling, ilethodiu Diftrentialit, p. 28.)
(18.) Show that 21/(m»-l) = l and 21/(<i*-l) = log2, where m and n
have all possible positive integral values difloring from unity, a is any even
positive integer, and each distinct fraction is counted only once.
(Qoldbacb's Theorem, see Li'our. 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)
= ir-/C; and, if d{n) denote the number of divisors of n, that 2 (d (n) - l)/r»
= 1; also that 2(71- l)/r = i;i/(r-l)>. {lb.)
CHAPTER XXXII.
Simple Continued Fractions.
NATURE AND ORIGIN OF CONTINUED FRACTIONS.
§ 1.] By a continued fraction is meant a function of the form
ai +
b.
aa+-
a, + ^... (1);
the primary interpretation of whicii is that Ih is the ante-
cedent of a quotient whose consequent is all that lies under the
line immediately beneath h^, 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.
In the most general case the component fractions — , -^ ,
a^ a^
— , , . ., 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),
^+AAA.... (2),
a, + a, + a4 +
the signs + being written below the lines to prevent coufLi.sion
with
■t2-i SIMPLE CONTINUED FUACTIONS CU. XXXM
61 b, bt «
«h + — + — +— + ... .
a-, a, a.
Examples have already been given (see clia]). in., Exercises
in., 15) of the reduction of terminating continued fractions;
and from these e.\ami)les it is obvious that «ivry terminating
continued fraction whose constituents a,, a^ f>t, b„ . . . 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 importiint kind
of continued fraction, that, namely, in which each of the nume-
rators of the component fractions is +1, and each of the
denominators a positive integer. When di.stinction is necessary,
this kind of continued fraction, namely,
111
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 i)y,
the numbers a,, a,, a,,. . . 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 a, the greatest
integer which does not exceed X; then we may write
-i'=«.+^^ (1).
where -rT',> 1, but is not necessarily integral, or even commensur-
able.
Again, let a, be the greatest integer in Xi, so that «»,■< 1 ;
then we have
where A'j> 1, as before.
A'. = a, + -J. (2).
The DOUkUoD Oi-h— -I- — + --)-. . .u frcaueutl; uied bv CoDtinental
^ Oj «» "4
writers
§§ 1-3 CONVERSION OF ANY NUMBER INTO S.C.F. 425
Again, let ^3 be the greatest integer ia Xt ; then
Xi=ch + ^ (3);
and so on.
Tiiis process will terminate if one of the quantities X, say
Xn-i, is an integer ; for we should then have
Xn-\ = (In
Now, using (2)
we get from (1)
X-a + ^
1 .
Thence,
using
.3), we get
X= »! +
/T 4-
T
U2 T
1
•
fls
"X,
>
and so on.
Finally,
then.
. . .
(a).
It may happen that none of the quantities X comes out
integral. In this case, the quotients «i, aa, . . - 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 ,„.
*i "^ :rT ;rr: • • • = "■ "^ w^ ;r^ • • • ^i^i'
02+03+ 03+03 +
then Oi = Oi', Oa = 05', 03 = 03', &c.
Now, since O3 and o^' are positive integers, and — ... and
03 +
—7— ... are both positive, it follows that . . . and — ; —
03+ *^ 03+03+ 03 +
—7— ... are both proper fractions. Hence, by chap, m., § 12,
Oj +
426 CONVERSION UNIQUE Cll.
XXXI
we must liavo
a, = a,'
and
(y).
11 11
a, + rt, + * ' ■ a,' + a,' + ■ ■ '
(«).
Again, from (8), we have
11 ,11
a,+ . . . =a, +— ; r- . . .
(')•
From (t), by the same reasoning as before, we have
0^ = 0,'
(0.
ind 111 111
o, + a, + a, + ■ ■ * ~ a,' + 04' + a,' + ■ ■ ■
(v).
Proceeding in this way, we can show that eacli partial
([uotient in the one continued fraction is equal to the partial
quotient of the same onicr iu the other*.
Tliis demonstration is clearly api)licable even when the
continued fraction does not terminate, provided we are sure
that the fractions iu (/?), (8), (v), &c. have always a definite
meaning. This point will be settled when we come to discuss
the question of the convergency of an infinite continue<l fraction.
Cor. If (i\, rt,, . . ., a„, 61, hi 6, lie all positive
integers, ar.+i and y,+, (nii/ positive quantities rational or irra-
tional each 0/ which is greater than unity, and if
^1 11,1 11
a,+ a,+ x,+i b,+ 6,+ y,+i
then must
a, = b,,a^-^bj o, = 6,, ania/so a-,+,=y,+,.
§ 4.] Afl an example of the general proposition of § 3, wo
may show that ereri/ commen.iurtdile numluT may be converted
into a termimiting continued fraction.
Let the number in question bo AjB, where A and B are
integers prime to each other. Let a, be the quotient and C the
remainder when A is divided by // ; a, the quotient and D the
* We Kupposo, as is clearly allowable, tbat, if the fraction terminates, th«
last quotient is > 1. It sboaUl nlno bo nnticcd tlint tbo firiit pnrtinl quotient
may be zero, but that noue of the olkers cau be zcio, as tbo process is
arraoRed above.
§§3,4
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. oi A and B. Since
A and B are prime to each other, the last divisor will be 1, the
last quotient a„, say, and the last remainder 0. We then have
A_ G _ 1
B~"^* B'^'^BjO
B D_ _1
G
E
1
-^-<h + j)-a, + ^j^
Hence
&c.
1
1
^ = a, +
B a^+ as +
It should be noticed that, if ^ <B, the first quotient a, ^\-ill be zero.
Example 1.
To convert 107/81 into a continued fraction.
Going tlirough the process of finding the G.C.M. of 167 and 81, we have
81)li;7(2
lljj
5)81(16
80
1)5(5
5
0
Hence
Example 2.
Consider -23 = 23/100.
We have
1G7
81'
2 +
16+ 5*
Hence
100)23(0
_0
23)100(4
92
8)23(2
16
7)8(1
7
1)7(7
7
0
42S CASE OF INCOMMENSURABLE NUMBER CH. XXXll
Cor. Jf we remove the restriction that the last partial quotient
shall l>e greater than unity, we way devtlop 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 namber of partial qnotients; but we
164* O
may write it 2+ -,„ — -r— r. wbich has an even namber.
' 10+4+1
§ 5.] Any single surd, and, in fart, any simple surd numlier,
such as A + i//>"" + CjP"" + . . . +71^""''", can be converted into
a continued fraction, although not, of course, into a terminating
continued frcKtion.
Tlie process consists in finding the grcat<?st integer in a series
of surd numbers, and in rationalising the denominator of the
reciprocal of the residue. Jlctliods for elTecting 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 happcus to
be a periodic continued fraction.
Example 1.
To convert JT3 into a continued fraction.
We have, 8 being the greatest integer < ./iS,
l/(sA3-8)
= 3+ -pj (1).
(Vl8+3)/4
Again, since the greatest integer in (J\.'i-\-3)H is 1, no have
5/ii+? = l + ">/i^^ = l+ 1
* "-T—^-4/(yi8-l)'
Similarly, we have
= 1 + — 7=*^ (2).
(yi3 + l)/S
= l+-7= (3);
§§ 4, 5 EXAMPLES
^ + 2 JI3-1. 1
3 ~ "^ 3 3/(Vl3-l)'
429
= 1+-7J: (4);
(n/13 + 1)/4
^ + 1_ ^/l3-3_ L
— J— -A+ 4 -^ + 4/(^/l3_3)'
=1+- pi— (5):
Vl3 + 3
^/l3 + 3 = 6+ Vl3-3 = G + -
l/(Vl3-3)
= 6 + -pi (C);
(Vi3 + 3)/4
after which the process repeats itself.
From the equations (1)...{6) we dei-ive
/T3_« 111111
V13-3+— j-j- — — _^^ ....
« «
where the * * indicate the beginning and end of the cycle of partial quotients.
Example 2.
To convert ^L — into a continued fraction.
We have
2 2/(V3-l)'
^/3 + l = 2 + J3-l = 2 + ^ ,
1/(^3-1)
= 2 + — =4 ;
(v/3 + l)/2
V3+1 73-1 1
~^~ ^""27^75^)'
^3+1
after which the quotients recnr. We have, therefore,
2 "^2+ 1+ ••• •
* *
It will be proved in chap. xxxm. that every positive number of the form
(iJP+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.
*30 EXERCISES XXVIIT CH. IIXII
EiEBCisra XXVIII.
Eipteas tha following u limpla oontiiiaed bmetiooa. tenniiutiog or
periodic as the etae mmj be: —
*^' 73- *^' 1193- <'■> S^- <*-) "i23-
(5.) 2-71628L (6.) 0079. (7.) ^'i. (S.) ^'5. (9.) ^(11).
(10.) V(10). (11.) ^(12). (12.) ^'}. (13.) ^/3 + l.
(15.)8howth.tl + -^g = l + ^^^-L... .
(16.) A line AB is divided in C, go that AB.AC=BC^. Expnn the
ratios ACjAB, BCIAB as simple eontinned fracaons.
(17.) Express ^'(a»+a) and ^{<^-a) as simple eontinoed bactions, a
being a positiTe integer.
(18.) If a be a positiTe integer, shov that
(19.) If a be a positive integer > 1, show that
(20.) Show that
2 a+ 3<i.f
^'"' — + 4+ 2.r 6+ • •• •
(21.) Show that ererj rational algebraical fonetioo of X QUI be expanded,
and that in one way only, as a terminating oontintwd fraction of the I
where Q, , Q, . . . .. Q, are rational integral functioos of x.
Exemplify with (*» + x« + r+l)/(x* + a««+fc»+i + l),
• •
•ad ,»A .f_
* 1+ iT-- ••
• •
•bow thai x-y = a-6i.
1 1 1
ftj + rtj + (If +
1
(1);
1 1 1
iTo = Wa H ... —
a3+ at+ as
(2);
1 1
a^s = fla + . . . -
(3);
§ G COMPLETE QUOTIENTS AND CONVERGENTS 431
PROPERTIES OF TUE CONVERGENTS TO A CONTINUED FRACTION.
§ 6.] Let US denote tlie complete continued fraction by Xi, so
that
and let
a,
and SO on.
Then ^j, 3*3, . . . are called the complete quotients corresponding to
a,, fls, . . ., or, simpl}', the second, third, . . . complete quotients.
The fraction itself, or x^, may be called the first complete quo-
tient. It wiU be observed that «!, a-i, a,, . . . are the integral
parts of a^i, ^^2, ^3, • ■ •
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 fractious thus obtained, when reduced (without simplifica-
tion, as under) so that their numerators and denominators are
integral numbers, by pjqi, pjq-i, • . . Then we have
Oi = — =-t- (a),
1 qi
where
1 UiUi + 1
»! + — =
?2
(/8),
_ , 1 1 aiaaaa + Oi + aa
02 + «3 a.«s + 1
-Pj
73
(y),
11 1
ffli + — — . . . — = kc.
(12 + rta + a„
2'.
(S),
and so on,
Pi = ai, 2-1 = 1
(«'),
^2 = aia2+l, qi = a^
(n
2h = aio-fl-i + a^+a3, g'3 = a-fl, + 1
(/),
and so on.
432 UECUimENCE-FORMTn.A FOR fONVEROENTS CH. XX XH
The fractions /),/7, , p.Jq^, ... are called ihi first, second, . . .
conrergents to the continued fraction.
Cor. If the continued fraction terminates, the last convergent
is, by its definition, the continued fraction itself.
§ 7.] It will be seen, from the e.xpressions for pu Pi, p, and
?., 3-3, ?j in § 6 (a), (yS'), (/), that we have
Pt = (hPi+Pi (1);
q> = a^i + qx (2).
This suggests the following general formulw for calculating the
numerator and denominator of any convergent when the num,rat",s
and denominatf/rs of the two preceding convergents are inuun,
namely,
/>» = a,^.-i+/>,-s (3);
g'. = a«!7»-i+5'«-j (4).
Let us suppose that this formula is true for the nth con-
vergent. We observe, from the definitions (a), (/3), . . ., (g) of
§ 6, that the n + lth convergent, Pn+Jg„+,, is derived from the
nth if we replace a, by a, + l/a,+,. Hence, since ^,_„ g„.,,
Pi-i, ^n-i do not cont<ain o,, and since, by hypothesis,
Pj, _a»Pn-l+Pn-2
?« Onqn-l + qn-i '
it follows that
P«+i_ (o,+ l/a..t.i)p.-i +/>._,
?.+i (a, + l/a«+i) qn-i + 7,-,'
or, after reduction,
P,+l _ g.-t-i (<T„/',-i •^P.-i)+Pn-l
ff.+i O1.+1 (a,y,-i + qn-i) + 7.-1 '
_ a%+lP» +Pn-l
by (3) and (4).
ITence it is sufficient if we take
P»+i = a»+\P»+p»-i ;
q»+\=<ty,+iq» + y.-i.
In othcr^ords, if the rule hold for the nth convergent, it holds
for the n+ 1th. 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 »o ou. That is to say, tlie rule is
general
§3 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 tliat the same is true
of the denominatois.
Cor. 2. From (3) and (4) it follows that
and
Pn-l
1 1
1^
(5);
(6).
For, dividing (3) by jo„_i, and writing successively n- 1, n-2,
. . ., 3 iu place of n, we have
Pn/Pn-l=an+--'T-—;
Pn-\IPn-2
Pn-llPn-2 = Cln-l +
PnSn-i '
P3/Pi=a3+Pl/P2;
1 1
= a3 + .
ao + flSj
From tbese equations, by successive substitution, we derive (5) ;
and (6) may be proved iu like manner.
Example 1.
The continued fraction which represents the ratio of the circumference
of a circle to the diameter is 3 + y- j-g— y- - Y+ 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 :-
It is
n
a
P
3
1
3
3
1
2
7
22
7
3
15
333
106
4
1
355
113
a
2fl2
103993
33102
6
1
104348
33215
7
1
208341
66317
where p^=3o5, for example, is obtained by multiplying the number over it,
namely H33, by 1, and adding to the product the number one place higher
still, namely 22.
28
c.
434 EXAMPLES CIL XXXIl
Tha saecossive conTergcnta are therefore
3 22 333 356 103993
r T' 106' 113* 33102
Example 2.
If PiNu rJ<},. . • • bo the convergent, to 1 + ^^ gTf rf * ' ' ."Tf ' • •
ad ao , show that
p, = (it-l)p.-, + (n-l);>,-, + ('«-2)p,.,+ . . .+3i., + 2p, + 2.
By the recurrence-formula we have
P.= n/'»-i+P«-s;
;>,-!=(" -iJf.-s+P.-j ■•
;>.-»=("- 2) p,-i+p.-«;
p,=3pj + p,;
and (eince />i = 1 . Pj = 3)
i', = 2i>i + l-
Adding all these equations, and observing that p^-,, r,-ji • • •> fi
each occur three times, once on the left muliiiilii^d by 1, once on the right
multiplied by 1, and again on the right multiplied by n-1, ii-2 3
respectively, we have
p.=(n-l)p.-, + (n-l)p,^, + (n-2)r.-,+ . • • +3p, + 2p, + (p, + 1),
which gives the required result since Pi = 1-
Example 3.
1111
In the case of the continued fraction a, + — - — — -— - — — . • • prove
that p,, = ?*,«, i'»,-i = <'i?»J<'i-
By the definition of a convergent, we have
9t.41 "3+ "l
gince every odd partial quotient is a, .
Again, by Cor. 2 above,
P^=.a,+^ .. .^ (P).
Pi» "5+ "i
?»»+i Pn
Hence
which gives
Also, since ri. = ''irni-i + 7'»-5i
9i.+i = ''lV«. + «»-!•
"jPs.-i+J>».-i=<'i9«« + 9>»-i (*)•
Now, if we writu n - 1 for n in (•>), wo have p»,-, = 7j,-i : hence («) giTO
'»ii'f«-i = '»i9«»-
Therefore
(7) loads to
§ 8 PROPERTIES OF CONVERGENTS 435
§ S.] Frdiu equations (3) and (4) of last section we can prove
the following important property of any two consecutive con-
vergcnts : —
Pnqn-l-Pn-iqn = {-'^T (l)-
For, by § 7 (3) and (4),
i'n+l'Zn -PnqrH-\ = (^n+lPn +Pn-l) Qn-Pn (^H+l^n + <7.i-i).
= - (j}nqn-l -Pn-iqn)-
Hence, if (1) hold, we have
= (-1)"".
In other words, if the property be true for any integer «, it
holds for the nest integer n + 1. Now
= 1,
that is to say, the property in question holds for w = 2, hence it
holds for n = 3 ; hence for w = 4 ; and so on.
Cor. 1. Tke convergents, as calculated by the rule of ^7, are
fractions at their lowest terms.
For, if pn and q„, for e.xample, had any common factor, that
factor would, by § 8 (1), divide (-1)" exactly. Hence p„ is
prime to qn] and F„/g'» is at its lowest terms.
Cor. 2.
qn qn-i qnqn-i ^ ''
Cor. 3.
qn qi \q2 qJ W3 qJ ' ' ' W» qn-J'
= a.,^-A^.....(^ (3).
q,q^ Ms q,,-iqn ^ '
Cor. 4.
Pnqn-3-p«-iqn = {-)''''a„ (4).
For
P«qn-1 -pn-".qn = {dnPn-X +Pn-2) 5'n-J -/'n-s (dnqn-l + ^n-s),
= (Pn-iqn-i -Pn-iqn-i) On,
= (-)"-'«„, by Cor. 1.
28—2
436 EXAMPLE CM. XXXIl
Cor. 5.
/'«/';■ -Pn-'Jl.-0 = ( - )'''njq„qn-t (5).
Cor. 6. TAe odd coiivergents continualli/ increase in xralue, the
even convergevts continually decrease; etery even convergent is
greater than every odd convergent; and every odd convergent is /ais
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, tee can always find two positive integers p' and q'
such that p'i -p'q = + 1 or = - 1, as we please.
For, by § 4, Cor., we can always convert pjq into a continued
fraction having an even or an odd number of partial quotients,
as we please. If p'/q' be the penultimate convergent to this
continued fraction, we have in the former c*se />/-/>'j = + l, in
the latter pq -p'q = - 1.
Example. If pjv. be the nth convergent to a, + . . .
— , and
to the
partial quotient a,, show that
P.7»-r - Pn-r^n = ( " l)""*^'.-,*.?.-
We have, by oar data,
^"=-.-. ' ..."
9. "»+ a.
(»).
""l ' . . • • • .
W:
9.-T <h+ <^-r
p» 1 1 1
hence £! = <!,+ . . . =— -^
9. "3+ a.-r+ ,-r+lPJm-r*xQm
Now
h)-
P.-r _ "n-rP.-r-l +P»-r-t
9«-T <'m-r9m-r~l + 9,-T-t'
Hence, by (a) and (7),
P._ (a.-,-l-.-w^|9J.-T4lP«)P»-r-i ■<•?.-,-<
9. <<'.-r + .-rtl<?Ji.-TH-l''j9»-r-l + 9.-T-«'
_ Pn-r-*- »-r*xQnPm-r-\lm-r+\Pn
9«-T + ii-r+l9»9.-r-l/.-t-+l^» '
_ i»-r4-lP»P»-r + i»-itl9«P«-r-l in
n-r+l "«9»-r + ii-r+lVii 9n-r-I
Now it is easy to boc that the nnmcrator and denominator of the fraction
Wt written are mutually prime; therefore
Pm = »-T^lPnP»-r + n-f*iQnP,-r-l-{ /.
9. = i.-r+lP«9.-r + .-r+lV.^.-r-i ■ <
§§ S, 9 APPROXIMATION TO CONTINUED FRACTION 437
From (c) we derive
Pn9u-r -Pn-rin = " (P,.-r'7,.-r-I " Pn-r-l In-r) i.-r+lQu.
= (-1)(-1)"-Vh-iQ™.
by (1) above,
as was to be shown.
§ 9.] The convergents of odd order are each less than the
ichole 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+l_C'n+lPn+Pn-l,
9'n+i (tn+i^ln + Qn-l
and the whole continued fraction Xi is derived from 7?„+,/'7„+i by
replacing the partial quotient a„+i by the complete quotient x^+i.
Hence
_a!n+lPn+Pn-l
From this value of a-j we obtain
Similarly
'in
X^
a-H+lPn+i'B-I
qn
Pn-iq,v-Pnqn
»
qn{i^n+\qn + qn
-0
Pn-
qn)
' q„-l qn-l{Xn+iqn+qn-i)
From (1) and (2) we deduce
_ q« _ gn-l
^ Pn-l qn^^n+l
(1).
(2).
(3).
Now 5',,-,, g„ aro positive integers ; a-„+, <t 1 ; and, by § 7,
Cor. 1, q„.i<qn. It follows, therefore, from (3) that Xi-pjq„
is opposite in sign to, and numerically less than, Xi-p„-,/qn-t.
In other words, pjq^ differs from .r, by less than p„-,/qn~i does ;
and if the one be less than a-, , the other is greater, and vice versa.
438 APPROXIMATION TO CONTINUED FRACTION ril. XXXII
Now tlie first convergent is obviously less than a-,, hence the
second is greater, the tliird less, and so on ; and the difference
between j-, and the successive convergents continually decreasefi.
Cor. 1. The difference between the continued fraction and
the nth convergent is less than i/(j,qn+\, and greater than
0|l+2/?ll?«+J-
For, by what has just been proved,
?« !7r.+J ?»+l
are, in order of magnitude, either ascending or descending.
Hence
— <- X] <. — ~
1
, by § 8 (2).
Again,
> ?^,by§8(5).
Vnyii+t
Since y„+,>y„ and since y.Wa.+a = (««+j7»+i + 7.)/a.+t
= q,+i + qn/a»+i<g.+i + q, (a,+5 being ^l), it follows that the
upper and lower limits of the error committed by taking the «th
convergent instead of the whole continued fraction may be
taken to be I/7,,* and 1/g, (7, + ?«+i). These, of course, are not
80 close as those given above, but they are simpler, and in many
cases they will be found sufficient.
Cor. 2. In order to obtain a good approximation to a
continued fraction, it is advisable to take that convergent vhose
corresponding partial quotient immediately precedes a very much
larger partial quotient.
For, if the next quotient be large, there is a sudden increase
in g,4i, RO that l/g.^.+i is a very small fraction.
The same thing apjtears fnnn thf consideration that, in
taking ^)„/(/, in.sUad of the wlndc fraction, we take a, instead of
§ 9 CONDITION THAT l^n/jn BE A CONVERGENT TO X^ 439
a„ + . . . , lliat is, we neglect the part ... of the
complete quotient. Now, if a„+i he very large, this neglected
part will of course he very small.
Cor. 3. The odd convergents form an increasing series of
rational fractions continualli/ appi-oacldng to the value of the
whole continued fraction; and the even convergents form a
decreasing series having the same property*.
Cor. 4. If Pnlqn-Xi<\lqn{qn + qn--i), vhcre q„-T, is tlie de-
nominator of the penultimate convergent to pjq,,. when converted
into a simple continued fraction having an even number of
quotients, then pjqn is one of the convergents to the simple
continued fraction which represents Xx; and the like holds if
a^i- p,Jq„<l/qn {q„ + q,i-i), where qn-i is the denominator of the
penultimate convergent to pJqn when converted into a simple
continued fraction having an odd number of quotients.
Let «!, Oj, . . ., On he the n partial quotients of i?„/g'„
when converted into a simple continued fraction having an
even number of quotients, and let pn-ifqn-i be the penultimate
convergent. Then pnqn-i -Pn-\qji. = 1-
Let 3-„+i be determined by the equation
1 1 1
Xi=ai + . . . — - — .
Then we have
Xl = {Xn-nPn +i',i-l)/(A+l?« + qn-\),
whence
(Th+X = {Xiqn-i -Pn-^)KPn - X.qn),
* The value of every simple oontinued fraction lies, of course, between
0 antl C30 ; and we may, in fact, regard these as the first and second con-
vergents respectively to every continued fraction. If we write 0 = J, and
oD = i , and denote these by — ' and -* , BO that we understand j)_, to be 0,
9-1 So
Pj to be 1, g_i to be 1, and q^ to be 0, then ■p_^ and pj will be found to fall
into the series p,, p.^, p.^, lic, and g_, and q^ into the series ?, , q.^, Jj, Ac.
It will be found, for example, that p, = ujPo+y_i, i;, = u,(/o + 9-i. ^uS-i -i'-Wo
= (-!)" = !, and so on.
440 CONDITION TIlAT/)„/«7„ BE A CONVEROEaiT TO a^ Cll. XXXII
or, if we put ( =pjqn - a:,,
ic^i = {(/>»?»-. -Pn-iqn)/q,-q,-i()/qni,
= (i/?.-?.-.^)/ynf.
Hence the necessary and suHicicnt condition that x,+, > 1 is tliat
ih»-gn-ii>qnt
that is,
i<ilqn(qn + qn-x),
which is fulfilled by the condition in the first of our two
theorems.
Let now 6,, Aj fc, be the first n partial quotients in the
simple continued fraction that represents x,. Then we have
_. 1 11
where y,+,>l.
Hence
1 11.^1 II
03+ a» + a-,+, bi+ 0. +y,+j
Therefore, by § 3, Cor., we must have
a,=0„ a^^L, . . ., a, = 6„ a-,+,=y,4.,.
Hence a, + . . . + — , that is, — is the nth convergent to
a-,.
The second theorem is proved in precisely the same way.
Since qn-i<qn, the conditions above are a fortiori fulfilled if
itt~Pn/qn<ll^'Jn.
§ 10.] The propositions and corollaries of last section show
that the method of continued fractions possesses the two most
iniportant advantages that any system of inimericul calculation
can have, namely, 1st, it furnishes a regular series of rational
approximations to the quantity to be evaluated, which increase
step by stop 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 -vnrtually 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 finite 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, then, in
the case of a simple continued fraction, however great « may be,
we have seen that the two convergents, p-M-ilq^n-i, P-mllin, in-
clude the fraction psjq, between them. Hence, if we can show
that p^n-ih-in-i and PmI^m 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, pjq,
approaches a certain definite value, which we may call the value
of the whole continued fraction. Now, by § 8, Cor. 5, Pin-il^tn-i
continually increases with n, and Pml^m continually decreases,
and p2nl7M>P2n-ill'>n-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, p^^lq^ -pM-i/g^n-i
= l/<?'>ii 731-1) and by the recurrence formula for q„ it follows that
q^n and q>a-i increase without limit with ;; ; therefore PmlqM
—Pin-ilqa-i may be made as small as we please by sufficiently
increasing n.
It appears, therefore, that every simple contintied fraction has
a definite finite value.
Example.
To obtain a good commeosurable approximatiou to the ratio of the
442 EXERCISES XXIX CII. XXXII
ciroumforcncc of a circle to the dinmctcr. Ilcferring to Example 1, § 7,
we have the followiog approximations in defect: —
3 833 103093 ^
&0.}
!•
IOC*
83102
and the following in excess :
—
22
855
101348
7 '
113'
83215
lie
Two of these*, namely 22/7 and 855/113, are distinguislicd beyond the other*
by preceding large partial quotients, namely, 15 and 2'J2.
The latter of these is exceedingly accurate, for in this case llq,q^f
= 1/113 X 33102 = -00(10002073, and <1h4.j'7,?,+, = 1/113 x 33215 = -0000002005.
The error therefore lies between •OOOOU02iie and •0000002ii7 ; that is to say,
355/113 is accurate to the Cth decimal place. In point of fuct, we have
T = 3-141592G.J358 . . .
355/1 13 = .S-14159202035 . ^.
Uiflorence= -00000020077 . . . .
Exercises XXIX.
769
(1.) Culcalate the various coDTcrgents to tttI' '^^^ estimate the errors
committed by taking the first, second, third, Ac, instead of the fraction.
(2.) Find a conyergent to the infinite continued fraction ^ — ;; . . ,
1+2+8 +
which shall represent its value within a millionth.
(3.) Find a commensurable approximation to ^/(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 3C5'25
days; calculate the various cycles in which transits of Venus may be cx|«cted
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.
(6.) Work out the same problem for Mercury, whose sidereal period it
87-97 days.
(6.) According to the Northampton table of mortality, out of 8G39
persons who reach the age of 40, 8559 reach the age of 41. Show that
this is expressed very accurately by saying that 47 oat 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, 1(>40
X.D.): it is in great favour, not only on account of its accuracy, but becauaa
it can bo easily remembered as consisting of the first three odd numbers
each repeated twice in a certain sucoossion.
§ 11 EXERCISES XXIX 443
(7.) Find a good rational approximation to s/0-^) which shall differ from
it by less than 1/100000; and compare this with the rational appi'oximation
obtained by expressing s,/{li)) 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 -7k</c, however small
It 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 j)„_, have any common factor it must divide a„ exactly.
(10. ) Show that the difference between any two consecutive odd convergents
to ^/(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 continaed fraction
is intermediate in value to the two preceding.
(12.) Prove that
9n'l-Pn=(-l)''+Vj2^3- • • 'n+I-
Show that pjq„ differs from x, by less than l/a^rta . . . a,.+i7„. Is this a
better estimate of the error than l/g„?„+i?
(13.) If the integers x and y be prime to each other, show that an integer
u can always be found such that
(x'+y'^u=z- + l,
where z is an integer.
(14.) Prove that
(i>„' - ?,') (i'„-l' - 9.-l') = {PnPn-l - 9„?n-l)' - 1 ;
Pn-1 + 9n-i {Pu-lPr>-i + 9-.-1 in-:)" + ^ '
(15.) Prove thatp„_ip„-7„_ig„ii^ is positive or negative according as n
is even or odd.
(It;.) If PjQ, P'jQ', P"IQ" be the nth, ?i-lth, n^th convergents of
J 1 1 J^
"■1+ "2+ "»+ "4 +
111
Oj-f o,-f O^-h
1 1
0-3+ O4 +
respectively, show that
P = a.J>' + P", (? = (a,a.,-t-l)F + ajF'.
(17.) If the partial quotients of x, =?„/?„ 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 p„-i = q„, and ((/„=il)/p„ is an 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, Air). Sup., i"' ed., t. i., p. 2'J.)
44+ EXERCISES XXIX C». XXXII
(I'J.) Showtbat
11 11
0,+ --...- a,+ -...-
Oi+ °i. _ a»-i+ <h
1 1" 1 I*
(20.) If X, = — — — . . . , Bhow thst p. = 9... .
(21.) The successive convcrgents of 2a + — ;- — . . . are
* ' " a+ 4a+ a+ 4a +
always double those of a + „ — ... .
■' 2a + 2a +
(22.) If the reduced form of the nth complete quotleut, f, , in
a, H ... be iJr),, show that
' a,+ 0,+ **' "
Vn = f n+1 •
(23.) Find the numerically least value of ox -by for positive integral
values of x and y, a and b 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
(commen.'surable 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 dois not exceed a ffirrn inteijer D, which shall most
closely approximate {by excess or by defect, as may be assigned)
to a given number commensurable or incommensurable. The
solution of thi.s problem is one of the most important uses of
continued fractions. It dejiends on a princijile of great interest
in the theory of numbers, which we proceed to prove.
Ijemma. — Ifpl'l andp'jq be two fractions such that nij'—p'q= 1,
then no fraction can lie between them unless its denominator is
greater than the denominator of either of them.
Proof. — Let ajb be a fraction intermediate in magnitude to
piq and p'lq. Then
q (> q q' ^''•
I' 0 'I <i ^
§§ 12-14 SIMPLICITY OF APPROXIMATION 445
^ " qb qq
pb — qa 1
qb qq '
Hence qb>qq'(pb-qa);
and b>(pb- qa)q.
Now piq — a/b is positive, hence pb - ga is a positive integer.
It follows, therefore, that b>q'.
Similarly it follows from (2) that b>q.
Hence no fraction can lie between plq and 2)'/q' unless its
denominator is greater than both q and q'. In other words, if
pq -p'q= 1, no commensurable number can lie between plq 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 Pnjqn must, a fortiori, be nearer than
p„-Jqn-i- Hence, since pjq^ and pn-\lqn-\ include the value of
the continued fraction between them, it follows that ajb must
lie between these two fractions. Now we have, by § 8, either
Pnqn-i-Pn-iqn='^, Or p„.^q„- p^q^--.^ I. Hence, by § 12, b
must be greater thau q^, which proves our proposition.
Example.
Considerthe continued fraction 11 = 3 + — oT 4T 2+ 5"
3 4 15 64 143 779 „ . ,
The snccessive convergenta are ^ , ^ , -j- , j^ > "33 > 2(57 ' " ^"^ '^''^
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 Xj 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-ilqn-^,
Pn-i/qn-i, Pnlqn to x>., aud kt M be odd, say. Then
Pn-2 Pn ^ Pn-1
are in increasing order of magnitude. We know, by last
446 INTERMEDIATE CONVERGENTS (11. XXXII
section, that no fraction whose denominator is less than q^-i can
lie in the interval />,_j;(/,_,,/>,_,/j,_,, and also that no fraction
whose denominator is less than q» can lie in the interval
pjq,, pK-\lq»-\\ but we have no assurance that a fraction
whose denominator is less than y, may not lie in the interval
r^-J'Jm-i, pjq», for j!J,'7,-a-/».--7. = a,, where o, may bo>l.
This lacuna is filled by the following proposition : —
r. The series effractions
/*.-» P<,-i-^r»-\ Ph-2 + ip»-\
P»-i •*•«!.- lp«-i /'■1-!
'«-a + 0.?.-i \ qJ
7«-s + 0,-l?.-i ?•
form {according as n is odd or eeen) an increasing or a decreasing
series.
2*. Each of them is at its lowest terms; and each consecutive
pair, say P/Q, FjQ\ satisfies the condition PQ - FQ=±\; so
that no commensurable quantity less complex than the more complex
of the two can be inserted between them.
The first and last of these fractions (formerly called Con-
vergents merely) we now call, for tlie sake of distinction, Principal
Contergents ; the others are called Intermediate Convergents to
the continued fraction. To prove the above properties, let us
consider any two consecutive fractions of the series (1), say PjQ,
Fiq; then
tP _ ^^ ^ JP.-» -^ rP— 1 />.-i-l-r+lp._,
Q Q qu-t + rqn-, ?,., + r+l7,-,
(where r = U, or 1, or 2, .... or o,- 1),
_ -(p«-i?»-i-i'«-j?«-i)
(7.-1 + rq,.t) (7-. + rTIg,.,) '
+ 1
(?.-. + rq,.t) {q,.t + r + 1 g,. J '
^iy if n be odd,
' Ofy '* " ^ even.
(2).
§§ 14., 15 COMPLETE SERIES OF CONVERGENTS 447
xl6nC6
PQ'-rQ = -li(nheoAd, \ ,g>
= + 1 if » be even. J
(2) and (3) are sufficient to establish 1° and 2°.
3°. Since P!Q-p„-i/qn-i = ±llln-i{qn-2 + rqn--,), and since
Xi obviously lies between PjQ and Pn-ilqn-i, it follows tliat t/ie
intermediate convergent PjQ differs from the continued fraction
by less than l/<7„-i Q, a fortiori by less than l/qn-i*
§ 15.] If we take all the principal convergents of odd order
with their intermediates wherever the partial quotients differ from
unity, and form the series
0 P, Pz Pn-^. Pn (KS
V ■ ■ ■' ,j,' • ' •' q/ ' ■ •' y„-/ • • •' q,r • • • ^'''
and likewise all the principal convergents of even order with
their intermediates, and form the series
1 P-i Pi Pj^ Pj^ /T!\
0' •• •' q.' • • ■' qi -/n-a' ' " " ?n-a' " ' ' ^'^^'
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 seriss of commensurable
quantities, increasing in comjylexity 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-ilq^-i and
pjqn be the last two convergents, the series of fractions
Pn-l Pn-1 +Pn Pn-l + ^Pn.
qn-l' qn-l+qn g„-i + 2y. '
• For a rule for estimating the errors of principal and intermediate
convergents to a continued fraction, see Hargreaves, Mess. Math., Feb. 1898.
4*8 CLOSKST KATIONAb APPROXIMATION CH. XXXII
forms cither a continually increasing or a continually decreasing
srrif.o, in tr/iirh no principal amvergmt occurs, hut whogf. terms
approach more and more nearly the value p J q^, that is, j-,*.
§ 16.] We are now in a position to solve the general problem
of § 12t. Suppose, for exaraple, that we are required to find the
fraction, whoso denominator does not exceed D, which shall
approximate most closely by defect to the quantity x,. What tee
have to do is to convert a", into a simple continued fraction, form
the series (A) of last section, and select that fraction from it trhose
denominator is either D, or, failing that, less than but nearest
to D, say P/Q. For, if there were any fraction nearer to x, than
P/Q, it would lie to the right of P/Q in the scries; that is to say,
would fall between P/ Q and the next fraction P'/Q of the series,
or between two fractious still more complex. Hence the denom-
inator of the supposed fraction will be greatc-r than (/, and hence
greater than D.
Similarly, the fraction vhich most nearly approximates to j",
by excess, and tchose denominator dues not exceed I), is obtained
* This may also bo Been from tbe fact that the continaed fraction
a, -I ... — may also be written a, + . . . : that u to
■ay, wo may consider the Ukst qaotient to be co , and tbe last coDvergent
(P.-i + *PJ/(7.-i + «?.)•
t The Brst general S'^lntion of this problem was given by Wallis (s«e
bis Algtbra (16m5), chap, x); Huj^hens also was led to discuss it when
deagning the toothed wheels of his Planetariam (see his Detcriptio Autowmli
Plamtarii. I6fi2). Que of the earlier appearances of continued fractions in
mathematics was the value of 4/t given by Lord Brouncker (about 1655).
While discussing Brouocker's Fraction in his Arilhmttiea Injinilorum (ICSfi),
Wallis gives a good many of the elementary properties of the oonvergenta
to a general continued fraction, including the rule for their formation.
SaundcrBon, Kulcr, and Lamlx'rt all helped in developing the theory of
the subject. See two interesting bibliographical papers by Oiintlier and
Favaro, BuUttino di BihUographia t di Storia drlle Scienie Mathematieht e
Fisielu, t. VII. In this chapter we have mainly follnwed Lagrange, who gave
the first full exposition of it in his a<lditions to theFrencli edition of Euler's
Algtbra (\lWt). Wo may hero direct the attention of the reader to a series
of comprehrnHiTO articles on continued fractious by Stem, CrflU'$ Jour., x.,
XI.. XVIII.
§^ 15, 16 EXAMPLES 449
by taking tlmt fraction in series (B) of last section whose de-
nominator most nearly equals without exceeding D.
N.B. — If tlie denomiuator in tlie (A) series wliicli most
nearly equals without exceeiling D be the denominator of an
intermediate convergent, the denominator in the (B) series which
most nearl)' equals without exceeding D will be the denominator
of a principal convergent.
Example 1.
To find tbe fraction, whose denominator does not exceed GO, which
779
approximates most closely to ,^— .
„, , 779 , 1 1 1 1
We have 207 = ^ + 1^ 3^ 4? 2T
1
5"
mu ^^ , 0 3 15 143
The odd convorgents are j , j , -j- , -g^ ;
1 4 64 779
the even couvergents ^ , j , p^ , ^^ •
The two series are
0 1 2 3 7 11 15 79 143 922
1701
2480
i' i' 1' I' 2' IT' 4 ' 21' 3S ' 245'
452 '
659 ' •
1 4 I'.l 34 49 64 207 330
493
636 779
0' 1' 5 ' 9 ' 13' 17' 55 ' 93 '
131'
169' 207
(A),
(B).
