1. Ряды (853737), страница 7
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3 b (!), (11.26), -" 6 *+ f (x), + (0 +1). 3 (11.27) * -" 6. 11.2. @ - - 3. " f (x) = e ;x x > 0, . " .< ,.#. "uZe u cos u du = e ( cos2 +u + 2 sin u) + C u ( sin u ; sin u)Zeu+Ce sin u dx =2 + 2( . ), " (11.24) (11.26) - -3. ":+1vvuu+1;uZuu22e(;cos!u+!sin!u);utt =a (!) = e cos ! u du = 21+!00vuu= t2 1 0 ! < +1C 1 + !2100(11:28)vu+Z1u2tb (!) = e ;u sin ! u du =0vuu= t2 !+1vu;uut 2 e (; sin ! u ; ! cos ! u) =1 + !20(11:29) 1 + !2 0 ! < +1:3 (11.25), (11.27) " (11.28), (11.29) .
" - - 3. ":vvuu+1+ZZ1 cos ! xuu22cos!x2;xtte = 1 + !2 d ! = d !C21+!v0uu++Z10 vZ1 ! sin ! xuu22!sin!x2;xtt 1 + !2 d ! = d!e = 21+!00 0 < x < +1: > 11.3. #. .8 >+>Z1< x 0 x < 1g(!) sin ! x d! = f (x) = > 05 x = 1(11:30)>: 0 x > 1:0< / vu+Z 1s u2f (x) = t g(!) sin ! x d!20s, " (11.27), , " 2 g(!) - 3.
" f (x). 8.,. "(11.26), vvsu+1ZZuu g(!) = t 2 f (u) sin ! u du = ut 2 u sin ! u du:(11:31)2 00& " @.-N, " (11.31): u=12sin!u;u!cos!u = 2 (sin ! ; ! cos ! ) :g (! ) = (11:32) u=0!2 !23 (11.32) # (11.30). >101 $ + "& $$ 3. + . # "". , . F -# # . 11.4. #. -#82>< @u = 2 @ u 2>: u@t(x 0) @x= ' (x); 1 < x < +1 0 < t < +1(11:33) u = u (x t), a { , ' (x) { ", 11.1.< (11.33) 3. x:2 2 3" @u #F @t (! t) = F 4 2 @@xu2 5 (! t):(11:34);, +Z1 ;i ! y1e u (y t) dy ; 1 < ! < +1F Gu] (! t) = U (! t) = p2 ;1 . I, IV N, " (11.34) "". t:U t0 (! t) = a2 (;i !)2 U (! t) = ;a2 !2 U (! t):- , .
. " (11.33), ++Z1 ;i ! yZ1 ;i ! y11e u (y 0) dy = pe ' (y) dy = R (!)U (! 0) = p2 ;12 ;1(11:35) R (!) { 3. . " ' (x).&, 3. U (! t) # u (x t) -# "". :102U t0 (! t) = ;a2!2U (! t) U (! 0) = R(!) ;1 < ! < +1: (11:36)! # (11.36) U (! t) = C (!) e ;a ! t, C (!) { . " !. #, . " (11.36), "U (! t) = R(!)e ;a ! t ; 1 < ! < +1:(11:37)&. 3., # (11.33):+Z11e i ! x U (! t) d!:u (x t) = p2 ;1,.#. (11.37) (11.35), # + . +Z1 i ! x1u (x t) = pe R (!) e ;a ! t d! =2 ;101+1+1ZZe i ! x;a ! t B@ p1e ;i ! y ' (y) dyCA d! == p12 ;12 ;123+Z1 6 1 +Z1= 4 2 e i ! (x;y);a ! t d! 75 ' (y) dy:;1;1!, e ;(xp;y) =(4 a t) 2 a2 t .
(11.33) ++Z 1 ;(x;y) =(4 a t)Z1 e ;(x;y) =(4 a t)1p 2 ' (y) dy = pe' (y) dy:u (x t) =2at2at;1;12222222222222222*#+, "!,1. $%"" " %"< G" % (. " 11.1).2. G" ( %"<) %"<( ). 5; %"" G" (. 11.1 " 11.10).f x1033. 6 G" %"< ( ) = ;j x j. 5; %"" G" - %"<.4. $%"" G":) 4) G" 4) G" %"< -"4) G" (. I { IV 11.1).5. $%"" " %"< G" % (.
