1. Ряды (853737), страница 8
Текст из файла (страница 8)
Fn=1n=1., n + 1 = 1 C = lim pn n = 1D = nlimn!1!1 n 1=(n + 1)2 = lim n2 = 1D = nlim!1 1=n2n!1 (n + 1)2v0 12uu1 A = 1:nt 1@p=limC = nlim!1 n2 n!1 n n115 2.1. &. . :1 11 nnXX1) n! C2):n=1n=1 n!< F.1) . an = n1! , an+1 = 1=(n + 1)! = n! = 1 ! 0 n ! 1:an1=n!(n + 1)! n + 18., D = 0 <n 1 .2) . an = nn! , an+1 = (n + 1)n+1 n! = (n + 1)n(n + 1)! = n + 1 !n =an(n + 1)! nn(n + 1)! nnn!n1= 1 + n ! e n ! 1:8., D = e > 1 .
> 2.2. &. . :1 n51XX1)! :1) 5n C2) (n5+nn2n=1n=1 F.< 51) . an = n5n , an+1 = (n + 1)5 5n = 1 n + 1 !5 ! 1 n ! 1:an5n+1 n5 5 n58., D = 1=5 < 1 .1)! , 2) . an = (n5+n n2an+1 = (n + 2)! 5nn2 = (n + 2)n2 ! 1 n ! 1an 5n+1(n + 1)2 (n + 1)! 5(n + 1)2( . .
n, { ). 8., D = 1 . >116 2.3. &. . 1 1 3 : : : (2n + 1)X:22nn!n=1< F.. an = 1 3 : 2: :2n n(2! n + 1) , an+1 = 1 3 : : : (2n + 1)(2n + 3) 22nn!an22n+2(n + 1)!1 3 : : : (2n + 1) =32+3 =n ! 1 n ! 1:= 4(2nn ++ 1) 4 + 4 2n8., D = 1=2 < 1 . > 2.4. &. . 1 1 !nX1;n :n=1 -#.< !n. an = 1 ; n1 , 22pa = (a )1=n = 1 ; 1 !n = 1 ; 1 !(;n)(;1) ! e;1 n ! 1:nnnnn8., C = 1=e < 1 . > 2.5.
&. . :1 p1XX1) n2n arcsinn n1 C2) ( n)n sinn 21n :n=1n=1< -#.1) . an = n2n arcsinn n1 , pa = (a )1=n = n2 arcsin 1 nnnn117 , 0. arcsin n1 n1 n ! 1, 1 = lim n2 1 = lim n = 12C = nlimnarcsin!1n n!1 n n!1., .p2) . an = ( n)n sinn 21n , pn a = (a )1=n = pn sin 1 nn2n , 0. sin 21n 21n n ! 1, pn sin 1 = lim pn 1 = lim p1 = 0C = nlim!12n n!1 2n n!1 2 n., . >-, $ !!#$#+6 5$1. $ 0 !:11 !1XXX1 4a))4)4nnn=1 (2 + 1)!n=1n=1 2111 2XXX(2 + 1) 4) 11 43 (3)tg4)sin 2n 4+ 1)2n+1n=1n=1n=111 n1 ( !)2XXX ) 3 n ! 4) (2) (3 8n+21)! 4)! 4n=1n=1n=11 1 5 (4 + 1)X) 2 5 (3 + 2)n=1nnnn:::n:::nnnnnnn:::n:::nnn:2.
$ 0 :; !:1 ; 1 n11XXX1n 1)4)arctg4) lnn ( + 1) 42 +1nn=11 1 n2X)1+4n=1)n=1nn1 1 1 n2X1+4nn=1 2n) )1Xn=1narcsin 1 4nn1p Xn=1n=1nnn4 ;3n118n1 1 1 n2X) 3n 1 +4n=11X) 2n n sinn 1n2n4n=1nn:1. )))2. )) )44444.) 4 ) 4) 4) 4) .) 4) 4) 4) 4) 4 ) 4) 4 ) 4|||||# $ , " .1X.