Hence, of the fractions whose denominators do not exceed GO, 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
= 3-763G3 . . . , and 779/207 = 376328 . . . For a rule enabling us in most
cases to save calculation in deciding between the closeness of the (A) and (B)
approximations, see Exercises xxs., 10.
Example 2.
Adopting La Caille's determination of the lenj-'th of the tropical year as
365'' S'^ 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 intei-calating 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,
o64UU
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. 11. 29
450 EXAMPLES CH. XXXIl
If wo turn WilOO/20929 into a continned fraction and form the (A) and (B)
■eric* of convcrgcnta, wo have (omitting the earlier terms)
4 33 161 2865 8434 14003 .
i' ¥' 89' "694 • 2043' 8392' "■ * '"
6
9
a*
13
¥'
17
4 •
21
5''
25
6 •
29
7'
62
15'
9o
23'
128 289 450 611
31 ' 70 • lOa ' 148
772 933 1094
1x7' 226' 205 ' '
(to. (D).
Hence, if we take npproximfttions which err by excess, we may with increas-
ing accnrncy intercalate 1 day every 4 years, 8 every 33, 39 every 101, and
so on * ; 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 iioxt of the series.
The (B) series may be used in a similar mannert.
Example 8.
An eclipse of the sun will happen if at the time of new moon the earth be
within aliont 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 34C-6196 days. It is required to
calculate the simpler periods for the recurring of eclipses.
Suppose that after any the same time from a now moon the moon and earth
have made respectively the multiples z and y of a revolution, then z x 29 '5306 =
y X 3400196. Hence y/x = 295306/3466196 = 0 + .-^ fT oT A IT 5T • • •
The Bocccssive convergcnts to this traction are 1/11, 1/12, 3/35, 4/47, 19/223,
61/716.
Sup]ioBO we take the convergent 4/47, the error incurred thereby will be
< 1/47 X 223 in excess, and we may write on the most unfavourable supposition
X 47 47x223'
* Tho fraction 4/1 corresponds to tlie Julian intercalation, introdnccd by
Julius Cicsar (45 n.c). 33/8 gives the so-called Persian intercalation, laid to
be due to tlie mathematician Omar Alkhayaui (1079 a.d.). Tho method in
priiu'nt use among most Euro|>ean nations is the Gregorian, which corrects the
Juliitn intercalation by omitting 3 days every 4 centuries. This corrcn|>onill
to the fraction 400/'J7, which is not one in the above scries; in fact, 70 day*
every 2H9 years would be more accurate. The Gregorian method han, how-
ever, the advantage of proceeding by multiples of a century. The Greeks and
Ilusaiani still use the Julian intercalation, and in consequence there is a
difference o( 12 dayH botwt'«;u their calendar and ours. Sec art. " Calendar,"
Encyciipcrdia TIritannicn, Uth ed.
t See (<ngrnni,'i''g additions to the French edition ot Euler'i Algtbrj (Parie,
1807). t. II., p. 3rj.
I
§ 16 EXERCISES XXX 451
Hence, it X = 47, »/ = 4- 1/223. But 360°/223 = l°-61. Hence, 47 lunations
after total eclipse, new moon will happen when the earth is less than 1°-G1
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 22:j
lunations, which amounts to IS 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 3607710, 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
convergenta.
Exercises XXX
(1.) Find the first eight con vergents to l + ^g— J— =— . . ., and find
the fraction nearest to it whose denominator does not exceed GOO.
(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 = 271828183 ....
(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. 87 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 AG 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 to 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 thuy grow broader and sparser as they recede
from A.
(8.) Show that between two given fractions p/j and p'jq', such that
pq' -p'q = i, 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.
2'J— 2
452 kxehcises xxx cii. xxxii
(9.) In tho series of (rnctions vliogo douomiuatorti arc 1, 2, 3, ... , n
there ia at least one wlioso denominator is r, say, sach tliat it diOfers from a
given irrational quantity x by less thiin 1/mk. (For a proof of this theorem,
duo to Diriclilct, nut depending on the theory of continacd fractions, aeo
Sorrct, Alff. Sup., 4~ id., t. i., p. 27.)
(10.) If tho ncnreiit rational approximation in excess or defect (sec § IG)
be an intemirdittte convergent I'lQ, where Q = X9,-, + g,_,, show that the
Bpproximntion in defect or excess will be nearer unless Q>i?, + 7,-i/'-T|,+,.
(11.) If ?.eto partial quotients be (contniry to the usual undcnitaiidinK)
admitted, show that cveiy continued fraction muy bo written in the form
Qj . . ., where a,, a,, a,, . . . are each cither 0 or 1. Show
the bearing of this on the theory of the so-colled intermediate convergcnta.
(12.) xz„=0, a?, = 1, arr=<'.+r o^r-i + Cr-j". show th.it ;>.+,/?.+r-Pj7.=
c'r/'/.'/.+r; 'i -P»/?» = (i^r+.^» 1 ^'^.-l)/7,.(9l,+T+/l•+r^l.^r-l). wliero /»='»- a,-
(nortrcavcs, Ilea, ilalh., rtb. 18Ub.)
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, xxxn., § 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 (Pi + Jl{)/Qi. We
suppose that its value is positive ; and we arrange, as we always
may, that Pi, Qi, E shall be integers, and that \/B shall have
the positive sign as indicated. It will of course always be
positive ; but P^ and Q, may be either positive or negative. It
is further supposed that B - Pi' is exactly divisible by Q^. This
is allowable, for, if ^-Pi^ were, say, prime to Qi, then we might
write (P, + JP)/Q, = {P,Qi + slqm/Qr = (P/ + ^o')/Q,',
where E - P,'= { = Q,= (P - Pf) = (P - P,') Q>'} is exactly divisible
by Qi'.
For example, to put 7(2- */ ^ ] into the standard form contemplated,
we must write
BO tliat in this case Pi= - 16, (?, = - 32, JJ = 96 ; iJ - Pi' = 9e - 236= - ICO,
which i3 exactly divisible by Q, = - 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 nECUKKENCE-FORMULA FOR P„ AND Q„ CH. XXXIII
§ 2.] If we adopt the process and notation of chap, xxxii.,
g 3 and 5, tlie calculation of the partial and complete quotient*
of the continued fraction which represents {Pi + s^Jt)jQi proceeda
as follows : —
P, + JR 1
ar,= - r, =«!+ _ ;
Pt+-JR 1
Pn + >/T{ ^ 1
kirn JJ;»+1
(1),
where it will be remembered that Oi, a^, . . . are the greatest
integers which do not exceed Xi, Xj, . . . respectively; and
Xf, Xt, . . . are each positive, and not less than unity.
It should be noticed, however, that since we keep the radical
\^ unaltered in our arrangement of the complete quotients, it
by no means follows that P,, Q,, Pt, Qt, &c., are integers, much
less that they are positive integers.
The connection between any two consecutive pairs, say /\,
Q, and jP,j.i, <2»+j, follows from the equation
1
Q. "'^iP^^i + jRVQ,
(2).
or
\(P,-a,Q.)P.^i-Q,Q,^,->-Ii\-^{P.-a,Q, + P„i)^ = 0
(3).
It follows from (3). by chap, xi., § 8, that
(Pn -a,Q,) P.+, - Q.Q.+, + 7? = 0,
whence
P.+, = a.(?.-P. (4).
/\.,' + <?,<?,.. = ■« (5).
If we write n - 1 for n in (5), we have
§§ 2, 3 EXPRESSIONS FOR P„ AND Q„ 455
From (5), by means of (4) and (6), we have
SO that Q„+i = Q,i- 1 + 2a„ P„ - a„= Q„ ,
= Qn-i + a«(P.-Pn+i) (7).
The formula3 (4) and (7) give a convenient means of cal-
culating Po, P3, Qs, Pi, Qi, &c., and hence the successive
complete quotients Xr., X3, . . .
Q2 is given by the equation
namely, Qa = ^-^ '-,
From this last equation it follows, since by hj'pothesis
{M-Pi")IQi is an integer, that Q2 is an integer. Hence, since
Pi, Qi are integers, it follows, by (4) and (7), that Po, P3, . . .,
Pn, Q3, ■ ■ •> Qn are also all integers.
§ 3.] We shall now investigate formula3 connecting P„ and
Q„ with the numerators and denominators of the convergents
to the continued fraction which represents (Pi+v'^)/Qi.
We have (chap, xxxn., § 9)
^ j9„-l Pn + Pn-lQv. + j^n-l ^
g-n-i P„ + g„-2 Q„ + qn-i -fR
Hence
(P, + JE) {qn-, P„ + qn-, Qn + ?»-. ^B)
= Ql (j}n--i Pn +Pu-1 Qn+Pn-i V^) (1).
From (1) we derive
qn-i Pn + qn-1 Qn = QlPn-1 - PlQu-l (2) ;
R-P^
Pn-l Pn ^ iA.-2 Qn = PlPn-i + —Q-^ 1"-i (3)-
450 Pn<\fR, <?„<-2\'7?, o„<2s/7? en. xxxm
I'roin (•-') Miiil (.'!) wc obtain, since 7^«-i <7n-a -;'»-» y»-i
= (-1)-,
( - 1)" ' P. = y, (;»«-, 7.-1 + /J.-, qn-x)
It- F''
+ — g— ^ ?»-! 7i.-a- <?i ^"-1 /'»-» (4) ;
(-l)"-'Q. = -2;;.-,?„-,i',-^^' ?.->'+ «./>.-.' (5).
The formulae (4) and (5) give us tlie required expressions,
and furninh another proof that Pj, P, Pn, Qi, Qs, • ■ ■, Q»
are all integral.
§ 4.] If in equation (2) of last paragraph we replace /*, by
its value (li(j\-xXn+p»-^l{qn-iXn + qn-i)- ^^, dcrivcd from
equation (A), we have
q.-J\^q.-.Q. = ^^'^'- ^qn-.^ (I).
Also, since Xn = (/\ + •SR)IQn, we have
Pn-^nQn^-^i (2).
From equations (1) and (2) we derive, by direct calculation,
the foUowing four : —
Pn =
7 ^r r, {7.-. (7-.^. + 7,-0) 2 v^ + ( - !)«-■ Q.} (4) ;
(7«-i^« + 7»-w
sfTt-P,=
, "^ «{27-.(7»-.+^-) v^-(- l)"-'<?.} (5);
(qn-l-Tn + qn-i) I \ ^i, / J
, ^- ^,{(^V^7-.+7.-^)('7-.^« + <7.-^)2^/S-(-l)"-'<^.}(6)•
The coclTicionts of Jit and 2%^// in these four formula; are
positive, and inerease without limit when n is increased without
limit Hence, since Q, is a fixed quantity, it follows that fur
§§ .3, 4 CYCLE OF (P, + -s/R)lQi 457
some value of n, say n = i', and for all gi-eater values, P„, (j„,
JR-Pn, iJH—Qn will all be positive. In other ivords, on
and after a certain value of n, n-v say, P„ and Q„ icill he
positive; and P„<Jll, and Q„<2jK
Cor. 1. Since F„ and Qn are integers, it follows that
after n = v P„ cannot have more than JR different values, and
Qn cannot have more than 2 J7i different values; so that ar„
= (P„ + jR)IQn cannot have more than Jli x 2 JB = 2i? different
values. In otlier words, after the ith complete quotient, the
complete quotients must recur within 2R steps at most.
Hence the continued fraction which represents {Pi + Jll)/Q,
must recur in a cycle of 2R steps at most.
Since ever after n = v P„ and Q„ remain positive, it is clear
that in the cycle of complete quotients there cannot occur any one
in which P„ and Q„ are not both positive.
It should be noticed that it is merely the fact that P„ and
Q„ ultimately become positive that causes the recurrence.
If we knew that, on and after n = v, P„ remains positive, then
it would follow, from § 2 (4), that Qy and all following remain
positive ; and it would follow, from § 2 (5), that Py+i and all
following are each <JR ; and hence, from (4), that Q^+i and all
following are each <2jR; and we should thus estabhsh the
recuiTence of the continued fraction by a somewhat different
process of reasoning.
Cor. 2. Since a„ is the greatest integer in {Pn + •JR)/Q„,
and since, if n>v, P„ and Q„ are both positive, and Pn<jR,
and Qr,>i, it follows that, if n>v, an<2jR.
It follows, therefore, that none of the partial quotients in the
cycle can exceed tJie greatest integer in '2jR.
Cor. 3. By means of (3) and (4), we can show that idtimately
Pn+Qn>jR (7).
Cor. 4. From § 2 (5), we can also show that ultimately
I\+Q„.,>JR (8).
458 PURE nECUlUUNO C.K. CU. XXXIll
Cor. 5. Since JJ{>P„, it follows from Cor. 3 and Cor. 4
t/iat ultimately
<(,?.-! (9).
EVERY RECUURINQ CONTINUED FRACTION IS EQUAL TO A
SIMPLE QUADRATIC SURD NUMBER.
§ 5.] We shall next prove the converse of the main pro-
position wliieli has just been established, namely, we shall show
that every recurring continued fraction, pure or mixed, is
equal to a simple qua<lratic surd number.
First, let us consider the pure recurring continued friction
ar =a, + ... — ... (1).
•
Let the two last convergents to
1 1
o, + , . . —
Oj+ a,.
be p'lq and pjq.
From (1) we have
1 _1_ 1
Q, + ■ ■ ■ a, + a-, '
_ pXx + p' _
" qx, + q '
whence
qx,' + {q-p)x,~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 Xi rctjuired.
We have, therefore,
a:, = a, +
a-,- -^ (3).
L + JA
= j^f .say;
which proves the proposition in the present case.
§§ 4, 5 MIXED RECURRING C.F. 459
It should be noticed tluit, since aj + O, p/(j>l; so that
p>q>q'- 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 ,,.
^'='^ + «-
+ ' ' ' ar+ OLi
+ 02 +
Also let
1
1
•
(5).
' « «2 +
«, +
*
Then, by (3).
L +
JN
^■- M •
From (4) we have
1
1 1
a^i = tti H .
' «o +
«r+2/l
whence, if FjQ
and F/Q be
the two
.ast
couvergents
to
1 1
Oi + . . . — ,
Oa + ttr
Py, + P
Xi
Qy: + Q"
a:, = -
_PL + P3I+PjN ,g,
QL + Q-M+QJW
Hence, rationalising the denominator, we deduce
U+VJN
W '
Example 1.
Evaluate X, = 1 + 2^ jl- A
The two laBt oonvergeuts to 1 + q t ^^^ ^/^ i&nd 4/3 ; hence
« + 1
_4xi + 3
'^'3x^ + 2'
We therefore have
3xi2-2x,-3=0,
the positive root of which is
460 CF. FOR -JiCjD) Cll. XXXIII
Examplo 2,
^ , „ 1 1 1 I 1
Tho two lost coDvergonts to 3 + - nre 3/1 and 13/4 ; nnd, by Example 1
above,
Wo have, therefore,
11 _l+^/lO
^■^2+1+ 8~-
"'-^■'4+ (l + yi0)/3'
_13(l + ^in)/3 + 3
"ill + 7111)^/3+1 •
22 + 13V10
ON THE CONTINUED FRACTION WHICH REPRESENTS ^/(CID).
§ 6] The square root of every positive rational number, say
J(C/D), where C and D are positive integers, and C/D is not
the square of a comuiensunible number, can be put into tlie form
JN/M, where NCD and M=D. Since X/M = C is an
integer, we know from what precedes that JN/M can be
developed, and tliat in one way only, as a continued fraction of
the form
1 111 1
x,<=a, + — ... ... ... (1).
a, + Or + Oi + o, + a, + '' '
We have, in fact, merely to ])ut P, = 0, R = N, Q, = M in our
previous formula:.
We suppose that JN/M is greater than unity, so tliat a, + 0.
If JN/Af were less than unity, then we have only to consider
if/jN = JAPXIN, which is grait<!r than unity.
The aiyclic part (/, + ... must consist of one term at
I
(2).
§§ •"), 6 ACYCLIC PART OF VN/M 461
least, for we saw, iu g 5, that a pure recurring continued fraction
caiiniit represent a surd number of the form JNJM. Let us
suppose that there are at least two terras in this part of the
fraction ; and let P'jQ', PjQ be the two last convergeuts to ,
«i + . . . — ; and p'lq, pjq the two last convergents to
1 111 1 ,,„ .,
a, + — ... . . . — . Then, if
1 1
we have *
1 1 1
Xi = ai + . . . ,
a., + ar+ !/i'
1 1 1 1_ J_ J_
U.,+ ' ' ' «r + "l + °-2 + ' ' ' «a + yi '
Hence
^ ^Pyi + F ^pyi+p
' Qyi + Qf qyi+q
Eliminating ?/i from the equations (2), we have
(.Qq - Q'q) *v - {Qp - Q'p + Pq - Pq) x, + {Pp' - rp) = o (3).
Now, if Xi = jA'jM, we must have
M-.T,- -N=0 (4).
In order that tlie equations (3) and (4) may agree, we must
have
Qp'-Q-p + Pq--P-q = 0 (5);
and
Qq-qq- ip (^^-
It is easy to show that equation (6) cannot be satisfied. We
have, in fact,
Pp-PpTpP/P-p/p'
Qq'-Qq Q'q Q/Q'-q/q ^^'-
But, by chap, xxxn., § 7,
P p 11 11
^v - . = «r + . . . o« . . . - ,
P p ttr-t + «i a,_i + Ui '
= ar-a, ±f,
where/ is a proper fraction.
462 CYCLK OF QUOTIENTS FOK ViV/iJ/ CH. XXXIII
Similarly
Q q 1 1 11
V q flr-i + «-j a,_, + M.J
= Or-a.±/',
where/' is a proper fraction.
Now ar-o, cannot be zero, for, if that were so, we should
have ar = <*«> that is to say, the cycle of partial quotients would
begin one place sooner, and would be o,, o,, a,, . . . , a,_,, and not
o,, a,, . . . , a,, as was supposed. It follows then that a, - «, is
a positive or nc;,'ativc integral number. Hence the signs of
PjF - pIp' and QjQ - qjq 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
^, = a + — . . . ~ — ... (S),
11 11
= a +
a, + a, + ■ * ■ o, + l/(x, - a) '
Hence
^ _pK^i-a)+p'
,., . ' q!(-r,-a) + q"
which gives
qW-(p' + q'a-q)ari-(p-ap') = 0 (9).
From (9) we obtain
_p' + q'a-q Jjp + q'g -qy+i(p- gp-f^
2q' * §7
In order that (10) may agree with Xi=Jn/M, we must have
p' + q'a-qO (11);
and
q"N/AP^(p-ap')^ {12).
Cor. 1. By equation (11) we have
F'/q' + a = q/q'.
(10).
^ 6, 7 CYCLE OF QUOTIENTS FOR -jNjM 4G3
Hence, by chap, xxxii., § 7, Cor. 2,
11 1^1 1
a, + 02 + o,-i o«-i + "i
It follows, therefore, by chap, xxxn., § 3, that
a, = 2a, aj-i = ai, o.-2 = "a, • • •> "i = <'»-i-
In other words, the last partial quotient of the cyclical part of
the continued fraction ivhich represents jNjM is double the
unique partial quotient which forms the acyclical part; and the
rest of tlw cycle is reciprocal, that is to say, the partial quotients
equidistant from the tivo extremes are equal.
In short, we may write
JN 11 1111 , „v
« " *
Cor. 2. If we use the value of q'a given by (11), we may
throw (12) into the form
q-NlM" =pq -p {q -p) ;
wltence
q'^NIM'-p"=pq'-p'q,
= ±1 (14),
the upper sign being taken if pjq be an even convergent, the lower
if it be an odd contergent.
§ 7.] All the results already established for {Pi + Jli)IQi
apply to J^jM. For convenience, we modify the notation as
follows : —
«, =«, x,=^{P, + JR)JQ, = (0 + JN)!M;
a, =a„ x, = {P, + jR)IQ,={L, + JF)IMr,
a, =cu, x, = {P, + JTt)IQ,= {L, + jN)IM,;
a, =«._„ x.={P, + jR)IQML,-, + jN)IM,^,;
a.+i = 2a,
a.+j = "i.
From § 2 (4), we then have
Ln = <^-iMn-l- In-l (1);
and, in particular, when n=\,
4C4
CYCLES OK DIVIDENDS AN'D DIVISORS CII. XXXIH
From § 2 (5), we have
and, ill particular,
(2);
(-'■)•
From § 3 (4) and (5), we have
( - 1)-Z, = (A7'l/)?.7-. - ^fp.Pn-i (3) ;
(-)'M,^3W-iNIM)q,' (4).
Tlio^e formulro are often useful in particular applications.
It will be a good exercise for the student to establish them
directly.
§ 8.] Let us call ij, Zj, &c., the Jiatiotial Dividends and M,
3f, , 31,, &c., the Divisors belonging to the development of jNjM.
Then, from the results of § 4, we see that
None of the rational dividends can ejrceed JN; none of the
partial quotients and none of the divisors can exceed ^JN.
All the rational difidend.s. and all the dirisors, are jMsitive.
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
and
Zj+i — /<i, L,+i- Lj, (1),
J/.« = JA. il.v, = 31„ (2).
We can also show that the cycles of the rational diviileuds
and of the divisors have a reciprocal property like the cycle of
the partial quotients ; namely, we have
L. =i,. 31. =31;'
i,_i = L,, 31,^1 = J/i ; _
L,-t = Zj, J/,-j = J/j ;
For, bv ? 7 (-2),
L.^,' + J/. H M, = W + ^1 -'^;
but Z.t, - Li and 3I,^.^ -- 31,, hence
3J.-3f (4).
(3).
§§ 7, 8 THE COLLATERAL CYCLES 4C5
Again, by § 7 (1),
Ls+\ = 0;Ma - L, ;
but Z,+i = Zi , a, = 2a, Ms = M, hence we havo
Li = 2aJ/- A-
Now, by § 7 (1'), Li = a3f, hence
Zi = 2ij - La,
therefore L, = Li (5).
Again, by § 7 (2).
L:- + MMs-, = L,' + MJT,
whence, bearing in mind what we have akeady proved, we have
il/.-i = i»/i (6).
Once more, by § 7 (1),
L2 = a-iMi — Xj.
Now 31,-1 = 3Ii and a,_, = a^, hence
Z, — ij = x/i - Zj_i.
But L,= Li, hence
Zs-l = Z.2.
Proceeding step by step, in tliis way, we estabhsh all the
equations (3).
It appears, then, that we may write the cycles of the rational
dividends and of the divisors thus —
Li, Zj, Z3, . . ., L~, Z2, Zi;
M„ M,, iV„ . . ., 3f„ M„ M„ M.
Since 31 precedes J/j , 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 :—
o-u <H, «3. • • •> "3> °2> "i. 2a; a^.
Zi, Zj, Z3, . . ., Zj, Z2, Zi ; Zi.
31, 3Iu 3L, 3I„ . . ., 3L, 31, ; 31, 31,.
That is to say, the cycle of the rational dividends is collateral
with the cycle oj'tlie jjartial quotients, and is completely reciprocal;
c. II. 30
4C6 TESTS FOR MIDDLE OF CYCLE CH. XXXIII
the cycle of the ditisors beging one step earlier* (tluil is, from th«
very iM-if inning), ami is rcciprucal after the first term.
§ D.] The following theorem forms, in a certain sense, a
converse to the propositions just established regarding the cycles
of the continued fraction which represents o/N/M.
If Z„ •=£,+,, M^ =3/,, o_ =a„
thm j[,„., = i,+j, 3/„.i = 3/,+,, 0^-1 = 0,^., (1).
We have, by § 7 (2),
Z.' + J/,3/„., = i.„' + M.^,M„
whence, remembering our data, we deduce
3/.., = iA,+. (2).
Again, by § 7 (1),
X,„ + Z„_, = a„_, 3/«_„
Z.+i + i,+i = o,+, 3/,+i,
whence, sini-i- />„ = £,+, by data,
= (a— ,-<!,+,) 3/,+, (3).
If 2i„-i>i,+j, we may write (3)
(Z„_, - X,+j)/3/,+i = a„-, - a,^., (4) ;
if /,>-,< Z,+„ we may write
(Z,+, - Z„.,)/3/„_, = Q,+, - o._, (5).
But, by § 4 (9), the left-hand sides of (4) and (5) (if they
dillor fnun 0) are each <1, while tlie right-hand sides are each
positive integers (if they difl'er from 0).
It follows, then, that each side of equation (3) must vanish,
80 that
Zr-1 = Z,« (6),
<»«-i=<»ii+i (7),
which completes the proof.
* The fact tliat t)ir cjclc of the divisore begins one step earlier than the
cjcle* of the partiitl i)Uoticiit« anJ rational dividends is true for the general
recurring continued fraction. Several otlier propositioni proved for the
special coKC now under consideration liavo a more general ajipUcaliuu. TU*
eircumslauces ar« left fur the nador hiusslf to disouvcr.
§§ 8, 9 TESTS FOR MIDDLE OF CYCLE 467
Cor. 1. Stiu-ting with tlie equations in the second line of {I)
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
Ln = Xn+i , Mn = Mn, a„ = a„ ;
in other words, the conditions reduce to
J-'n — J-iii+i i
and the cnnckision becomes
Hence, if two consecutive rational dividends be equal, tfiri/ are
the middle terms of the cycle of rational dividends, which must tliere-
fore he an even cycle ; and the partial quotient and divisor cor-
responding to the first of the two rational dividends will he the middle
terms of their respective cycles, which must therefore be odd cycles.
Cor. 3. If we put m=n + 1, the conditions in (1) reduce to
ilf„+i = Mn, a„+i = a„ ;
and the conchision gives
Using this conclusion as data in (1), we have as conclusion
and so on.
Hence, if two consecutive divisors (Mn, il/»+i) be equal, and also
the two corresponding partial quotients («„ , a„+j) be equal, these two
pairs are the middle terms of their respective cycles, which are both
ei>m ; and the rational dividend (in+i) coiresponding 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 -JN/M. In certain cases they are useful also in
reducing surds of the more general form (L + sIN)!]^! to con-
tinued fractions.
Example 1.
Express ^8463/39 as a simple continued fraction ; and exhibit the cycles
of the rutioual dividends and of the divisors.
30—2
3,
1,
2,
4;
63,
63,
a.
79;
42.
107,
CI;
468 EXAMPLES CII. XXXIII
Wc have
>163^ -78+78'463_g ^ •
3'J 89 (78+>/M63)/6r
78+^5/8403^2 + -4^+>/&'M^g^ ^
61 CI (I4 + J8IC3)/107
44+^8468^^ -63+ v/8163_^ 1
107 " 107 (63 + ^8463)/42
63+^/6463_g -63+s/84M^g 1
42 ~ 42 " (63 + ,yH463)/107
63+JiM = i^*e.
Since we have now two successive rational dividends each eqnal to 63, we
know that the cycle of partial quotients has culminated in 3. Hence the
cycles of partial quotientfl, rational dividends, ond divisors arc —
Partial quotients . . 2, 1,
Itational diridends . 78, 44,
I)i\-isora . . . 39, 61, 107,
and we have
78463_„ J_ i_ i_ _L J. _L
89 ~ ■'■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 ^SJM, prove the following formalo : —
lfc = 2r,
Pc_Pm9i+Pi9i-i ,j ..
9. ?i(9i+i + «i-i) * ■''
ifc = 2( + l,
9. 9.;.'+9i'' ^ '•
if m bo any positive integer.
For brevity we shall prove (III.) alone. The reader will 6nd that (L)
and (II.) may be proved in a similar manner. For a different kind of demon-
stration, sec chap, xxitv., § 6.
Wo have
^*'^ = a+-- . . . _L J- . . . i (2m cycles),
= a-i . . . — „ — - . . . — ; — (m cycles),
o,+ a,+ 2a+ o,+ o+p^,^» ■' "
_ ('»+P»J»«c);'«+Pi«-i
{<'+Pm^lln^)<lm<+9mr-\'
_ {"Pm,^ Pm,'t)jm,J-Pm,*
9m.{'>9m. + 9m^l+PwJ
(«).
§ 9 EXERCISES XXXI 40!)
Now tho cqufitioua (2) and (3) of § 3 give us
Pmc Pmc+\ +Pmc-i Qmc+1 = (^1^^ 1<,J
In tho present case,
Qm.+l=<?c+l = 'U. = ilZ.
The equations (j3) therefore give
<'9mc+<!mc-l=Pmc 1 (^\
aPrr^+P^l^im^qJ ^^''
From (a) and (7) (III.) follows at once.
The formulm (I.), (II.), (III.) enable us, after a certain number of oon-
vergents to Jn/M have been calculated, to calculate high convergents
without finding all tho intermediate ones.
Consider, for example,
V84G3_ _L_LJ_J_J_J-
89 ~ "'"2+ 1+ 3+ 1+ 2+ 4+ •
» •
Here c=G, t = 3, and we have for the first four convergents 2/1, 5/2, 7/3,
26/11; hcnco
P6_P_i!h±JVh
?« Qilli + Si)'
26x3 + 7x2^92
~ 3(11 + 2) ~39'
Pr.^Pi'±i^!B^hl^
3i2 ^Peie
92° + (8463/39°) ■ 39- _ 16927 _
~ 2 X 92 X 39 "■ 7176 '
P^^P_^±ME^l±l,
16927- x39°+8163x 7176"
~ 2 X S'J' X 16927 X 7176 '
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 fractious, and
exhibit the cycles of the partial quotients, rational dividends, aud divisors:—
(1.) V(lOl). (2-) W(G3)- (3-) V(B)-
. ) JL_. (5.) 2-±^). (6.) 1 + V*.
(7.) Express the positive root of i> - 1 - 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 tho two fractions.
Also
470 EXEKCISES XXXI CM. XXXIII
(U.) Show that
^(a. + 6) = „ + ^. ....
i^/(a'-6) = o-2;j— . . . .
(10.) Express ,^'(ii' + 1) as a simple continacd fraction, and find nn
cxprcssiou (or the nib convcrRcnt.
Evaluate tbo following recurring continued fractions, aiid find, where you
can, closed expressions for their nth convcrgents; also obtain recurring
formuliD for simplifying the calculation of high oonTergents :—
1
(U.)
o + — -. . . .
o +
•
(12.)
1
a- ' ' '
•
(18.)
1 1
• •
Show, in
this case
that
Pi»+i
-2p„+J'„-,=aipsn
(U.)
1
1 +
1 1
1+ • • • 2+ • • ••
where the cycle consists of n units followed by 2.
(15.) Show that
f J._L . . W 1_ . . \
\*+ 4*+ J \2x+ J
• • a
is indc{>endont of x.
(IC.) Show that
. c+ a +
1 1
a+ b+ ' ' '' , a + b + e+ ' '
show that
2(x-^■y + t)-(a^-^)^-f) _ 1 1 1
•iu-{a + b + r)-abc ~ be + 1 ea + l'*' ab + l'
(IM.) bhow that
/_o_ \«_ n'
{b+ ■ • •) -2u + 6'- • * • •
§ 9 EXERCISES XXXI 471
(i;i.) If p. be the numerator of any convergent to a^/2, then 2j)'±l will
also be the numerator of a convergent, the upper or lower sign being taUen
according as jtjq is an odd or an even convergent; also, if q, q' be two oon-
seontive denominators, q' + q'" will be a denominator.
(20.) Evaluate
J_ J_ 1
1+1+" ••«+••• •
* ft
where the cycle consists of /i- 1 units followed by n,
(21.) In the case of = — t — . . ., prove that
ft ft
P2n= ?=„« = {(v/2 + 1)="+' + (v/2 - l)=»+i}/2^2,
l>2»-i = i?en = {(v/2 + iP'- W2-lp}/V2.
(22.) Convert the positive root of ax- + al)x-b = 0 into a simple con-
tinued fraction ; and show that y„ and g„ are the coefficients of a;" in
{x+ bx'-x*)l{l - ab + 2.x- + x*) and (ax + ah + l.x- + x*)l{l - ab + 2.x' + x^)
respectively.
Hence, or otherwise, show that if o, /3 be the roots of 1- (((6 + 2)2 +i- = 0,
then
a" -8'^
P:r,+l = Izn
_ (a"+i - ^+1) - (a" - /S")
(23.) If the number of quotients in the cycle of
show that
JN 11 111
■^ =(H . . . ^ ... be c,
il a, + 112+ a„+ ai+ 2a +
1 111 1 1, „„„„,„.,_ iV<?^
a-\ , . . i . . . (m cycles)— ^,„ -.
Oi+ aj+2a+ai+ Oj+a^ •' ' M-p,„c
(24.)* If c be the number of quotients in the cycle of ^/NjM, show that
if c = 2« + l,
p'(-^i+yV^ N
Tl-r-l + ri+r ^^■'
r=0, 1 t-1;
andif c = 2f,
Pl-T-iPl-T-l+Pl+r-lPl+r _ ^
9(-r-2?(-7-l + ?(+r-l!(+r •'^'"
(25. )t If JZ = a-\ . . . = — .... and if the convergent
* ' ^ a,+ 0.,+ 0^+01+ 2<i+ "
• •
• 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 eee Serret's Cour$
d'Algebre Supgrieure, 3<°° ed., t. I., chaps, i. and ii.
gun-Bis XXXI
xxxm
' A -.-..■
'•ail'
3X^j VaoB S»% fBom Bdtapae; :
•mi At Tp»fc^*» veeat of Ab smxgBttal piuri
<f lAe «mHr«» 4f cwr 3ifli9Ba «imc& an pc
1 11
I»i ^ iTj— ■'■iM- Sir d tte
pwt if tar cjmflt (rfAflir h^iobb
ij 10, 11 DIOPHANTDTE PROBLEMS 473
APPLICATIOXS TO THE SOLUTION OF DIOPHANTINE PROBLEMS.
§ 10.] When an equation or a sj'stem 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 Diophantcs, who,
so far as we know, was the first to systematically discuss such
problems, and who showed extraordinar)' 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 firactions : its complete development belongs to the
higher arithmetic, on which it is beyond the purpose of the
present work to enter t.
Equations of the \st 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'B Diophantot of Alexandria (Camb. 188.5).
t The reader who wishes to purstie the study of the higher arithmetic
Bhonld first read Theory of Numbtrt, Part I. (1892) by G. B. Mathews,
M.A-; then the late Henry Smith's series of Eeports on the Theory of
Numbers, published in the Annual Beports of the British Association (1859-
60-61-62) ; then Legendre, Thiorie da Sombret ; Dirichlet's VorUtungen
uber ZaMentheorie, ed. by Dedekind; and finally Gauss's DUquiritiorvt
Arithmetica. He will then be in a position to master the various special
memoirs in which Jacobi, Hermite, Summer, Henry Smith, and others have
developed this great branch of pure mathematics.
474 tix-bt/ = c CH. XXXIII
roinoveii. We may obviously confine ourselves to the cases
where a is prime to b ; fur, if x and y be integers, any factor
common to a and b must Le a factor in c. In other words, if a
be not prime to b, the equation ax±by = c has no integral solution.
§12.] To find all the integral solutiuii-i of ax- by = c; and to
separate the positive integral solutions.
We can always find a particular integral solution of
ax-by = c (1).
For, since a is prime to b, if we convert ajb into a continued
fraction, its last convergent will be a//). Let the penultimate
convergent be pjq, then, by chap, xxxii., § 8,
aq-pb = ±l (2).
Therefore
a{±cq)-b{±cp)=c (3).
Hence
x=±cq, y'=^±cp (4)
is a particular integral solution of (1).
Next, let (j-, y) be any integral solution of (1) whatever.
Then fix)m (1) and (3) by subtraction we derive
a{^-{±cq)\-b{y-(±cp)\ = 0.
Therefore
{i!-(±cq)]l{y-{±cp)\=bla (5).
Since a is prime to b, it follows from (5), by chap, in.. Exercises
IV., 1, that
x-{±cq) = bt, y-(±cp) = at,
where t is zero or some integer positive or negative. Hence
every integral solution of (1) is included in
x = ±c/i + 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 jwsitive.
r. If aib>plq, then the upiKT sign must be taken in (2),
and we have
x-cq + t/t, y=-cp + at (ti;.
§§11-13 ax + hy = c 475
There are obviously an iutinity of integral solutions. To get
positive values for x and y we must (since cp/a<cq/b) give to
t values such that - cp/a > < :t» + » . There are, therefore, an
intinite numlier of positive integral solutions.
2°. If a/b<p/q, so that cp/a>cq/b, 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/a^fjp- + co .
We see, therefore, that ax — by=^c has in all cases an infinite
number of positive integral solutions.
§ 13.] To find all the integral solutions of
ax + by = c (7),
a7id 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
( ± eq) a + { + cp)b = c (8),
that is, x' = ± cq, y'=+cp is a, 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, 1/= + cp + at (9);
so that there arc in this case also an infinity of integral
solutions.
To get the positive integral solutions : —
1°. Let us suppose that a/6 >j3/2', 80 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 cpjal^t
>cq/b.
2°. Let us suppose that ajb<plq, so that cpla>cq/b, then
x = -cq-bt, y = cp + at (9").
Hence for positive integral solutions we must have - cpja 1^ t
>-cqlb.
476 EXAMPLES CI(. XXXIII
In both these cases the number of positive integral solutions
is limited In fact, the number of such solutions cannot excce»l
l + \cq/b-cp/a\; that is, since 10^-^1 = 1, the numl>er of
positive integral solutions of the equation <ix + by = c cannot
exceed 1 + cjab.
Example 1. To find all the integral and all the positive iutegral Bolationi
o(ar + 13!/ = 159.
We have
A-_L J- JL J_l
13~1+ 1+ 1+ 1+2*
The pcnnltimate convergent is 3/5; and we have
8x6-13x3 = 1,
8 (795) + 13 (-477) = 159.
Hence a particular eolation of the given equation is i' = 796, y'= -477; and
the general solution ia
z = 795-13(, y=-477 + 9«.
For positive integral solutions we mnat have 795/13 ■!«•< 477/8, that is,
eiiS-^t^S'JI- The only admissible values o( t are therefore 60 and 61;
these give i = 15, y = 3, and x = 2, !/ = ll, which are the only positive integral
solutions.
Example 2. Find all the positive integral solutions of 3x + 2i/ + 3» = 8.
We may write this equation in the form
3j + 2y=8-3i,
from which it appears that those solutions alone are admissible for which
« = 0, 1, or 2.
The general integral solution of the given equation is obviously
i = 8-3j-2«, y=-8 + 3r + 3«.
In order to obtain positive values for z and y, we must give to ( integral
values Ij-ing between +4- ji and +2J-». The admissible values of t are
8 and 4, when < = 0; 2, when 2 = 1; and 1, when z = 2. Uenoe the only
positive integral solutions are
1 = 2, 0, 1. 0;
y = l, 4, 1, 1;
« = 0, 0, 1. 2.
In a similar way we mny treat any single equation involving more than
two variablee.
§ 14.] Any system of equations in which the number of
variables exceeds the number of equations may be treated by
mcthixla which depend ultimately on what has been already
done.
§§ 13, 14 SYSTEM OF TWO EQUATIONS 477
Consider, for example, tlie system
ax + by + cz = d (1),
a'x + b'y + c'z ■= d' (2),
wliere a, b, c, d, a, &c. denote any integers positive or negative.
This system is equivalent to the following : —
-{ca')x + {bc')y = {dc') (3),
ax + by + cz — d (4),
where {ca') stands for ca - c'a, &c.
Let S be the G.C.M. of the integers {ac\ {be). Then, if S
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" + {bc')t/&, y = y' + {ca')t/8 (S),
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 {ab') t/8 = d- ax" -by' ( G ),
where c {ab')/& 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)bes = s', ^ = i!'. Then
z-z' _ {ab')
t-i" 8 ■
Hence, if € be the G.C.M. of {aU) and 8, that is, the G.C.M.
of (ic), {cd), {ub'), then
z = z' + {ab')uji, t = t'+hilf. (7),
where u is any integer.