" 11.2).6. "- G" ; %"" "- G" (. 11.3).7. "- G" ; %"" "- G" (. 11.4).8. 6 " ( G" %"<0 ( ) = sin0 ; %"" "- G".9. 6 " G" %"<(0 ( ) = sin0 ; %"" "- G".10. 7; " (+R1 0 1( ) cos= ( ) = 1 ;010f xxf xg!xx! x d!x > xf xex > xf xxx >:s+Z 1 i x !112;jxj= ; 1+1.3. ( ) =21 + 241+;1sp0 5 40+1=614(1)=;8. ( ) = 2 sin21;+Z 1sin sin2 0 +1.( )=1; 20scos9. ( ) = 2 1 +1 ; 0 +16= 14 (1) = 042+Z 12(1 + cos ) cos( )= 0 +1.1; 20210.
( ) = 4 sin (0 5 ) 01.S!!x!! <!!2!!< x <!bx <!!f xd!!d!!! <!!a !g!!b !f xee!xd!< ! <104!x <aII. 1 . -./0* 12 4 @, . " -an = f (n)" . .. .. . a1 a2 : : : an : : : :2 ". 1Xan = a1 + a2 + : : : + an + : : :n=10 0 .. 8 0 , , a1 { , a2 { , . ., an { n-, . , . 1.1. . .1X< 1) n = 1 + 2 + : : : + n + : : :C an = n.n=11X2) sin n1 = sin 1 + sin 12 + : : : + sin n1 + : : :C an = sin n1 .n=11X3) arcsin n2 1+ 1 = arcsin 12 + arcsin 15 + : : : +n=1+ arcsin n2 1+ 1 + : : : C an = arcsin n2 1+ 1 . >&, { 0 "" .;. , , +.
0 , S . 0 1X . F , ., ann=1.. sn = a1 + : : : + an n = 1 2 : : : :1057 sn, n , n- .7 , .. , . . nlim!1 sn < 1:/ 0 .E + .. , . 1.2. @ + 1X1n=1 (2n ; 1)(2n + 1) . ..< 1!111an = (2n ; 1)(2n + 1) = 2 2n ; 1 ; 2n + 1 " 1 ! 1 1 ! 1 1 !1sn = a1 + a2 + a3 + : : : + an = 2 1 ; 3 + 3 ; 5 + 5 ; 7 +!# 1 ! 111+ : : : + 2n ; 1 ; 2n + 1 = 2 1 ; 2n + 1 :. nlim!1 sn = 1=2, 1=2. > 1.3.
@ + 1 2n + 3nXn=1 5n . ..< .n + 3n 2 !n 3 !n2an = 5n = 5 + 5 2 2 !2 2 !n3 2 3 3 !2 3 !n32sn = 4 5 + 5 + : : : + 5 5 + 4 5 + 5 + : : : + 5 5 =106" 2 !n# 3 " 3 !n#n 3 1 ; (3=5)n1;(2=5)22= 5 1 ; 2=5 + 5 1 ; 3=5 = 3 1 ; 5 + 2 1 ; 5 :; nlim!1 sn = 2=3 + 3=2 = 13=6, 13=6. > 1.4.
@ + 1 pppXn+2;2 n+1+ nn=1 . ..< , . , :pp sn = a1 + a2 + a3 + a4 + a5 + : : : + an = 3 ; 2 2 + 1 +pppp p p p p + 4;2 3+ 2 + 5;2 4+ 3 + 6;2 5+ 4 +pp pppp+ 7; 2 6 + 5 + :::+ n + 2 ; 2 n+ 1+ n =p (n + 2) ; (n + 1)p ppp= 1; 2+ n+2; n+1 = 1; 2+ p=pn+2+ n+1p1p= 1; 2+ pn + 2 + n + 1:p.!1 sn = 1 ; 2, p nlim 1 ; 2. > 1.5. @ + 1Xnn=1 . ..< ; an = n, sn = a1 + a2 + : : : + an = 1 + 2 + : : : + n = n(n2+ 1) :.