an n=n01Xn=n01Xan = n=n f (n)0 f (n) . x = n +. " f (x), x n0.; 0 +Z1f (x) dx a n0:a 2.6. &. . 1 1X:n=2 n ln n< " f (x) = x ln1 x x 2. ! 1X . . 8., f (n)n=2 ++Z1Z 1 dxf (x) dx = x ln x :22119.A+Z1 dxZA dxZA d ln x =+1=lim=lim=limln(lnx)2x ln x A!+1 2 x ln x A!+1 2 ln x A!+12 , . > 2.7. &. . 1X1:n=3 n ln n(ln ln n)21< " f (x) = x ln x(lnln x)2 , x 3 . . ; 1Xf (n) n=3++Z1Z1dxf (x) dx = x ln x(lnln x)2 33 #+Z1ZAZA d ln xdxdx= A!lim= A!lim:222+1+1xlnx(lnlnx)xlnx(lnlnx)lnx(lnlnx)3338 y = ln x, +ln AlnlnZ A dyZ A d ln y1d ln x ====;222ln x(ln ln x)y(ln y) ln 3 (ln y)ln y ln 33ln 31 ; 1 ! 1 A ! +1:= ln ln3 ln ln A ln ln 38., , . > 2.8.
&. . 1X1:2n=1 n ln (3n + 1)ZA120<.13n + 1 ! 3 n ! 11:=nn ln2(3n + 1) (3n + 1) ln2(3n + 1) . " 1X12n=1 (3n + 1) ln (3n + 1) + . . ..1XF., f (n), f (x) { "n=1, . x 1.1X0 f (n) n=1 ++Z1Z1f (x) dx = (3x + 1) dx2ln(3x+1)11 #+ZAZ1dxdx=lim=22A!+1(3x+1)ln(3x+1)(3x+1)ln(3x+1)11A!AZ11dln(3x+1)= A!lim=lim;=2+1 3A!+1 3 ln(3x + 1) 1ln(3x+1)10111@A= 1 := A!lim;+1 3 ln 43 ln(3A + 1) 3 ln 48., , . >-, $ !!#$#+6 5$$ 0 !:1 11 1XXp 442)1)2lnn=2n=2 ln11XX113)44).n=3 ln (ln ln )n=1 (2 ; 1) ln(2 + 1)nnnnnnnnn1) 42) 43) 4|||||1214) . 3./0*-2// )& F , , . +.
. ;. , . .. F .1X an , n=11X janj.n=11X@ + janj .n=11X an, 0 . .n=11X an, . . +. 1 n=1Xjanj. F 0 . . ,n=1 . @+ . . + 1X, . janj n=11X. an. & , + n=1., F -#, ( , ) . , .. 3.1. F. . 1 sin nX:n=1 n2< . j sin nj < 1 . n, j sin nj < 1 :n2n21X n12 , ., n=11Xnj , . .
. > j sinn=1 n2122 3.2. F. . 1 cos nX(ln 3)n :n=1. j cos nj < 1 . n, j cos nj < 1 = 1 !n :(ln 3)n (ln 3)n ln 31 1 !nX, n=1 ln 3 q = ln13 < 1 (. 1.6). 81 cos njX., j(ln3)n , . .n=1 . > 3.3. &. . 21Xnn(;1) 2n :n=1< 2 F.. an = (;1)n n2n , <!2 an+1 = (n + 1)2 2n = 1 n + 1 ! 1 n ! 1: an 2n+1 n2 2 n28., D = 1=2 < 1 . > 3.4.
&. . 3n + 2 !n1Xn(;1) 2n ; 1 :n=1 -#.< 3n!n+2n. an = (;1) 2n ; 1 , 23+3n+2janj = janj1=n = 2n ; 1 = n1 ! 32 n ! 1:2; n8., C = 3=2 > 1 . >123qn1Xan () , 1X, janj .n=18 . # +, + ., . . { +. 0 . , , . .n=1 '( $ ;" 1X(;1)n+1 bnn=1bn > 0:; , { 0 , +., . .-34 7;. E , . .b1 > b2 > : : : > bn > : : :nlim!1 bn= 0 . 3.5. &. . 1 (;1)nX 2 IR:n=1 n< & (.
1.9), > 1 . E 0, n ! 1, 0 . 0 < 1 + .. N. ,0 bn = 1=n { ..,124 n ! 1, 0 N.1 (;1)nX&, 0 > 0, n=1 n0 > 1 . > 3.6. &. . :1 (;1)n11 (;1)nXXXlnnn1) n ln n C2) (;1) n C3) n ; ln n :n=2n=1n=1< 1) . bn = n ln1 n { .. nlimbn = 0, N. !11X0 n ln1 n .