From (5) and (7) we now have
x = x' + {bc')u/€, y = y' + {ca')u/t, z = !/ + {ab')u/f (8),
where x' = x" + {be) t'jh, y = y" + {ca') t'/S.
If in (8) we put u = 0, we get x = x',y = y', z = !! ; therefore
{x , y , z) is a particular integi-al 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) glees all
478 FEUMAT'S I'UOULEM CU. XXXIll
thr Integral »>lulioii.<t of (1), (2), (j-', i/', z) being amj particular
integral solution, t the G.C.M. of (be), {ca), (al/), and u any
integer whatever.
The positive integral solutions can be found by properly
limiting «.
Example.
3x + 4y + 27r = 34, Sx + Sy + 21t = 20.
Here (6c')= -51, (<;a') = 18, (a6') = 3. Hence « = 3; a particular integral
solution is (1, 1, 1) ; and we have for the general integral solution
i=l-17«, t/ = l + Cu, X=:l + U.
The only positive integral solution isx = l, y = \, » = 1.
Equations of the 2nd Degree in Two Variables.
§ 15.] It follows from § 7 (4) that, if pjq» be the nth con-
vergent and Mn the (M + l)th rational divisor belonging to the
development of J{C/D) as a simple perioilic continued fraction,
then
/>/>,'-<?</.' = (-)- 3/. (1).
Hence the equation Dx^ -Cf= + H, where C, D, U arc po.^itiv0
integers, and CjD is not a perfect square, admits of an infinite
number of integral solutions provided its right-Itand side occurs
among the quantities ( - )" J/, belonging to the simple continued
fraction which represents JiCjD) ; and the same is true of the
equation D^ - Ci/' = -II.
The mo.st important case of this proposition arises when wo
Biii)poi?e /)= 1. We thns get the following re.sult : —
The equation x'-Ci/' = ±II, where C and II are positive
integers, and C is not a perfect square, admits of an ii\finite
number of integral solutions provided its right-hand side occurs
among the quantities ( - )" J/, belonging to the development qf JC
as a simple continued fraction.
Cor. 1. The equation x'-Ci/'=l, where C is positive and not
a perfect square, always admits of an infinite number qf solutions'.
* By what seems to bo a historical misnomer, this equation is commonly
ppnken of as tho IVIIiau Equation. It was oriRinally proposed by Fcrmat
a* a obollcugc to the Eugliah mathematicians. Solutious wen obuiucd b/
§§U-1G LAGRANGE'S THEOREM REGARDING «"- (7?/'= ±^^ ^~^
For, if llie number of quotients in the period of JC be
even, =2s say, then {-)'^]\T^ will be + 1 (since here J/=+ 1).
Therefore we have
■where t is any positive integer ; that is to say, we have the
system of solutions
a;=ihu, y=q-t, (A),
for the equation a^ - Cif = 1.
If the number of quotients iu the period be odd, = 2s - 1 say,
then ( - r- Wo,_, will be - 1, but ( - )"-W„-,, ( - ^-^M^-,, . . .
will each be + 1. Hence we shall have the system of solutions
a:=Pit,-a, y = qits-'.t (B),
for the equation x^ — Cy- = 1.
Cor. 2. The equation a?-Cy- = -\ admits of an infinite
number of integral solutions jn-ovided there be an odd number of
quotients in the period of JC.
% 16.] In dealing with the equation
ar-Cf=±n (1)
we may always confine ourselves to what are called primitive
solutions, that is, those for which a; is prime to y. For, if .r and y
have a common factor 0, then &- must be a factor in II, 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'^ = ±II/B- as 5^ 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 //.
With this understanding, we can prove the following im-
portant theorem : —
If II<JC, all the solutions of {\) are furnished by the
conmr gents to JC according to the method of § 15.
This amounts to proving that, i{ x = p, y = qhe any primitive
integral solution of (1), then pjq is a convergent to JC.
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 (Euvra, t. u.,
p. 377.
480 GENERAL SOLUTION OF a?-Ci)''= ±1, Utt ± // ClI. XXXIII
Now WO have, if the upper sign be taken,
j> - cy = //.
Hence plq- JC =n/</(p + JCj),
<JCIq{p^JCq).
<WiPl<lJG^\) (2).
Now piq - JC is positive, therefore piq JC> 1. Hence
p/q-JC<\l-2,f (3).
It follows, tlierefi)re, by chap, xxxii., § 9, Cor. 4, that p/q is
one of the convergeuts to JC.
If the lower sign be taken, we have
q'-illOp'-II/C.
where niG<J{ljC). We can therefore prove, as before, that
qlp is one of the convergonts to J(l/C), from which it follows
that p/q is one of the convergents to JC.
Cor. 1. All t/ie solutions of
^-C,f=\ (4)
are furnished by tlie penultimate contergentg In the successive
or alternate jKriods of JC.
Cor. 2. If the number of quotients in the period of jC be
even, the equation
£'-Cf = -l (5)
has no integral solution. If the number of quotients in the
period be odd, all the integral solutions are furnished by t/ie
penultimate conrergents 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
ir+yJC=(p + qJC)'\ ..>
'r-yJC={p-qJC)'i ^ '•
we have
Hence (fi) gives a solution of (4).
In like manner, if « be any integer, and (/>, q) the first
Bolutiou of (5), a more general solution is given by
x^yJC = (p-^qJCr-\ ...
a-yJC=(j>-qJcM
§§16.17
EXAMPLES
481
Finally, if {p, q) be the first solution of (1), we may express
all the solutions derivable therefrom* bj' means of the general
solution (6) of the equation (4). For, if (r, s) be any solution
whatever of (4), we have
p'-Ccf = ±U,
{f-C<f){r'-Cr) = ±n,
{pr± Cqsf -C(j)s± qrf = ±U.
Therefore
x=pr+Cqs\ .g>
y=ps±qr j
is a solution of (1).
The formulae (6), (7), (8) may be established by means of the
relations which connect the convergents of JC (see Exercises
XXXI., 25, and Serret, Alg. Sup., § 27 et seq.). This method of
demonstration, although more tedious, is much more satisfactoiy,
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 i' - ISy'' = 1.
If we refer to chap, xsxii., § 5, we find the following table of values
for ^/13 :—
n
«»
Pn
,»
^n
1
2
3
3
4
1
1
i
3
3
7
2
3
4
11
3
i
5
18
5
1
6
7
110
137
33
38
4
3
8
2.56
71
3
9
393
109
4
10
G49
ISO
1
11
6
4287
1189
4
Hence the smallest solution of x'
in fact,
13i/» = l isx = 649, ?/ = 180. We have,
649«- 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 a?-Ci/ = ±IT, wiiev H > ^C en. xxxm
From (6) above, wo see that the Rcneral eolution is given bj
x = i {(649 + 180^13)" + (049 - 180V13)*}.
y = 4 { (M9 + 180 v/13)» - (C49 - ISO ^13}') I J 13,
where n is any positive integer.
In particular, taking n = 2, we get the solntion
a = 64'J» + 13.180'=8424ni,
y= 2.649.180=233640.
Example 2. Find the integral solutions of x*- 13y«= - 1.
The primary solution is given by the 5lh convergent to ^/13, u may be
seen by the table given in last example.
The general solution is, by (7),
* = ^{(18 + 5V13)*'-' + (18-5s/13)>»-'}.
!/ = 2^13 {(18 + 5V13)*-» - (18- 5^13)«-'}.
where n is any positive integer.
Example 3. Find all the integral solntions of x*- 13y'=3.
The primary solution is x = 4. y = l, as may be seen from the table abova.
The general solution is therefore, by (t),
i = 4r±135, y = 4<±r,
where (r, i) is any solution whatever of x' - 13y' = 1.
In particular, taking r=649 and » = 160, we get the two solutions, z = 256,
y = 71, and x = 4936, y = 1369.
§ 18.] Let us uext consider the equatioa
x'-Cy' = ±H (9).
where C is positive and not a perfect square, and 11 is positive
but >JC.
We propose to show that the solution of (9) can always be
maile to depend on the solution of an equation of the same form
in which H<JC\ that is, upon the ca.se already completely
solved in §S 15-17.
Let (x, y) be any primitive solution of (9), so that x is prime
to y. Then wo can always determine (x,, y,) so that
ayi-y^i = ±l (10)».
lu fact, if piq be the penultimate convergent to xjy when
converted mUi a simple continued fraction, we have, by § 12,
ar,=-tj:±p, y, = ly±q (11).
* Thuto ii no connection between the doable tigni beie and in (9).
§ 18 LAGRANGE'S CHAIN OF REDUCTIONS 483
If we multiply both sides of (9) by x{ - Cy^, and rearrange
the left-hand side, we get
{XX, - Cyy,y -C{xy,- yx,f = ±B (x,^ - Cy?).
This gives, by (10),
{xx^-Cyy,r-C=±R(x^-Cy^) (12).
Now
xx^ - Cyy, ^t^x'-ChD + ixp- Cyq) (13).
But we may put xp - Cyq = SH± K,, where Ki'^hU. 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
xx-,-Cyy, = Kx (15),
where
zi>izr. (16).
We therefore have, from (12),
K:--C^±U{x,'-Cy?) (17).
Now, by hypothesis, ^G<H, therefore C<E:' and K^-G
<E\
Since (ar,, ^i) are integers, it follows from (17) that, if (9)
have an integral solution, then it must be possible to find an
integer Kil^^H such that
{K^-C)IH=n, (18),
where H, is some integer which is less than H-jH, that is, < H.
If no value of Ki < \H can be found to make {K^ — C)IH
integral (and, be it observed, we have only a limited number of
possible values to try, since Ki1:^\H), then the equation (9) has
no integral solution.
Let us suppose that one or more such values of Ki, say K,,
Kx, K", . . ., can be found, and let the corresponding values of
Hi be Hi, Hi, Hi', . . . 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^-CyC- = ±H \
xi'-Cy,' = ±H'
x,'-Cy,' = ±H"
(lU
where H, Hi, H", ... are all less than H.
31—2
484
PRACTICAL MKTIIOI) OF SOLUTION CM. XXXIII
If it also hajipens tliat iu all the equations (19) the numerical
value of the ripht-liand side is < JC, then these equations can
all be conii>letcly solved, as already explained.
If (•Til Vi) he a solution of any one of them, wo see, by (10)
and (15), that
or ar = (ir.'a-. + Cy,)/iy.'. i' = (^.'y. + a^)//7.',
If iu 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.
where Hi and IT, are each less than /T, ; and we have
X, = {K,.r, + Cij..)III, , y, = {K,y, +jr,)//I,
Xi = (A"; X., + Ci/.)/II.;, y, = ( AVy, + x^yii^
(21).
(22).
Since the fTs are all integers, the chain of successive operations
thus indicated must finally come to an end in every branclL
Thus we sec that any integral solution o/{9) must be deJucibh
from the solution of one or other of a finite group of equations qf
tite type
x'-Cf=IW^ (23).
where II^^^^KjC.
The practical method of solution thus suggested is as
folhiws : —
Find all the integral values of A',<i// for which {K^*- 0)1 II
is an integer. Take any one of those, say A', ; and lot //, be
the corresponding value of {Ki'-C)/H. Then, if II,<JC, solve
the equation x^-Cy* = ±IIi generally; take the formula (20);
au<l find wliioh of the solutions (j,, »/,), if any, make (j", y) integral
We thus get a group of solutions of (9). If IIt>JC, then we
find all the values of A'j< J//, for which (A,' - C)/II, is integral,
* Since tliv bIkdh of x and y are imiiilcront in tlic Bolulions of x*- Cy' =
*//, it ia unnccfRiiary to tako ncconnt of the doublo oiRna of //,. //,', *o.
Fur the eaiuu rcunuii, lliv uiubiguitieit uf 8i(;n iu (20) and [22) arc indciH-'udunt.
§ IS EXAMPLE 485
= U<i say, and, if Il2<JC, solve the equation x?—Cy2=±Hi;
then pass back to x through the two transformations (20)
and (22) ; and, finally, select tlie integral values of x and y tluis
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.
Esample. Find the integral solutions of
x«-15»/-=61 (9').
Let (fi'i=-15)/61=J7, (18'),
where ffjt. 30.
Then iri==15 + 61Hi.
Since K^ t> 900, we have merely to select the perfect squares among the
numbers 15, 76, 137, 198, 259, 320, 381, 412, 503, 564, 625, 680, 747, 808, 869.
The only one is 025, corresponding to which we have A', = 25 and Zf, = 10.
Since Hi>^15, we must repeat the process, and put
(AV-15)/10 = iJj (18"),
where i'jt>5, and therefore ii.'j'>25.
Since A"2-=15 + 10//2, the only values of K.^- to he examined here are 5,
15, 25. Of these the last only is suitable, corresponding to which we have
K3=5, Ha=l.
We have now arrived at the equation
Xj'-15!/.,='=1 (21'),
the first solution of which is easily seen to be (4, 1). Hence the general
solution of (21') is
^J=^{(^ + v'15)» + (4-^/i5)"} ]
(24).
The general solution of (9') is connected with this by the relations
x^ = {5x,^15ij^)ll, yi = {5y„TX,)ll (".22');
x=(25x,=Fl5i/,)/10, »/ = (25i/,TX,)/10 (20').
Hence x = lix.j^i'iy,, y=^'Sx, + liyr,\
x=11x2=f30i/3, y= =f2X3 + 11(/2 j
where Xj and y, are given by (24). The question regarding the integrality of
X and y does not arise in this case.
As a verification put Xo = 4, y^=l, and we got the solutions (11, 2),
(101, 20), (14, S) and (74, I'J) for (9'), which are correct.
:! ^''^^
486 REMAINTNO CASES OF BINOMFAL EQUATION CH. XXXIII
§ 19.] TLcre remain two cases of the binomial equation
1^ - Cy' - ± II wliicli are not covcreil by the above analysis —
x'-Ctf=±II (26).
where C is a perfect square, say C = l{'; and
x' + Ci/' = + n (27).
The equation (26) may be written
(x-ll!/){x + l{y) = ±fl.
Hence we must have
r-Iii/-
'="} (28).
where « and v are any pair of complementary factors of + //.
We have therefore simply to solve every such pair as (28), and
select the integral solutions. The number of such solutions is
clearly liniitoil, and there may be none.
In the case of equation (27) also the number of solutions is
obviously limited, since ejich of the two terms on the left is
positive, and their sum cannot exceed //. The simplest method
of solution is to give y all integral values :^^/(///C'), and
examine which of these, if any. render II- Cf a perfect square.
)j 20.] In conclusion, we shall brictiy indicate how the
solution of the general equation of the 2nd degree,
aj:» + Ihxy + bi/' + 2gx + 2fy + c = Q (29).
where a, b, c, /, g, h are integers, can be made to depend on the
solution of a binomial equation.
By a .slight modilication of the analysis of chap, vii., § 13,
the reatler 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
{Oy + Ff-C{a.r + hy + gy = -a\ (30);
or (Gx-*-G)'-C(lur+by+/y = -biL (31),
v\)ere^=af>c + 2/gh- q/''-bg'-c/i\ C=/i'-fih, F=gh-(\f,
G = /{/'- bg ; any into the form (30). If, then, wo put
rtj-
Cy^F=i)
+ % + r/ - 7 J
(32),
§§19, 20 GENERAL EQUATION OF 2XD DEGREE 487
(30) reduces to
$^-Cv'=-aA (33),
which is a binomial fonn, and may be treated by the methods
already explained.
If h->ab, then C is positive, and, provided C be 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 a = 0 or 6 = 0, or both
a = 0 and 6 = 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).
If P<ab, then C is negative, and we have case (27).
inr = ab, then C=0, and the equation (29) may be written
(ax + hyf + 2agx + 2«/j/ + ac = 0 (34),
which can in general by an obvious transformation be made to
depend upon the equation
V'^Q^ (35),
which can easily be solved.
Example 1. Find all the positive integral solntions of
3x' -Sxy + -n/-ix + 2ij = 109.
This equation may be written
(3x-42/-2)»+5(!/-l)'' = 33G,
Bay f'+57;2=336.
Here we have merely to try all values of tj from 0 to S, anj find which of
them makes 336 - 5ir a perfect square. We thu3 find
J=±16, ii=±4;
{=i4, i;=±8.
Hence
Si-4!/-2=±16, v-l=±4 (1);
3x-4i/-2=±4, y-l=±8 (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 j/ = 5 or
y=9. To get corresponding positive integral values of x, we mii=t take the
lower sign in the first of (1), and the upper sign in the first of (2). Hence
the only positive integral solutions are
x-2, y = j, and J=14, y = 0.
483 EXAMPLES CU. XXXIIl
ILxamiilo 2. Find the positn-o iDtrgral eolations of
3xj/ + 2y'-4i-3y = 12.
Tliia is a CAPO where the terms of the 2nd degree break up into two rational
(actors. We may put the equation into the form
(9x + 6y-l)(3y-4) = 112.
Since 3i/ - 4 i^^ ohvionsly less than 9x + 0y-l when both z and y aro
positive, 3y-4 must be equal to a minor factor of 112, that U, to 1, 2, 4, 7,
or 8; the second and the last of these alone give integral values for y, namely,
y = 'i and y = 4. To get the corrcsjonding values of x, we hiive 9x + C.y- 1
= 5C and ttx + C;/ -1=14, that is to say, Ox = 45 and 9x= -9. Uence the
only po;>itive integral solution is x = 5, y = 2.
I^xample 8. Find all the integral solutions o(
9x» - 12xy + 4j/' + 3x + 2y = 12.
Here the terms of the 2nd degree form a complete square, and wo may
Trrito the equation thus —
(3x - 2i/)» + (3x - 2y) + 4y = 12,
or 4(3x-2y)» + 4(3x-2y) + l + 16j/ = 49;
that is, (Cx - 4y + 1)» = 49 - ICy.
Uencc, if
u = Cx-4y + l (1),
so that u is certainly integral, we must have
y = (40-u')/16 (2).
Now we may put u = lG^±», where « is a positive integer >8.
It then appears that y will not be integral unless (49 - «')/10 be integral.
The only value of ( for which this happens is » = 1. Therefore
u = 16/i=>=l (3).
Hence, by (1), (2), and (3), we must have
i=2 + 4/i(1-8m)/3. y = 3-2/i-lG^« (4),
or
* = 4m + (5-32ai')/3, y = 3 + 2,i-16M» (5).
It remains to determine /i so that x shall be integral.
Taking (4), we see that ^ (1 - 8;i)/3 will be integral when and only when
lt. = %r or /i = 3r- 1.
Uting these forms for pt, we get
i = 2 + 4r-96»', y = 3-C»-14li." (6);
x= -10 + C8»-9Gr', y= -n + 90»-144»' (7).
Taking ('>), we find that (5-32;i*)/3 is iutcgrol when and only when
^=3r+l or /i = 3i'-l.
Using these forms, we get from (6)
«=-5-C2»-9Gk', y=-n-9nr-144r« (8);
x= -18 + 76»-9G»«, y=-15 + 102»-144r« (9).
The formnliD (G). (7), (H), (fl), wherein r may have any integral valno,
positive or negative, coulaiu all the integral solutions of the given equation.
§ 20 EXERCISES XXXII 489
Exercises XXXIL
Find all the integral and also all the positive integral solutions of the
following equations : —
(1.) 5i + 7y = 29. (2.) iex-17i/ = 27.
(3.) lli + 7y = U03. (4.) 13G7x-ioi% = lC24G.
(3.) If £x. ys. be double £;/■ is., 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 ninny 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 fiuit 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 slie 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, aud 5 for remainder when dirided 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. 6d., and histories 5«.,
show that the cheapest way of doing it will cost £129. 15s.
Solve the following systems, and find the positive integral solutions: —
(IG.) x + 2y + 3z = 12b.
(17.) x+y + z + u= 4,1 (18.) 2x + 5y+ 32 = 324,1
5i/-l-Gj-h9u = 18.| 6x-4i/H-14z = 190.f
(19.) Sx-Gy-h 72 = 173,1 (20.) 17x -H9i/ -I- 21^ = 400.
17x-4j/-f3j = 510.|
(21.) x+ y+ 1+ u=2G,-\
3x-i-2y + iz+ «=63, |.
2x-l-3y-h2j-(-4«=74.J
(22.) Show how to express the general integral solution of the system
OuXi-hflioXj-f. . . + ai„x„=d„
OjiXi + aj^Xj-h. . , + a^x„=d,.
by means of determinants, wiieu a particular solution is known.
490 EXERCISES XXXII ClI. XXXIII
Find the valacs of x which make the values of the foUomng functions
intcgrnl 8i|uarcs : —
(23.) 2x' + 2x. (24.) (j^-x)/5. (23.) i + 11 and 1 + 20, simultaneously.
(26.) Tf + fl and 1]: + 3, simultaneously. (27.) i' + i + 8.
Solve the following cqaations, giving in each cose the least integral
solution, and indicating how all the other integral Bolutions may bo found: —
(28.) i«-44j/'=-a
(20.) i»-44y»=+5.
(30.) i'-44i/»=-7.
(31.) i«-44y»=+4.
(32.) *« + 3y'=628.
(33.) i'-6V=-ll.
(34.) *'-47y'=+l.
(35.) i'-47y'=-l.
(3i;.) x»-2(Ji/'=-1103.
(37.) z«-7y«=18fi.
(3ij.) i'-(a3 + l)y' = l.
(39.) T»- (a'- !)!/' = !.
(40.) x^-{a^- + a)y^=l.
(41.) i'-(a'-a)y'=l.
(42.) x> + 5xy-2x + Ay = i
r,3.
(43.) ry-2x-Sy = lS.
(44.) i>-!/» + 4x-5y = 27
(45.) 3j= + 2jy + 5y»=390.
(4G.) i» + 4jy-lly' + 2x
-H&y
-140
= 0.
(47.) x>-x!/-72y- + 2x-
iiOy-
-659 =
= 0.
(48.) i' + 2jri/-17y' + 72y-75=0.
(49.) 61i' + 28xy + 25l!/» + 204x + 526!/ + 260=0.
(60.) Show that all the primitive solutions of d'-Cy'=±/T are
furnished by the convergents to y/{ClD), provided II<J(CO). Sh1)w also
how to reduce the equation Dx'- (7i/'= ±H, when 11>^(CD).
(51 .) Find all the solutions of
4x»-7y'=-3,
and of 4x>-7v' = 53.
(52.) If D, E, F, II be integers, and II<J{E*-DF) (real), show that all
the Bulutions of
Dx'-2Exy + ty=i=n
are famished by the convergents to one of the roots of
Di»-2£* + f=0.
(See Scrret, Alg. Sup., $ 36.)
(63.) If tT,=p,-zg,, where z is a pcriodio fraction having a cycle of
e quotients, and ]>, and 9, have their usual mcaningn, then
where *H^,=a^, + - — -
<h^ +
and
; = ",-
In particular, if i = ^'(C/D), then
DP.^ - s/{CD) 9,^= {oAf, - pL,-p^(CD)]'{Dpr - ^{CD) 7,)/Jlf,».
Point ont the l>oaring of this result on the solution of I>x*- Cv'= * 11.
CHAPTER XXXIV.
General Continued Fractions.
FUNDAMENTAL FORMULA.
§ 1.] The theory of the general continued fraction
, bi b} ...
^' = '^ + ^^"- ('^)'
whore «,, ffl., (T,, . . ., &2, ''s, • ■ • are any quantities whatever,
i.s 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, x.x.xii., as exercises for the
reader. Tiiere 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
the third convergent of 1 + :j — - — t— . . . we are led to
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 convergcnts of this
fiactiun formed by the direct process of chap, xxxii., ^ 6 ("), (/J),
402 C.FF. Ay FIRST AND SECOND CLASS TTI. TXMV
(y). It must also lio romeinliored tliat no piece of reasoning
that involves the use of the value of a non-trrminating continued
fraction is legitimato till we have shown that the value in
question is finite and definite.
Jii casfn where any dijficuUy regarding the meaning or conver-
gennj of the continued fraction taken in its primary sense arises,
ve regard the form on the right of (A) merely as representing the
assemblage of convergent s /),/<7,,W7a. • • •./'«/?« w'^"^* denomi-
nators are constructed by means of the recmrence-formulw (2) and
(3) below.
That is to say, when the primary definition fails, wo make
the fonnula) (2) and (3) the definition of the continued fraction.
In what follows we shall be most concerned with two varieties
of continued fraction, namely,
a, + ttj +
and Ci + - — - — • • • (^)>
wherc a,, a,, a„ . . ., 6„ b„ . . . are all real and positive. We
shall speak of (B) and (C) as continued fractions of the first and
second class respectively.
§ 2.] Kpi/qi, pjqt, &c. be the successive convergents to
bi bt / , \
cUi+ at +
then
Ph = a«P.-i + bnPn-t . (2) ;
g, = a^qn-i + t„7,-i (3),
with the initial conditions Pt=l,pi = ai; qi = l, qt = <h-
Cor. 1. In a continued fraction of the first class p» and q»
are both positive ; and, protided a,-d;:l, each qf them continually
increases tcith n*.
In a continued fraction of the second class, subject to the
restriction a,«tl + i'„, j», and q, are positive, and each of them
Continually increases with u*.
• It doca not necessarily follow that Lp, = eo and Lq, = a) , for llie «uo-
ec'Bivc incrcmcnti here arc not positive inU^ral numbers, aa in Iho caw of
■imi'lc coutioucd fracliuua.
§§1-3 PROPERTIES OF CONVERGENTS 493
These conclusions follow very readily by induction fiom such
formulae as
Pn -Pn-l = («n - 1 )i^H-l + KPn-t (4).
Cor. 2.
F^=«„+.A- A-_..> (5);
Pn-i a„-i + a„-j + (h.
2^ = a.^-^^' ...^ (6).
§ 3.] From (2) and (3) we deduce
Cor. 1. The convergents, as calculated by the recurrence-rule,
are not iiecessarily at their lowest terms.
Cor. 2.
Pn _ Pn::! = /_)-. ^2^3 . . -h /2\
qn <7n-l 9nqn-l
Cor. 3.
^" = „^ + A _ ^3 ^ . . . (_)'>¥i^^jA (3).
q„ qiq-i qiqi q^-iq^
Cor. 4.
p„qn-2- Pn-'.q,, = (-)"'' a,fiA ■ • • ^n-1 (4);
Pn Pn-1 ^ , yi-i a-nbA ■ ■ ■ ^«-l /gX
Cor. 5.
= ^;i%5 (G).
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 tlie second class, subject to the
restriction a„<t:l + b„, all the convergents are positive, and foi-tn
an increasing series.
494 CONTINUAVr DEFINED CH. XXXIV
These conclusions follow at onco from (2) and (j), if we
remember that, for a fraction of the second class, wo have to
replace b„ . , .,b» by -b„. . ., -<<«.
CONTINUANTS.
g 4.] The functions p,, g, of a,, Oj, . . ., o«; 6i, ftj, • . m K
which constitute the numerators and denominators of the con-
tinued fraction
bt b, 6,
belong to a common class of rational intej,TTvl functions*.
In fact, />, is determined by the set of equations
Pt=<hPl + f>tP», Px = <hPl + b,Pl P, = anPm-l+bnPn-t
together with the initial conditions ^o = li Pi = ai; while g, is
determined by the system
qi = (hqi + btqi, qi = atq, + btqi, .... ?« = a«7«-i + *-.7.-i
(2),
together with the initial conditions ^i = 1, ?« = o^•
Iti8obvious,therefore,thatg„tsM«sa/n#/u7lcri<>no/o,,flb
o»; bt, 6«, . . ..b^aspn is of Uu a-i, • • ■ . ««; ^i. ^. • • •. ''••
We denote the function />, by
^" \a,,(i,, . . ., a,/
(3).
and speak of it as a continuant of the nth order whose denomin-
ators are ch, <*, ««, and whose numerators are b , t,.
We have then
/ b„...,b.\ y
^" \flt„ *,, . . . , a J
* This was firat pointed out by Enter in his memoir entitled " Specimen
Algorithmi Siugulariii," Sov. Comm. I'rtrop. (1764). Elegant demoD«tr«tioni
of Eulcr's results were given by Mobius, CrtlU't Jour. (1.S30). The theory
has been trvatcd of late in oonnoction with determinants by Sylrester and
Muir.
^ 3-5 FUNCTIONAL NATURE OF CONTINUANT 495
When the numerators of the continuant are all unity, it is
usual to omit them altogether, and write simply Jr(a,, (u, . . . , 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 ^••■'^)andi^(a„ «„.,.,«„)
\aj, »3, . . ., «n/
into K{1, n). In this notation we should have, if r<s,
^<'.'>=^(„„l:;:::;:f3 (^^
^(•'■••)-'^C.,,.';:::;;f;*') («'■
In particular, K{r, r) means simply Ur, so thatj^i = K{\, l) = a^.
To make the notation complete, we shall denote p^ and q^ by
K(^ ), which therefore stands for unity ; and, in general, wlien
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 hy K{ ) 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 g •! (1),
K{1, n) = a„ K{1, n-\) + KK{1, » - 2), ^
K(l, n-\) = a„.iK{l, n-2) + K-,K{1, n - 3),
K{l,l+\) = ai^,K{l,l) + h^,K{ ),
K(l,l) = a„ K{ )=1.
(7).
The following rule of Hindenburg's gives a convenient
process for writing down the terms of a series of continuants,
say K{1, 1), A'(l, ■>). K{1, 3), . . . :-
4!)G
EUI.Kll's fONSTIUUTION FOR CONTINUANT Cll. XXXIV
12 3 4 5
1 - ,
Oi
t.
<h
a.
U.J
<h
"4
«5
«!
«4
O,
b.
/'4
«.
rtj
^
a.
<h
b.
a.
b»
b.
b.
1st. Write down a,, and enclose it in the rectangle 1, 1. The
termin 1, 1 i8Z(l, 1).
2nd. Write a, t<) the right of all the rows in 1, 1 ; and write
bt underneath. Enclo.se all the rows thus constructed in the
rectangle 2, 2. Then the rows in 2, 2 give the products in
A'(l, 2), namely, rt,a.j + 6j.
3rd. Write a, at the ends of all the rows of 2, 2 ; repeat
under 2, 2 all the rows in 1, 1, and write b, 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 A'(l, 3), namely,
0,0.^0, + bid, + <hb,.
The law for continuing the process will now be obvious. The
scheme is, in fact, merely a graphic representation of tlie con-
tinual application of the recurrence-formula
A'(l,n) = o.l'(l, n-l) + 6,A'(l,»»-2) (8).
By considering Ilindenburg's scheme we are led to the
following rule of Euler's* for writing di)wni all the terms of a
continuant of the «th order.
Write down a,a.,(h • ■ • (i>,-i«ii. T/ih ts the Jir.ft term. To
get the rest, omit from this product in ervnj possible way one or
more jrairs of consecutiw as, ahcaifs replttcing the second a oj
the pair by a b of the same order.
' EiiUt (I.e.) pavr tlic rule for the Kiiniilc contiDUuut mcrrly. Cayley
(PhiL itiig., lSo3) gave the luoro general form.
§§ 5, 6 PROPERTIES OF CONTINUANTS 497
For example, to get the terms of K{1, i). The first is a^a^fisa^. By
omitting from this, first fljOj, then njflj, then a^a^, and replacing by b.,, 6,, 64
respectively, we get three more terras, b«a^a^, a^b^a^, a^ajt^. Then, omitting
two pairs, we get b^b^. We thus get all the terms of A' (1, 4).
It is easy to verify this rule up to K(l, 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,
^f h, . . ., ^"\^g;( in, . . ., h,\
Voi, ff.„ . . .,aj \a„, a„_„ . . ., aj
We could obviously form the continuant ir(l, n) by starting
with a„a„-i . . . aMi instead of aiU^ . . . «„-!««, and replacing each
consecutive pair of a's in every possible way by a b 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{mJ) (10).
Cor. 2. We have the following recurrence-formula : —
K{1, m)=aiK{l+ 1, m) + bMK{l+2, m) (11).
For, by Cor. 1,
K{l,m) = K{m,r),
= a,K(m, l+l) + b,+iK(m, /+2), by (7),
= a,K{l+l, vi} + b,+iK{l + 2, m),hy Cor. 1.
§ G.] The theorems (1) and (4) of § 3 may be written in
continuant notation as follows : —
E(l,n)K{2, n-l)-K{l, n-l)K{2,n)
= (-)''bA...b„K{)K{ ) (12),
.£"(1, n)K{2, n-2)-K{l,n-2)K{2, n)
= ( - )"-' bh . . . b„., 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, Mnir, and others.
c. II. 32
408 eui.er's continuant-theorem en. xxxiv
A'(l, «) /v'(/, m) - A'(l, m) K{1, n)
= (-r-'*'tA«. . . t-+iA'(l, /-2)A'(m + 2. fi) (14),
where l</<m<?j.
Tliis tliPorcm is easily rcmcmborM by mcnni; of tlie fuUowing elegant
mcmuria tcchnica, given by itn discoverer : —
1, 2, . . ., f-2, t-1, H, ■ ■ ., wi. |m + l, 111 + 2, . . ., K.
Draw two vcilical lines enclosing the indices belonging to A'(/, m); then twn
horizontal linos as above; and put dots over the indices immodiatcly outsiil'
the two vertical lines. The indices (or the first continuant on the lert of (14)
are the whole row ; those of the second arc inside the vertical lines; those of
the third and fourth under the npper and over the lower horizontal lines;
those of the two continuauts on the right outside the two vertical lines, the
dotted indices being omitted. The b'a are the i's of A'((, 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 tlie recurrence-formula; (7) .iiid (11), that, if the formula
hold for /, m + 2, ami for /, m + 1, or for /- 2, m, and for l-\,m,
it will hold for /, m. iS'ow (12) asserts the truth of the theorem
for /=2, m=n-l; and it is easy to deduce from (12), by
means of (7) and (11), that the theorem holds for / = 3, »i = n- 1,
and also for /=2, m = w - 2. The general case is therefore
establi.shed by a double mathematical induction based on the
particular case (12).
Tlie theorem (14) might be made the basis of the whole
theory of continued fractions ; and it leads at once to a variety
of important i)articular 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
rcciprocnl 8im])le continuants : —
■*•'(«!. <h <?<, a,, . . ., «,, a,)
= A'(o,.n„ . . .,a<)' + A'(rt,, ff, a,.,)' (A);
■S'(oi,o> a,.,, a(,rt,.,, . . ., a,, a,)
= ir(<i„ ffj <T,.,) {A'(«,, o,, . . ., rti) + A'(o„ a, a, ,)}
(B).
I
^§ 6, 7 smith's proof of a theorem of fermat's 499
Example. Show that every prinie p of the form 4\ + 1 c:in be exhibited as
the sum of two integral squares*.
Let /xj , /i^ , . . . , M, he all the integers prime to j) anil <c Ap ; and let simple
continued fractions be formed iov pjfi^, pjfji^, . . ,, pl/i,, each terminating so
that the last partial quotient > 1. Then each of these continued fractions has
for its last convergent the value K(a^, Qj, . . ., a,^jK(a„, Oj, . . .,a„), where
the two continuants are of course prime to each other, and aj>l, a„>l.
From this it appears that there are as many ways, and no more, of
representing p by a simple continuant (whose constituents are positive
integers the first and the last of which are each gi-eater than unity) as there
are integers prime \o p and <^^p.
Now, since A'{«i, a„ a„) = 7C(a„, . . . , a„, Oj), and a„>l, it is
obvious that A'{(7„, . . ., a^,a^) must arise from one of the other fractions pjpi.
Hence, given any fraction pjn, 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 4X + 1, then the greatest integer iu Jp is 2X,
which is even. Since, therefore, the number of continuants which are equal
to p must be even, and since A' (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(a^, a„, . . .,
"i-i' "•' "i-i' • • •' "s> ''i)' ^""^ '' follows from (B) that such a continuant
cannot represent a prime, unless j = l, or else i = 2, and aj = l, all of which are
obviously excluded.
We must therefore have an equation of the form
p = K(a^, flj, . . ., oj, Ui, . . ., Oj, Oj),
K(a^, a„, . . ., flj)'' + A'(«,, a^, . . ., a^^jP,
by (A), which proves the theorem in question.
As an example, take 13 = 3x4 + 1.
„ , 13 ,„ 13 . 1 13 , 1 13 „ 1 13 „ 111
We have -r-=13; — = 6 + ^; -^ = i + ^; -r = ^+i' — = 2 + ,-,— s!
1 2 2 3 34 4o 1+1+2
-r = 2 + l. So that 13 = A'(13) = ir{6, 2) = A'(4, 3)=ii:(8, 4) = A'(2, 1, 1, 2)
I) u
= A(2, 6); and, in particular, 13 = A'(2, 1, 1, 2) = K{2, l)2 + A'(2)==3' + 22.
§ 7.] By considering the system of equations (I) of § 4, it is
easy to see that, if we multiply ar, br, K+i hy c^, the result is
the same as if we multiplied the continuant ^'(1, n) (n>r) by
Cr. Hence we have
= cx.
\a,, oj, . . ., aj
* The following elegant proof of this well-known theorem of Fermat's was
given by the late Professor Henry Smith of Oxford ^Crellc'i Jour., 1855).
32—2
600 REDUCTION TO SIMPLK CONTINUANT CH XXXIV
We may so detenuiue c,, c,, . . ., c, that all the nuuierutorB
of the coatiuuant become equal. In fact, if we put
c-A = \ e^A = \ . . ., c,.,cA = \
we get
Ct = \bibjbjjtb,, . , , ,
Hence
k( * M
=(i/xy b,b^-A-A . . .^ £^[^^^ xaX. «3*A. ^/Ai..' • • -y
(16).
where p is the numher of even integers (excluding 0) which do
not exceed n.
Cor. Every continuant can be reduced to a simple continuant,
or to a continuant each of whose numerators is - 1.
Thus, if we put X = + 1 aud X = - 1, we have
k( *• ''-)
\a,,a„ . . ., nj
= hjb^-i ... X A' (a,, ajbi, ajbjb,, aj>jbj>t
anbn-ibn-, . . ./^A-t • . ■) (17).
= (-)''M.-, . . .xA-(^_ _^^' a^Jb,',-a,bJbA. . . .',
(-)»-> aA-A-..../*A-....) ^**^-
§ 8.] The connection between a continuant and a continued
fraction follows readily from (11). For we have, provided
K{'Z, n). A' (3, n), A' (4, «),... are all dillerent from zero,
-g(l.n)_^ . ft.
r(2rJr)-""*A'(2, n)/A'(3.n)'
^(2. w) ^6j
A'(3.«) '^'"A'(3,n)/A'(4.«)-
Ilonco
K(\,n) „.hj_b, br «...
A'CA ») ^ a, + a, + ■ • ' A'(r, n)/A'(r + 1, n) ^^^''
^§7-10 C.F. IN TERMS OF CONTINUANTS 501
If in tliis last equation we put r = n, and remember tliat
here K{7i + 1, n) = K{ ) = 1, we get
a result which was obvious from the considerations of § 4.
§ 9.] When the continuant equation
^(1, n) = a„K{l, n-l) + b„K{l,n- 2),
or Pn = UnPu-i + h„pn-i,
which may be regarded as a finite difference equation of the
second order, can be solved, we can at once derive from (20) au
expression for
' Oa + a3 + ■ ■ ■ a„ ■
When «„ aud 6„ are constants, the problem is simply that of
finding the general term of a recurring series, already solved iu
chap. XXXI. , § 7.
Example. To find an expression for the nth convergent to
F=X-\ — . . . — ....