limn!1 sn = 1, . > 1.6. ! #, . , a + aq + aq2 + : : : + aqn + : : : a 6= 0:107< & 0 .. 8 + sn.) q 6= 1 " n n1;q2n;1sn = a + aq + aq + : : : + aq = a 1 ; q :) q = 1, , sn = na.;. nlim!1 sn.n = 0 , ., lim s = a .) E jqj < 1, nlimq!1n!1 n 1 ; qn) E jqj > 1, nlim!1 q = 1 , ., nlim!1 sn = 1.n) E q = ;1, sn = a 1 ; (2;1) , . . sn = 0 n sn = a n.
% , nlim!1 sn .) E q = 1, nlim!1 sn = 1.1 n;1X!. , aq jqj < 1 (n=1a 1 ; q ) jqj 1. >-, $ !!#$#+6 5$6 ! "! . B " ! "" .1 2n + 5n11XXX1142)443)1)n3n=1n=1 (3 ; 2)(3 + 1)n=1 ( + 1)1 11 s +21 10 + (;1)n;1 3nXXX4)45) ln 1 + 46) ln + 17nn=1n=1n=1Sn nnnnnn:1) $, = 144) , = 59 304SS=2) 45) 4|||||1083) , = 1 346) .S= $ 1X : ann=1, nlima=0.!, ,.&,!1 n , 1 .X. , p1n . ! ,n=11 nlim!1 pn = 0. F n- psn = 1 + p1 + : : : + p1n p1n + p1n + : : : + p1n = n p1n = n2., nlim!1 sn = 1, .
.F . . ( ): " , . 1.7. 8 . ., + .211XXn+12n1) p 2C2) arctg 2n2 + 3+n 1+ 7 Cn=1 n + 3n + 5n=1p 211XX3n;n+53) arcsin 2n + 1 C4) arctg n2 1+ 1 :n=1n=111+n + 1 = lim sn = 1 6= 0p< 1) nlim!1 n2 + 3n + 5 n!131 + n + n521Xp 2n + 1. .n+3n+5n=101222n+12n+12) ; nlim!1 2n2 + 3n + 7 A =!1 arctg 2n2 + 3n + 7 = arctg @nlim011BB2 + n2 CCBCC = arctg 1 = 6= 0= arctg B@nlim!1 3 7 A42 + n + n22+11X2n arctg 2n2 + 3n + 7 .n=1109p0p12;n+52;n+53n3n@3) ; nlim!1 arcsin 2n + 1 = arcsin nlim!1 2n + 1 A =v01uu51t3 ; + CCBpB2BCn n CC = arcsin 3 = 6= 0= arcsin BBBnlim!12 3@2 + n1 CAp 21X arcsin 3n2n;+n1+ 5 .n=14) 8 , .1 = arctg 0 = 0: >limarctgn!1n2 + 1-, $ !!#$#+6 5$$ 0 !, . 111 p2 2 + 3 ; 11 1XXX42)ln1+1) p3 343)sin 4+5 2+1n=1n=1n=111 1=n 1XXX6) ln 1 44);1 45) ( 2 + 1) sin 2 1+ 1 4n=1n=1n=11 p3 5 + 2 3 + 31XXp7)48) arcsin 4 25+ 15 7+3 +1n=1n=1nnnnnennnnnnnnnnn:6 ! " : 1, 2, 4, 5, 6, 7.||||| $ $: $ .
1Xn=1an 1Xn=1110bn +. .E 9n0 2 NII , an bn ,11 8n n0 XX bn . an (n=1n=111XX, an . bn).n=1n=1% " " . 1.8. &. . :1 2n1 ln n1 1XXX2) n + 2n C3) pn :1) pn + 5n Cn=1n=1n=1< 1) . . n 1 !n11pn + 5n < 5n = 51X (1=5)n (. 1.6), n=1 .2) . . n 2n > 2n = 1n + 2n 2n + 2n 21X 12 ( an = 1=2 n=11 2nX n ! 1), n + 2n +n=1.3) . . n > 2 lnpnn > p1n1X, , p1n , n=1 . >11XXanE 9 nlim!1 bn = const 6= 0, n=1 an n=1 bn , .
