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Full text of "Cours Math"

^j* (1 

.( A-a5j-a J i .-il V. ) Aj-HC AjUaLo ^j-ajjJ E ir-.j*-< "^ J^J N lil *J*^ "" (j^s "Ub lK 

U:N— >R Tq U{n) = 5n + 1 . 
K:N^E Tq V{n) = n 2 + 3n - 1 . 

W:N^E To Win) = — . 

' 2n+l 

t: N -^ E Tq tin) = e 2n+3 . 

S: N — > E Tq 5(n + 1) = Ln(S(n) + 1) et 5(0) = 1 . 

f:N^R Tq f{n + 2) = f{n+ 1) + f{n) et /(0) = 0, /(l) = 1. 



f/(n + 1) = f/(n) + 5 auec 1/(0) = 1 
t{n + 1) = tin) ■ e 2 avec t(0) = e 3 



.n>k JS <J?J i> U n+1 >U n (jS^j bj SjjI ji« l^si iU n ) n > k aJUI* t> dj£ 
.n > /c lK <_M i> f/ n+1 < f/ n ij£^ b) A^UI* l^ii ( U n ) n > k <&Z* i> J_£ 

■ 1 J&» 

(jo^j 1 oV^jj^ l n = i2n-3) 2 >V (Z n ) n >o ^^ 

M n+1 <M n jV^SlH- M n = ^ ^ (W n ) n > M**M 

Gi<S ° V ^ JJ; ^ *n = (-2) n C^ (i?J nS0 ^Ul 



.2 J&* 
.UU3 s^l> (Lf n ) n > M**ll ol* l/„+i - U n = 5 > ji L« (1 

4 i a3l lLa i— lib a 1@ uil ml 4-uUn-s <Jl2La JS j Sjjljp-o ' ■ la j" 1@ uil ml 4_ul m-s 4-ulolo JS 

UUi sajI> (K n ) n > M**H l4* K n+1 - V n = In + 4 > J ^ (2 

.U-5*~as- (VK n ) nS0 aJUUI^ VK n+1 - W n = (2n+3 j ( 9 2n+1) < 0M3 

.U-5fc*l> (t n ) ns0 AJLnJI^ t n+1 -t n = e 2n+3 (e 2 -l) > ui^(4 

A^l^jll j ji' ail (jjj jjj-ai-a 1 g ml uil 4jmYlft AjjIaLo (J£ j oAjl jia A^l^all qa j&\ 1 ^ ml ml A jmYlft 4_l]l2La (J£ 

.4jjjj jjE- ^j^a L-lIL "II (JjLjiVI Clllj 4_lujAJ^J| AjlLtLall Lai m A ■ afll Jla 

5"(x) < J ^ 5"(x) = Ln(x + 1) -x Alhlt cjljjii a^Ijj .i*j (5 
■*-»** (Sjnao M2-M J^L J S n+1 -S n = g(S n ) < 0* 
UUi s^!> (/ B ) na0 aJLuJI <# / n+1 - /„ = /„_! > (^W) J ^ (6 



jVUJi ^ ojitiii- upft Jul j lim n ^ +O0 f/ n = 6 £ E u 1 ^ ,:i ! ^J&* M {u n ) n > k <JLu* t> Jj£ 

Jlla 
.lip] t_>ujJI AjI^J (J ojj^laJI AJlaVl jlfitVI jJ*J ia-L 

.&JC.U* (f/ n ) n > M**M lP] Iim n ^ +O0 y n = lim n ^ +O0 5n + 1 = +oo • 
.s^Lic (K n ) n > M**H CPJ lim n ^ +O0 K n = lim n ^ +O0 n 2 + 3n-l = +oo • 