Here we have to solve the equation Pn=Vn-\+Vn-it \i\lh the initial con-
ditionspi) = l, pi = l. The result ia
£•(1. n) =p„= {(1 + V5)"« - (1 - v/5)»+i}/2"+\/5.
Hence
f„ K(l, n) {(1 + V5)"«-(1-^/5)"-^'}/2""n/5
ql'KCi^n) {(l + V5)"-(l-x/5)»}/2V5 '
_ (i+^/5)''+i-(i-.y5)"+'
-* (l + ^/5)»-(l-V5)'' ■
From the expression for K(l, n) (all the terms in which reduce in this caso
to + 1) we see incidentally that the number of different terms in a continuant
of the nth order is
2n+l ;g — on ln+l^l + "n+l^S+0 n+1^3+ • • •/•
§ 10.] When two continued fractions 7^ 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, n)-equiTalence, that is, Prmllrm=Pn'hrn-
See Exercises xxxiii., 2, 17, &c.
602
BEDUCTION TO SIMPLE C.F.
CH. XXXIV
It follows at once from g 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 o^, b,,
and br+i by any quantity »»( + ()) without disturbing the e<iui-
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,+
hi b, bt
a, + a, + a, + '
bn
0.+ --
1 1
<
1
chjbi+ Uibjbft- aJ),lbA+ ' ' ' a,6,_,6,_j . . . /6,6,
(21).
§ 11.] If we treat the equations (1) as a linear system to
determine K{1, 1), £"(1, 2),
miuaut notation, wc get
A'(l,«) =
., IC{1, n), and use the deter-
a,
b.
0
0
0 .
. . 0
0
0
1
«a
b.
0
0 .
. . 0
0
0
0-
-1
aj
b.
0 .
. . 0
0
0
0
0-
1
at
bu.
. . 0
0
0
0
0
0
0
0 .
. .-1
"n-
/.
0
0
0
0
0 .
. . 0-
-1
a
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.
EXBRCISES XXXIIL
(1.) Asaaming tliat both tlio fractions
x =
b ^
'a+ b+ c +
arc convprRCDt, show that
_ a b e
x{a + l + y) = a + y.
* See Mail's Tlieory of Determinant; cba)). iti.
§§ 10, 1 1 EXERCISES XXXIII 503
(2.) If piq and p'jq' be the uUimnte and pennltiraate convergents to
a+ ; — • . . . Ti bIiow that
b+ k
a + , — . . . ; — ... to 71 pcriocIs = - p^—, — ; ...—.,
, b+' k+ ^ qL q+PT q +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 pIq is an even or an odd convergent.
(3.) Evaluate OH . . . to n quotients, o being any real quantity
(I + (1 4"
positive or negative. Show from your result that the continued fraction in
question always converges to the numerically greatest root of j;^ - ax - 1 = 0 *.
(4.) Deduce from the results of (2) and (3) that a recurring continued
fraction whose numerators and denoiuinators are real quantities in general
converges to a finite limit ; and indicate the nature of the exceptional cases.
(5.) Evaluate 2- r — ^— -— . . . to n terms.
14 2 2 2
(6.) Show that the nth convergent to g— .3 — .r — j— r — . . . , every sub-
o— o ^ o — o — o ^
2
sequent component being - , is (2" - l)/(2" + 1).
(7.) Show that z ; — ... to n terms = —vn — =-.
*' x + l-x + 1- x"+'-l
(8.) = — — -, ^ — . . . (h + 1 components)
^'l-a + l-o + 2- ^
= l + a + a(a + l)+ . . . +u(a + l) . . . (a + n-l).
(9.) If 0(") = . . . n quotients, then
^ (m + n) = {,p (hi) + 0 (n) - 00 (m) (p (u) }l{l + <p [m] <j> («)} .
(Clausen.)
(10.) Show that
A'(0, Qj, 03, . . ., «„) = £■ (oj, . . ., a„);
K(. . . a,b,c,0,e,/,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, . . .);
A'(. . . a, 6, c, 0, 0, e, /, . . .) = K{. . . a, b, c, e, f, . . .).
(Muir, Determinants, p. 159.)
(II.) Show that the number of terms in a continuant of the nth order is
I4.r„ n,(»-2)(»-3) , (K-3)(»-4)(«-5)
i + in-ij-t- 21 + gi T. . . .
(Sylvester.)
(12.) Ifp„=ir( '^ »■•■•' "), show that there exists a relation of
the form
^Pn^ + •«i>„-i'+ Cp„_,' + Di)„_,«=0,
where A, B, C, D are integral functions of a„, b„, o„_i, 6„_j.
* This is a particular case of the theorem (due to Euler?) that the
numerically greatest root of x--px + (j = 0 is p . . . •
j4 EXERCISES XXXIII CII. XXXIV
(13.) SliowUiat
jf/ i..(fc. + a,)6,.{i, + <.J5... . •) = (t, + a.){6, + oJ(6, + «,) . . .;
\i. <h> "t. "s- • • •/
and afduco the theorem of § 19. (Muir, I.e.)
Taking (<i, b, c, . . ., A) to donoto the continued fraction — ;— —
... J. and [a, b,c /.]■ or, wbcn no confusion is likely, [a, k], to
denote A' I ~,'~ ''"''", I. prove the following theorems*: —
\a, 0, c */
(11.) If i = (a, b.c e, y), then !/ = («, . . ., c, 5, a, i) ;
xy-(e a)x-(a, . . .,e)y + {e, . . .,a){a <f)=0;
(a,. . .,e){e 6) = (< a)(a, . . ., d);
{*-(" «)}{y-(< »)>
= (<• aYid a)'(e <i)'. ..(«)'.
(15.) (« c)-{/t d) = (« a)(d aj'C- a)'. ..(.!)•.
(16.) [a, b, c, d, f]=l/(a, b, e, d, e)(b, e, d, <•)(<•. d, e)(d, e) (<r).
(17.) Prove the following equivalence theorem :—
(a «,/, a' t'.f, a" «",/". a'" «'",/'")
- L«T7i V"' 'J "^ [«, «'] - [a'. «"] - la". «'"] - [a"'. «'"]/'- - [a'", cf-jf '
118.) (a,f, a',/', a",/", a'"./'",. . .)
_1 I g' aa[[ a'a" )
~a [ ■'"a/a'-a-o'- o'/V-a'-a"- a'/V" - a" - o'" - ' ' "I *
1 J_ _1 1 1
( 9.) a + ^,^ j^ OT+ c+ m+ ■ ■ ■
'il
1 1 )
2 + 6m -2 + cm- )
(20.) ,/2 = l + ^-L^-L...4J7 + ji^ji^ji^...}.
(21.) (a f, /, rt «, /', o, . . ., e, /", . . . ad oo)
-(«•,.. .,.!,/, ? a,/', <,...,<"./"..• -ftdoo)
= (" <)-(< a).
(32.) Show that the rucccssive constituents a, p, y Kt^,' may bo
omitted from the continued fraction {. . . a,b,a, p,y \, m, i*, c, d, . . .)
without altering its value, provided [fi, . ., n]=il, o==t[y, . . ., fi],
and i>=±[p \]; and construct examples.
• •
(23.) If x=(<>, . . ., e, f, . . .), the other root of the quadratic equation
to which this leads is x = (/, e a, . . .}.
(21.) If 6+ , — . . . ; ... ... bo one root of a qusdratio
• •
* The notation and the order of idcns used in (14) to (23), as well as
some of the siivcial rcbults, uro duo to Mobius {CrclU'$ Jour., 1830).
§ 12 CONVERGENCE OF A C.F. 505
equation, the other is
1 111 111
6 +
*1+ >>m-''m- <hn-i+ <'m-i+ ' ' ' a+ a„+ ' ' ' a+ ' ' ' '
» •
(Stern, Crclle's Jour., 1827.)*
(25.) ir q >p, show that
i_g -ppq (g -p)pg (q^pf
Pik-P)'
{,q-p)q = 1--f
r-p'
CONVERGENCE OF INFINITE CONTINUED FRACTIONS.
§ 12.] By the v.ilue or limit of an infinite continued fraction
is meant the limit, if any such exist, towards which the con-
vergent pjqn approaches when n is made infinitely great. It
may happen that this limit is finite and definite ; the fraction is
then said to be convei-gent. It may happen that L p„/qn fluctuates
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 pjQn tends constantly towards
n=aD
+ 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 ; — , ,_ - z — z — :; — . . . diverges to - co , for = — :; — -—
-i + W5-l+l+l+ 1+1+1 +
. . . converges to -4 + Jv'5> "^ ™*y ^^ easily seen from the expression for
its nth convergent given in § 9.
The last example brings into \ieyi a fact which it is important
to notice, namely, that the divergence of an infinite continued
fraction is sometliing quite different 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 (24) are generalisations of an older theorem of Galois'. See
Qergonne Ann. d. Math., t. jlii.
50(5 PARTIAL CRITERION FOR C.F. OF FIRST CI-ASS CH. XXXIV
not safe iu general to arguo that a continued fraction does not
diverge because the cuntiuued fraction formed by taking all its
constituents after a certain order converges.
With the exception of simple continued fractions and recur-
ring coLtinucd 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 thefo we .shall confine ourselves iu what follows*.
§ 13.] A continued fraction of the first class cannot be
divergent; and it will be convergent or oscillating if any one qf
the residual fractions x, a:, j-,, . . . converge or oscillate.
The latter part of this proposition is at once obvious from the
equation
/*, /', bn
x,=ai + . • . — .
Oj + a, + X,
Again, since (§ 3, Cor. 6) the odd convergents continually
increase and the even convergents continually decrease, wliile any
even convergent is greater than any following odd convergent, it
follows that Lpt„l<hn = -A and Lp»-^lq^.i = B, where A and li are
two finite quantities, and A-^li. U A -B, the fraction is con-
vergent ; if .4 >^, it oscillates ; and no other case can arise.
§ 14.] A continued fraction of the first class is convergent if
the series 2«,.,a^ii, be divergent.
We have, since all the quantities involved are positive,
q» = anqn-i + buqn-i;
Q»-l = aH-iqm-i+ bn-iqn-t, qn-i>ci,-\qm-ti
gn-t = aM-tqm-3-*- b,.jq^.t, y,-j>o,-57,-s;
• • ■■■.•••
?4 = 04(?i + ^45'!. qtXttqs ;
q, = a,qt + b,qi , q, > 0,7, ;
qt = <hqi-
• Our knowledge of the convcrRcnoo of oontinuod fractiona ii chiefly dae
to SchlOmilcU, Handb. d. Atgebraiichen Analysis (1845) ; Amdt, Disquisitione*
Nonnulla de FraetionibuM Continuis, Sundiffi (18'1.5) ; Scidcl, Untersiu-hungen
Ubfr die Converijou und Divergent drr Kcttenhriiche |HiibilitAtinuiL'<chrift
MfiDchcD, IH40) ; nlHn AhhntuHungen d. .Mnth. Clatte d. K. liayerischen Akad.
d. H'lss., lid. vu. (1855); and Stern, Crtlte'i Jour., xxint. (1H48).
g 12-15 COMPLETE CRITERION FOR C.F. OF FIRST CLASS 507
Hence
qn>{an(>n-l+bn)qn-1,
g'„_l> (a„-ian-3 + ^n-O^n-S,
g'i.-3>(an-5«n-s + bn-i)qn-i,
• • ■ • •
qt>{aia3 + bt)q2,
q% = {fh(h + b^qi-
Therefore
qnqn-i>qiq2 (h + (h(h) (*« + a^a*) • ■ ■ U>n + dn-ia^,
and, since q^ = 1, ga = Oj,
srl^>F:('-^)('*°f)---(-°=if)»
Now, since 2ff„_ia„/ft„ is divergent, n (1 +an-ia„/b„) diverges
to + 00 (chap. XXVI., § 23), therefore Lqnqn-i/b^bj . . . 6n= + <»•
Hence
r ('^ Pm-i\ ^ j^ bjb, . . .bin ^ Q
that is, the continued fraction is convergent.
Cor. 1. Tkefraction in question is convergent i/La„-ia„/bn>0.
Cor. 2. Also i/La„/bn>0, and 2a„ be divergent.
Cor. 3. Also ?yia„+i6„/a„_i^„+i> 1.
The above criterion is simple in practice ; but it is not
complete, inasmuch as it is not proved that oscillation follows
if 2a„_ia„/i„ be convergent. The theorem of next paragraph
supplies this defect.
§ 15.] 1/ a continued fraction of the first class be reduced to
the form
,111 1 /,x
dt+ d3+ di+ dn +
so that
di = ai. d,-j^, cfa- ^^ , *~bA' ■'"
J Cinbn-\"n~t • • • /^k
"" = bb o ^^^'
then it is convergent if at least one oftlie series
ds + di + d^ + . . . (6)
d^ + dt + dt+ . . . (7)
be divergent, oscillating if both tJtese series be convergent.
508 COM PLETECRITEUION FOR r.F. OF FIRST CLASS CH. XXXIV
This proposition depends on the following inequalities be-
tween the q'& and d'a of the fraction (4) : —
0<7.<(l+rf.)(l+rf,). . .(l+<^,.) (8);
q„>d^■^d^+ . . . +rf» (9);
<?»-.>! (10).
These follow at once from Enler's law for the formation of
the terms in q^, wliich, in the present case, runs as follows : —
Writo duwu rfjrf, . . . dn and all the terms that can be formed
therefrom by omitting any number of pairs of consecutive d'a.
We thus see that y» contains fewer terms than the product
(1 + d,) (1 + (/,) . . . (1 + dn) ; and, since the terms are all positive,
(8) follows. Again, in forming the terms of the Ist degree
in q„, we can only have letters that stand in odd phices in the
succession rf,*/,^, . ■ ■ d.„; hence (9); aud (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
If we suppose rf, 4= 0, neither q^ nor q^-i can vanish. Hence,
if both Lqm and Lq»-^ 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
rf, + rf, + (/« + . . . + d, will converge ; and the product on the
right of (8) will be finite when n = oc . In tliis ca.se, therefore,
both q^ and qn-\ will be finite ; and the fraction (4) will
oscillate.
If the series rfj + </, + </, + . . . diverge, then by (9) Z^», » ao,
and the fraction (4) will converge.
By the same reasoning, if the series dj + d,-*-d,+ . . . diverge,
then the fraction
.11 1
^ </,+ </,+ d,+
will converge ; and consequently the fraction (4) will converge.
§§ 15, 16 EXAMPLES 509
Remarh — We might deduce the criterion of last paragraph
from the above. For we have
did,i = Cha-ijbi, d.id3 = aM^lbi, ••^ d„-id^ = a^^iajbn.
Now, if the series 2(^„ coBvefgeTthe series formed by adding
together the products of eVery possible pair of its terms must,
by chap, xxx., § 2, converge : a fortiwi, tlie series l,dn-idn, that
is, ^a„-ia„/b„, must converge. Hence, if this last series diverge,
'S.dn cannot converge. 2c?„ must tliercfore diverge, since it cannot
oscillate, aU its terms being positive. Therefore either (6) or (7)
must diverge, that is to say, the fraction (4) must converge.
Example 1. Coasider the fraction
^■•■2+2+ 2+ • • • •
_2(2n-l)'(2n-3)''. . . 3M»
Here a^n+i- (2„)» (2n - 2)« . . . 4= . 2» *
It may be shown, by the third criterion of chap, xxvi., § G, Cor. 5, that
the series ^d.^,^^ is divergent. Or we may use Stirling's Theorem. Thns,
when n is very great, we have very nearly
d„.+, = 2(2nip/2'" (»!)*,
= 2 [{V(27r2n) {2n/eP'}/{2=» (2^1.) («/«)=»}]',
= 2/7rn.
The convergence of ^d^n-n '^ therefore comparable with that of 21/h, which
is divergent.
Hence the continued fraction in question convergea.
Example 2.
a + ...
a+ a+ a +
oscillates or converges according as x>l or >1.
Example 3.
12 3
2T3T4T--- •
Here La„_,aJ6„=L(n-l)n/(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
jjT^A-A ...A..,. (1)
510 CRITEIUON FOB C.F. OF SECOND CLASS Oil. XXXIV
be suck that
a^Zhn^X (2)
for all valufis o/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 = abne occur, then F=l- 1/6', where
S=l + bt + btb, + bihbi + . . . + bjb, ... 6, + ... ad co (A),
so that F= or <l 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-i S:btb,. . .b, (3);
;), ^bi + bA + b-Al^t + . . . + b-A ■ . .b, (4) ;
qn - g.-i S bjbj .../>. (5) ;
q^S.! +b3 + bA+ ■ ■ • + f>A ■■■(>» (6) ;
qn -Pn S qn-l - Pn-1 S • • . ^ ?. "i^a S 1 (7).
To prove (3) we observe that
Pn -Pn-l = (a« - i)Pn-l - bnPn-J.
Hence, since ;>„, q^ are positive and increase with n (§ 2,
Cor. 1),
Pn - Pn-l S bn (pn-l - Pn-i),
Pn-l -Pn-i 2 t.-i (Pn-%-Pn-t),
acc. as a. £ 6. -)- 1 ;
ace. as a,-, S 6,., + 1 ;
p, — p2 = bA- acc. as a, = 6, + 1.
Tlicrcfore/?, -^,-1 ^bA • • • l>n, where the upper sign must
be taken if it occur auywliere among the conditions to the riglit
of the vertical line.
To prove (4), we have merely to put in (3) «— 1, « - 2,
. . ., 3 in place of n, adjoin the equation Pt = bt, and add all
the resulting equations.
(5) and (G) are estahli.siicd in preci.sely the same way.
It follows, of course, that p^ and ^„ both remain finite or
both become infinite when » = oo , according as the series in (6)
is convergent or divergent.
§16 CRITERION FOR OF. OF SECOND CLASS 511
To prove (7), we have
?« -lK=rin{qn-l -Pn-l) " ^n (qn-i-p„-i),
= (in-l -Pn-l) + b„ {(qn-l - Pn-l) " (<7n-s - Pn-2)],
according as a„^b„+l, provided q„-i-p„.i is positive.
This shows that, if any one of the relations in (7) hold, the
next in order follows. Now q2-p2 = a2-bo = l, according as
a.3 ^L_ + l; and q3-p% = (^(h - ^3 - ha^ = {(h - ^a) (^3 + 1) - &s
S (a.2 - bi) + 63 (eh - 62 - 1), according as a^~bi+ 1 ; hence the
theorem. It is important to observe that the first > that occurs
among the relations «f.2 = ^2+l, a3=bs+l, . . . 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-n-jOn S 1, it follows that
Pn/q„ £ 1 - l/g-n (8).
Therefore, since <7„>1, 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, F<1.
If the sign = occur throughout, we have
LpJq„ = l-L\lq„=\-llS,
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^, b, 6„ be
positive, but still put a„ = 6„ + l, a„_, = 6„_i+l,. . ., 03 = 63+1,
05 = 62+ 1, we get by the above reasoning
Wg„=l-l/gr„ (8');
q„=l+bi + bibs + . . . + bibs ... 6, (G').
512 INCOMMENSURABLE C.FF. CH. XXXIV
Now (8') gives us ?,= 1/(1 -/>«/?»). Hence the following
remarkable transformation theorem : —
Cor. Ifh ib^he any quantities whatsoever, then
1 + i, + 6,6, * . . .-^btb,. . .bn
_ 1 fc. b,
l-6j + l-6,+ l
from which, putting «i = i,, «, = ia6, «, = i,6, . . . 6,+i,
we retidily derive
1 + «1 + tta + . . . ^ «n
_ 1^ U, H?_ "' "' <<:l«4
~1- 1 + ttl- «! + «,- Ml + «J- «> + "«- ' *
«■.-»"»-! «»-»»» /JQ\
an important theorem of Euler's to which we shall return
presently.
INCOMMENSURABILITY OF CERTAIN CONTINUED FRACTIONS.
§ 17.] Tf a,, a„ . . ., a„ b,, b, b, be all positive
integers, then
I. The infinite continued /raction
hi b, bn f.K
0^+ a,+ o, +
converges to an incommensurable limit provided that after soms
finite value of n the condition o,-<6, be always satisfied.
II. The infinite continued fraction
b, b, bn ,n\
• — • • • ' — • • • V*/
Ot- Oj- a, -
converges to an incommensurable limit provided that (ifler tome
finite wtlue of n the condition a, S i, + 1 6« always satined, where
the sign > need not always occur but must occur infinitely often'.
To prove II., let us first suppose that the condition
o. £6. + 1 holds from the first Then (2) converges, by § 16,
* TbfM tbGorcma are dae to Lugendni, iUmenU dt QiomStrit, DOto it.
^ 16, 17 INCOMMENSURABLE C.FF. 513
to a positive value < 1. Let us assume that it converges to a
commensurable limit, say X/Xi, where Xi, X„ are positive integers,
and Ai>.'V2-
Let now
Pi = . • . ■
03- a4-
Since the sign > must occur among the conditions 03 S ^3 + 1,
Ui^bi+l, . . ., P3 must be a positive quantity < 1. Now, by
our hj'pothesis,
X,/Xi = bj{ai - ft),
therefore P3 = {(f2\t- bi^i)/^,
= yA„, say,
where X3 = ai\2-h,Xi is an integer, whicli must be positive and
<A3, since pi is positive and < 1.
Next, put
k I.
Pt = ... .
at- tti —
Then, exactly as before, we can show that p^ = Xt/X, , where A4 is a
positive integer <A3.
Since the sign > occurs infinitely often among the conditions
«« S 6n+ 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
^i> h, A3, Aj, . . . such that Ai>A2>A3>A4> . . . ; but this is
impossible, since K is finite. Hence (2) is incommensurable.
Next suppose the condition a„S6„ + l to hold after 11 = m.
Then, by what has been shown,
y — — ■• • •
is incommensurable.
Now we have
bi 63 bm
F=
consequently i' = ) -"".-^ ,■ ,
^ ^ {am-y)qm-i-bmqm-a
Qm - yim-l
(3),
33
514 EULER'S transformation CH. XXXI V
where pJin, Pm-J^m-i are the ultimate aiid penultimate cou-
vergenta of
a,-<h-' ' ■ «« ■
It result* from (3) that
y (F'Jm- 1 - p»-i) = i'V- - Pm (4).
Now Fq^-i-p^-i and Fq„-p^ cannot both be rero, for
that would involve the equality pjq^=p^.jq^.i, which is
inconsistent with the equation (2) of § 3. Hence, if F were
commensurable, (4) would give a commensurable value for the
incommensurable y. F must therefore be incommensurable.
The proof of I. is exactly similar, for the condition a^-^b,
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' 3' 5'
Brouncker's fraction, for example, 1 + - — — — - — ....
^ 2 + 2+ 2 +
converges to the incommensurable value 4/n-, and yet violates the
condition of Proposition I.
CONVERSION OF SERIES AND CONTINUED PRODUCTS INTO
COXTINDED FRACTIONS.
§ 18.] To convert the series
u, + «,+ ...+ u, + .. .
tnto an "equivalent " continued fraction of the form
o,- a,- a,-
A continued fraction is said to be "equivalent" to a series
vhen the nth convergent of the former is equal to the sum of %
terms of the latter for all values of n.
Since the couvergents merely are given, we may leave the
denominators ji, q^, ■ • ■ , '/• arbitrary (we take q,= I, a*
usual).
%n, 18 euler's transformation 515
For the fraction (1) we have
Pn/qn -Pn-l/qn-l = ^1^2 • • • bjQn-iqn (2) ;
qi = (h, q2 = (hqi-h, ■ ■ ., qn = a„q„.i-bnqn-i (3);
Since
Pi/qi = bi/qi
Pnlqn = Ml + Kn + . . , + U„
we get from (2) and (5)
Un = bih . . . l>,Jqn-iqn,
«n-l = ^l*2- • ■ K-llqn-iqn-l,
(4).
(5).
(6).
th = hhlqiqi,
From (6), by using successive pairs of the equations, we get
bi = qilh, h = q2lk/Ui, Z'3 = 2'3«3/g'l«2, . • ., bn = qnU„/q„-^Un-l
Combining (3) with (7), we also find
01 = 21, a-i = q'i{ih + u.i)lq,ih, a3 = qs{ik+U3)/q2n^, . . .,
a» = g'a(««-l +«»)/2'»-l«n-l (8).
Hence
Sn=Ui + n^+ . . . + tu,
_g'i«i q-.ua/ih
qjihlqiih
qi- g'aK + MsVa'iWi- gaith + uayquh-'
(9).
qn{iin-l+ U„)/qn-lU„.i
It will be observed that the q's may be cleared out of the
fraction. Thus, for example, we get rid of ^i 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 q^ ; and so on. We thus get for (S„ the
equivalent fraction
" 1- {th + «2)/Mi - ("2 + «3)/«2- ' * ■ («»-l + ««)/«n-l
which may be thrown into the form
Ml U, ttl«, Mn-jMn
(10),
s„^
1 - Ui + U^- t/j + !/3
I'n-l + Un
(11).
33—2
516 EXAMPLES — nROUNCKEIl'S FRACTION CII. XXMV
Thi.s formula is practically tho same as the ono obtained
incidentally in § 16 ; it was first given, along with many applica-
tions, by Euli-r in his memoir, " De Transforui.itione Serierum
in Fractionea Continuas," Opuscula Analytka, 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 tho
scries converges, and that to the same limit.
By giving to «,, u.,, . . ., tu 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-iX' + . . . + VnOf
VlX ViX v,v,x
^«^^»ii«
~ I — v, + VjX — V.J + VlX — ■
' «,., + v^x
ar x" x^
-+ - + . . . + —
X V,'X Va'x
rin-iX
r,- ViX + Vj- ViX+Vj-
' I'.-iX+l',
'^ X + "'-"^ x'+ ^. «l«i^ • • "' ^
bi bibi ' ' ' bibf . . 0„
UiX b^cujX bi<hx
bn-x(tnX
~ bi- b^ + a^x - b, + OjX - '
b^+UnX
(13);
(14).
Exaniplo 1. II -iir<x<iir, then
tan-'i = z-i'/3 + i»/5-x'/7+ . . .,
~l+8-x»+ 5-3x'+ 7-6i'+"'*'
and, in particular,
i~l+ 2+ 2+ 2+ ■ ■ ■•
which is Brounokor'a formula for the qaadrature of the circlo.
Example 2. Ifx<l,
(l+x\"-l4."" M'»-lif_ 2(m-2)x 3{m-8)x
' "^ ' 1- 2 + (ii»-l)«- 8 + (i»-2)*- 4 + (m-3)x-
§§ 18-20 REDUCTION OF INFINITE PKODUCT TO C.F. 517
Also, if »i> - 1,
2m= 1 + ^ iL"* - 1) 2(m-2) 3 (m-3)
1- nt + 1- ?» + l- m+1- ' ' ''
and, if m > 0,
_ ^ l(m-l) 2 (m - 2) 3 (m - 3)
" 1+ 3-m+ 5-m+ 7-7n+ ' ' ' '
§ 19.] T//e analysis of last jyaragraph enables us to construct
a continued fraction, say of the form (1), whose first n convergents
shall he any given quaiititiesf^f, . . .,fn respectively.
All we have to do is to replace Ui, «.,, . . ., «„ in (10) or (11)
by/i,/2-/i, . . .,/„-/«-! respectively.
The required fraction is, therefore,
/. A-AAifs-f) iA-A)(f-f)
1— fa~ J3~fi~ fi~/i~
{.In— 2 ~Jn-3/ \Jn ~Jn-V
Jn Jn—1
Cor. Hence we can express any continued product, say
d-id^ . . . dn
e^e-i. . . Cn
as a continued fraction.
We have merely to "gnt fi = d,/ei, A^dido/eie^, . . ., effect
some obvious reductions, and we find
p di eiidi-e^) d.^.i{d3-e3) d^ddi - e^{di-et) d^et{di-ei){di-ei)
"~ei- di— d^3 — e^3— d^i — esfit— did^-e^t —
dn-\en-l{dn-'i-en-i){dn-e^ /,/.\«
. . . J — J (16) .
§ 20.] Jnstead 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/(oo , x)) be reduced to a fraction of a given form, say
1_ I — I - ' ' ' I - ' ' ' V '»
where /3o, /3, /^n are all independent of .r.
There is a process, originally given in Lambert's Beytrdge
• A similar formula, given by Stem, CrclU'i Jour., x., p. 2C7 (1833), may
be obtained by a slight modification of the above process.
518 lambekt's transformation ch. xxxiv
(til. II., p. 75), for reducing to the form (1) the quotient qf two
conrergmt series, my F{\, a-)/F{0, jr).
We suppose that the absohitc terms of F(l, a:) and F{0, t)
do not vanish, and, for .simplicity, we take each of these tcnn.s to
be 1. Then we can establish an equation of the form
F(l,x)-F{0,x) = l3,:rF{2,x) (2,),
whore F(2, x) is a convergent series whose absolute term we
suppose again not to vanish, and y3, is the coefficient of x in
/''(I, a:)-F{0, x), which also is supposed not to vanish*.
In like manner we establish the series of equationa
^(2, a;) - F{1, x) = P,xF(3. x) (2,),
F(3, x) - F{2, x)=li,xF{i, x) (2,),
F{n + l,x)- F(n, x) = Pn*,xF(n + 2, x) (2,+,).
Let us, in the meantime, suppose that none of the functions
/'' becomes 0 for the value of x in question. We may then put
G{n,x) = F{n+l,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 (2,+,) may now be written
G(n,x)-\^ Pn^.xG (n +l,x)G (n, x).
that is, G(»i,a:)=l/{l-/3.„a:G(n+l, .r)} (4).
If in (4) we put successively n = 0, n = 1, . . ., we derive
the following : —
^^"''^'"f^ r^- • •l-(l-l/G(n,a:)) ^*''
^ G{n,x)~ I- ' ' ' l-{l-l/G{n + tn,w)) ^''
* Tlic TaniR)iiDg of one or more of tlicso coeOioioiiU woulil lead to a mora
general form than (1), namely,
1- 1^ •
Oeneral cxpronaiona have been foond {orfi,,ft by Heilcrmann, CrtlU'i
Jour. (1846), and by Muir, Proc. L.il.S. (1U7G).
(9),
§ 20 LAMBERT'S TRANSFORMATION 519
In order that we may be able to assert the equality
(^ (0, .^) = j-^ Y3 . . . '^ _j ... ad 00 (7),
it is necessary, and it is sufficient, that it be possible by making
m sufficiently great to cause 1 - 1/(? (w, x) to differ from the mt\\
convergent of the residual fraction
1 - 1 _ • • • 1 _ • • • yy)
by as little as we please.
Let us denote the convergents of (8) hy Pijqi, p^jqi
Pm/qm- Then, from (6), we see that
{1-1/G(n, x)}-pjq„
^Pm-Pm-l{l - 1/G(7l + m, X)} Pm
qm-qm-i{l-l/G{n + m,a;)) q^'
_ {1 - 1/g {n + m, X)] (Pmlqm-Pm-Jqm-i)
qm/qm-i -{1-1/G {n + m, «)}
_{l-l/G{n + m, x)} P„+, ff„+a . ■ ■ /?„+„.a;'" .
qm[qv,-qm-i{l-l/G{n + m,x)]] ^
The neccsmry and sufficient condition for the subsistence of (7)
is, there/ore, that the right-hand side of (9), or of (10), shall
vanish when m = <a.
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{n, 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's would vanish, and
(?(0, x) would not be determinate. Suppose, however, that
F{r+\, x') = Q, so that G{r, x') = 0; then (5) furnishes for
G (0, x') the closed continued fraction
620 EXAMPLE Cll. XXXIV
lu order that tliis may be identical with the value given by
(7), it is necessary and sufiicient that G(r+1, x), as given by
(C), should bocome ao , that is, it is necessary and sufficient that
the residual fraction
Pli^ ^ . . . a.l 00
should convcr^jo to 1 ; but this condition will in general be
satisfied if the relation (4) subsist for all values of w, and the
condition (9) be also satisfied when n-^r-Vi.
% 21.] As an example of the process of last paragraph, let
Fill, x) = l + -r, X + „-,-7— — w ,\ + • • • (' !)•
^ ' l!(y + n) 2!(y + n){y + n+l) ^ '
Then
Fin . 1, .) - Fin, .) = - (^,„)J,„,,) F^n . 2. .) (2') ;
and
G{n,x)=\l[\^- ^ -G(f. + 1. a:)| (4'),
I y (7 + n)(y + H+ 1) ' ')
where G{n, x) = F{n+ 1, x)lF(», x).
Hence
rtn -r^- ' ^/Y(y+l)^/(y+0(Y-<-2) a-/(y + n-l)(y + n)
0(V,x)-^^ 1+ ' 1+ • • ■ 1-{1-1/G(w, J-)}
(5');
and
1 x/{yi-ii)(y + n+l)
^ G(n,x)~ 1 +
a-/(y + n + m-\){y + n + m) ...
l-{l-l/G(n + m, a-)} ^^'•
Tlie 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 + tn, jr) = 1 when m = oo .
Let us suppose that x ia positive. Then the residual con-
titiued fructiuu
;^ 20, 21 EXAMPLE 521
xl{y + n) (y + n + 1) x/{y + n + l)(y + w + 2)
1+ 1 +
a:/{y + n + m-l){y + n + m) , ,.
is (by the criterion of § 14) e^ndently coavergent. Hence the
factor Pmlqm-Pm~-ilqm-i in the expression (9) vanishes when
«»=oo.
Also, since the ^''s continually increase, Lq„lq„--i -^ 1.
Therefore we may continue the fraction to infinity when x is
positive.
Nest suppose x negative, =-y say ; we then have
r(n ,.\ ^ y/y(y + i) My + i)(y + 2)
^yj>-y)=Y^ — ^ Yz • • •
y/(y + «-l)(y + ») , „. .
l-{\-llG{n,-y)\ ^^^'
and
1 _ ^ _ y/(y + «)(y + « + i)
G{n.-y) 1-
y/(y + n + m. - 1) {y + n + m) . „.
~l-{l-\IG{n + m,y)} ^^ ^
The fraction (8) in this case is "equivalent" to
_J_ f_^ E__ . . . V . . . \ (8"),
y + nly + M+l-y + « + 2-'''y + w + OT— J
■which is obviously convergent (by § 16), if y have any finite
value whatever. Hence the factor pjqm -pm-i/qm-i belonging
to the equivalent fraction (8) must vanish.
Again, by § 2 (6),
<?m-l
y/{y + n + m -l){y + n + m) y,'{y + n + m-2){y + n + m-l)
-1 i- 1-
y/(y + »)(y + w + l)
... J
= !__!_ / y y _.^(i2).
y + 7i + 7n [y + n+ m-l- y + n + m-2- y + n)
(14),
622 C.FF. FOR TAN X AND TANH x CU. XXX IV
If only » be taken large enough, the fraction inside the
bnickcta satisfies the condition of § 16 throughout: its value is
therefore < 1, however great m may be ; and it follows from (12)
that Lqjq^.i = 1 when m = oo .
Since LG (n + m, — y) = 1 when ot = x , it follows that all the
requisite conditions are fulfilled in the present case also.
We have thus shown that
F(h^^ ± xly(y+l) xl(y+l){y + 2)
J>\0, x) 1+ 1 + 1 + ■ • •
whence, by an obvious reduction,
F(l, x) y_ X X X
F{0, a;)~y + y+l + y + 2 + ' ' ' y + n+' '
a result which holds for all finite real values of x, except such
as render i^(0, x) zero*, and for all values of y, except zero
aud negative integers.
If we put ±x'li in place of x in the functions F(0, x) aud
^(1, x), and at the same time put y = i, we get
/'(O, - ar'/4) = cos a-, /'( 1 , - a:»/4) = sin xjx ;
F(<d, a^ji) = cosh x, F{1, ar'/4) = sinh x/x.
Cor. 1. Hence, /rom (14), toe get at once
OC it? 3^ St
Cor. 2. Thf numerical constants tt and ■n' are incomnwnsurahU.
For, if TT were commensurable, ir/4 would be commeusunilile,
Bay =X//i. Hence we should have, by (15),
* In a seDM it will hold even then, for the fraction
7 1^^7 + 1+ 7 + 3+' -f
«'hioh represents F(0, x)IF(\., x) will conTer|;e to 0. Of oonme, two eonseou-
tiva funolioDi F(n, x), /'(ii + l, x) caoDOt Taniah for the same Taluc of x\
otbeiwiM ws aboulJ have >' (ao , x) = 0, which U impossible, sinos f (ae , x) = 1.
§§ 21, 22 INCOMMENSURABILITY OF V AND e 623
>/^ X7^' \7/.»
1 =
1- 3- 5-
\ \» X'
(17).
/i- 3/i- 5/* — ' ' *(2?i+l)/A-'
Now, since X and ^i are fixed finite integers, if we take n large
enough we shall have (2?j + l)/x>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 pmcer of e is incommensurahh* .
§ 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 : —
K
T7/ o X , «/8 a(a + l)/3(/J+l) „
j'(a,/3,y.^)=i + ^^H- \,,,;^;^i) •^-+--..
and
G (a, /3, y, x) = F{a, /? + 1, y + 1, x)/F{a, p, y, x),
then
^^"■'^'■>'''^^~1- 1- 1-' • •ljG{a. + n,p + n,y + 2n)
(18),
_a(y-i3) „ _ (/?+l)(y+l-°)
^•-7(7^' ''^- (y+l)(y + 2) '
_ _ (a + l)(y + l-)S) „ _(/? + 2)(y + 2-a)
'*»- (y+2)(y + 3) • • ''' (7 + 3)(y + 4) '
(a + w-l)(y + w-l-;3) „ (/? + w)(y + n-a)
^*'-'~ (y + 2»-2)(y + 2w-l) ' '^'" (y + 2«- 1) (y + 2«)"
• The reeulta of this paragraph were first given by Lambert in a memoir
which is very important in the history of continued fractions (IlUt. d. I'Ac.
Roy. d. Berlin, 17(il). The arrangement of the analysis is taken from Legendre
(I.e.), the general idea of the diBCUssion of the convergence of the fraction
from Schlomilch. t Werke, Bd. in., p. 134.
524 GACSS'S C.F. FOR HYPERGEOMFTRIC SERIES CH. XXXIV
After what has been done, the proof of this theorem should
present no difficulty.
Th>' ' Q of the question of convergence is also com-
parativ when x is p<jsitive ; but presents some difficulty
in the case where x is negative. In fact, we are not aware that
any complite elementary discussion of this latter point has been
given.
Cor. If in (18) we pat /3 = 0, and write y - 1 in place qf y,
wa gat the tran^ormation
. tt_ . a(a-H)_, . a(a-f-l)(«-t-2)_,,
y y(r + i) r(y + i)(r + 2)
where
1- 1- 1- '
(19).
a „ _ y-'
y r(y*i)
f,_ ('^i)y o 2(r_+i-a)
'^ (y+i)(y + 2)' '^•'(yVsHT^S)'
^-1=
(g-m- l)(y -»• n - 2) ^ _ w (y -i- » - 1 - a)
(y + 2B-3)(y + 2»
-2)' '^ (y + 2«-2)(y + 2(.-l)-
Gauss's Theorem is a very general one ; for the h}'pergeometric
series includes nearly all the ordinary elementary series.
Thus, fur example, we have, as the reader may easily verify,
(i+xr = /'(-in,AA-*);
log(l+x) = xf(l, 1. 2, -x);
sinh X = * Z. L F{k, k\ \ , x»/4tfc') ;
*— • k — «
sin X = X I L F(k, i,i,- x'lWc) ;
8in-'x = xF(i.i,f,x^;
= xV(l-x»)/'(l. l,i,x^;
tan->» = xf(J, I. |,-x»).
§22 EXERCISES XXXIV
Exercises XXXIV.