. , . % " . & . " .111 1.9. &. . 1 1X 2 IR:n=1 n< 8 1 !> 1 ... ln 1 + n1 n1 n ! 1, 1 1 X1 1!X n ln 1 + n .1 n=11 ! n=1Xln 1 + n . (F., n=1 sk ! XkkX1sk = ln 1 + n = (ln(n + 1) ; ln n) = ln(k + 1)n=1n=11 1X. . klim!1 sk = 1.) 8., n=1 n ., < 1 .1n 1 1 1 : 0 Xn n1 1n=1 nX . n .n=11X+ ., > 1 n1 . F 0n=1 . 110 1X1@; n;1 A :;1(n+1)n=1! , . 10kX11sk = @ (n + 1);1 ; n;1 A = (k +11);1 ; 1n=1 klim!1 sk = ;1. N+, " f (x) = x1;1 Gn n + 1], # ( 0 < < 1)11 = f (n + 1) ; f (n) = f 0 (n + ) = 1 ; :;(n + 1);1 n;1(n + )112.1 ; 1 ; n ! 1(n + )n1110 1XX1 @ (n + 1);1 ; n;1 A 1 n; n=1n=1 . 1 , 1 .
1XX n1 1 n; + ,n=1., + .n=1 1X!. , n1 > 1 n=1 1. % . . > 1.10. &. . :11XX11) arctg n2 + 1 C2) arcsin n +1 1 Cn=1n=1!1X113)ln n2=5 ; ln sin n2=5 :n=1< 1) . arctg n2 1+ 1 n2 1+ 1 n ! 1, 22 + 1)narctg1=(n= nlim!1 n2 + 1 = 1:nlim!11=n28., . " 11XX arctg n2 1+ 1 n12 . 1 n=1n=111 1XX1 .(.1.9),arctgn2 + 1n=1n=1 n22) . arcsin n +1 1 n +1 1 n ! 1, arcsin 1=(n + 1) = lim n = 1:limn!1n!1 n + 11=n8., 11 1 . " XX1 arcsin n + 1 n . 1 n=1n=11 11XX1 .(.1.9),arcsinn+1n=1 nn=11133) an !112=52=5an = ; ln n ; ln sin n2=5 = ; ln n sin n2=5 :F, .#.
+3sin x = x ; x6 + o(x3) x = n12=5 ! 0 n ! 1!! 116=52=5an = ; ln n n2=5 ; 6n6=5 + o(1=n ) =!14=5= ; ln 1 ; 6n4=5 + o(1=n ) 6n14=5 n ! 1:8.,." !1 1 1XX ln n2=5 ; ln sin n12=5 n14=5 . 1n=1n=11 1X n4=5 (. 1.9), 1 1 n=1 1 !Xln n2=5 ; ln sin n2=5 . >n=1-, $ !!#$#+6 5$3 !.11 lnXX2) p1) ln1 43 24n=2n=13)nnn1Xn4) ( +2 1)3n 4n=1p1X7) (3 + 1)(2p + p3 ) 4n=11 p3X2 arctg 1 410)3nnnn=1n1) 45) 49) 4nnn5)8)1 1 + sinXn=11X2n2n4sin 4n=11 1 p +1X11) p ln p ; 1 4n=2nnn2) 46) 410) 4n3) 47) 411) 4|||||1146)9)1 ;n2X4n=11Xn=11Xn=312)e2+34n3 + 5n 4n2 tg6 4n1Xn=1nsin 2n2:4) 48) 412) . 2-./0* 12 !"1-34 .
+. nX an=1an+1# a . F, 0 # (n )an+1 = D:limn!1 an; D < 1 , D > 1 . , , D = 1, 0 +. . .1-34 *5. +. nX=1 an + pn an. F, 0 + ( )pn a = C:limnn!1; C < 1 , C > 1 . , , C = 1, 0 + +. . ., D = 1 C = 1 + . . . !, , D = 1 C = 1, ,1 . @,1 2 D = C =1XX n 1=n .
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