.^jsii (M/ n ) n > ^Lu-iiu^! lim n ^ +00 W n = lim n ^ +00 ^= lim^^^f = ^ • 

.s^ul. (t n ) na0 M2*M«^J lim n ^ +O0 t n = lim n ^ +O0 e 2n+3 = +oo • 



4ii.!>U 



_ < _ s JLojI (JULaJI ^aj ojj^AaII (j^al^aiJI (JJ1J l_kjjxj]l I^A JLaxiailjj 4jmYlft AljjJaJ l_i^)XJ AjlllLa ^_jLg_J 



.Jlia 

lim n ^ +00 n 2 — 5n + 3 lim n ^ +O0 VrTTT — n lim n ^ +00 Vn 3 + In — 3 — In 



lim n ^ +0O Vn 2 + 6n+2-n lirrin^+oo -— -— r lim n ^ +O0 e n+1 -n 



3 



1 <u-ali. 

lim n ^ +0O f/ n -K n = f lim„_4.«K. = 

d n 

2 Ajj-ali. 



'lim n ^ +00 i/ n ■ V n - c|j m v = o 



.4-jjlaIa Igjji Likll i—ljlj-all dlli JjJaJI <Jaj (j-a ^ylc-Vl (jo S^jAa^a j 0.1)1 jla Ajlljla ( U n ) n>Q C±i\£. LI • 
_4jjlala L^-ili LJaJI i_jj|ja]| Cjlj JljAaJ! (Jaj (j-a (JiuiVl (j-a o^j^a-a j 4j_aiiL)ia Ajlliila ( {/ ) - Cjj\£ LI • 

lM 6- U n >V n ciii=. Saj! jl* 4JII1* (l4 l ) n > c^l£ j 4^UI* 4JLti* ({/ n ) n > ^^ bj • 
lij AjI^jII ^ La$J ^^ j (jjijlai^ (l^) n > * ( U n ) n > dw^^ 1 U^ W^ 1 c-u'j^ 1 ^ -^J-^ 1 

.lim n ^ +00 (f/ n -7 n ) = 0^i£ 



R ^ R D-^ / ^ U n+1 = f{.U n ) &»2 V^l j^ ^»3l. (l/ n ) na0 o^ 

f/i>U uKbU^l> (y n )n> ^^ • 

t/i<r/ <^IJ]*— »H. [U n ) n > M2-M • 

(^2n+l)n>0 ' (^2n)n>0 OH^ 1 U^ (!/„)„*) M^» >-^tfJH ^ >> *--^ / *JU» ^ &] 

.^jVl aJLJ! ^j iiiit ^jja .SAjlJLa AJb g = f°f £u* \ Kn+ ' a jUa^S 

[^2(n+i)+i = #'^2*1+1) 

2 4jj-aLiJI (JLaxlui! (J^l (j-a V al S ij la a ^^A (JA Aijx-a (Jjl a 13 Igjjlii 4_uilj.l] Lai 4j]LiLa 4jLij <Lui!j.i j_^aaj Laj3 I.1A 

.(_J^i.l 4_l]LlLaJ IgJiJjlLaJ La AjlLlLa AjtjjJa 4_uil jj (j^-aJ Laj-aC 



4ii.!>La 

U n+ i=f(U n ) c^^^lJiAJLtl. (f/ n ) ns0 jSd 

f(p) = P 0\* P U^^^jliLa (t/ n ) n > *Jl2Ull ^l£ IJJ 



.lJli. 



.2<Jt5- 



11 1 v" n 1 f x ~ , 

U n = 1 + -r + 7T +■■■ + ^ <=>!/=> TT^I ! 



n 2 



Vr, 



_1_ _1_ _1_ V" ^ ( ^ 1 = 1 1 

" 1 + V2 + V3 + '" + V^ ** ^ 2j fc=1 Vn ** K = tf»-i + -p 

1-3-5 (2n-l) Ilfc-i(2fe-l) y i = ? 

*■ 2-4.6 (2.) ~ *■ = -fe- ~ „.„ 2„-l 

v 2n 



/"(x) 4JUI ^'j^ Jj-^h ^n'""Vl u^-oj U(n) = / (n) p^l -^ *jW*ri 4ij*-a <JIj1o AjJj^I