Exjunine the conTergencc of the following : —
111 I' 2^ 8»
1' \*.2> 2».3^ ,. , , 1 1.2 2.3
<5> i+iT 1- iT • • • • <^' ;rr ^.Tf- — n-+" • • •
1* 2* 3* „, , 1.3 3.5 5.7
(7.) i+- ... . (8. 1 + rr T- TT • • • •
' ' Z+I+X+ 1+1+ 1 +
,„, 2 1».3 2».4 3>.5 ,,„, 2 2» 2» 2«
<^) iTXTTTTT--- • (i«) mTiTiT--- •
6, 6,
(11.1 Show that the fraction of the second clasa, o, — . . . , con-
a,- a,-
Tergea to a positive limit if, for all Taloea of n,
(4/6,6, + a,/6,i>, + . . .+a»+i/*.i'»+i>-l.
(Stem, Gdtt, Naeh., 1845.)
fl2.) Showthat -?1- -^ -^'-. . .,where<i.>0,coDTerge8ifa,^,>a, + l.
<h- "i- <H-
(13.) Show that the series of fractions (p,-p,-i)/(?,-?»-i) forms a
deteending series of coDTcrgents to the infinite continacd fraction of the
second class, provided ». ^ {'»+ !< and the sign > occurs at least once among
these conditions.
(11.) Show that
Z X X
i + 1- x + 1- x + 1- ' ' ■■
where z>0, is equal to x or 1 according as x< or < 1.
12 3
(15.) Evaluate o^ 3Z iT ' * • "
■I M + l in + 2
and
where m is any integer.
1 + 1- ■1+2- m + 3-
Show that
<^®> ^ + 6+6TtTT]'"- • •=^'^6~r+6T2^ a + 6 + 4- ••' '
X z* 2.3r» 4. ox'
(17.) ""'=1:^: 2.3-x'+ 4.S-x»+ 6.7-x»+ ' * " '
X I'x 2»x S'x
(18.) iog(i+x)=,- ^-^ 3^:^^ i^:^ ....
Eiercises (5) to (10) are taken from Stem's memoir, CrelU't Jour., xxxtiu
526 EXERCISES XXXIV CH. XXX IV
„„, , 1« 2» 3>
(19) 1 = 3-5-7-- •• •
(20.) log^^' ' ''' ''^ «''
I
(2n -•!)'(»' -1)»
*("»' + » + ")-
,„. . 1 X s 2x 3x
(22.) e'=j- j-^^^ ^;^;^ g^j:^ ^^j:^. . . .
Evalaate the following : —
,„„, ,112 3 4 _.,„,. 1 2 3 4 -
(23) 1 + 1-3-43 5^6-- •• •-*■ (^J rf2-T3-;:4-:f- " ■*-<
,25) J-iL^il -Log Z ,26)-?-^-?-^ =^-^
(25) f^i— 1+1+- • • • "^e (-'^•) 2+3+4+6+ ••- -e-^
(27.) Show that tanz and tanhx arc incommonsarable if z be commcn-
BQrable.
Establieh the following transformations : —
m) ^-± JL J. JL JL J- ± ±. . .
*'"'■' '^"l- 1+ 2- 3+ 2-5+2-7 +
,„„,,„ , I I'x I'x 2»x 2-^1 3>x 3'x
(29.) log(l + x) = j^ ______... .
„« , . . ' 1'*' 2'x' 3'x>
(30.) tan-x = ^-3^^ sT 7T - " ' '
I l'x> 2'x« 3»i»
tanh-'x=j- ^_— — . . . .
ntunx (n>-l*)tan*x (n>-2^tan>x
(31.) tan nx = -y— * gi— — *- ^~ .
(Eoler, Hem, Acad. Pet., 1813.)
Bin(n + l)i - 1 1
(32.) -. ^=2C08X-s 5 — — • • ;
> ' suinx 2oosx- 2ooax-
where there ore n partial quotients.
(33.) If
0 (o, ft, y, ')
_j,(g'-l)(7''-l)j|(?'-l)(g""'-»)(/-l)(/^'-')^,
(J - 1) (,» - 1) (8 - 1) («»- 1) (,» - 1) (,»+' - 1)
then
»(a./g + l. 7-Kl. *) _ J_ /?if ftf
^(a,p,y.x) -I- 1- 1- •• ••
I
§ 22 EXERCISE!- XXXIV 527
where
_(g-+^-l)(g>+-^-l) B4r
(Heine, Crelle's Jour., mii.)
(34.) Show that
„_( _i,_^^ 3' 5' \
" l" "^2(a-l)+ 2(a-l)+ 2(a-l)+ ■ • •)
( _1 3» 5' 1
" l"''" """2(0+1)+ 2(a + l)+ 2(a + l)+ ' f'
WaUis (see Muir, Fhil. Mag., 1877).
CHAPTER XXXV.
General Properties of Integral Niunbc -.
NUMBEKS WHICH ARE CONGRUENT WITH RESPECT TO
A GIVEN MODULUS.
§ 1.] Jfmbe any positive integer whatever, which we call the
modulus, ttco integers, M and N, which leave the same remainder
when divided by m are said to be congruent with respect to the
modulus TO*.
In other words, if M=pm + r, and N=qm + r, M and N are
said to be congruent with respect to the moduhis m. Gausa,
who made the notion of congruence the fundamental idea in his
famous Disquisitiones Arithmeticw, uses for this relation between
M and N the symbolism
M=N{moAm);
or simply M s N,
if there is no dnultt about the raoduhis, and no danger of con-
fusion with the use of h to denote algebraical identity.
Cor 1. If two numlters M and N be congruent with respect
to modulus m, then they differ by a multiple of m; so that «w
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 be prime
to m, the other must be prime to m also.
In the present chapter we sliall use oidy the most elementary
conseiiucuces of the theory of congruent numbers.
* To »avo repetition, let it bo aodprntocHl, when nothing else in indicated,
tliat Ihroughuut thin chapter every Idler BtimcU for a posiiliTe or ncgatira
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 wliich 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 " m, 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, . . ., (7«-2), (m-1) (A),
or, if we choose, of the set
0, 1, 2, . . ., -2, -1 (B),
where there are m integers.
Another way of expressing tlie above is to say that, if we
take any m consecutive integers whatever, and divide them by m,
their remainders taken in order will be a cyclical permutation of
the integers (A).
Example. If 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 7p, or
7j) ± 1. Bearing in mind that every integer N has one or other of the forma
7m, 7m±l, 7m±2, 7m±3,
alsothat (7Hi±rp=(7n!)3i3 (7m)=r + 3(7m))-±i-3,
= (V-m? ± 2lm-r + 3mr-) 7 ± »■*,
= Jl/7±r3,
we see that in the four possible cases we have
A-3=(7m)s = (7%3)7;
A'3=(7m±l)»=il/7±1; ^
^.•3= (7m ±2)3,
= iir7±8 = (J/±l)7±l;
:js=(7m±3)'=(3/±4)7=r].
c. IT. 34
5:iO EXAMPLES CH. XXXV
In every case, then-fore, Hie cube has one or other of the forma 7p or
7f±I.
Kxam])1c 2. Prove that 8*"+' + 2"+» is divisible by 7 (Wolsteubolme).
Wo liftvo 3»»+' + 2"+' = (7 - 1)"*' + 2"+'.
Now (sec above, Example 1, or below, § 4)
(7 - 4)"+' = 3/7 - 4»»+'.
Ucnce 3'»+' + 2»+'=iV7-4»»+' + 2"+',
= Jlf7-2««(2'»-l).
But a*" - 1 is divisible by 2' - 1 (see chap. v. , § 17), that is, by 7. Heoce
2»+>(2'»-l) = A'7.
Finally, therefore, S*"** + 2«+'= ( J/ - A') 7,
which proves the theorem.
Example 3. The product of 3 sacccssive integers is always divisible by
1.2.3.
Let the product in question be m (m + 1) (m + 2). Then , cinco m must have
one or other of the three forms, Sin, 3m + 1, 3m- 1, we have the following
cases to consider : —
3m(.Sm + l)(3m + 2) (1);
(3m + l)(3m + 2)(3in + 3) (2);
(3m-l)3ni(3m+l) (3).
In (1) the proposition is at once evident ; for 3m is divisible by 8, and
(3m + 1) (3m + 2) by 2. The 6.imo is true in (2).
In ca.sc (3) wo have to show that (3m - 1) m (3m + 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 3in-«-l are
divisible by 2.
In all casc.s, therefore, the theorem holds.
Example 4. To show that the product of p successive integers is alwavi>
divisible by 1 . 2 . 3 . . .p.
Let as suppose that it has been shown, 1st, tliat the product of any p - 1
successive integers whatever is divisible by 1.2. 3. . .p-1; 2nd, that the
product of p successive integers beginning with any integer np to x is divisible
by 1.2.3 . . . p-l.p.
Consider the product of p successive integers beginning with x+1. We
have
(i + l)(x + 2)...(x+p-l)(i+p)
=p(i+l)(x + 2)... (x+p-l)+i(i + l)(x + 2).. .(x+p-1)... (I).
Now, by our first sapposition, (x + 1) (x + 2) . . . (i+p— 1) is divisible by
1.2. . . p-1 ; and, by oar second, z (x + 1) (x + 2) . . . (x +p - 1) is diriaibl*
by 1.2.3 . . .p.
Hence each member on the right of (1) is divisible by 1 . 2 . 8 . . .p.
It follows, therefore, that, if our two suppositious be right, then the pro-
duct of p BUCcrsHivc intogiTS beginning with x + 1 is divisible by 1 . 2 . 3 . . .p.
Dot we have shown in Kxamplc 3 that the product of 3 conHCCUtive integer*
is always divisible by 1.2.3; oud it is sclf-«vi Juut that the product of 4 con-
§ 3 PYTHAGOREAN PROBLEM 531
siicntivc integers boginning with 1 is divisible by 1 . 2 . 3 . 1. 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 y = 5 ; and so on. Hence it holds generally.
This demonstration is a good example of " mathematical induction."
Example 5. If a, b, c be three integers such that a' + b-=c', then they are
represented in the most general way possible by the forms
a = \{m''-n'}, 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 = \c', where a', b', c'
are prime to each other, and we have
o'»+6'2=c'» (1).
No two of the three, a', i', c', therefore, can be even ; also both a' and 6'
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^), 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';
bnt c' and a' have by hypothesis no common factor.
Now we have from (1)
whence
C-^)K^>^= ^^■
(3),
Therefore, since (c' + a')/2 is prime to (c' - a')/2, each of these must be a
perfect square ; so that we must have
p=mn (o),
where m is prime to n.
From (3) and (4), we have, by subtraction and addition,
a'=m'-n', e' = 7n'+n';
and, from (5), i'=2/3 = 2mn.
Eetuming, therefore, to oar original case, we most have generally
a = \{m'-n^), 6 = 2Xmn, c = \(m^ + n').
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 PllOI'ERTY OF AN INTEOllAL FUNCTION Cll. XXXV
§ 4] Tlie following theorem may be deduced very readily
from the priiK-ii)le3 of § 2. Let /(j-) stand for />o+/>,x+/)jjr' +
. . . +p,x", where po. Pi, - • ■, Pn are positive or negative
integers, and a: any positive integer; then, if x be congruent
ifith r with respect to the modulus m, f{x) will be congruent with
f{r) with respect to modulus m.
By the binomial expansion, wc have
(ym + r)" = (</w)* + ,C,(?m)-'r+. . . + .C,., (ym) r""' + r",
= M»m + r*;
whore 3/, is some integer, since all the numbers ,(7,, ,Ci, . . .,
nCn-i are, by § 3, Example 4, or by their law of formation (see
chap. IV., § 14) necessarily integers.
Similarly
(qm + r)""" = M,., m + r""',
• • « ■
Hence, M x = qm + r,
/{x)-=p,+pir^Pir^ + . . .+|j,r" + (p,il/,+p,3/, + . . .+/),3/,)»».
=/(r) + 3fm.
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 dirisiliiliti/ o//(x) hij m for all intetjral
values of x, we need only test the divisibility by m <'//(0), /(I),
/(2) /(»'-!).
Examiilcl. lj(itf(x) = z{x + l){2x + l); and let it be rcqnircd to find when
/(i) is divisible by 6. Wo havo/(0) = 0,/(l) = 6,/(2) = .SO,/ (3) = 84,/(<) = 180.
/(5) = 330. Each of these is divisible b; C ; and every integer is ooDRruciit
(mod 6) with one of the tix numbers 0, 1, 2, 3, 4, 5 ; bcnce x{x + l)(2x + l)
is alaayi divisible by 6.
Cor. 2. f\qf{r) + r} is always divisible by f{r); for
/{q/(r) + r| = Mf(r) +f{r) = (M + l)/(r).
Hence an infinite number of values of x can always be found
which will vuike f(x) a composite number.
^ 4, 5 DIFFERENCE TEST OF DIVISIBILITY 533
Tliis result is sometimes stated by saying that no integral
function of x can furnish prime numbers oily.
Example 2. Show that ar» - 1 is divisible by 5 if s be prime to 5, but not
otherwise.
With modulns 5 all integral values of x are congruent with 0, ±1, ±2.
If /(i) = i<-l,/(0)=-l,/(±l) = 0,/(±2) = 15. Now 0 and 15 are each
divisible by 5 ; but - 1 is not divisible by 5. Hence x*-l is divisible by 5
when X is prime to 5, but not otherwise.
Example 3. To show that a;- + x + 17 is not divisible by any number lesa
than 17, and that it is divisible by 17 when and only when x is of the form
17m or 17m -1.
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
x=ml7 or x=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 /„ {x) denote an integral function of the «th degree.
/„ (a; + 1) -/„ (a;) =p„ +^1 (« + 1) + . . . +^„., {x + l)""' +_p„ (a; + 1)»
-Po-Pix-. . .-p„.iaf-^-p„af (1).
Now on the right-hand side the highest power of x, namely
a;", disappears ; and the whole becomes an integral function of
the M-lth degree, fn-i(x), say. Thus, if m be the divisor,
we have
m m ^ ''
It may happen that the question of divisibility can be at
once settled for the simpler function fn-\{x). Suppose, for
example, that it turns out that /„_i {x) is always divisible by m,
whatever x may be ; then/„ {x + 1) — /„ {x) is always divisible by
m, whatever x may be. Suppose, farther, that /„ (0) is divisible
by m ; then, since /„ (1) — /„ (0), as we have just seen, is divisible
by m, it follows that/„(l) is divisible by m. Similarly, it may
be shown that f^ (2) is divisible by m ; and so on.
634 KXERCISES XXXV en. XXXV
If tlip divisibility or non-divisibility of /,-i (x) 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. /, (x) = i* - x is always diviBiblo by 5.
= 6i* + 10x» + 10x» + 5x,
= iI5.
Now /.(1) = 0,
therefore /, (2) -/, (1) = .V<,.5.
and /.(2) = ,V„5.
Similarly. /, (3) - /. C^) = J/,5,
therefore /,(3) = (.V, + JI/,)5:
and BO on.
Thus we prove that/, (1), /, (2), /, (3), Ac, are all divisible by 5 ; in other
vrords, that z* - x in always divisible by a.
Exercises XXXV.
(I.) The enm of two odd squares cannot be a sqnara
(2.) Every prime greater than 3 is of the form &n ± L
(3.) Every prime, except 2, has one or other of the forms 4ii^l.
(4.) Every integer of the form 4n - 1 which is not prime has an odd
number of factors of the form in - 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 game
form.
(7.) If two odd primes differ by a power of 2, their snm is a multiple of S.
(8.) The difference of the squares of iiny two odd primes is divisible by 24.
(9.) None of the forms (3m + 2)n* + 3, 4mn-m- l,4mn-m-ncan repre-
sent a square integer. (Goldbach and Eulcr.)
(10.) The nth power of an odd number greater than nnity can b« 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 tliat N -xy is a sqaare.
(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 consccatJTa nneven nombera,
of which n' is the middle one.
(14.) There can always bo found n consccutlTS integers, each of which is
not a prime, however great n may bo.
§
KXERCISES XXXV 535
(15.) In the scale of 7 every square integer mnst have 0, 1, 2, or 4 for its
nnit digit.
(16.) The scale in which 34 denotes a square integer has a radix of the
form ?t(3;! + 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 Sp + l. If this condition be satisfied, tliere 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, I-" - 4x3 + 5x= - 2x is divisible by 12.
(19.) x'/5+x</2 + x3/3-x/30, and x«/C + x=/2 + 5x-'/12 - x=/12 ore both in-
tegral for all integral values of x.
(20.) If X, y, 2 be three consecutive integers, (Sx)'-3Sx' is divisible
by 108.
(21.) x' - X is divisible by 6.
(22.) Find the form of x in order that x' + 1 may be divisible by 17.
(23.) Examine iiow far the forms x- + x + 41, 2x- + 29 reiiresent 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^=-^ is divisible by 22.
(27.) Show that the remainder of 2-'^'^''+ 1 with respect to 2=^ + 1 is 2.
(28.) 3=^-2=^" is divisible by 5,ilx~y = 2.
(29.) Show that 2'-^+' + 1 is always divisible by 3.
(30.) 43^*' + 2"=t> + 1 is divisible by 7.
(31.) x^'" + x-"' + 1 never represents a prime unless x = 0 or x = l.
(32.) If P be prime and =w' + b-, show that F" can be resolved into the
sum of two squares in ^n ways or J {n + 1) ways, according as n is even or odd,
and give one of these resolutions.
(33.) If 2^ + y- = 2', X, y , 2 being integers, then xyz = 0 (mod CO) ; and if x
be prime and >3, !/ = 0 (mod 12). Show also that one of the three numbers
= 0 (mod 5).
(34.) The solution in integers of x' + j-=2;= can be deduced from that of
x^ + i/'=2^. Hence, or otherwise, find the two lowest solutions in integers of
the first of these equations.
(35.) If the equation x' + y^ = :^ had an integral solution, show that one of
the three x, y, z must be of the form 7m, and one of the form 3ik.
(36.) The area of a right-angled triangle 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.
(10.) If n be odd, show that l + „C4 + „Cg + „Ci8+ ... is divisible by
53G LIMIT AND SCHEME FOE DIVISOUS OF N CU. XXXV
ON THE DIVISORS OF A GIVEV INTTOER.
§ C] Wc have already seen (chap, m., § 7) that every
composite integer N can be represented in the form a*b^cy . . .,
where rt, i, c, . . . are primes. If iV be a perfect square, all tlio
indices must be even, and we have N=a^U'^<fy . . , ; so that
jN=a''b»'cy' .... _
In this case N is divisible by JN.
If N be not a perfect square, then one at least of the indices
must be odd ; and we have, say,
JV=o'"+'6'^c'»' . . . =a"7/*cr' . . . a'+7>8V»' . . .,
80 that N is divisible by a'b^cy .... which is obviously less
than JN. Hence
Eieri/ composite number has a/actor 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, ouce 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 Burekhard, Dase, and the British Association*.
§ 7.] The divisors of any given number N = a'l/'cy ... are
all of the form a'b^cy .... where a', fi', y, . . . may have any
values from 0 up to o, from 0 up to /?, from 0 up to y, . . .
respectively. Hence, if we include 1 and ^V itself among the
divisors, the divisors of N=u'0^cy . . . are the various terms
olAained by distributing the product
(1+0 + 0-+ . . . +a*)
x(l +t + t'+ . . . +6*)
X (1 +c + c" + . . . + rT)
^ (1).
* For an intercBting account of the oonitruction aiid dm of thate tables,
see J. W. L. GlainliiT's Boport, Rfp. liril. Anoc. (1877).
i^ C, 7 SUM AND NUMEEU OF FACTORS 537
Cor. 1.
Since
!+« + «'+. . .+«'■ = i
a- 1
is+i - 1
1 + 6 + «*= + . . . + is =
6-1
and so on,
It follows that the sitm of the divisors of N= a'l^c-i . . . is
(«'+'- l)(&g+^-l). . .
(a-l)(6-l). . . •
If in (1) we put a= 1, 6 = 1, c= 1, . . ., each divisor, that is,
each term of tlie 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, /3 + 1 iu
the second, and so on, it follows that
Cor. 2. The numhcr of the divisors of N= a'b^cy . . . is
(a+l)(/J + l)(y+l). . . .
Cor. 3. T/te number of tvai/s in which* N'= n°-h^cy . . . can
beresohedi7itotwofactorsis^{l + (a+l){/3+l)(y + l). . .}, or
|(o + 1) (^ + 1) (y + 1) . . ., 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 = JN X iJN^ corresponds to only one divisor.
Cor. 4. The number ofv:ays 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 Ducoiirse of Combinations, Alterna-
tions, and Aliquot Parts (1G85), chap, ni., § 12. In the same work are given
most of the results of §§ b and 7 above.
53S EXAMPLES CH. XXXV
were each equal to unity. Hence the number of ways is
J(l + l)(l + 0(1 + 1). . . (n factors) =i. 2" = 2-'.
Example 1. Find the different divisors of 360, their tain, and their
naiuber.
WohaTo860 = 2'3'5.
The divisors are therefore the terms in the distribnted prodact
(l + 2 + 2» + 2')(l + 3 + 3')(l + 5); that is to say,
1, 2, 4, 8, 8, 6, 12, 21, 9, 18, 30, 72, 5, 10, 20, 40, 15, 30, 60, 120,
45, 90, 180, 360.
Their8nmi8(2«-l)(3>-l)(5'-l)/(2-l)(3-l)(5-l) = 1170.
Their number is (1 + 3)(1 + 2)(1 + 1) = 24.
Example 2. Find the lea^t number which has 30 divisors. Lot the
nuniber be N='t°b^cy. There cannot be more than three prime factors : for
30=2x3x5, which has at most three factors, must =(o+ 1) (^ + l)(y + 1).
There mi);ht of course be only two. and then wo must have30 = (a+ 1) {ji+ 1);
or there nii(;ht bo only one, and then 30 = a + 1.
In the first case a = l, /3 = 2, 7 = 4. Taking the three least primes,
3, 3, 5, and putting the larger indices to the smaller primes, we have
JV^ 2*. 3^.5 = 720.
In the second case we should get 2>* . 3, 2* . 3*, or 2* . 3*.
In the last case, 2".
It n-ill be found that the least of all these is 2* . 3* . 6 ; so that 720 is the
required number.
Example 3. Show that, if 2'> - 1 be a prime number, then 2*-' (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 nomber
are 2*'> and 2*- 1. Heuce the sum of its divisors, excluding itself, is, by
Cor. 1 above,
(2Tinj2i^ripif -a"-'(2--l) = (2--l) {(2--l) + ll - a«-'(2-- 1),
= (2«-l){2--2-'J,
= 2«-'(2»-l){2-l},
as was to be shown.
0.\ TUE NU.MIiEU OF INTEfiERS 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 th&ie 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 snob a nomber waa ealled a Perfect
Nnmber. 6, 28, 496, 8128 are perfect numbers.
§§ 7, 8 euler's theorems regarding <}> (N) 5S9
by <t> (N). Thus <^ (N) is taken to denote the number of integers
(including 1) trkic/i are less than N and prime to N.
We have the following important theorem, first given by
Euler :—
Zf N'=ai'uu''Ht,'' . . . ffn*", then
*<^=-'('-J;)('-|)(-l>--('4.) w
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 N, 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 ai which do not exceed N are
loi, 2ai, 3ai, . . ., {Nja^On, iV/oi in number (3);
all the multiples of a.^ which do not exceed N are
lOa, 202, 3».i . . •, {Nja-^ai, i\7aa in liumber (4);
and so on.
All the multiples of ctiOs which do not exceed N are
Ifflifla, 2aia3, 3«ia2, . . ., {NJata^) aiO^,
iV7ai«2 in number (5) ;
and so on.
Similarly, for Oia^as we have
laifljOa, 2aiajaa, Saia^a^, . . ., {N/Uiaias) asanas,
N/aia^ai in number (6).
Let us now consider the number
N N N
— + — +— +. . .
N jsr N
a^a^
ai«3
c^at
OiOoOSs
Oicuai
N
+ + . . .
alasa^
a^a^a^at
• •
• • •
(7).
540 EULER's theorems regarding <^(A'^) CH. XXXV
The number of terms in the first line is ,<7i. Tlie mimbor
in the second line is ,C,, since every possible group of 2 out of
the n letters a,a, . . . rt, occurs among the denominators. The
number in the tliinl line is »Cj for a similar reason. And so on.
Now consider every ujultiplo of the r letters a^a^at ... a,
which does not e.xceed N \ in other words, every number, not
exceeding N, that has in common with it a factor of the form
Oi" 'Oj*' • . ■ Or*'- This multiple will be enumerat^id in the first
line, once as a multiple of a,, once as a multiple of a,, and so
on ; that is, once for every letter in it, that b, rCi times.
In the second line the same multiple will be enumerated once
as a multiple of OiCCj, once as a multiple of OxOi, and so on ; that
is, once for every group of two that can be formed out of the r
letters a,aa . . . Or, that is, ,<?» times. And so on. Hence,
paying attention to the signs, the multiple in question will in
the whole e.\pression (7) be enumerated
rC,-rC, + rO,-. . . ± r^- + r^ = 1 - (1 " 1 ^
times ; that is, just once. This proof holds, of course, whatever
the r letters in the group may be, and whether there bo 1, 2, 3,
or any number up to n in the group.
It follows, therefore, that (7) enumerates, without repetition
or omission, every integer which h;us a factor in common with N.
But, from formula (1), chap, iv., § 10, we see that (7) is simply
^'-^(-s)('-i)---('-i) <*
To obtain the number of integers less than N whkh are
prime to iV, we have merely to subtract (8) from A'. Wo thus
obtain
,(„,^(._i)(,_i)...(,-i).
which establishes Euler's formula.
Example. N=100 = 2>.6>; 0(1OO)=2>.6»(1-J)(1 - J) = 40.
§ 9.] //■ if= PQ, where P and Q are prime to each other, then
«(.V) = <^(/>).^(Q) (1).
§§8,9 ,piPQR...) = 4>(P),l>(Q)4,{R)... 541
For, since P auJ Q are prime to each other, we must have
F = a,''a^'^. . .,
Q = b,i>'b./'. . .,
wliere none of the prime factors are common ; and therefore
i!'/=a,"'a/'. . . b,»'b/'. . .,
where ai, a^, . . ., bi, b^, . . . are all primes.
But, by § 8, we then have
-K-.')(-i3---(-^)(-,l)---.
Cor. If FQRS . . . be 2}>-{?ne to each other, then
<f>{PQRS. . .) = ^(P)<t>{Q)<l>(R)4.{S). . . (2).
For, since P is prime to Q, R, S, . . . , it follows that P is
prime to the product QRS . , . Hence, by the above proposition,
<I>(PQRS. . .) = <i>(P)<l>(QRS. . .).
Repeating the same reasoning, we have
<I>{QRS. . .) = .t>(Q)<t>{RS. . .);
fmd so on.
Hence, finally,
<I>{PQRS. . .) = <l>{P)4>{Q)<t>{R)4>(S}. . . .
Remark. — There is no difficulty in establishing the theorem
4>{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 4> (PQR , . .) =
0 (P) <^ (0 <^ (R) . . . The course followed above, though not
so neat, is, we think, more instructive for the learner.
Example. 56 = 7 x 8,
0(7) = 6,
0(8) = 4;
0(56) = *.(7)x,^(8).
542 OAUSS'S TBEOREM REGARDING DIVISORS OF N CH. XXXV
§ 10.] 1/ di, (/,, d,, . . ., (tc, denote all the divis(/rs of the
integer N, then*
*W) + 0(</,) + *(rf.) + . . . = iV. . . (1).
(Giiuss, Disq. Arith., § 39.)
For the divisors, rfi, rf-j, d, are the terms in the
distribution of the product
(l+a,+0|'+. . . +o,°')(l +a, + rt,' + . . .+a,*i). . . .
If we take any one of these terras, say rf^ = a,*''a/'' . . .,
then, by § 9, Cor.,
= .^(«,v) </.«.')• • •;
since «!, a.j, . . . arc primes.
It follows that the left-hand side of (1) is the same as
{l+<^(a,) + ./.(a,») + . . .+^(0)}
x{l+^(a,) + <^(a,') + . . .+*(«,••)}
(2).
But <^ (a,--) = a,"- A - - ^ = rt/ - a,'--'.
Hence
= l + Oi-l+ai'-ai + . . . + a,*' - o,"i"',
and so on.
It appear-s therefore, that (2) is eriual to a'>a^ . . ., that
is, equal to N.
Example. A'=315 = 3>.5.7.
The divisors arc 1, 3, 6, 7, 9, 15, 21, 35, 45, 63, 105, 315, and we bavo
*(l) + ^(3) + 0(6)+ . . . +^(31.5)
= 1 + 2 + 4+ i; + 6 + 8 + 12 + 24 + 24 + 86 + 48+ 144 = 816.
* Here and in what follows 1 is incladed among the divisors, and, foroon-
vonicncc, ^ (1) is token to stand fur 1. Strictly speaking, ^ (1) has no meaning
at all.
§§ 10, 11 PRIME DIVISORS OF ml 543
PROPERTIES OF m!
§ 11.] The following theorem enables ns to prove some
important properties of ml : —
TAe highest power of tlte prime p which divides m\ exactly is
where -^(— ), ^i^)' • • • denote the integral parts of tn/p,
m/p", . . . ; and the series is continued until the greatest power of
p is readied which does not exceed m.
To prove this, we remark that the numbers in the series
1, 2, . . .,»»
which are divisible by p are evidently
\p, 2p, Zp, . . ., Ip,
where kp is the greatest multiple of pl^m. In other words,
k = I{mlp). Hence I{mjp) is the number of the factors in to!
which are di\nsible by p.
If to this we add the number of those that are divisible by
p^, namely I (m/p"), and again the number of those that are
divisible by p^, namely J{7n/p^), and so on, the sum will be the
power in which j? occurs in ml.
Hence, since p is a. prime, the highest power of p that will
divide m\ exactly is
/©*<l)^^0)
It is convenient for practical purposes to remark that
'(?)=^K-)A}-
For, if
then
m/p'--' = i + k/p"-' {k Kp'-') (1),
m/p^ = ilp + k/p^ (2),
=j + l/p+k/p^l<p) (3).
544
exampt.es
CU. XXXV
Now
l/p + klp'Xp- i)/p + (j^-'-i)/if,
<1.
Hence, by (3),
But, since i/p =j + l/p,
^•='(^'{'C-)A}- '•'<^>-
We may thercfdre proceed as follows : — Divide m l/y p: tale
the iutiijral quotient and divide again by 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 10001 exactly.
In dividing sncccssively by 7 the integral quotients ara 142, 20, 2 ; the
anm of these is 164. Hence 7'" is the power of 7 required.
Example 2. To decompose 251 into its prime factors.
Write down all the primes less than 2S ; write under each the snoocssive
quotients ; and then add. We thus obtain
1 2
8
6
7
11
13
17
19
23
12
8
6
8
3
1
1
1
1
6
3
1
8
1
33
10
6
S
3
1
1
1
1
Hence 251 = 2". 8". 5». 7'. 11'. 13. 17. 19. 23.
Example 3. Express 89!/25t in its simplest form as a produot of prime
factors.
BcBult, 2'» . 3' . 5' . 7* . 11 . 13' . 17 . 19 . 29 . 31 . 87.
Example 4. Find the highest power of 5 that will divide 27 . 28 . 29 ... 100
exactly.
Besnlt, 6".
Example 6. If m be expressed in the scale of p, in the form
">=Po + PiP+f5;''+- • •+P«P*-
the highest power of p that will divide ml exactly is the
■'-fo-Pi-fi- ■
5 th.
p-1
ExAinplo 6. If m=2* + 2' + 21'+ . . . (* terms), where a<p<.y<,
the gnateot power of 2 that will divide ntl is the (m - kjih.
^11,12 PROPERTIES OF mlf/lglhl . . . 54:5
§ 12.] 1/ /+ g + h-*- . . .>«?, thm m\!/\g\h\ . . . is an
integer*.
To prove this, it will be sufficient to show that, if any prime
factor, p say, appear in f\g\h\ . . ., it will appear in at least
as high a power in m\ In other words (§ 11), we have to
show that
+ . . . . (1).
Now, if d be any integer whatever, we have
//d=/'+/"/d (/">d-l),
g/d=g' + g"/d (g":i>d-l),
h/d = h' + h"/d (/i'>(^-l),
• . , ',
and we obtain by addition
/+ q + h+ . . . J., , ,, f"+q" + h"+...
Hence, if /" + ?" + ^" + . . .<d,
^ff+n + h+ . . .\ J., , ,,
/(^ -^ ^ )=/ +9+h +. . .,
If, on the other hand, /" + g" + h" + . . . >d,\ then
/r— ^-^ j>/ +g +k + . . .,
• This theorem might, of course, be inferred from the fact that
m!//I</!AI . . . represents the number of permutations of 7n things / of
which are alike, g alike, h alike, &c.
+ If 71 be the number of the letters f,g,h,.. ., the utmost value of
f"^g" + h" + . . .isn((i-l). Hence the utmost difference between the two
Bidesof (•2)is/{n((J-l)/<i}.
C. II. 35
546 EXERCISES XXXVI CIl. XXXV
It appears, therefore, that, even '\{m^/+g + h + . . .,
A fortiori is this so if m >/+ g + h+ . . . .
If uow we give d the successive values p, y', . . . , and com-
bine by a«lditiou the inequiJities tlius obUined from (3), the
truth of (1) is at once established.
Cor. 1. If /+g + h+ . . . >m, and none of the uumbcrs
/,g,h,... is equal to m, the inteytr m\lf\g':h\ . . . is divisible
by m if m be a prime.
Cor. 2. The product of r success! iv integers is exactly
divisible by r\.
The proofs of these, so far as they require proof, we leave to
tlie reader. Cor. 2 has already been established by a totally
dillcrent kind of reasoning in § 3, Example G.
Exercises XXXVL
(1.) What is the least multiplier that will convert 915 into a complete
sqnare ?
(2.) Find the number of the divisors of 2100, and their sum.
(U.) I'ind the iiilco-ral solutions of
Ty = 100i + 10y + l (a);
xy = 12« {?);
y'=10ai (7)-
(1.) Ko number of the form i* + 4 eioept 5 is prime.
(.3.) No number of the form 2**+' + 1 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 divieors (itself excluded).
(7.) To find a number H of the form Vabc ... (a, 6, e 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 2^7. 2*r,
where p, q, r arc primes. (Two numbers arc amicable when each is the sum
of the divisors of the other, the number itself not being reckoned aa a
divisor.)
(fl.) To find a cube the sum of whose divisors shall be a square.
(One of Fcrmafs challenges to Wallis and the EngUsh mathematicians.
I'ar. Op. Math., pp. 1««, 190.)
(10.) If S be any integer, n the number of its divisors, and P the product
of them all, Uiiu .V" = i".
^^ 12, 13 EXKRCISES XXXVI 547
(11.) The sum and the sum of the squares of all the numbers less than
N and prime to it aru m (a - 1) (6 - 1) (c - 1) . . . and i^P (1 - 1/a) (1 - 1/6)
. . . + jJV(l-a) (l-l) . . . respectively. (Wolstenholme.)
(12.) If p, q,T, . . . be prime to each other, and d (N) denote the sum of
tlie divisors of N, show that
d{pqr...) = d(p)d{q)d{r)... .
(13.) If N=abc, where a, 6, c are prime to each other, then the product of
all the numbers less than N and prime to N is
(abc - 1)1 n {(a - l)l/(6c - 1)! a«>-'l(c-i|}.
(Gonv. and Caius Coll., 1882.)
(14.) The number of integers less than (r^ + 1)" which are divisible by r
but not by r- is (r- 1) {(r^ + l)"- l}/r=.
(15.) Prove that
(IG.) 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). .. 2a X 6(6 + 1). . . 26/(a + 6)! is an integer.
(19.) The product of any r consecutive terms of the series i- 1, x'- 1,
a;'- 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 iuteger in {71 - S (n)]l{p - 1)"', 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 (m - n) «t S (m) - S (n) for any
radix. Hence show that (;i + 1) (h + 2) . . . (n + m) is divisible by m!.
(Camb. Math. Jour. (1839), vol. i., p. 226.)
(21.) If/(n) denote the sum of the uneven, and ii'(n) the sum of the even,
divisors of n, and 1, 3, 6, 10, . . . be the "triangular numbers," then
•/{n)+/(n-l)+/(n-3)+/(n-6)+. . .
= F(n) + F{n-l) + F(n-3)+F{n-6)+. . .,
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 tlie series of numbers
k, k + a, k + 'ia, . . ., k + {m-l)a
with respect to m, where m is prime to a, are a permutation oftho
numbers of tlie series
0,1,2,. . .,(«»-!).
35—2
548 pnorEiiTiEs ok an inteoral a.v. en. xxxv
All the remainders must be difTerent ; for, if any two
different numbers of the series had the same remainders, then
wo should have
k + ra = ixm + p, and k + sa = ii'm + p,
whence
(r - s) o = (fi - fx') m, and (r - s) a/m = /a - /i'.
Now tliis is impossible, since a is prime to m, and r and s arc
each < m, and therefore r-s<m. Hence, since there are only
m possible remainders, namely, 0, 1, 2, . . ., (m- 1), the proposi-
tion follows.
Cor. 1. J/ the remainders of k and a with respect to m Im>
k and a', the remainders will occur as follows: —
X', k' + a, k' + 2o', .... A' + ra',
until we reach a number that equals or surpasses m ; this we must
diminish by m, and then proceed to add a at each step as before.
Thus, if 4 = 11, 0 = 25, m = 7, the series is
11, 36, Gl, 86, 111, 136, ICl.
Wc have k' = i and o'=4, hence the remainders are
4,4 + 4-7 = 1, 5, 5 + 4-7=2, tc;
in fuct,
4, 1, 5, 2, 6. 3, 0.
Cor. 2. If the progression of numlters be continued beyond
m terms, the remainders will repeat in the same order as before ;
and in this jyeriodic series the number of remainders intervening
between two that differ by unify is always the same.
Cor. 3. T/wre 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-1).
That is, the number of terms in the scries in que.<ition which are
prime to m is 4> (m). See § 8.
Tiiis follows from the fact that two numbers which are
congruent with resiwct to m are either both prime or Ixith non-
prime to »H.
Cor. 4. If out cf the scries of uumbi-rs
0, 1.2,. . .,(w-l)
§§ 13, 14 PROPERTIES OF AN INTEGRAL A. P. 549
we select those that are less than m and prime to it, say
(the number n being 4> {tn)), then the numbers
k + r,a, k + r^a, . . ., k + r„a,
where k = 0 or a multiple of m, and a prime to m as before, are
all prime to m : and their remainders with respect to m are a
permutation of
r,, n, . . ., r„.
For, as we have seen already, all the 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 = nm+p-k
would have a factor in common witli m, which is impossible,
since r, and a are botli prime to m.
Hence the remainders must be the numbers ri, /-j, . . ., r„
in some order or otlier.
§ 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 ivill recur in a shorter cycle of m.
Consider any two terms of the series out of the first »»', 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'/ni 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 (»»' + r)th,
then, since
{k + (»»' + r)a}-{k + ra} = m'a,
= gm'a,
= ma',
it follows that the {m' + r)th term has the same remainder with
respect to ?» as the »-th.
In other words, the first m' remainders are all diflferent, 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 /f = ll, n = 2.5, m = 15, we have the series
U, 30, 61, SG, 111, 136, 161, 186, 211, 236, 261. . . . ;
550 FERMAT'S theorem rn. XXXV
anti liprc g = iii a'=5; m' = 3; a" = 2; fc'=ll; pa" = 10. HcDco the re-
mainders are
11, 6. I, 11, G, 1, 11, C, 1, 11, 6
Cor. I/tie G.C.M., g,nfa and m divide k exactly, and, in
jHirticulnr, if k = ii, the remainders of the series
k, k + a, k + ia, ...
are the numhers
0,j, lg,2g.3g (m'-l)g
continually rejHuted in a certain order.
For, iu this case, since k = gx, we have {k + ra)lm = (« + ra')/m',
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 i: = 10, a = 26, m=:15 ; then the Bcries of numbcrg is
10, 35, 60, 85, 110, 135, ICO, 185
'Wehave(;=6; a' = o; m' = 3; ic = 2; and the remainders ore
10, 5, 0, 10, 5, 0, 10, 5, ... ;
that is (o say,
2x6, 1x6, 0x6, ... .
§ 15.] From § 13 we can at once deduce Fermat's Th&irkm*,
which is one of tiie corner-stones of the theory of numbers.
If m be a prime number, and a be prime to m, a"'' - 1 i$
divi-sible by m.
If a be prime to m, then we have
la=/i,f7i + p,,
• • > . •
where the numbers p,, p, p«_, are the numbers
1, 2 (ffi-1) written in a certain order.
* Great historical interest attaclies to this theorem. It was, witu lerenl
othiT striking result.-) in the theory of numbers, published without demonstra-
tion among Fi'rmat's notes to an edition of Bachct dc Mcziriac's DiophantuM
(IGTO). For many years no demonstration was found. Finally, Euler (Con.
iTurrit. Acad. Prtrop., viii., 1741, and Commrnt. Sov. AcniL Petrnp., m., 1761)
gave two proofs. Another, due to Lagrange (.Vour. Mem. dt V Ae. de Berlin,
1771). is rrprodnced in § 18. The proof h-ivcn alioTo is akiii to Kuler'sMoood
aud to that given by Uauss, Duq. Arilh., § I'J.
^5^14-17 EULER's GENERALISATION OF FERMAT's THEOREM 551
Ileuce
1.2. . . {m - 1) a"'-^ = (mm + pi) {jum + p.,) . . . {f;„~im + p,„-i),
= Mm + P1P2 ■ ■ ■ pm-i,
= Mm + 1.2 . . . {m- 1).
We therefore have
1.2. . . (?»-l) (a"'-^-l) = il/OT.
Now, m being a prime number, all the factoi-s of 1.2 . . . (?»- 1)
are prime to it. Hence m must divide a"'~'- 1.
It is very easy, by the method of differences, explained in § 5,
to estabHsh the following theorem : —
If m he a frime, aP'-a is exactly divmble by m*.
Since »"'-« = «(«"•"' -1), if « be prime to /«, 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 <i> (m) denotes, as usual, the number of integers which
are less than m and prime to it.
For, if r,, 7-2, . . ., »"„ be the integers less than m and prime
to it, we have, by the corollary in question,
r„a = fi-nm + p,,,
where the numbers p,, P2, • • -i Pn iii"e simply r,, Vn, . . ., »•„
written in a certain order.
We have tlierefore, just as in last paragraph,
n''2 . • • »'n (a" - 1) = -3/'»,
whence, since rj, ra, . . ., r„ arc 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 estabhshed
by means of the principles of § 13.
* For another proof of this theorem see § 18 below.
5o2 WII50N'S TIIKOHEM — gauss's PKDOF CH. XXXV
Any two integfi-s whose i)ro(iiift lias tlie rciuaindfr + 1 with
respect to a. given luodulus m may be called, after Euler, Allied
NumbiTS.
Consider all the integers,
1,2,3 (w-1),
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 exhaustivi-ly arranged in allied pairs.
For, take any one of thctn, say r, then, since r is prime to m,
tlie remainders of
r.l, r.-l r(m-l)
are the numbers
1, -2 (w-1)
in some order. Hence, snnie 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 = »i-l;
for, if we have
i^ = ixm + 1,
then r"- l=/am; 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-l, we can arrange the series
2, 3 (w-2)
in allied pairs. Now every product of two allies is of the form
/im + 1 ; hence the pmduct 2.3 . . . (w - 2) is of the form
(ti,tn + 1) {fLttn + 1) . . ., which reduces to the form Mm + 1.
Ill ine
2.3. . . (»n-2) = J/m + l;
and, multii>lying by w - 1 , we get
I.-'. a. . .(m-2){m-l) = Mm{m-\) + m-l.
Whence
l.i'.;> . . . (m- I)* I -^Aw*.
§§ 17, 18 THEOREM OF LAGRANGE 553
Tliat is, {/' m be a prime, {m - 1)! + 1 is divisible by m, which is
Wilson's Tueorem*.
It should be observed that, if m be not a prime, (»»— 1)!+ 1
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 (ot-1)!, 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 + \) (.r + 2). . , (x-vp-l)
= x^-'' + AiXP-^ + . . .+Ap-.x + Ap.„
and p be prime, then A-^, A-^, . . ., Ap.^ are all divisible by p.
We have
{x+p){a^-'^ + Ayjf-- + . . .+Ap-iX-^A,,-^)
= {x + \){{x + l)''-' + A,{x+\f-^ + . . .+A,-.{x+\)-\-Ap-,\.
Hence
px'''^+pAiOfl~^+pAiX''~' + . , .+pAp-^_x + pAp^-i
= {(x+Vf~xP\ + A^{{x+\f-'-x''-'\ + A.A{x+\)''-'-.i^--\+. . .
Therefore
pAi=pCi + p-iCxAi,
pAi =pCi + p-iCj^i + p-iC«Ai + p-iCiAi.
Plence, since p-iC,, p-iC^, p-sCi, ... are not divisible hy p
Up be prime, we get, by successive steps, the proof that Ai, A.,
A-i, . . . are all divisible by^.
* This theorem was first pnblished by Waring in his Medilationef Atge-
bmiae (1770). He there attributes it to Sii- Jolin Wilson, but gives no proof.
The first demonstration was given by Lagrange {Nouv. ilim. de I'Ac. de
Berlin, 1771) ; this is reproduced in § 18. A second proof was given by Euler
in his Opuscula Anabjtica (1783;, vol. i., p. 329, depending on the theory of
the residues of powers.
The proof above is that given by Gauss (DUq. Arith., §§ 77, 78), who
generalises the theorem as follows: — "The product of all the numbers less
than m and prime to it is congruent with -1, if m=:p'^ or =2^**, 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 quadratic residues.
654 EXERCISES XXXVII CH. XXXV
V*)T. 1. Put X = 1, and we get
2.3. . . p=\ +(Ai + Ai + . . . + Af.,) + Ap-t.
Therefore A^-i + 1, tkat is, (p - 1)! + 1, w divisible Inj p.
Cor. 2. Multiplying by x and transposing, we get
a*-x = x{x+\). . .(x^-p-\)
-(1 +^p-,)a:-(^,x»'-' + yl,a'-' + . . .■k-Af-.r).
But x(x+\) . . . {x+p- I), being the product of p con-
secutive integers, must be divisible by p. Also, if ^ be prime,
1 + Ap-t is divisible by p.
Therefore, x' — x is divisible by p i/p be prime. From which
Fcnnat's Theorem follows at once if x be prime to p.
Exercises XXXVIL
(1.) x" - X is divisible by 2730.
(2.) If X be a prime greater than 13, x" - 1 is divisible by 21R40.
(.3.) If tbc nth power of every number eud with the same di);it as the
nnmber itself, then n = ip + l.
Give a rule for detemiiuing by inspection the cnbe 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 z prime to n, then either x'"""''- I or x'*~'i^+I
is divisible by k,
(6.) If n be prime, and x prime ton, then either i" I"-'!/'- 1 or x*i*"'l'' + l
ii dinsible by n'.
(7.) If m and n be primes, then
m*"' + n"*"' = 1 (mod. mn).
(8.) If o, ^, 7, . . .be primes, and >?=o^7 . . ., then
S (A'/a)"-' = 1 (mod. afiy . . .).
(9.) If n be an odd primo, show that
(a + 1)« - (o" + 1) = 0 (mod. 2n).
nencc phow that, if n be an odd prime and p an integer, then any int<»gpr
cxpri'sspd in the scale of 2n will end in the same digit as its {pn-p + ljlh
power. Uednce Format's Theorem. (Math. Trip., 1H79.)
(10.) If n be prime and >x, then
x«-» + x"-'+. . . ^x + lsO (mod. n).
(11.) If n bo an odd prime, then
l+2(n + l) + a'(n + l)'+. . .+2"-«(»' + l)"~'s0('nod. n).
(12.) If n be odd, l" + 2" + , . .+(n-l)*BO (mod. n).
§§ 18, 19 NOTATION FOR NUMBER OF PARTITIONS 555
(13.) If n be prime, and p<n,
(p - 1)1 (n-p)\ - ( - 1)P=0 (mod. n),
and, in particular,
[{ i (n- l)}'P + ( - 1) '"-"" = 0 (mod. n).
(Waring.)
(14.) Find in what cases one of the two {i(n-l)}l±l is divisible by n.
What detennines which of them is so ?
(15.) If p be prime, and n not divisible by p - 1, tlicn
11 + 2"+. . . + (p-l)» = 0(mod.i)).
(IG.) If ^ be any odd prime, TO any number >1 which is not divisible by
p - 1, then
/n-lN-"'
1="' + 2='" +...+ ( ^—^ j = 0 (mod. p).
(17.) If neither a nor h be divisible by a prime of the form in-\, then
a4Ji-; _ jjn-s ^f m not be exactly divisible by a prime of that form.
Hence show that a*"-- + h*'''- is not divisible by any integer (prime or not)
of the form in - 1.
Also that a? + lP is not divisible by any integer of the form 4«-l which
does not divide both a and 6. Also, that any divisor of the sum of two
integral squares, which does not divide each of them, is of the form in + 1.
(Euler.)
(18.) Show, by means of (17), that no square integer can have the form
4m.n - m - n", where m, n, a are positive integers. (Euler. )
PARTITION OF NDSIBERS.
EuUr's Tlmory 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 dasses of partitions, namely, (I.) those in
which the parts may be equal or unequal ; (II.) those in which
the parts are all unequal. Wlien 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 {\ 1) and P?« ( | | ) are
used to denote partitions of the classes (I.) and (II.) 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
656 EXPANSIONS AND PARTITIONS Cn. XXXV
the pjirUi It is alwaj's iinpliod, unless the coutrar}' is stated,
tliat the le:tst part admissible is 1 ; so that :^ m means any
integer of the series 1, 2 m. An asterisk is use<l to mean
any integer of the soriCvS 1, 2, ...,», or that no restriction is
to be put on the number of tlie part« other than what arises
from the nature of the partition otherwise.
Thus F{n\p\ q) means the number of partitions of » into p
parts the greatest of which is q\ /*(«!/>,> 17) the number of
partitions of n into p parts no one of which exceeds q ;
P {n\* \1p-q) the number of partitions of n into any number of
parte no one of which is to exceeil q\ P«(n j:^/? | » ) the
number of partitions of « into p or any less number of unequal
parts unrestrict<>d in nia^itude ; Pu{n\p\r>AK\) tiie number of
l>artiti()ns of n into p unequal parte each of which is an odd
integer; P(h|*|1, 2, 2', 2', . . .) the number of partitions of
« into any number of parte, 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 tlie tlicory of invariants. It is also closely con-
nected with the theory of series, as will be seen from Euler'a
enumeration of certain species of partitions, which we shall
now briefly explain.
§20.] If we develop the product (1 +rx) (1 +sj:') , . .
(1 + zj^), it is obvious that we get the term s'j" in as many
difl'erent ways as we can produce n by adding togetlier p of the
integers 1, 2, . . ., q, each to be taken only once. Hence wo
have the following equation : —
(l+sj:)(l + j:jr»). . . (1 +rj^) = 1 + 2/'«(n |/>|>g) £»a- (1).
Again, if to the product on the left of (1) we adjoin the
fjictor l + c + c' + s'+. . . adoo (that is, 1/(1 - s) ), we shall
evidently get z'jf as often as we can produce w bj' adding
together any number not exceeding /> of the integers 1, 2, . . ., q.
Therefore
(l+c.r)(l+rjJ). . .(1 +s^)/(i-j)
= 1 +S/'« (»!>;,, ^j);^ (2).
§§ 19-21 KXPANSIONS AND PARTITIONS 557
In like manner, we have
{I + a-) (I + or) . . . (l+af) = l+-%Pu(n\*\:!f>q)x'' (3);
(1 + sx) (1 + zx-) . . . ad 00 = 1 + 2P« {n \p\*) s^x"- (4) ;
(1 + a;) (1 + is=) . . . ad 00 = 1 + 2Pm (m j * i * ) x" (5).
Also, as will be easily seen, we have
ll{\-zx){l-za?). . .{l-z3fl) = l + ^P{n\ij\1Sr'q)z''x'' (6);
ll{l-z){l-zx). . .{l-Z3fl)=l + %P{n\-ii>p\-^q)s?'x'' (7);
1/(1 - a:) (1-^)- ■ .(l-a;'')=l+2P(M|*|>>!7)a;" (8);
1/(1 - zx) (1 - zar) . . . ad 00 = 1 + 2P (» I J9 1 * ) «"«" (9) ;
\l{\-z){\-zx){l-zx-) . . .ad CO = 1 + SP \n |>;> !*)«?«'' (10) ;
l/(l-a;)(l-a.'=). . .a.A^ = l + tP(n\*\*)x" (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 recur rence-foi-mula 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\ * \^q).
By (8), we have
1/(1 -a;) (1- a-). . .{l-x'')= 1 + ^P{n\^\:!f>q) x\
Hence, multiplying on both sides by 1 — xf, and replacing
ll(l—x){l-ar). . .{l-x"'^) by its equivalent, we derive
1 + 2P(m|*1>- (/-!)«"
= 1 + 2 1P(«| * \>q)-P{n-q\* \>q)}x- (12),
where we understand P(0, | * |:j>2') to be 1.
Hence, if M^^gf,
P{n\* \>q) = P{n\ * \1f>q-l)+P{u-q\ * l^^q) (13) ;
and, if n<q,
r{n\*\1f>q) = P{n\*\::^q-l) (14).
By means of (13) and (14) wo can readily calculate a table of
double entry for P{n[ * l^*^), as follows : —
558
EULEU'S TAULK KOR P(n | * | :j- <;) CH. XXXV
13 8 4 6 6t 7c 8 9 10 11 12 13 14 15 16 17 18 19 20
1 1 1 1 1 I
6 6 6 6 7 7
8 "lo] 12 U 16 19 21
11 15 18 [23 27 3^1 89
13 18 23 301j7 47 57
I J 20 26 85 44 J>8 71
15 21 28 38 49 Ca 82
. 22 29 40 52 70 89 [ 116
. 30 41 54 73 94 123
42 55 75 97 128 164 [212 267 340
. 56 76 99 131 169 2191278 358
F 1
Take a rectangle of squared paper HA C; 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 .si)aces blank, the last entry in the
column being understood to be repeated indefinitely.
Ne.xt, write the values of /-"(l | » |:^1), /'(2|*|>1), . , .,
that is, 1, 1, . . ., in the row headed 1.
To fill the other rows, construct a piece of paper of the form
abed. 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 tho
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 ab in the same line as
the blank. Again, in the ninth line 157 = 146-»-ll; and
so on.
By sliding abed backwards and forwards, so that be always
lies on .'1 1>, we can fill in the t.-iMe rapidly with little chance of
crmr. Wo shall speak of the table thus couslructtd iu> Euler's
§§ 21-23 ENUMERATIONS REDUCIBLE TO EULER's TABLE 559
Table. It will be found in a considerably extended form in his
Introductio, Lib. I., cliap. xvi.
A variety of problems in the enumeration of partitions can
be solved by means of Euler's Table, as we shall now show.
§ 22.] To find by means of Euler's Table the number of
partitions of n into p parts of tmrestr-icted magnitude.
Let ns first consider P (n \p\i). By (9) above, we have
l + :S.F{n\p\*) .r"5P = 1/(1 - zw) (1 - zx''} . . . ad co ,
= 1 + •S,x''sf'l{l -x){l-ar). . .{l-x^),
by Exercises xxi. (18).
Hence
lPiii\p\*) x" = ^x^lil - x) {1-x'). . . (1 - a*),
= 2P(«|*l^;')-»"^", by (8).
ThGrGfoPG
P(n\p\*) = F{n-p\*\1f>2y) (15).
Again,
l + ^Pu{n\p\*)x''z'' = {l+zx){l+Zir). . . ad co,
= 1 + 2a;JP('+i) aV(l -x){l-^r). . . (1 - x^),
by chap, xxx., § 2, Example 2.
Hence
%Pn(n\p\*)x'' = xVf'">-'y(l-x)(l-x-). . .{1-xp),
= 2P (» j * \:lf>p) .i-''+5!'(^+i), ^y (g)_
Therefore
Pu(n\p\*) = P{n~ip(p+l)\*\::f>p) (16).
Example 1. P(20 | 5 | »)=/'(15 | . |>5) = 84.
Example 2. P«(20 | 5 | .) = P(5 | » 1 1>5)=7.
§ 23.] If we take any partition of 7i into p parts in which
the largest part is q, and remove that part, we shall leave a parti-
tion of n-q into ])-! 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 j» infinite, as a particular case, we have
P{>i\*\q) = P{n-q\*\:!t>q) (18).
It will be observed that (18) makes the solution of a certain
class of problems depend on Euler's Table.
5G0 TIIKOREMS OF CONJUOACY CM. XXXV
By comparing (l.">) and (18), we have the theorem
P{n\*\q)^F{n\q\i,).
which, however, is only a particular case of a theorem regarding
conjugacy, to be proved presently.
§ 24.] T/ieorems regarding conjugacy.
(I.) P(»|>;,!>7) = P(7i|>y|>;,) (19).
(II.) P{n-p\q-\\>p) = P(n-q\p-\\-:(>q) (20).
(III.) P{n\p\q) = P{n\q\p) (21).
To prove (I.) we observe that, by (7), we have
1 + •S.P{n i>/>l>g)3'^= 1/(1 -z) {\-zx). . . (1 -ly),
_, , v^(l-^^')(l-^^*)---(l-^*'')
'^ (l-x){l-a^)...{l-j,f) '
Hence
vp(„,:c,.. -)^, (l-^-')(l-a^-'). . .(l-^->)
il (n\:f>p\:f>q)X (l-arXl-x"). . . (l-;r') ■
(l-a:)(l-f')...(l-j^*'')
(l-a:)(l-x')...(l-j^)(l-j-)(l-aJ)...(l_;H')-
Since the fiiuction last written is symmetrical as regards p
and q, it must also be the equivalent of 2P(n l>gj>/>);r".
Hence Theorem (I.).
Theorem (II.) follows from (6) in the same way.
Since, by (17), we have
P{»\p\7) = P{n-q\p-l\>q).
P(n\q\p)^P{n-p\q-l\1i>p);
therefore, by (II.),
P(n\p\q) = P{r,\q\p).
which establishes Theorem (III.).
The following particular ca-scs are obtaine<l by making p or
q infinite : —
P{H\>p\»)^P(n\*\1i^p) (22);
P{n\p\») = P(n\*\p) (23).
§§ 23-26 FURTHER REDUCTIONS TO EULKr'S TABLE 561
§ 25.] The followiug theorems enable us to solve a number
of additional problems by means of Euler's Table :—
P(n\p\::f>q) = F(,i-p\*\^p)-'S,P{n-^-p\*\:!f>p)
+ 2P(«-/i/,-^l*|>>p)
-^P{n-fH-p\*\>p)
(24).
Here the summations are with respect to /*,, //n, . . . ; and
/x, is any one of the numbers q, q+1, . . ., q+2) - 1, Ma the sum
of any two of them, /j.^ the sum of any three, and so on. The
series of sums is to be continued so long as n — fj^-p'^O. If
P{n\p\^q) come out 0 or negative, this indicates that the
partition in question is impossible.
P{n\:^p\>q) = P{n\*\:!fp)-^P(n-v,\*\:!pp)
+ 2P (« - 1*2 I * I ^1})
-^P(n-y,\*\^p)
. . . . (25).
Here v^, v„, . . . have the same meanings with regard to
q + 1, q + 2, . . ., q+p as formerly /jj, ft-j, . . . with regard to
q, q+1, . . ., q+p-1.
P{n\*\*)
= P(«-11*|>1) + P(h-2|*|>2)+. , .+P(Ol *!>.») (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 pai-titions 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 tliis 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. (1832),
c. II. 36
5C2 DRAPH OF A PARTITION CH. XXXV
theory \vc cannot witliiu our present limits enter ; but we desire,
before leaving the subject, t*> call the attention of our readers to
the graphic method of dealing with partitions, which is one of
the chief weajMins of the new theory.
By the graph qf a partition is meant a series of row., if
asterL^ks, each row containing as many asterisks as there are
»uuit8 in a corrcisponding part of the partition. Thus
• • •
• • • • •
is the graph of the partition 3 + 5 + 3 of tlie number 11.
For many purposes it is convenient to arrange the graph so
that the i>arts come in order of magnitude, and all the initial
afiterisks are in one column. Thus the above may be written —
The graph is then said to be reoiilar.
The direct contemplation of the graph at onc«
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 erery partition of n into p parts the greaUst cj
which is q, there is a conjugate partition into q parts the
greatest of u-hich is p. Hence
s • •
• • •
P{n\p\q) = P{n\q\p),
an old result.
Again, to evert/ i>artition of n into p jxirfs no one qf trhick
efceetis q, there will be a conjugate partition into q or fewer parts
the greatest of which is p. Hence
Pin\p\>g)-P{n\:i>q\p) (27).
a new result ; and so on*.
* According to Sylreatcr {Le.), thU wa; of proving the theoreroa of
§3 2G, 27 EXTENSION AND CONTRACTION OF GRAPHS 563
§27.] The following proof, given by Franklin*, of Euler's
famous theorem that
{l-x){l-x'){l-aP). . .ad<x>=^(- )Pa:«*'*P' (28)t,
is an excellent illustration of the peculiar power of the graphic
method.
The coefficient of x" in the expansion in question is obviously
Fu{n\evm\*)-Fu(n\odLd\*) (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
* *
* * *
* * *
* * * *
*****
******
# * * * Vf »
*******
* * # * * * #
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.
■je. -jt -jt jt jt
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 —
„, We then have a regular graph R 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 Rcndus (1880).
+ Euler originally discovered this theorem by induction from particular
caries, and was for Ions unable to prove it. For other demonstrations. Bee
Letjendie, Xklorie des Nombrcs, t. u., 5 13, and Sylvester (i.e.).
3G— 2
564 rUANKMN's PROOF OF EULER's EXPANSION CH. XXXV
to tlio number in tlio lust slope or exceeds it only by one,
a.s, for example, in
« • • • • • •
• ••• •••••
Contraction or extension in the first of tiicse would produce
an irregular graph ; contraction in the second would produce an
irregular graph ; aud extension would produce a graph which
corresponds to a partition having two {mrts etpial. These two
cases may be spoken of as uiiomjutjute ; they can only ari^e when
tlie p parts of the partition are
p, p^\, p + 2 -J;'-!,
and the number
n=/> + (p+l) + . . . +(2;^-l) = |(.V-;));
or when the p parts arc
p+l, p + 2, p + 3, . . ., 2p,
and
n = {p + l) + (p + 2)+. . . +2p = i{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 respectively, we see
that, unless n bo a number of the form j(3/>'+/>),
I'u (n I even | # ) = Pm (n 1 odd | » ).
When n hiis one or other of the forms i (iip'tp), there will
be one unconjugate partition which will be even or odd
according asp is even or odd ; all the others will occur in pairs
wliich are conjugate in Franklin's transformation. Hence
J>i, (J (3/>'±|>) I even \»)-Fu (hA^p'lp) | o<ld | .) = ( - 1)" (30).
Euler's Theorem follows at once.
ExF.ncisEs XXXVIII.
(1 .) Show how to evaluate Pu(n\ >p\,) by moaim of Knlcr'n Table,
Eraloate
(2.) /'(13,fij^3). (3.) /'(13|>6|).3I.
(4.) i'(10|.l.). (0.) /'^a)|'Jl».8).
§ 27 EXERCISES XXXVIII 565
Establish the following : —
(6.) P«(n|.|.)=P(7i-49(? + l)|»| >q), where i 7 (7 + 1) Ji^st > n.
(7.) Pu{n\p\,) = P{n-ip{p-l)\p\*).
(8.) P (n I p I . ) = Pu (n + Jp (p - 1) Ip I . ).
(9.) PH(n|p|l>g)=P{n-ip(p-l)|p| t"Z-p + l).
(10.) Is the theorem P(n-p | g -1 1 .) = P(n-?|p- 1 1 ♦) universally
trne?
(11.) Show how to form a table for the values of P {« | . | 2, 3, . . . , q).
(See Ptoc. Edinb. Math. Soc., 188.3-^.)
(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(»|.|l, 2, 2=, 23, . . .) = 1.
(14.) Pu(nlp|l,3, . . .,23-1) = P(«-P=+p1p11,3,. . ., 2(7-1).
(15.) P(nlp|2, 4, . . .,2g) = P(n-p|pll, 3 2^ - 1).
(16.) P(K|.|odd) = P«(nl.|.).
(17.) P{n\>p\2,i 2?) = P(n|t-?|2, 4 2p).
(18.) P(n+p|p|l,3 2gr + l) = P(n + 3|3|l, 3 2p + l).
(19.) Pu(n + p'\p\l, 3, . . .,23 + l)=P«(» + 3»l3|l, 3, . . ., 2p + l).
(20.) P{7i + 2plp|2, 4 23 + 2) = P(n + 23|7|2, 4, . . .,2p + 2).
(21.) Show that P {>i\p\ .) = P (n -l\p-l\ *) + P {n-p\p\ *); and
hence construct a table for P (/i \p [ .). (See Whitworth, Choice and Chance,
chap. lu.)
CHAPTER XXXVr.
Probability, or the Theory of Averages.
§ 1.] An elementary account of the Theory of Prohahility,
or, as we should prefer to call it, the Theory of Averages, haa
usually found a place in English text-books on algebra. This
custom is justified by several considerations. The theory in
question afl'ords an e.xcellent illustration of the application of the
theory of permutations and combinations which is the funda-
uieutal part of the algebra of discrete quantity ; it forms in its
elementary parts an excellent lugical exercise in the accurate use
of terms and in the nice discriniinatiou 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 modem life.
The student is proliahly aware that there are certain occur-
rences, or classes of events, of such a nature that, although we
cannot with the smallest degree 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 ca.se may be) as the number N increa-ses.
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 m.ay assert
with con^idorable 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 woidd be necessary to state the limits both of
habitat and epoch within which the men are to be taken ; and,
even with a million ca.ses, we must not o.xpect to be able to assign
§ 1 DEFINITION OF rROBAlULITY 567
the proportion of those who survive for 5 years with absohite
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 18th century. The "event,"
as it is technically called, here in question is the living for 5
yejirs 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 hving of all the men
of 20 in England during the 18th century we may, following
Mr Venn*, call the sei-ies 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 : —
If on taking any very large number N out of a series of cases
in u-kich an event A is in question, A happens on pN occasions,
tlie prohability of tlm event A is said to be p.
In the framing of this definition we have, a.s 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, in N out of a series of cases, Ac."
But, from the point of view of the ideal or mathematical theory, nothing
would thus be gained.
fifiS REMARKS ON THE DEFIXITION CH. XXXVI
How fiir tlie possible (luctuations of p, tlic extent of the seriea,
and tlio luagnitiide of iV will affect the bearing of any con-
clusion on pnictice must be judged by the light of circurastancos.
It is obvious, for instance, that it would be unwise to ajipl)- to
the 1 1th century the probability of the duration of human life
deduced from statistics taken in the 18th. This leads us also to
remark tiiat the application of the theory of probability is not
merely historical, as the definition might suggest Into most of
the important practical applicJitions 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
18tli centurj-. The warranty for this extension of the series of
caises by induction must be sought in experience, and cannot in
most cases he obtained a priori.
There are, however, some cases where the circumstances aro
80 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
liazard played with cards, dice, &c. If, for example, we assert
regarding the tossing of a halfpenny that out of a large numljcr
of trials heads will come up nearly as often as tails — in other
words, that the probability of heads is J, 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 ma<le that this counterbalancing of causes takes place. Th.it
this is really the right point of view will be best brought homo
to us if we reflect that undoubt^-dly a machine could bo con-
structed which would infallibly toss a halfj^nny so as alwavs
to land it heid-up on a thickly sjinded floor, provided the coin
were always i)laced the same way into the machine; also, that the
coin might have two heads or two tails ; and .so on.
In cflscs where the statement of probability rests on grounds
80 simple ns this, the difficulty regarding the extension of tiio
series by induction is less prominent The ideal theory in such
• In tlio proper, logical scnae of the word.
§§1,2 COROLLARIES ON THE DEFINITIOX 5G9
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
i.ot 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*.
Tliis 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 ia -96 x .5000 = 4800.
Cor. 2. If the probability of an event be p, the probability of
its failing is \-p.
For out of a large number N of cases the event will happen
on pN occasions ; hence it wiU fail to happen on N-pN
* It is essential that pN also be a very large number. See Simmons,
Pruc. L. J/. S., XXVI., p. 307 (16^5).
570 COROLLARIES ON THi: nEFINITION Cn. XXXVI
= (1 -p) N occasions. Hence, by the definition, tlie probability
of the failing of the event is 1 -jo.
Cor. 3. J/th4i un'mTse of an event be made up ofn alt'
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
vaijsbep,,p, p,,, then pi + p, + . . .+/>,= !.
For on every one of N occasions the event will happen ; and
it will happen in the first way on piN occasions, in the second on
p^N occasions, and so on. Hence N=piN + p,N+. . .+/>,A'';
that is, 1 =^1 +P3+ . . . +p„.
Cor. 4. Ifati event is certain to happen, its pritlntbility is I ;
if it is certain not to happen, its probability is 0.
For in tlie former case the event happens on 1 . N cases out
of N ca.ses ; iu the latter on 0 . A' cases out of N.
The probability of every event is thus a positive number
lying between 0 and 1.
I'or. 5. Jf an event must happen in one out of n ua;/s all
equally probable, or if one out of n events must happen and alt are
e</ually probable, then the probability of eitch way of happening in
the first case, or of each event happeninij in the second, is \jn.
This follows at once from Cor. 3 by making />i =/>, = . . . =/>,.
As a particular case, it follows that, if an event be equally
likely to happen or to fail, its probability is A.
Definition. — The ratio of the probability qf the happening of
an eient to the probability of its failing to happen is called the
odds in favour if 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 favmir
Mop-.l-p; the odd.s ajjainst I -p.p. Also, if the odds in
favour be m : n, the probability of the event is m/{m + n). If the
probability of the event be J, 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. ^/'the unliyrse of an event can Im atialysed into m + n
cases each of which in the long run will occur equally often*, and
• This u oiiuU; exproMed by saying that all the o*iM an Aqnally likely.
§§ 2, 3 DIRECT CALCULATION OF PROBATilLITIES 571
if in m of these cases the event will happen and in the remaining
71 fail to happen, the probalilifi/ of the event is m/{7n + 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 in 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 outpust
duty, what is the probability that all of them desert ?
The 3 men may be chosen from among the 20 in ^^C^ ways, all of wliieh
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
„ , „ 5.4.3 /20.19.18 ,,„ .
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 ^ together along with the («-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(ii- 2)!/(n- l)! = 2/(;i- 1).
When Ji = 3, this gives chance =1, as it ought to do. The odds against
the event are in general ;i - 3 to 2 ; the odds will therefore be even when the
number of people is 5.
Example 3. If a be a prime intogcr, and 7i = a', and if any integer 1 1> 74
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, xixv., § 8, the number of integers
less than a'"' and prime to it is n''"'(l - 1/a). Also the number of integers
> n is a*". Hence the required chance is u'-" (1 - l/a)/u''=a~' (1 - 1/a) = 1/a"
-1/aM-'.
Example 4. Find the probability that two men A and ZJ of )k 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, »th, jn + ^th, n+pth
years. Let these numbers be !,„, i„, 2,„+p, J„+p. The probabilities that A
and B live to be m+p and n-^p years of age respectively are Im+plhi^ '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
672 niUECT CALCULATION OF PROBABIUTIES CH. XXXVI
from tlio fcpond. lu thin way we couM Boloct altogctlicr MS pairs of men
wlio mny Iw rUvo or deail aftur p yenra linvc clapwH. The niimbor out of
the M m-'U living after p years is .V/„+r/'„, by § 2, Cor. 1. Similarly the
number livint; out of the N men is A'/,+p//,. Out of Ihenc we could form
HSI„+pl^pll„ln Pft'". This last number will be the number of pairs
of Burvivors out of the MS pairs with which wo started. Hence the
probability required is J„+pJ,+p/'m'»=('m+i./'J C+r/'-): '" o'ber words, it
is the product o( the probabilities that the two men singly each surrive 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 m respectively in „C,, ,C, ,C, ways
resjioctively. Hence we may consider the universe of the event as consisting
of ,C, + ,C'j + . . . + ,C„ = (1 + 1 )" - 1 = 2* - 1 equally likely cases. The number
of these in which the drawing is even is ,C, + ,C,+ . . .=^{(1 + 1)"
+ (1- l)"-2} = J(2»-2) = 2"-'-l. The number of ways in which an odd
drawing can be made is .C', + ,C,+ . . . =4 {(l + l)«- (1 - 1)"1 = J2" = 2«->.
Hence the chance that the drawing bo even h (2»-' - l)/(2"- 1), thot it bo
odd 2"-'/(2*-I). 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 block 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 powns to the chance that either or both of the pawns can take the rook.
Let us look at the board from the side of while ; and calculate in the first
place the whole number of possible arrangements of tlii' pieces. No block
pawn can lie on ouy 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 j,Cj = 8 X 50 X 55 arrangements. Again, we may have the rook on any one
of the 60 squares and the two pawns on any two of the remaining 55 squares;
in all, 60x65x54 arrangements. The universe may therefore be supposed
to contain 02 x 50 x 55 equally likely cases.
Instead of calculating the chance that the rook can take either or boUi of
tho pawns, it is simpler, as often happens, to calculate the chance of the
oomplcmentary event, namely, that the rook can take neither of the pawns.
If the rook lie on one of tho front row of squares, neither of tho pawns can
lie on the corresponding column, that is, the pawns may occupy any two ont
of 49 squares ; this gives 8 x 4U x 48 arrangements. If the rook lies in any
one of the remaining 50 iu|uares, neither of tho pawns must lie in tho row or
oolnnin belonging to that square; hence there are for the two pawns 42 x 41
positions. Wo thus have 50x42x41 arrangements. Altogether wo havo
§§ 3, 4 DIRECT CALCULATION OF PROBABILITIES 573
8x49x48 + 56x42x41 = 56x49x42 arrangements in which the rook can
take neitlier 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 = 076/1705.
Consider now the attack on the rook. If he is on a side sciuare, he can
only be attacked by either of the two pawns from one square. For the side
sqiiares we have therefoi'e 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 50 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, xxni., § 15. Let there be r bags,
the first of which contains Oi, bi, c,, . . ., k^ counters, marked
with the numbers oj, /3j, yi, . . ., k^; the second, a.,, i/o, Cj, . . . ^2,
marked <u, p^, y„, . . ., xj; and so on. If a counter be drawn
from each bag, what is the chance that the sum of the numbers
drawn is « ?
By chap, xxm., § 15, the number of ways in which the sum
of the drawings can amount to n is the coefhcient, A „ say, of x"
in the distribution of the product
(«,.r"' + 6iir*' + . . . + kiX'')
y. {a^.v'^ + b^a^' + . . .+k.x")
X (arX'^ + brX^' + . . . + kr3f').
Again, the whole number of drawings possible is the sum of
all the coefficients ; that is to say,
(a, + 61 + . . . + ^1)
X (t/j + ^2 + . . . + il-j)
y.{ar + hr+ . . . + kr) = D, say.
Hence the required chance is A„/D.
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 nwnber of ways in which 12 can be thrown with three dice is the
coefficient of i'- in
574 DIKECT CALCULATION OF PROBABILITIES Oil. XXXVI
that is to lay, of x* in
(I+i + i»+i»+*« + r')*.
Now the coefficients in (1 +x+ . . . +i*)' up to the term in i* are (»ee
chap. IT.,§ 15) 1 + 2 + 3 + 4 + 5 + 6 + 6 + 4 + 3 + 2. Hence the coeiTicient of x»
in tlic cabe of the multinomial is 6 + C + 5 + 4 + 3 + 2 = 26.* The required
probability is therefore 1/2.5.
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
ino!it probable throw with the two dice, and what the chance of that throw?
The numbers of wn.vs in wliich the sums 2, 3, 4, 5, 6 can be made arc the
cooClicicnts of x», i», x*, x», «* in the expansioDof (3z + 2x'+z*)(x + 2x'+8x').
Naw this product is equal to
3x« + Sx* + 1 Jx« + 8x» + 3x«.
Tiic sum that will occur oftenest in the long run is therefore 4. The
whole number of dillciont wajs in which the different throws may turn out
is (3 + 2 + 1) (l + 2 + ;jJ = 36. Hence the probability of the sum 4 is 14/36
= 7/18.
Example 3. An nm contains m counters marked with tlie 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 d fferent drawings is w'.
The number of those which give the sum n is the coefficient of x" in
(x + x»+. . . + 1"")', that is to say, of x*"' in (l + x+. . . + x"'-')'. Now
1 + X + . . . + x"'-' = (l-x'")/(l-x). We have therefore to find the coefficient
of X*-' in
(l-i'")'-(l-x)-'={l-,C,x"« + rC5X»»-rCji*» + . . .}
* V*i'* 1.2 "^^ i.a.8 '^^- • •]•
The coefficient in question is
_ r(r+l) ._^ .(n-l) r(r + l). . ■(n-m-l)r
■-'" (n-r)! " (n-r-m)lll
r(r + l). . .(n-2m-l)r(r-l)
(n-r-2m)!21 -• • • •
The required probability is A^^jm''.
Example 4. If m odd and n even integers (n<tm-l) be written down at
random, show that the chance that no two odd integers are adjacent ia
nl (n + l)l/(m + II)! (n - m+ 1)1.
In order to tind in how many different ways we can write down the
intei^t rs 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 omsider the
m - 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 mij^ht also have found the coefficient of x* by expanding
(1 x")*(l-x)"', as in Example 4 below.
t Thi> is gcnernlly called Di'moivre's Problem. For an interesting account
o( ltd hiitory sec Tudlmntcr, llitt. yrob., pp. C'J, 85.
§§ 4, 5 ADDITION RULE 575
74 eveu intoKcrs into the spaces so that tlioro ehaU 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 s" in
(l+j + j;= + . . . ad 00 )2(x + i3 + . . . ad co )'"-';
that is, of s»-"'+i in (l + x + x2 + . . .)^{l + x + x- + . . .)'"-»;
that is, of a;»-^+' in (1 - x)-l"'+').
This coefficient is
(m + l)(Hi + 2). ■ ■ (» + !)_ (« + !)!
(n-m+l)I ~ml(n-m+l)l'
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
m! 77i! (n + l)I/m! {n-m + iy. = nl (» + l)I/(n - 7n + 1)1.
The whole number of ways in which the 771 + 7t integers can be arranged is
(m + 7i)I, hence the probability required is 7il{tt + l)!/(7t-7a + l)!(m + n)!.
ADDITION AND MULTIPLICATION OF PROBABILITIES.
§ 5.] In many cases we have to consider the probabilities of
a set of events wliich are of such a nature that the happening of
any one of them upon any occasion excUides the happening of
any other upon that particular occasion. A set of events so
related are said to be nnitualhj 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 caU the
Addition Rule, applies : —
If the prohabilities of n mutually exclusive events be pi, p^,
. . ., p„, the c/uince that one out of these n events happens on any
ixirticular occasion on which all of them are in question is pi+pi +
. . .+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 wiU happen on piN^, PiN", . . ., p„N 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
N occasions, therefore, one or other of the n events will happen
GO. piN + p^N + . . . +pnN=(pi+Pi + . . . +Pn)N' occasions.
Hence the probability tliat one out of the n events happens on
any one occasion is 2h + I>i + • • • +2'»-
676 MULTiPMCATinv nm-E en. xxxvi
It should be ohserved that the reasoning would lose all force
if the cvcnt.s were not mutually exclusive, for then it might be
that on the />, ^ ocatsions on which the first event hapjeus one
or more of the others hapfien. We shall give the proper formula
iu this case presently.
As an illustratiou of the application of this rale, let as snppose 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 que»tion
may hnppon, namely, the throw may be 2, 3, 4, 5, 6, 7, or 8 ; and these ways
arc of course mutually exclusive. Now (see § 4, Example 1) the probabilities
of these 7 throws are 1/30, 2/30, 3/36, 4/3G, 5/30, 6/30, 5/36 rei^pvctivcly.
Hence the probability that a throw with two dice does not exceed 8 is
(1 + 2 + 3 + 4 + 5 + 6 + 5)/36 = 2G/3C=13/18.
§ 6.] When a set of events is such that the happening of
any one of them iu no way affects the happening of any other,
we say that the events are mutually independtnt. For such a set
of events we have the followiug Multiplication Rule : —
i/' tlie respective probabilities of n independent events be />,,
Pi< • • •, P«, the probability that they all happen on any occasion
in which all o/thim are in question is pip^ . . . p^.
In proof of this rule we may reason as follows : — Out of
any large number N of cases where all the events are in qtiestion,
the first event will happen on ^i A'' occasions. Out of these />,-V
occasions the second event will also happen on Pt(piN^ =PiPi^
occasions ; so that out of N there are pip,N occasions on
which both the first and second events happen. Coutinuing
in this way, we show that out of N occasions there are
p,p, . . . pnN occasions on which all the n events happen.
The prolability tliat all the n events happen on any occasion
is therefore />,/>, . . . ;>,.
It should be tioticed that the above reasoning would stand
if the events were not independent, provided />, denote the
probability tliat event 2 happen after event 1 has happened, />,
the probability that 3 happen after 1 and 2 have happened, aud
so on. *
It must be observed, however, that the probability calculated
is then that the events happen in the order 1, 2, 3, . . ., «.
Hence the followiug conclusion : —
i;^ 5-7 EXAJrPLES OF ADDITION AND SIULTIPLICATION 577
Cor. Ij the eceiUs I, •!,..., n be inierdepmdent and pi
denote tfie probabiliti/ of l,2h the probabilifi/ that 2 hapjjen after
1 has happened, p, the probabiliti/ that 3 happ)en after 1 and 2
have happened, and so on, tJien the probability that the events
1, 2, . . .,n luippen in the order indicated is p^p^ . . . />„.
As an illustration of the multiplication rule, let U3 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 5/0. 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 probabihty of the event in question
is 7/12.
§ 7.] The following examples will illustrate the application
of the addition aud multiplication of probabilities.
Example 1. One urn. A, contains m balls, pm being white, (l-p)»K black;
another, B, contains n balls, qn white, {l-g)n black. A person selects one of
the two urns at random, and draws a baU ; calculate the chance that it be
white ; and compare with the chance of drawing a white ball when all the
ni + K 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.) ip. Similarly the chance that a white biiU
is diawn from B is i,q. The whole chance of drawing a white ball is there-
fore {p + q)l2.
If all the balls be in one urn, the chance is {pm + qii)l{in+n).
Now (pm + 5;i)/(m + n)> = <(p + g)/2,
according as 2{pm + qn)> = ■<{p + q) (m + n),
according as (m-n) (p-g)> = <0.
Hence the chance of drawing a white ball will be unaltered by mixing if
either the numbers of balls in A aud £ be equal, or the proportion of white
balls in each be the same.
C. II. 37
578 EXAMPLES OF MULTIPLICATION AND ADDITION CIL XXXVI
If Oie nambcr of balls Iw unoqnal, and tlio proportions of whito bo an-
cqiml, then the miring of the balls will incrca«c the chnnco of drawing a
white if the urn which contains most balls hnro also the larger projiortioD of
white; and will dimiuish the chance of drawing a white if the urn which
ooutains mogt balls have the smaller proportion of white.
De Morgan* has used a particular case of this example to point out tho
danger of a (iiUacious use of the addition rule. Let us suppose the two cms
to be OS follows: A (3 wh., 4 bl.) ; h (-1 wh., 3 bl.). We might then with
some plausibility reason thus: — The drawer most select cither il orU. If he
select A, the chance of white is 8/7 ; if ho select B, the chance of white is
4/7. Hence, by the addition rule, the whole chance of while is 3/7 + 4/7 = 1.
In other word^, white is certain to be drawn, wliich is absurd. The mistake
consists in not taking account of the fact that the drawer has a choice of urns
and that tho chance of his selecting A must therefore bo maltiphed into hit
chance of drawing white after be has selected A. The chance should there-
fore be 8/14+4/14=1/2.
The nec<^ssity 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 N of times. In the long run the one urn
will be selected as often as the other. Hence out of H times A will be selected
A/2 times. Out of those A72 times white will be drawn from A (3/7) (A/2)
= A (3/14) times. Similarly, we see that white wUl be drawn from h A'(4/14)
times. Hence, on the whole, out of A trials white will be drawn
(3/14 + 4/14) N times. The chance is therefore 3/14 + 4/14.
Example 2. Four cards are drawn from an ordinary pack of 62 ; what is
the chance that they be all of different suits?
We may treat this as an example of § G, Cor. The chance that tho
lir»t caril drawn be of one of tho 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 ol cards to choose
from, 3.13/51. After two cards of differeut suits are drawn, the chance that
the next is of a different suit is 2.13/50. Finally, the chance that the last
caid is of a different suit from the first three is 13/49. Uy the principle justf
mentiuned tho whole chance is therefore 8.18.2.13.13/51.50.49 = 13*/17.2o.49
= 1/10 roughly.
Example 8. How many times must a man be allowed to toss a penny in
order that tho odds may be 100 to 1 that he gets at least one head?
Let z 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 iudc|>cndeut of the result of every other, the chance ot
"all tails" in x tosses is (1/2)*. The chance of one head at least is therefore
1 - (1/2)*, Ijy the conditions of the question, wo must therefore have
1-(1/2)«=100/101;
• Alt. "Theory of rrobabiLty/'iiicy. J/etru/). lUipublislicd A'ncy. Purti
Uaih. (Ibl7), p. U'J'J.
§ 7 EXAMPLES OJ)' MULTU'LICATION AND ADDITION 579
hence 2== = 101,
x=Iogl01/log2,
= 2-0043/-30in,
= 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 liead 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 may 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 np 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 {1- (1/2)'}"'. By the conditions of the problem we must therefore
have
{l-(l/2)'}i»=l/2;
(1/2)'= 1 - (l/2)iAo = -06097 ;
x= -log -06697/108 2,
= 3-9 very nearly.
Hence he must liave 4 trials to secure an even chance.
Example 5. A man is to gain a shilhng on the following conditions. He
di-aws 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 shilhng? (Laurent, Calcul des Probabilitis (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 - 1/3*, . . . , 1 - 1/h-, . . . Hence the chance of failing in all the trials
is (1 - 1/2=) (I - 1/3-) ... (1 - 1/«-) ... ad X ,
Now
,i.('4.)('4.)-('-^.)
_ {1.3}{2.4} . . . {(n-3)(»-l)}{(n-2)n}{(n-l)(» + l)}
~,^. P.2^..«2 »
- r M?_+i)
n-.2 V nj 2
The chance of failing to gain the shilling is therefore 1/2. Ilence tlie chance
of gaining the shilling is 1/2.
We might have calculated the chance of gaiuing the shUling directly, by
37—2
580 EXAMPLtSOFMULTlPUCATION AND ADDITION CH. XXXVI
observinK thnt it is tlio sum of the clianoes of the following CTcntR : 1°,
gaining in the first trial; 2°, foilinR in let and fc-aining in 2nd; 3°, failing
in Ist and 2iid and gaining in the 3rd; and so on. In this way the chance
proseuU iUilf a* the following infinite series: —
i.H'-^0^---H'-^.){'-r.)-{'-i),v',,----
Tlie f um of this scries to infinity must therefore be 1/2. That this is eo may
be easily verified. The present is one example among many in which the
theory of probability soggcsts interesting algebraical identities.
Ejample 6. A and fl cast altcrnntcly with a pair of ordinary Hiee. A
wins if he throws 6 bcfure I) throws 7, and I> if he throws 7 before A throwi
6. If J bigin, show that his chance of winning : i)'s=30 : 31. (Duyghens,
De Hatiocinii' in Ludo Alta, 1G57.)
Let p and q be the chances of throwing and of failing to throw 6 at a
single cast with two dice ; r and $ the corresponding chances for 7.
A may win in the following ways: 1°, A succeed at Ist throw; 2°, A fail
at 1st, B fail at 2ud, A sDcceed at 3rd ; and so on. His chance is thurcfuie
represented by the following infinite scries: —
ji + 5il) + 9«.;»j) + . . .=p{l + (?<) + (9«)' + . . .},
=j./(l-j.).
B may win in the following ways: — 1°, A fail at Ist, B sncccod at 2nd;
2°, A fail at 1st, U fail at 2nd, A fail at 3rd, B succeed at 4th; and su on.
Ilis chance is therefore
jr + j».ir + g»7«jr+. . . = ?r{l + (}») + {j*)' + . . .},
= 9r/(l-g.).
Ilcnce A' a chance : B's=p : qr.
Now (see § 4, Example l)p=S/36, g = 31/36, r = G/36; hence
A's chance : B's=S/36 : 6 . 31/3C»,
= 30 :31.
For Hnygliens' own solution see Todliunter, Hut. Prob., p. 21.
Example 7. A coin is tossed ni-t-n times (m>n). Prove that Uie chance
of at least m c>insc<:utire head^ apiwaring is (n + 2)/2"+'.
The event in question happins if there apptar — Ist, exactly m ; 2nd,
exactly m -t- 1 ; . . .; (n + l)th, eiactly m + n consecutive hiads.
Now a run of exactly m consrcutive heads may commence with the Ist,
2nJ, 3rd, n-ltb, nth, n + lth throw. Since m>n, there cannot be mure
than one run of m or more consecutive heads, so that the complication duo
to re|>«tition of runs docs not occur in the |)resent problem. The chance*
of the first and last of those cases are each 1/2"'*'', the chances of the other*
Il'tm-M^ Hence tlio chance of a run of exactly m consecative heads is
2/2-+' + (h - 1 )/2"" = (n + 3)/2»'-".
In like manner, we sec that the chance of a run of m-t-l consecutive
heads is (n + 2)/2"*+' ; and so on, up to m + n-2. Also the chances of a mn
of exactly m-f n- 1 and of exactly in-i-n consecutive head* are 1/2— 1^>~' and
lyomf. icaiKtIivcly.
§5 7, 8 PROBABILITY OF COMPOUND EVENTS 581
lloiicu the cliance 2> of a run of at least m heads is given by
_n + 3 n + 2 _5_ ^ 1
P ~ 2m+2 "■" 2'»+3 + • • • + 2">'H> "'' 2'"+»+i '*' 2"»+" ■
The summation of the series on the left-hand side is effected (see
chap. XX., § 13) by multiplying by (1 - 1/2)== 1/4. We thus find
_w + 3 71 + 2 n + 1 4
iP~- 2m+i "*■ ym+S ''' 2"'+^ + . . . + 2m-hi+i
_ 2(« + 3) _ 2(» + 2) _ 2.5 _ _2 ^
2*n+3 2"*''"* • • • ~ 2m-Hl+l 2"*"^"^
n + 3 . 6
' nm+l T • • • T o,«4._4.| + om+-i»4.Q •
gn'+l T . . . T 2m+i>+l ^ 2"''H»+> ^ 2"*+^' 2'''+"''"- '
J ."+3 " + 4 S 2 1
4^^2"»+3 2"'+-* 2*'*'*''*'*'^ 2"''^'*"- 2"*"^'*"*"-*
_« + 2
-2m+3'
Hence i) = (n + 2)/2'"+i.
GENERAL THEOREMS REGARDING THE PROBABILITY OF
COMPOUND EVENTS.
§ 8.] The probaliilit!/ that an event, whose probability is p,
hap2)en on exactly r out of n occasiotis in which it is in question is
uPrp^q^''', where g= 1 -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 multiphcation rule
ppqpqq . . . where there are rp's and n — r q's, that is, p''q''~''.
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 « things r of which are alike and n—r alike,
that is to say, «!/»•! (w — »-)! =„Cy. There are therefore „Cr
mutually exclusive ways in which the event with which we are
concerned may happen ; and the probability of each of these is
p^'q'"''. Hence, by the addition rule, the probability in question
is .CrpY-''-
It will be observed that the probabilities that the event
happen exactly «, 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 PRonAniuTY of coMrotrjm events ch. xxxvi
or 1, <'r 2, . . ., or fi times, the sum of all these prohahilities
ought to be unity. Tliia is so ; for, since p + q=l, (p + q)''= I.
It will be seen without further demonstration that the pro-
]>osition just establislied is merely a particular case of the
following general theorem : —
If there be m eirnts A, B, G, . . . one but not more qf which
muit happen on every ocrasioti, and if their probuhilities he p, q, r,
. . . re^ectivilij, the probability that on n occa,*i')ns A happen
exactly o times, B exactly /3 times, C exactly y times, . . . is
n\p'q'^n. . ./al^ly!. . .,
where a + ^ + y+. . .=n.
It should be obf^erved that the expression just written is
the general term in the expansion of the multinomial
(/> + 7+r+. . .)"•
Exiinipio 1. The facea of a cnbical die are marked 1, 2, 2, 4, 4, 6;
required the probability that in 8 throirs 1, 2, 4 turn op exactly 3, 2, 3 tinics
resiwotively.
By the general theorem just stated the prolmbility is
81 /ly/iy/iy 7.5.2
81 21 31 \6 J V3/ VS/ " »' '
~Qi 'PProiJniately.
Exainplc 2. Out of n occasions in which an event of probability p is in
question, on what number of occasions is it most likely to happen?
We have here to determine r so that „CtP^9''~' """y ^ * maximum.
Now «C,y<j"-'/.C^,p'-'g— ^• = (i.-r+ l)p/rj.
Hence the probability will increase as r increases, so long as
(n-r-\-\)p>rq,
that is, (n + 1) p > r (p + ?),
that is r<(n+l)p
If (n + l)p bo an intcRer, =» say, then the event will be equally likely to
Imppi n on • - 1 or on < occa'iions, and more likely to happen < - 1 or < times
than any other number of times.
If (n-t- l)p bo not an integer, and t be the greatest integer in (n-i- l)p, then
the event is most likely to happen on • occasions*.
• When n is very large, (n + l)p differs inappi-eciably from np. Henoe
out of a very large number n of occasions an event is most likely to hap|>en
on pn occasions. This, of course, is simply the fundamental principle of g 2,
Cor. 1, arrircd at by a circuitoas route starting from itself in the first
instanoe.
§§ 8, 9 pascal's problem 583
As a numerical instance, suppose an ordinary die is thrown 20 times,
what is the niimber of aces most likely to appear?
Here »i==20; p = l/6; (n + l)p = 3i.
The most likely number o£ aces is therefore 3.
§ 9.] The probabilittj that an event happen on at least r otit
of n occasions where it is in question is
nCrpY-'' + nCr+lP'^Y'"-' + ■ • ■+ nCu-^f'-'q + p" . . . (1).
For an eveut 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 n 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 he 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-'^.-.C^^rPY-"-
Hence the probability that the event happen at least r times in
n trials is
p^ + rC^'-q + ,+,ap'-q'+. . . +,.,C„_,j3V"'"
=^^••{1 + rC,q + r+,C..q^ + . . . + n-iC,..,.^''-'-} (2).
As the two expressions (1) and (2) are outwardly very different, it may be
well to show that they are reaUy identical. To do this, we have to prove that
oat OKNtHAI, lUKML I.K KoR COMPOUND EVENT CIl. XXXV I
The cxprcKnioii ]a»t written is, up to tlio (n - r)tli power of j, identical with
(l-,)"---!! + 5/(1 -9)}» = (l-7)"-^/(l -?)• = (! -8)-'.
Now, as may bo readily verified,
(1-9) '•=1+,C,7 + ^,C,?»+ . . . +,_,P.-r9"-'+ • • • .
The rciiuircd identity is therefore establislud.
Example. A and li play a game which muBt be either lost or won; the
probability that A piins any game is p, that li gains it l-p = q; what is the
chance tliat A gains m games bofure B gains n? (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 chanco that A gains m games at least out
of m + n- 1, that is, by (1) above,
P'^-' + m-H.-iC, ?■»+-'?+ . . . +„,+,-,C„p»«'-' (1').
We might adopt tlie second way of looking at the question given above,
and tUUB arrive at the expression
P"*{l + mC,g + „^,C,«'+ . . . +„+,-,C..,<j->} (2').
for the required chance.
§ 10.] The re.sults just aiTivcd at may be consideralily
generalised. Let us consider n independent events .^i, At,
. . ., An, whoi?e respective probabilities are p,, ji^, . . .,/),.
In tlie first place, in contrast to §§ 8, 9, let us calculate the
chance that one at Ifaxt of the n events happen.
Tlie complementary event is that none of the « events happen.
The probability of this is (1 —p^ (1 -/>,,) ... (1 -/'■). Hence the
probability that one at least happen is
1-(1 -/>.)(! -;'.)• • . (!-/'»)
= ~pi - ^PiPi + "S-ptPiPi - . . . ( - )"-V'i/'i • • • r« (!)•
Next let us find the probabiliti/ that one and no more qf the n
events happen.
The probability that any particular event, any A,, and none
of the others happen is p, (1 -/>,) (1 -p,) ... (1 -/>,). Hence
the reijuired probability is
5/>. (1 -p.) (1 -p,) ... (1 -Pn)
= 5;>,-jC,5;»,/>j + ,Cj2/>,;>,;3,-. . . (-)"-',(7,-,;>,/>j. . .j», (2).
* riiraouB in the history of mathematics. It was lirst solved for the
particular case p = </ by Pascal (l(i.'i4). The more general result (1*) above
vi.i- i-ivi'n by Jnhn IVrnmilli (1710). The other formula {2") wcms to be dua
U< .MMiiliiiurt (171 1). See Toilhuiitcr, Ih/t. I'rob., p. 98.
§ 10 GENERALISATION OF PASCAL'S PROBLEM 585
For the products two and two arise from - 277, (/>., +2h+ ■ • •
+Pn), and eacli pair will come in once for every letter in it. Again,
the products three and three arise from 2pi {P2P3 +PiPi + • • • ) >
lience 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 tlie probability
that r and no more of the n events happen is
^PiPi ' • i'r (1 -Pr+i) (1 -Pr+2) ... (1 -p„)
= '^PlPi- . . Pr- r+lC{S.lhPl • . .jCr+l
+ T-i^Ci^PlPi ■ . ■ Pr+1
(-Yr+sC.'S.p.p. . . .pr+,
(-)"'\Cn^rPlP-2- ■ -Pn (3).
We can now calculate tJ/e probabi/iti/ that r at least out of tlm
n events happen.
To do so we have merely to sum all the values of (3) obtained
by giving r the values 7; r+1, r+2,. . ., n successive!}'.
In this summation the coefficient of %PiP2 ■ ■ . Pr+i is
\~y {r+s(^s ~ r+sCj-l + r+sCj_2 — . . .(-)'" r+sCj + ( — 1)*}.
Now the expression within the brackets is the coefficient of
af in {l+xy+'x{l+x)-\ that is to say, in {l+xy+'-\ This
coefficient is r+s-iO,. Hence the coefficient of 'S.p,p.2 . . . pr+, is
( ~ )'r+«-lL's-
The probability that r at least out of the n events happen is
therefore
"^PlPi- ■ -Pr-rCi^Pip.. . .Pr+l
+ r+lColjhp. . . .pr+2
{-yr+,-lC,1pip.i. . .prv,
{-y-\-,C^-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
/
686 THIRD SOLUTION OF PASCAL'S PROBLKM CH. XXXVI
probabilities are all equal, the formulae (3) and (4) above ought,
when p,=p,= . . . =/>■ eacli = p, to reduce to ■Crjj'g""'" and
the expression (I) or (2) of § 9 respectively.
If the reader observe that, when pi =/>,= . . . =Pn=p,
^PiPt . . . Pr = mPrp', &C. , he wiU have no difticulty in showing
that (3) is actually identical with jCrP^<f-^ iu the particular
case in question.
The particular result derived from (4) is more interesting.
We find, for the probabilitj- that an event of probability p will
happen r times at least out of n occasions, the expression
(-)'-V.C.-,;>- (5).
Here we have yet another exjiression 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.
Examplp. The probabilities of three independent events arc p, 7, r;
required the probability of happening —
1st. or one of the events bat not more;
2nd. Of two but not more;
8rd. Of one at least ;
4th. Of two at least ;
6th. Of one at most;
Cth. Of two at most.
The results are as follows : —
Ist. p+j + r-2(p9+pr+jr) + S;>7r;
2nd. I>g + pr + ?r-3p^;
8rd. p + 9 + r-(pg+pr + gr)+j)5r;
4th. pj+;)r + gr-2p5r ;
6th. l-(|'?+pr + gr) + 2p}r;
6lh. 1 -pjr.
The first four are particular oases 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 iu the solution of some of the most difficult questions in
the subject The spirit of the method may be e.xjilained thus.
§§10, n RECtTRRKNCE METHOD 587
Suppose, for simplicity, tliat 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 Mj, for a number of successive values of x ; say the
relation
/(%:+3, Mx+l, «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 Uj, 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 /3 respectively, and the probability that the game be drawn
is 7. To start with, A has vi 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 Uj and v^ denote the chances that A and B win in the end when each
has X counter."!. If we put m + ;t=/), the respective chances at any stage of
the game are u^ and ip^^..
Consider A's. chance when he has x + 1 counters. The next round he
may, 1st, win ; 2nd, lose ; 3rd, draw the game. The chances of his
ultimately winning on these hypotheses are an^j^^ ; ^Uj. ; yWx+i respectively.
Hence, by the addition rule,
If we notice that a + ;8 + 7 = l (for the game must be cither won, lost, or
drawn), we deduce from the equation just written
a";c+2-(<' + ^)«x+i + |3"x = 0 (li-
lt is obvious that ■Uj.=A\^, where A and X are constants, will bo a
solution of (1), provided
oX2-(a + /3)\ + /3 = 0 (2),
that is, provided X=l or X = (3/a. Hence u^^A and «i=B(i3/a)* are both
solutions of (1) ; and it is further obvious that u^=A + B (§\af 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 u, and «■,, then all other
* First proposed by Hnyghens in a particular case ; and solved by
James Bernoulli. See Todhunter, Hist. Prob., p. 01.
/
588 PROni-EM nEOARDINa DURATION OF PF.AT CM. XXXVI
values cnnlJ be ctiltiilatoil liy llic rocurroiicc furmnla (1) ilwlf. The nolutioD
v, = A+Ji{fila)*, containing two undptcrmiiicd constants A and B, is
therefore gafficiently general for our purpose*. Vie may in fact dctcmiino
A and B most simply by remarking that when A has none of the counters hia
chance is 0, and when he has all the counters his chance is 1. We tlioa have
A+D=0, A + B{fila)i'=l,
whence A = aPI(a''-p''), B = -o''/(o''-/J'').
We therefore have
«,=aP-'(a'-/S')/(aP-/3'');
and, in like manner,
r,=;J''-»(a'-/S')/(a''-/S").
The chances at the bopinning of the game are given by
„„ = a-(a">-/S")/(aP-^P),
r,=/3"(a»-/S")/(ai'-^i').
Cor. 1. Ifa = p, then (see chap. «v., § 12)
«m = '"/P. »,="/?.
The oddt on A in thit particular case are m to n.
It might be supposed th:it 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 countert cannot reduce the odds in A'$
favour to la* than a-p to /S.
For, if we give A 1 counter, and B n counters, the odds in A'b favour are
Q"(o-)3)//}(a"-/J»):l; that is, (o-/3)//S(l-(^/a)»| : 1. Now, if a>p. this
can be diminished by increasing n; but, since L (/J/a)" = 0, it cannot become
less than (o - /3)/^ : 1, that is, o - /3 : ;3. """
Hence we see that, if A be twice as skilful as B(o = 2;S), we cannot by
any disparity of counters (so long as wo 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 name.i a card; and that card and all above it are removed and shown
to him. He then names another ; and so on, nntil none ore left. Acquired
the chance that during the operation he names the top card once at leastt.
Let u, be the chance of succeeding when there are n cards; so that u,.,
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 ntli card, the
chance of each of these events being l/ii. Now his chances of ultimately
euccoeding in the n cases just mentioned are 1, u,.,, u,_,, . . . , n,, 0
respectively. Denco
u,=l/fi + u,_^n + u,.,/n+ . . . +ujii + u,/n.
We have therefore
nu,= l-mi + i/,+ . . . +u,-, (1).
• Thia piece of reasoning may be replaced by the considerations of
chap. XXXI., § H.
+ Urprint of frolilfiiu from the Kd. Timet, vol. iLii., p. G'J.
EVALUATION OF PROBABILITIES INVOLVING FAGTOKIALS 589
From (1) we deduce
(n - 1) »„_, = 1 + 7-, + «, + ... + ,/„_, (2).
From (1) aud (2)
. ''"»-{n-l)"„-, = "„-5.
that 13,
"("»-«„-!)= -("„-! -"„-:) (3),
Ileuco
(" - 1) ("n-l - ",.-2) = - (''„-2 - «„-j),
('' - 2) {«„-2 - «„-3) = - (h„_3 - !t„_ J,
3(K3-ig=-(H„-Uj).
Hence, multiplying together the last n-2 equations, we dedace
4«!("n-''„-i) = (-l)''-=("-.-«i).
Since «j = l, «2 = 5, this gives
«„-«„-i = (-l)''-V"l W.
Hence, again,
''„-j- "„-2= (-!)"-=/(»- 1)!.
«,-«, = (-1)72!,
Ui-0 = 1.
From the last n equations we derive, by addition,
«„ = l-l/2! + l/3!-. , . + (-l)''-i/i!! (5).
Introducing the sub-factorial notation of chap, xxni., § 18, we may write
the result obtained in (5) iu the form «„=1 — «;/«!.
From Whitworth's Table* we see that the chance when n=8 is "632119.
When n=oo the chance is 1 -l/c= •632121 ; so that the chance does not
diminish greatly after the number of cards reaches 8.
EVALUATION OF PROBABILITIES WUERE FACTORIALS OF
LARGE NUMBERS ARE INVOLVED.
§ 12.] In many cases, as has been seen, tlie calculation of
probabilities depends on the evaluation of factorial functions.
When the numbers involved are large, this evaluation, if pursued
directlj', 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 a:'
* Choice and Chance, chap. iv.
t lu some cases the process of chap, xixv., ^ 11, Examples 2 and 3 is
asefui.
590 KXEKCISES XXXIX CU. XXXVI
wlieu X is large. In it^ luoderu foriu this tbeurum luay bo
sUted thus —
(see chap, x.vx., § 17).
From tliis it ajjpeare that, if x be a large number, x\ may
be replaced by J(2'rx)x'e''', the error thereby committed being
of tlie order 1/V2ut\i of tlie value of x\.
As an example of the ase of Stirling's Theorem, let us consider the follow-
ing problem : — A pack of 4n cards consista of 4 suits, each cousistinK of n
cards. Tiie pack is »liuQli'd and dealt out to four players ; required the
choiice that the whole of n particular i'uit falls to one particular player. The
chauce in question is easily found to be given by
p = (3n)ln!/(4ii)I.
Bcnce, by Stirling's Theorem, we have
V(2ir3n) (3n)»'t-*'^(2»n) n"«-«
^■^ ^/(2ir 4..) (4;i)«" <-*»"' "'
the error being comparable with l/llnth of p. Hence, approximately,
y = ,y{3irn/2)(27/25C)".
Example. Let In = 52, n = 13, then
/) = V(3 X 3-1416 X 13/2) (27/256)".
This can be readily evaluated by means of a table of logarithms. Wo
find
p = 156/10'«.
The event in question is therefore not one that would oocur often in the
experience of one individual.
Exercises XXXIX.
(I.) A startH at half-piist one to walk up Princes Street; what is the
probability tliat he muct li, who may have started to walk down any time
between one and two o'clock ? Given that it takes A 13 minutes to walk op,
and B 10 niinutt'S 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 chauce of drawing at least one white — 1st, when the first
ball is replaced; 2nd, when it is uot replaced?
(4.) If n persons meet by chance, what is the probabiUty that they all
have the same birthday, nuppoHing every fourth year to be a leop year?
(S.) If II queen and a knight bo placed at random on a cheu-board, what
is the chauce 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 J.
(7.) There are three bags, the first of which contains 1, 2, 1 count«rs,
marked 1, 2, 3 respectively ; the second 1, 4, 6, 4, 1, marked 1, 2, 3, 4, 5 ro-
Bpectively; 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, . . ., (2n-l), 2 are drawn; find the
probability that tlie 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 particukar 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 ;i dice, wheu n > G ?
(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 t\vice. 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, 8,
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' Diary 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 tiU 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.) Wliat 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.
502 EXKROISKS XXXIX CIl. XXXVI
(22.) A and B throw fiir a 8lako; A's die in marked 10, 13, Ifi, 20, 21, 25,
and /I's 5, 10, ir>, 20, 25, 30, The liipliest throw is to win and equal throws
to Ro fur nothiDR; show tliat A'a chance of winninK is 17/^13.
(23.) A pack of 2fi cardH, n red, n black, is divided at random into 2 eqn I
parts and a card is drawn from each ; find the chance that the 2 drawn are
of the same colour, and comparo with the chance of drawing 2 of the same
colour from the undivided pack.
(24.) Am cards, numbered in 4 sots of m, are distributed into m stacks of
4 each, face np ; find the cliance that in no stack is a higher one of any set
above one with a losver 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)(in-5)/(m-l)(m-2).
(20.) If m things be given to a men and h women, prove that the chance
that the number received by tlie group of men is odd is
{4(6 + a)"'-4(6-a)'»l/(6 + a)"'.
(Math. Trip., 1881.)
(27.) A and 7? e.ich take 12 counters and play with 3 dice on this condi-
tion, tliat if 11 is thrown A gives a counter to U, and if 14 is thrown B gives
a counter to A ; and he wins the game who first obtains all the counters.
8how that A's chance is to li'a as
244,140,625 : 282,42!l,536,481.
(Iluyghens. See Todh., Ilitt. Proh., p. 23.)
(2fi.) A and B play with 2 dice; if 7 is thrown A wins, if 10 B wins,
if any otiicr number the game is drawn. Show that A'a chance of winning
is to B's as 13 : 11. (Huygbcns. See Todli., Uitt. Prob., p. 23.)
(2'.t.) In a g.ame 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
beatK 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 A'a winning
8 games running from C is 103/332.
(30.) There are m posts in a straight line at equal distances of a >a i
apart. A man starts from any one and walks to any other; prove that the
average distance which ho will travel alter doing this at random a great
many times is ^(in-fl) yards.
(31.) The chance of throwing/ named faces in n casts with a j>-t lituxd
die ia
j(p + l)._Zp- + /</-J)(p_i)» j j(p + l)«.
(Dcmoivre, Doctrine oj Chanctt.)
(33.) 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
ii
1-1/21 + 1/3!- . . . (-I)*-'/"!-
(This is known as the Trrixe I'fubUm. It wus originally solved by
Moutmurt and Bernoulli.)
§ 13 VALUE OF AN EXPECTATION 593
(33.) A and B play a game in which their respective chances of winning
are o and /S. They start with a given number of counters p divided between
them ; each gives np one to the otlier when he loses ; and they play till one
is ruined. Show that inequality of counters can be made to compensate for
inequality of skill, provided a//3 is less than the positive root of the equation
xP - 2xP-' + 1 = 0. If J) be large, show that, to a second approximation, this
. . « 1 P-1
rootis2-2j=j-25^j.
MATUEMATICAL MEASURE OF THE VALUE OF AN EXPECTATION.
§ 13.] If a mail were asked what he ^yollld 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 Mim 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 notliing ;
and no more can be said.
If, however, the player be allowed to repeat the game a large
numher 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 N of
times, the holder of a single ticket would get £a on pN
c. n. 33
i
594 ADDITION OF EXPECTATIONS CII. XXXVI
occasions. £/» on qN occnsiona, £c on rN occai^inna, . . . Henco
the lioldiT of a single ticket in each of the A' lotteries would get
£{pya + giVb + 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 + qb + rc + . . . .
Hence the price of his ticket is made up of parts corresponding
to the various prizes, namely, pa, qb, re, . . . 'i'hese parta are
called the values of the expectations of the respective prizes ; and
we have the rule that the viilue of the expectation of a sum of
money is that sum multiplied by the chance of yetting it.
The student must, however, remember the understanding
upon which this definition has been based. It would have nu
meaning if the lottery were to be held once for all.
Example. A plnver throws a six-faced die, and is to receive 20t. if be
tbrows ace the Crxt throw ; half that sum if he thrown ace the srcond throw;
quarter that eum if he throws aco the third throw ; and so on. Ittquired 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, 6/C', 6'/C', 5'/G*, . . . Hence the respective
values of the corrcxpondiug parts of his expectation are 20/6, 20.5/C'.2,
20 . 6'/C . 2', 20 . 5'/C' . 2', . . . shillings. The whole value of hia expectation
is therefore
that is, C(. 8i<i.
§ 14.] It is important to notice that the rule which directs
us to add the component parts of an e.vpectation applies whether
the separate contingencies be mutually exclusive or not Thus,
if P\> Pit Pi, ■ • ■ be the whole probabilities of ol/taining the
sejHtrate sums a^, 02, a,, . . ., then the value of the expectation
is }>ia, + p/i, + p/t, + . . .,evcn if the expectant may get more
than one of the sums in question. Observe, however, that /», must
be the whole jirobability of getting o,, tliat is, the probability of
getting the sum a, irrespective of getting or failing to get tho
other sums.
If the expectant may get any number of the sums O], a,,
§§ 13-15 ADDITION OF EXPECTATIONS 595
. . ., a„, we might calculate his expectation by dividing it into
the following mutually exclusive contingencies: — ai, Wj a„;
Oi + Oa, «! + «s, &c. ; Oi + Ui + a,, &c. ; , , .; Oj + f/j + . . . + a„.
Hence the value of his expectation is
2a,p,(l-^,)(l-^3) . . . (l-jo„)
+ ^{ch+ai)piP3{l-ps) • • ■ (i-Pn)
+ 2 («! + Oa + ai)p,p.vp., (1 -pi) ... (1 -p,)
+ («, + «. + . . . + a„)piPiP3 . . .pn-
By the general principle above enunciated the value in
question is also Saj/),. 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 b.
The chance of getting a is p, and of getting b is q. Kcquired the value of
his expectation.
He may get a alone, or 6 alone, or a + i ; and the respective chances are
p0--9)< l(^-p)t Vi- Hence tlie value of his expectation is ap(l-q)
■^hq(l-p)-¥(a + h)pq, which reduces to ap + lq, 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 ; but p and q 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 rec^uired 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 ou good and easily convertible security ; 2ud, 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
696
MORTALITY TAni.K
cn. XXXVI
Tlif Tl" Tnhlt ofth/i TtutUiUe of Aehuirin.
Age.
Number
Decre-
Age.
Kamber
Decre-
Age.
Nun) Iter
iHcro-
Uring.
ment,
UTing.
ment
LIrlng.
roenu
X
i.
</.
X
Im
d.
X
I.
d.
10
100,000
490
40
82,284
848
70
33,124
2371
11
90,510
397
41
81.436
854
71
35,753
2433
12
99,113
329
42
80,582
865
72
83,320
2497
13
98,784
283
43
79,717
837
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
2.5,691
2527
16
97,942
818
46
76,969
990
76
23,164
2464
17
97,624
379
47
75,973
1041
77
20,700
237*
13
97,245
466
43
74,932
10S2
78
18,320
2258
19
96,779
656
49
73,850
1124
79
16,068
2138
20
96,223
609
50
72,726
1160
80
18,980
2015
21
95,614
643
51
71,506
1198
81
11,915
1883
22
94,971
650
1.2
70,373
1235
82
in, 032
1719
23
94,321
638
63
69,138
1286
83
8,313
1545
21
93,6?3
622
54
67,852
1339
84
6,768
1346
25
93,061
617
55
66,513
1399
85
6,422
1138
26
92,444
618
50
65,114
1462
86
4.284
941
27
91,826
634
57
63,0.52
1627
87
3.343
773
23
91,192
654
68
62,125
1692
88
2,570
615
29
90,533
673
59
60,633
16C7
89
1,955
495
30
89,865
694
60
68,806
1747
90
1,460
408
31
89,171
706
61
67,119
1830
91
i,or.2
829
32
S8,465
717
62
05,289
1915
92
723
264
83
87,748
727
63
63,374
2001
93
469
195
34
87,021
740
64
51,373
2076
94
274
139
35
86,2?1
757
65
49,297
2141
95
135
86
36
85,5'J4
779
06
47,156
2196
96
49
40
37
84.745
802
67
44.900
2243
97
9
0
38
83,H43
821
68
42,717
2274
98
0
39
83,122
838
69
40,443
2319
In the first column are entered the age."? 10, 11, 12, . . .
Opposite 10 is enterc<i an arbitrary nuiulier loo.oOO of diildreu
that reach their tenth birtliday; opposite 11 tlie number of these
that reach their eleventh birthday ; opposite 12 the nnmber that
reach their twelfth birthday; and so on. We shall denote these
numbers by /,,,, ^,i, /„, ... In a third column are entered the
differences, or "decrements," of the numbers in the second
column ; these we shall denote by (/,„, <f„, rf,„ ... It is obvious
that dt gives the n\imber out of the 100,000 tliat die between
their xth and a;+ 1th birthdays. It is impo.s,Mible here to discuss
the methods cmi)ioycd in constructing a table of mortidity, or
§§ 15, 16 USES OF MORTALITY TABLE 597
to indicate tlie limits of its use ; we merely remark that in
ajiplying it in any calculation tlie 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 dx die between the ages of
X and x+\, 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 /,„+.„/?„, = 1/2. Hence lm^=l,^2. 1^1^ will in
general lie between two numbers in the table, say Ip and Zp+, . Hence m + n
must Ue between p and p + 1. We can get a closer approximation by tho
rule of proportional parts (see chap. xxL, § 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 N (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 tho („, 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 ;„,- /,„^, die. If we suppose these deaths to bo equally
distributed tlirough the year, as many of the lm~^m+i ^''^^ ''^<^ ""y assigned
amount over lialf a year as wiU live by the same amount under half a year.
Hence the l^ - Z„,+i lives that have failed will contribute \ (?„, - ;„^,) years to
the united lite of the /„, specimen hves. Again, each of the 7„^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 i('m~'nn.i)
+ '»»+! = i Cm +'m+i)- Similarly, the contribution during the second year is
\ {'m+i + 'm+!!) ; ^'"^ ^^ °°- Hence the united life is
i('m+'m+l) + 4('m+l + 'm+2)+ • • ■ = i 'm + 'm+l + 'm+, + • • • (1),
• Demoivre's hypothesis.
598 EXAMPLES CH. XXXVI
t)io horios continuing so long as tlic numbers in the tabic have any significant
valtio.
If wc now divide the united life by the number of original lives, we find
for tliu iuc:in duration
^V=i + (f^, + f„4i+ . ■ •)/'» (2).
Owing to oar assumption rogarding the uniform distribution of deaths over
the intcr^-als between the tabular epochs, this expression 'n 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 11.
The compound event whuse chiince is required may be divided into
mutually exclusive contingenoics as follows: —
l^t. B may die in the first year, and A survive ;
2nd. „ second „ ;
and so on.
The 1st contingency may be again divided into two : —
(o) A and B may both die within the year, B dying Crut ;
(/3) 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 ('a-/„4,)
{'»- '*+i)/'o'6- Since the deaths are equally distributed through the year, if
A and B both die during the year, one is as likely to sun-ive as the other ;
hence the chance of A surviving B on the pnsent hypothesis is ^. The
chance of the contingency (a) is therefore ('«- 'a+ij('»- 'm-i)/-'.'»- The
chance of (/3) is obviously 'a+i ('* - 'ih-i)/'o'6 •
Hence the whole chance of the Ist contingency, being the sam of the
chances of (a) and (jS), is ('a + 'o+iH't-'w-i)/-''.'»-
In like manner, we can show that the chance of the 2nd contingency is
('aH + ',.,)('(H.,-'l^,)/2/a'k.
Hence the whole chance tlmt A survive B is given by
S.,»={(/. + /„+,)('t-'w-i) + (',^n + ',+i)(Wi-'»+j)+- • •]rit,lk (1).
The reader will have no diflioulty in seeing that (1) may bo written in the
following form, which is more convenient for arithmetical computation :
S*6 = i + {^"'^('6»r-) - 'w^,)-'.'w-,l/2U (8).
where » stands for the greatest age in the table for which a significant value
of f, is given.
U we denote by S^, the chance that B survive A, we have, of course
If a = 4, it wUI be found that (2) gives S,^»=l/2 ; as it ought to do.
§ 17.] Let US now consider the following money problem in
life contingenoics :— ]V/,at shmUd an Insurance Offirf ask Jbr
undeitakinij to pay an annuity of £1 to a man of m yiars oj age,
§ 17 ANNUITY PROBLEMS — AVERAGE ACCOUNTING 599
the first payment to be made n + \ 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 tlie use of
the shareholders' capital, for managemeut, and for " margiu " 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 iu different offices, as is well known. We
suppose also that the rate of interest on the invested funds of
the office is £«' 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 „!(«,„ 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 /„ men of m years of age supposed to
be all living at the preseut date.
The office receives at once „|(am^m pounds. On the other
hand, it will be called upon to pay
£'m+n+l> £^m+ll+2) • • •) £'m+n+()
n+ I, » + 2, . . . , 11 + t
years hence respectively. Reducing all these sums to present
value, and balancing outgoings and incomings on account of the
C lives, we have, by chap, xxii., § 3,
Hence
nil'^m— (^' 'm+n+1 + y A»+n+2 + ... +V lm+n+t)/'m)
= «"2'u„+,i;7C (1).
r— 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, tiiat is, the oomlitiona under
which it is to last.
GOO PROBLEMS SOLVABLE BY ANNUITY TABLE CO. XXXVl
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 noniiuee live so long, we must put « = 0. Then, using the
actuarial notation, we have
ua„=i'L»^IU (2).
r— I
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 / = oo . Then, according as the
annuity is or is not deferred, we have
„1<7,„ = r" 5 /„+,+re7^« (3).
r-I
a„=TL^r^/L (4).
r-l
§ 18.] The function a„, which gives the value of an im-
mciliute complete auimity on a life of 7n years, is of fundamental
importance in the calculation of contingencies which depend on
a single Ufe. 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, «|am,
it««.. «|i«« can ^ ^ found from the annuity tables; for wo
have
,i«« = e" Ivn-n am+,/fm (5) ;
{ia^ = am-1^ L+tOm+l/lm (6);
■iia- = («"/-+» fflm-H. - v'*' /.+,+, a«+,+i)/C (7) ;
as the reader may easily verify by means of formula; (I) to (4).
The.se results may also be readily established a priori by
means of the theory of expectation.
§ 19.] Jjct us next find at,m '^« present valiu of an im-
mediate complete annuity oj £l on the joint lires of tteo nominees
of k and m years of age resptctiivly.
The understanding here is tiiat the annuity is to be paid so
§§17-19 SEVERAL NOMINEES — METHOD OF EXPECTATIONS GOl
long as both nominees are living and to cease wlien either of
them dies.
The present values of the expectations of the 1st, 2nd, 3rd,
. . . instalments are
Vlic+ilm+jlklm, «'"4+jL+2/4C 1^%+3lm+s/hlm, &:C., ... .
Hence we have
at,m=(^•4+l4+l + ■»'4+2A„+3 + . . .)/44-.,
= ^'v'lt^L^/U^ (1).
Just as ill § 18, we obviously have
» , "* , m = ''" ^*-Hl , m+n 4+ii 'm+n/ 4 4> >
\tClt,m = (I'll, m ~ ''flSt+J.m+I 4+« Im+tlh^m >
n|l<*t,m = \0 dk+n.m+n 4+n 'm+n
— V fljr+n+i , m+n+I 4+n+l 'm+n+l)/ 4 'm !
and it will now be obvious that all these formulse can be easily
extended to the case of an annuity on the joint lives of any
number of nominees.
Tables for «*,„ have been calculated; and, by combining
them with tables for a,„, 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, 7, be the probabilities 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-PAr-
Hence
a;r^ = -«''(l'r + 9r-Pr'7r).
1
= am+an-a„h«.
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^n,n of an annuity to be paid 6o
long as any one of three nominees shall be alive, the respective ages being
k, m, n.
If p,, q„ r, be the chances that the respective nominees be alive after t
years, then
ai:^=2t''{i-(i-p.) (1-9^(1-'-.)}.
= 'Lv'(p, + q, + T,-q,r,-r,p,-p^,-\-p,q;r,),
The numerical solution of this problem would require a table of annuities
on three joint hves, or some other means of calculating ai,,,,.,,.
602 LIFE INSURANCE PREMIUM CH. XXXVI
§ 20.] A contract of life insurance is of tho following
nature : — A man A agrees to make certain payments to aa
insurance office, on condition that the office pay at some stated
time after his ilcath 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,
80 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 A\ insurance is such that they shall not ultimately lose if
they enter into a huge number of contracts of the kind made
with A ; but, on the contrary, earn a certain percentage to cover
expenses of management, interest on sharchoKlers' capital, tic.
The usual form of problem is as follows : —
What annual premium P„ must a man of m years of a^e pay
(in advance) during all the years nf 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 I
P„ is to be the "net premium," that is, wo suppose no
allowance made for profit, &c., to the office. Suppose that tlie
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 £P„L, £P„,l„+i, ... at the beginning
of the 1st, 2nd, . . . years respectively. It pays out on lives
failed £(/»-/„+,), £(/„+,- 4.+)), ... at the end of the Ist,
2nd, . . . years re-spt-ctively. Hence, to balance the account,
we must have, when all these sums are reduced to present
value,
Pm{L-^L+,v + L+,v'+ . . .)
= (/»-/-+i)» + ('-+i-/-«)t;' + (/-+,-t.+,)f'+ . . . (1),
the summation to be continued as long as the table gives signi-
ficant values of t.
Since rf« = 4i - /«+! , we deduce from (1)
(2).
^ 20, 21 RECURRENCE METHOD FOR ANNUITIES 603
Dividing by Im, we deduce from (1)
-fm(l + {L+lV + l,„+l1^ + lm+3V' + . . .);/,„}
= V + v{U+iV + Im+iV- + . . .),'/,„
- {L+iV + L+i'o' + . . .)IL.
Hence
F,n (1 + Cm) = v + va„^~ a,a,
F„, = v-aJ{l+a^) (3).
The last equation shows that the premium for a given life
can be deduced from the present value of au 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 act>iarial calculations ; but the mathematical student
wiU 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.
Eecurrence MetJtod for Calculating Life Annuities.
The reader will have no difficulty in showing, by means of
the formulaj of § 17, that
«„ = «(!+ a„,+r)ln+i/im (1).
From this it follows that we can calculate the present value
of an annuity on a hfe of m years from the present value on a life
of TO + 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 U8 construct a table as follows : —
In the 1st column tabulate 4 ;
„ 2nd „ <4;
„ 3rd „ v'4 = 2),, say;
„ dth „ ir'+'ofj, = C^, say.
(504 COMMUTATION METHOD CU. XXXVl
Next form the 5th column by adding the numbers in the
3rd cohiinn from the bottom upwards. In other words, tabulate
iu the 5th column the values of
iVx = />x+l + D^t + ^«+. + ■ • • .
In like manner, in the 6th column tabuliito
J/x=C«+a+i + C'^i+ • • • •
All this can be done systematically, the main part of the
labour being the multiplications in calculating />x and C,.
From a table of this kind we can calculate annuities and
life premiums with groat ease. Referring to the formula; almve,
the reader will see that we liave
a„ = iV„/Z). (2);
.,a» = i\r„+./Z). (3);
l,a„ = (i^.-iN'«+,)/Z)- (4);
.l,«- = (iVm*. - N^*,)ID^ (5) ;
P^ = MJi\\-, (6).
§ 22.1 In the fnrc|;oing ciuipter the object haa 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 li
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 inteq)retation*."
* "An Eiuy on ProbBbilitics and on their Applicstion to Life Contin-
ganciea and Insurauoe OlBoei" (De Morgan), Cabinet Cyclopadia, Aff.,
p. xxvi.
§§ 21, 22 GENERAL REMARKS— REFERENCES 605
The rpader who wishes for further iufonnatiou slioulJ consult
tlic elementary works of Do Morgan (just quoted) and of Whit-
worth {Choice and Chance) ; also the following, of a more advanced
character : — Laurent, Traite du Calcul des ProbahiUtes (Paris,
1873) ; Meyer, Vorltsungen uher Wahrscheinlichkeitsrechnung
(Leipzig, 1879); Articles, "Annuities," "Insurance," "Proba-
bilities," Encijclopcvdia Britannica, 9th edition.
The classical works on the subject are Moutmort's Essai
d! Analyse stir les Jeux de Hazards, 1708, 1714 ; James Bernoulli's
Ars Conjcctdiidi, 1713; Demoiwe's Doctrine of Chances, 1718,
1738, 1756 ; Laplace's Theorie Amilijtique des Frobahilites, 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 tlie 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 auuljtical processes, as the theory of probability.
Exercises XL.
(1.) A bag contains 4 shillings and i 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. Wliich 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 ; all 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 dea Noyuux was played with 8 discs, black on one side and
G06 EXERCISPS XL CU. XXXVI
white on tlio othrr. A oUkc F! wax uamcil. Tlio diHca were tossed up by tlie
plajor; if tlio uuuibor of Macks turned np was odd the playiT won S, if all
wore blnckii or all vrhitcii be won 2.S', otherwiso ho lost S to his op|ionent.
Show thiit the expectations of the player and opponent are 1315/256 and
U5SI'2r,G rchi>eclively. (Montmort. See Todh., Hint. Prob., p. 95.)
(7.) A promises to give H a shilling if he throws C at the first throw
with 2 dice, 2 shillings if he throws 6 at the second throw, and so on, until
a G is thrown. Calculate the value of li'a expectation.
(8.) A man is allowed one throw with 2 ordinary dice and is to gain a
unmber 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 oortaiu number of balls, some of which are white.
I am to got a shilling for every ball so long as I continue to draw white only
(the balls drawn not being replaced). 13ut an additional ball not white
having been introduced, I claim as a compon^iation to be allowed to replace
every white ball I draw. Show that this is fair.
(10.) A per.-ion throws up a coin n time.s; for every sequence of m(m>n)
heads or m tails he is to receive 2'"-! shillings; prove that the value of his
expectation is n (n-i-3)/4 shillings.
(11.) A mannfacturcr has n sewing machines, each requiring one worker,
and each yielding every day it works q times the worker's wages aa 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, ore out of repair. The worker's
wages mii»t 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
ntxt to nql{q + 1) - J . (Math. Trip., 1875.)
(12.) A blackleg bets £5 to £1, £7 to £6, £9 to £5 agauist horaei whoie
chances of winning are |, \, ) 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 area : 1; a + l :1;
. . ., a + n-1 :1. Show that it is po.isible for a bookmaker, by properly
laying bets of different amounts, to make certain to win if n > (a -t- 1) (e .(- 1),
and impo.'isible if n < a (e - 1), where e is the Napierian base.
(14.) If A,, denote the value of an annuity to last during the joint Uvea
of p persons of the same age, prove that the value of an equal annuity, to
continue so lon^; as there is a survivor out of n persons of that age, may be
found by means of the formula
nA "<"-^)j ,n(n-l)(n-2)
(15.) M is a number of married couples, the husbands being m yean of
age, the wives n years of aec. What is the number of living pairs, widow*,
widuwors, and dead pairs after ( years?
Work out the cose where If = 500, m = 'tO, n = 30.
(lo.) If 5^1 have the meaning of § 16, show that
K 22 EXERCISES XL 607
(17.) Fiarl the probability tliat a man of 80 survive one or other of two
men of 90 and 95 respectively.
(18.) If «, , • • • 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 i men whose ages
are I, m, n, . . . respectively is »
2a(-2a,,„, + 2a,,^„- . . . .
(19.) What anunal premium must a married couple of ages m and n
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 he 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.
(92 ) 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 aUve, and is to go to the survivor when one of
them dies. Show that the present values of the interests of 4 and B are
"m - J "m, » - " m, P + 5 " m, n. P
and <'n-h''t>>,n-''n,P+h<'m.n,P
respectively. .
(23 ) If the population increase in a geometrical progression whose ratio
U r, show that the proportion of men of n years of age in any large number
of the community taken at random is CJi")/- dJ''").
RESULTS OF EXERCISES.
I.
(1.) 504000. (2.) 1210809600. (3.) 720. (4.) 12. (5.) 0. (3.) 5010;
64864800. (9.) 1235.520. (10.) 6188; 3003; 3185. (11.) 408688; 18 ways of
setting together on the front, 10 ways of setting at equal distances all round.
(12.) (igCj ijC^ + ijCj ijC, jCj + ,.,0^ ]„C„ jC; + jjCj j„C, 9C3 + ,5(74 o'^itiP^-
(13.) ,„Cj„(,C5s,Cio6„C2o. (14.) 172S00. (15.)2G7148. (16.) 1814400, if
clock and counter-clock order be not distinguished. (17.) 2{2n--3n + 2)(2n- 2)!.
(18.) 960. (19.) ,C,,C,,P,; sC, 7C3 ,P, 3P3. (20.) 52!/(13!)'; 391/(13!)'.
(21.) 32!/(12!)-8!. (22.) 64!/(2!)6(8!)232!. (23.) 26; 136. (24.) 286; 84.
(25.) (p + q)^lp^-q\; {p + qry.lp\{qry.; a little over six years.
II.
(1.) 448266240i3. (2.) -2093. (3.) 2". 1 .8 . . . (2n-l)/H!. (4.)
(-)"+'-(2n)!/(n + r)I(n-r)l. (B.) 2=».l.a.. . (4«-l)/(2n)!. (6.) If h be
even, the middle term is {H!/(in)l}x"/'; if n be odd, the two middle terms
are {nl/i(n - 1)! J(n + 1)!} {2x(»-il/2 + i^C+'W^}. (11.) (2,^3 + 3)2"
+ (2V3-3)=»'-l; (2V3 + 3)="'+'-(2V3-3)^»+>. (IB.) iu{n + l). (16.)
2"-! (2 + 71). (27.) r + 1. (28.) 10. (29.) Hn^ + Un). (32.) 100274064.
(33.) ZaJ + 7 2a«6 + 21 2a''b- + i2Za^bc + ■io1a*b^ + 105 ^a'b^c + 210 Za*bcd +
li0^a^b^c + 2lOZii:^b-c- + i202,a>b-cd + G30Za-lrc-d. (37.) 23!/(4!)^5'.
III.
(1.) 944. (2.) 20. (3.) (n + l)(n + 2)(n + 3)(7i + 4)(n + 5)/5I if the
separate numbers thrown be attended to; 5» + l if the sum of the numbers
thrown be alone attended to. (4.) 231. (6.) p^.lC„. (7.) 62. (8.) 15„Cg.
(11.) (2H)l/2"7i!. (IS.) {N + a + b + c-3)lja\b]d. (16.) 1 or 0 according as
«isevenorodd; {(l + ^/S)""- (1 - Vo)''+'}/2»+V5. (17.) 2,„_iC^i„.jC,.,.
(18.) 116280.
V.
(1.) xjy must not lie between 1 and b^ja'. (2.) x must lie between
4(7-v/53) and i{7 + J53). (3.) x between {dc - b'^)l{ad - be) and
{d''-ab)l{ad-bc), and ;/ between {ah- c')! (ad -be) and {or -cd)l{ad-bc).
(IB.) Greater. (17.) Less. (39.) 3'''.
C. 11. 3'J
CIO RESULTS OF EXERCISES
VI.
(1.) Habc. (a.) ahelZJS. (4.) (/"/S"*"' is a roinimam Talae if m do
Dot lio between 0 and 1, otherwise a maiiniam. (S.) Minimum when
apx''=bqy^ = rrz''. (7.) There is a maximnm or minimum when (x + f) logo
= {i/ + m)lof;b = {t + n) lope, according as logalogbloge is positirc or nega-
tive. (8.) i = («(</ma)'"»+"l. (9.) x= 1, x = 3S/15 give maxima ; « = 2, x = 8
minima. (10.) ^abc. (11.) Minimum when x = ni<:/(>ii-ii), y = nc/(iii-ii).
(18.) Minimum 2v'(af')/(a + 6)-
VII.
(1.) 3. CD. (3.) 9/4. (3.) log 13/7. (*.) ln(B + l). (5.) 0. (6)
a'^+^-i'-^mlp. (7.) a'^'mln. (8.) n', ao , h'' according as p> = <5. (9.)
(in«-mB + n»)/(m'+mn + ii'). (10.) l/2a. (11.) aH-«/«,y/p. (la ) o*.
(18.) ICki/9. (14.) 1. (IB.) p. (16.) -Jn(n -!).-•-«. (17.) a">+«-»^9»m(ni - n)/
n-y{p-q). (18.) (n-l)/-'a. (19.) log a. (M.) 1. (21.) 1. (2i.) 1.
(23.) 1. (24.) 00. (26.) x if x=l + 0, 0 if x = l -0. (26.) f*. (27.)
0 if n be negative, if n be pofitive 0 or oo according as a< >1. (28 ) 1.
(29.) 1. (30.) 0 or 00 according as mxn. (31.) oo or 0 according as
axl. (32.) 1. (33.) A (34.) A (35.) ^f{ab). (36.) Exp (24/3).
(37.) 00 or 0 according as X,(a,-i/,) is positive or negative. II a^=bf,
<7^, + J.,.„ the limit is /.(a-i-*'-.)/"'; Ac. (38.) IJe. (39.) 0. (40.) <i/6.
(U.) 2. (42) 1. (43.) 1. (44.) 1. (46.) Jt. (46.) 0. (47.) cos a.
(48.) 0. (49.) -8. (60.) X. (61.) J. (82.) 1. (63.) 1. (84.) 0. (66.) 0.
(56.) 1. (67) log m/log n. (68.) 1. (69.) 1. (60.) 1. (61.) 1. (62.) «-»'"'».
(63.) «--"•''"'. (64.) e^'. (66.) 2/t. (74.) See chap, iix., § 23.
VIII.
(1.) Div. (2.) Div. (3.) Conv. if x be positive. (4.) Conv. (6.) Div.
(6.) Div. if modxt-a; conv. if modx>a. (7.) Conv. if x < 4 ; div. if x •< 4.
(8.) Conv. (9.) Piv., (x<l). (10.) Conv. (11.) Div. (U.) Conv. if o>l;
div. ira>l. (13.) Div. (14.) Div. (16.) Abs. conv. (16.) Div.
IX.
(1.) (-)>■ '3.1.1.3. . . (2r-5)/2.4.6.8...2r. (2.) 1.8. .. (2r-l)/
2.4...2r. (3.) 3.7.11 ...(4r-l)/4. 8.12. .. 4r. (4.) 2. 1 .4. 7 . . .
(.Sr-6)2-/>/12. 24.36.48 ... 12r. (6.) ( - )'-'l .2 . . . (3r-4)a'/«-^/r!.
(6.) -1.2.5 . . . (3r-4)a>-*-/3.6.9 . . . 3r. (7.) -(n- l)(2n- 1) . . .
(nr - n - l)/rl. (8.) 1 . 4 . 7 . . . (.Sr/2 - 2)/(r/2)! if r be even ; 0 if r bo odd,
(9.) (-)*n(n + l).. .(n + J(r-n)-l)/{i(r-n)ll. (10.) 1 + 1 (x/a) + 1 (x/a)>
-t-}|(x/<0'. (11.) The 6rgt. (13.) The third. (13.) The fourth and filth.
(14.) The eighth. (16.) II n=l, the 2nd and 3rd; if n = 2, the 2nd ; ifn-(3,
the 1 St. (19.) If m = 0, .S = a; if in = I, .S = l; if n>l, S = 0: if «<1(*0)
the Bcricx is divergent. (23.) 1 - ^ i. (23.) If m-< 1, .'S = m(iii - Iji""-'; if
111 = 0. S = 0.
RESULTS OF EXERCISES 611
X.
(1.) Sl/rt'-(c-a)(n-6). (2.) 0. (3.) 21/a'-^-2/(c - o) (a - 1). (*.)
2r + l + l/2'-+i. (6.) )-,if rbeeven; r-l,if r = 4J + l; r+l,if r = 4?-l. (6.)
nHr<f-m'^l-mJir-iP<r^+^fi-i-Jir--p-r'-+ ■■■ {16-) H« + 1) (« + 2)(« + 3).
(19.) 1-1.3 .. . (2)i-l)/2";il. (20) 7 . 10 . . . (3n + l)/3.ti. . . (3n-3).
XI.
(2.) 27.5/I2S. (3.) 8G9Gnn/256. (4.) id; 0. (6.) 11989305/2048. (6.)
(-)'-{(r-l) + (r+.'>)/2'-i-}. (10.) 1-0001005084 ; 1-000400080-5. (11.) 2mx.
(12.) l + 2j;(l-r»)/(l-r). (13.) l + (-)"-ix/2".
XII.
(1.) -367879. (2.) -ones. (5.) (l-x)-e^. (6.) 3(e-l). (7.) e + l.
(8.) 1/f. (9.) 15e.
XIII.
(4.) 917. (5.) 21og{(r-l)/(.r + l)}+log{(x + 2)/{.r-2)}. (6.) log (12,.).
(7.) (l + l/x)log(l + x)-l. (8.) i(.r-x->)log{(l + .r)/(l-x)} + 4. (9.)
When x=l the sum is 18 -24 log 2. (10.) J. (12.) S {a-3»-=/(3n - 2)
+ i»»-V(3»-l)-2a^'V3«l-
XXV.
(1.) ^n{n + l) + l{r-2)n{n + l){n-l). (2.) Jh,(«+1) (K + 4)(n + 5). (3.)
3/4-l/2K-l/2(n + l). (4.) 1/1o-1/o(5h + 3). (5.) 1/12- l/4(2K + l)(2rt + 3).
(6.) l/18-l/3(n + l)(n + 2)(n + 3). (7.) «/2 + 6/4-a/(» + 2)-6/2(n + l)(n + 2).
(8.) l/8-(4!i + 3)/8(2n + l)(2K + 3). (9.) 7/36- (3H + 7)/(n+l) (h + 2) (« + 3).
(10.) ll/180-(C»i + ll)/12(2;i + l)(2« + 3)(2K + 5). (11.) 3/4 + n-(2« + 3)/
2 (n + !)(« + 2). (12.) u„ = (n + l)^(n + 3)(n + 5)/;i(n + l). . . (n + G); apply
§ 3, Example 4. (13.) sin d sec (« + 1)0 sec 0. (14.) cot («/2»)/2» - cot 9.
(16.) tan-'na". (16.) tau-'l + tau-'l/2- tan-U/n- tau-U/(H + l). (17.)
(m + H)l/(»i +!)(«- 1)!. (18.) {l/(m-l)l-(» + l)!/(m + n-l)!}/(m-2).
(19.) (-r^-iG„. (21.) {m-l-(n)l/"i"->'}/('»-2). (22.) {a>''+^'lc"'' -
„ir+ii)/(a_c + r + l). (23.) (oi"+-7ci»+'^-»i-a/ci'-i)/(n-c-r + l). (24.)
{(a-l)""-'7ci"'-'i-(a + H)"^iV(c + 7i + l)i'»-i'}/(m-l)(a-c-l). (25.)
Deduce from (24). (26.) Deduce from (24). (27.) 2m{l - (-)"2"(m- 1)
(m-2) . . . (m-«)/1.3 . . . (2k- l)}/(2w- 1).
XXVI.
(1.) 2"+' + 4(3"+' -3). (2.) ;{l + (-l)''} + C-3{!»+> + (-r)"+'}-
V{i"-(-i)"}. (3.) ll{l-{lx)»+'}/{l-4x}-9{l-(3x)»+i}/{l-3x};
(2 + 3t)/(1-7x + 12x5), x<J. (4.) 3 {1- (2x)"+>}/{l-2x} +2 {1- (3i)»+i}/
{1 - 3x } ; (5 - 13x)/(l - 5x + Gx''), x < J . (5.) J { 1 - (3x)"+'}/(l - 3x) +
J{l-(5.r)'-n}/(l-5x); (1 -4x)/(l -8x + 15x=), x<J. (6.) 3{l-(2x)''+i}/
{l-2xJ-2{l-x"+'}/{l-x}; (l + x)/(l-3x + 2x=), x^J.
G12 RESULTS OF EXERCISES
XXVII.
(1.) (l + 2jr')/(l-x')'. (2.) -[log{(l-i)/(l + x + T>)}-^/3tan->{V8T/
(2 + j-)}]/3i; H^ + 2^-'/'C09(^3j-/2)}. (4.) He-' + e"* {cos (^18x12) + J3
8iD(v'3j-/'2)}]. (8.) l(2"' + 2co9.mT/3): IS^'coB-mF/e. (6.) 1/2 - l/{n + 2)I.
(7.) {2'»+'-l-(ri + 3)(m+l)/2}/(m + l)(m + 2)(m + 3). (8.) 1/(1 + *)-
log(l + i). (9.) 4 =08* -1 cos 20. (10.) l-(2K + 3)/(r. + 2}'. (U.)2-41og2.
(14.) Bin mwlmw ; cosh mr.
XXVIII.
The partial qnotients arc as follows : —
(1.) 0, 4. 1, C, 2. (a.) 0, 2. 4, 8, 10. (3.) 1. I.i. 1, 1, 1, 3. 1, M, 1. 1,
6. (4.) 31, 1, 1, 1, 1, 1, 1, 1, 1, 3. (6.) 2. 1. 2. 1, 1, 4, 1, 1. 0. 3, 12, 3.
6, 1, 2. (6.) 0, 120, 1, 1, 2, 1, 1, 0. (7.) 1, 2. (8.) 2, 4. (9.) 3. 3, 0.
(10.) 3, 6. (11.) 3, 2. 5. (12.) 1, 4, 2. (13.) 2, 1, 2. (14.) 3, i, 6,
(16.) 0, 2, 1; 0, I. (17.) a, 2, 2*a; a-1, 2, 2 (.4-1).
XXIX.
(1.) The Ist, 2nd, 3rd, . . . convcrgcnts are 1, 2/3, 9/13, 20/29, 20/42,
78/113, . . .: the errors corresponding less than 1/3, 1/39, 1/377, 1/1218,
1/4740, 1/17615, . . . (a.) 972/1393. (3.) 2177/528. (4.) TransiU at
the same node will occur 8, 243, . . . years after : after 8 years Venus will
be less than l°-5 from the node. (S.) Transits at the same node will occur
13, 33, . . . years after.
XXXI.
(1) 10,2*0; (2.) 0, 1, 126. 2;
0, 10, 0, 0, 0*3, 0*3 ;
1. C4,C3, i.
(3.) 1, 5, 3, 1, 8, 1,3, 5. 2;
0, 12, 13, 8, 12, 12,8, 13, 12;
1*2, 5, 7, 20. 3, 20, 7, 5.
(4.) 0,7, 1,4,3,1,2,2,1,3,4, 1,14;
0,0. 7.5,7,5,4,6,4.5.7. 6, 7;
61, 1. 12. 3. 4, 9. 5. 5. 9, 4, 3, 12.
(B.) 1, 2, 10, 2, 1; (6.) 2,4;
1*0, IS, 2.-,, 25, 1*6 ; 2, 2 ;
25, 20. 6. 20. 25. 2, I.
RESULTS OF EXERCISES C13
* » « » *
(10.) a + ^ — ; a + (a"-i-j8"-')/(a'» -;?")» °- ^^^ ^ ^^i^g ^^6 roots of
*
x--2ax-l = 0. (11.) i{o + V(a= + 4)}; (a"+i-/3"+')/(a''-/3"), whereoand
|S are the roots o(x^-ax-l=0. (12.) § {a - ^(a'> - 4)} ; (a" - /3")/(a»+' - /3*<->),
where a and /3 are the roots of x= - oi + 1 = 0. (13.) {-ab + sj{ci%^ + iab)} 12a;
if o, p be the roots of i»- (a6 + 2)a; + l = 0, then pj„ = 6(a"-/3")/(o-;3),
9»n=(a"+'-(3''+»-a" + /S")/(a-yS), and ?,„-,= (p,„-Ps„-5)/6, 9.„-, = (gs„-
^.n-jVi- {14-) - 1 + v/ [{3 (a" - /3") + 2 (a»-> - /S"-') l/la"*' - |S"+')], where a
and ^ are the roots of X--X- 1 = 0. (20.) - in + V[{(in- + ")(«''"' -/3""') +
(in'+l) (a»-2-/3''-=)}/(a»-jS")], where a and |3 are the roota of r"- a; -1=0.
XXXII.
(1.) 3 + 7«, 2- 5f. (2.) 17e + 7, 16( + 5. (3.) 220G - 7f, IK - 3309. (4.)
1013«- 8021756, 13C7t- 4077746. (5.) 13. (6.) 2S0. (7.) G. (8.) It
25fr. = 20s., 41. (9.) Buy 300 of each and spend 1021<i. (10.) C9. (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 fc > 2. (8.)
Converges. (9.) Oscillates. (10.) Oscillates. (15.) Each of the fractions
converges to 1. (23.) e. (24.) 1/(1 -e). (25.) log, 2. (26.) (3-e)/(e-2).
xxxrs.
(1.) 11/30. (2.) 3/11, 29/44, 3/44. (3.) m(n» + 2n)/(m + n)',m(m+2n- 1)/
(m + n)(m + n-l). (4.) (365 .4» + l)/(1461)». (5.) 4/9. (7.) 55/672, 299/2G88.
(8.) 1/42. (9.) («-!)/« (2n-l). (10.) (39!)»/26152I, 4(391)»/26!521. (11.)
2 (r - l)/n (n - 1). (13.) 7n/2, or, if this be not integral, the two integers on
either side of it. (14.) "2 r(r-l)n(n-l) . . . (n-r + 2)/rt'. (18.) 16/31,
8/31, 4/31, 2/31, 1/31. (19.) The chances in A's 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 - 1/«)/2,
(l-l/n)/(2-l/n).
XL.
(1.) £1 : 11 : 6. (2.) His expectations are lis. Grf. and lOj. Hd. respect-
ively. (3.) £8:5:94, £2:4: 2i. (4.) n(l- 1/2--), (1-1/2T- (7.)75.2irf.;
{n + l)(4n-l)/6n. (12.) £0, £1, £4 : 2 : 24.
INDEX OF PROPER NAMES,
PARTS I. AND II.
The Soman numeral re/en to the part*, the Arabic to the page.
Abel, ii. 132, 13C, 142, 141, 152,
164, 287
Adams, ii. 231, 243, 251
Alkhnyami, ii. 4o0
Allardicf, i. 441
Arohiiucdca, ii. 99, 412
Argand, i. 222, 254
Arudt, ii. 500
Babhaoe, ii. 180
Bernoulli, James, ii. 228, 233, 276,
403, 405, 587, COS
Bernoulli, John, ii. 275, 298, 3GC,
403, 584'
Bortrand, ii. 125, 132, 183
Bezout, i. 358
Biermann, ii. 08
Blissord. L 84
Bombelli. i. 201
Bonnet, ii. 03, 132, lf»3
Boole (Moulton), ii. 231, 398
Hoarguot, ii. 183, 253
Briggfl, i. 529; ii. 241
Briot and l)ouc|uet, ii. 396
Brounckcr, ii. 351, 413, 479, 516
Burc-khardt, ii. 536
Biirgi, i. 558
Barnside, ii. 32
CiNTOB, iL 98
Cardano, i. 253
Catalan, ii. 132, 183, 220, 251, 353,
416
Cauchy, i. 77, 254; ii. 4-.>. 47, 83,
110, ll.";, 12:1. 132, i:w, 142, IM,
171. 188, 226, 239, 2S7, 340, 844,
896
Cavley, ii. 33, 312, 325, 871, 496
Clausen, ii. 340, 503
Clcrk-Maiwfll, ii. 325
Cossali, i. 191
Cotcg, L 247
Cramer, ii. 396
Dase, ii. 536
Dedekind, ii. 98
De Gua, ii. 396
De Morgan, i. 254, 346; ii. 125, 132,
381, 390, 417, 421, 578, 004
Dcmoivre, i. 239, 247; ii. 298, 806,
401, 4U3, 405, 407, 411, S74, 593.
697, 005
Desboves, iL 63
Dcscarte.'i, i. 201
Dinphautos, ii. 473
Pirichlet, iL 95, 140, 473
Du Uois lieymond, ii. 133, 147, 148,
Dur^e, ii. 396
Ei.T, ii. 231, 344
Kuclid, L 47, 272
Euler, L 254; iL 81, 110, 18fl, 231,
2.''>2, 280, 841, 342, 343, 344, 345,
348, 358, 363, 305, 306, 408, 41'J,
448, 494, 41)0, 512, 515, 616, 526,
539, 550, 551, 553, 555, 556, 668
Favabo, ii. 448
Format, ii. 478, 499, 640, 660. 591
Ferrers, ii. 6i'.2
Fihonaoci, i. 202
Fomyth, ii. 890
i'uiti ii. 77
INDEX
C15
Fourier, ii. 135
Franklin, ii. 83, 5G1
Frost, ii. 9G, 112, 30G, 397
Galois, ii. 505
Gauss, i. 46, 254; ii. 81, 132, 184,
333, 345, 473, 523, 542, 550, 553
Glaisher. i. 172, 530; ii. 81, 240,
313, 357, 371, 397, 410, 421, 53C
Goldbach, ii. 422
Grassmann, i. 254
Gray, ii. 243
Greenhill, ii. 313
Gregory, ii. 110
Gregory, James, ii. 333, 351
Grillet, ii. 59
Groiinu, ii. 313
Gross, ii. 541
Gudermann, ii. 312, 313
Guutlier, ii. 312, 448
Hamilton, i. 254
Hankel, i. 5, 254
Hargreaves, ii. 447, 452
Harkness and Morley, iu lOG, 148,
163, 396
Harriot, i. 201
Heath, ii. 473
Heilermaun, ii. 518
Heine, ii. 95, 98, 527
Heis, ii. 313
Herigone, i. 201
Hermite, ii. 473
Hero, i. 83
Hindenburg, ii. 495
Horner, i. 34G
Houel, ii. 312
Hutton, i. 201
Huyghens, ii. 448, 580, 587, 592
Jacobi, ii. 473
Jensen, ii. 184
Jordan, i. 76; ii. 32
KoBN, ii. 125, 133
Kramp, ii. 4, 403
Kronecker, ii. 237
Eummer, ii. 133, 184, 473
La Caille, ii. 449
Lagrange, i. 57, 451; ii. 396, 448,
450, 453, 479, 5.50, 553
Laisaut, ii. 313, 336, 358
Lambert, i. 176: ii. 312, 345, 448,
517, 523
Laplace, ii. 50, 605
Laurent, ii. 184, 579, G05
Legondre, ii. 473, 512, 523, 503
Leibnitz, ii. 333, 403
Lionnet, ii. 249, 252
Lock, ii. 271
Longcbamps, ii. 110
Macdonald, i. 530
Maohin, ii. 333
Malmsten, ii. 80, 132
Waseheroni, ii. 81
Mathews, ii. 473
Mayer, F. C, ii. 312
Meray, ii. 98
Mercator, ii. 312
Mortens, ii. 142
Metius, ii. 442
Mever, ii. 605
Mobius, ii. 397, 494, 504
Montmort, ii. 405, 407, 584, 592,
605, 606
Muir, i. 358; ii. 334, 471, 494, 4D7,
502, 504, 518, 527
Napier, i. 171, 201, 254, 529; ii. 78
Netto, u. 32
Newton, i. 201, 436, 472, 474, 479;
ii. 14, 280, 330, 335, 351, 373,
386, 392, 396, 401, 591
Nicolai, ii. 81
Ohji, 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
Pniseux, ii. 396
Purkiss, ii. 61
Pythagoras, ii. 531
Eaabe, ii. 132, 372
Eecorde, i. 216
Keifif, ii. 145
Keynaud and Duhamel, ii. 49
Kiemann, i. 254; ii. 110, 265, 325
Eudolf, i. 200
Salmon, i. 440
Sang, i. 530
Saunderson, ii. 443
Scheubel, i. 201
CIG
INDEX
Bchl6milcl), ii. 46. SI, 80, 111, 1R4,
210. 35'J, 373. r,lt6, 523
Scidel, ii. 143, 60t>
Serret, i. 76; ii. 32, 443, 453, 471,
481. 490
Shanks, ii. 334
Sliarp, ii. 333
Simpson, ii. 417
Smith, Ueary, ii. 473. 499
Spragoe. i. 531 ; ii. 83
StA-nville, ii. 335
BUudt, ii. 231
Btern, ii. 312, 418, 497, 505, 600,
617, 525
Stevin, i. 171, 201
Stifel, i. 81, 200
Stiriing. ii. 308, 401, 404, 422, 589
Stokes, ii. 145
Stolz, ii. 93. 163, 181, 185, 396
Sutton, i. 531
Sylvester, i. 48, 176; ii. 312, 494,
603, 556, 561
Tait, ii. 253
TnrtBRlia, i. 191
Tchobiclipf, ii. 183
Thomo, a. 184, 396
Todhunter, ii. 271. 276, 674, 680,
584, 587, 592, 605
Van Ceclev, ii. 833
Vandermondf, ii. 9
Venn, ii. 567
ViMe, i. 201; ii. 270
Vlncq, i. 530
Wallace, ii. 312, 314, 315
AVallis, ii. 351, 44S. 479. 527, 537
W.iring, ii. 132, 417, 653, 555
Weber, ii. 98
Weierslr.is.s, L 230; ii. 98, 151, 100,
168, 185
Whitworth, ii. 22, 26, 33, 665, 589.
005
Wilson, ii. 551
Wol.stcnholmc, i. 413: ii. 17 83
372. .'547
Wiuui>ki, ii. 213
THE END.
CAMBniDGE : PBINTED IIY W. I.EWLS. itjl., ATTUE CNIVERSITV PBESS
I
BINDING SECT, JjJM 1 4 1982
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152
Algebra
C^
1889
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