(. n=22.6). 8., .2) +, bn = lnnn { .ln x . nlimb=0.F0+,"f(x)=n!1x xlimf(x)=0..!1xf 0 (x) = 1 ;xln2. x > e N0lnx(lnx)1 = 0lim=lim=limx!1 xx!1 x0x!1 x " f (x) x ! 1. 8., bn { .. nlim!1 bn = 0.0 N.. . n > 2 ln n > 1n n11XX n1 , lnnnn=1n=1+ . 8., .1253) +, bn = n ;1ln n { .. nlim!1 bn = 0.
F 0 ., .. cn = n ; ln n nlim!1 cn = 1.., , .. lnnn n ! 1, .. cnn = 1 ; lnnn , ,cn + .. nlim!1 cn = 1.8., bn = c1 { .. nlimb=0.0 N.n!1 n. . n 1 1n ; ln n n1X n1 , + n=11X n ;1ln n . ; , . > n=1-, $ !!#$#+6 5$3 0"0 ""0 .11 (;1)nXX1) (;1)n 5 ; 2 42)3 ; 14nn=13)n=1n1 (;1)nXp 4n=1n1X1 (;1)nX4n=2 ln1X+ 1) 46) (;1)n 11 43 (3(2 + 1)n=11nX(;1)q8)ln ln(ln )n=34)n + 1 n(;1)n 32 ;2 4n=11X7) (;1)n 1 3 !(2 + 1) 4n=15)nnnn:::1) 45) .4n:::n:::n:nn2) ".46) 43) o .47) .4|||||126nn4) c ".48) ".( $ 1X (;1)n+1 bn, n=1 N.
E + . .. F 0 n- sn = b1 ; b2 + b3 ; : : : + (;1)n+1 bn n- 1Xrn =(;1)k+1 bk = (;1)n (bn+1 ; bn+2 + : : :):F k=n+1rn :jrnj < bn+1:; , S 1 +X (;1)n+1bn +. S sn, # n=1 rn bn+1.n+11 3.7. ,. X (;41)n2 . 0 01.n=1< F N, bn = 41n2 { .. nlim!1bn =0.F n- rn 0 jrnj < bn+1. @+ n, . jrnj < 0 01.F .
n = 4. ;jr4j < b5 = 4 125 = 0 01:; , S . 0 01 s4, . .1 + 1 ; 1 = 115 : >S s4 = 14 ; 1636 64 5761 (;1)n+1X 3.8. ,. . 0 001.n=1 2n!< F N, bn = 21n! { .. nlim!1 bn = 0.8., n- rn 0 127jrnj < bn+1. @+ n, . jrnj < 0 001.F . n = 5. ;1 < 0 001:jr5j < b6 = 2 16! = 1440; , S . 0 001 s5, . .1 ; 1 + 1 = 19 : >S s5 = 12 ; 14 + 1248 240 60-, $ !!#$#+6 5$B! "" 0 .11n+1n+1XX= 0 0142) (;1)31) (;31)2n=1n=111n+1n+1XX= 0 0144) (;1)n3) (;1)!n=1n=1"nn""nn"= 0 014"= 0 001:1) 5 = 3019 410800Ss2) S4=s1549 417283) 4 = 58 4|||||128Ss4) 4 = 54136912 .Ss 4<8/*=/0>/ -( .
.., 0 "f1(x) f2(x) : : : fn(x) : : : + x, + D:6+ + 1Xf1(x) + f2(x) + : : : + fn(x) + : : : = fn(x):(4:1)n=18, , .E (4.1) x . + D, . , . , .$+ x D, (4.1) , *+ .; , . ". . ". . 4.1. @ . 1 nXx:(4:2)n=0< ; x, + (;1 1), (4.2) ,, , , . x (4.2) (.
1.6), . ". (4.2) (;1 1).; x, + (;1 0), (4.2) , (;1 1) . ". (4.2). >129 4.2. @ . 1 1X:(4:3)n=1 nx< ; . x ". (4.3) , , x > 1, , x 1 (. 1.9), . ". (4.3) (1 +1).; + x (1 +1) (4.3) , (1 +1) . (4.3). > 4.3. @ . 1X(3 ; x2)n:n=1< ;1 x, j3 ; x2 j< 1X (3 ; x2)n n=1 ( ), . x (. 1.6), .1X (3 ; x2)n j3 ; x2j < 1 :n=1j3 ; x2j < 1 () ;1 < 3 ; x2 < 1 () ;1 < x2 ; 3 < 1 ()ppp() 2 < x2 < 4 () 2 < jxj < 2 () x 2 (;2 ; 2) ( 2 2):&,p .p P (;2 ; 2) ( 2 2):!, . 0 . .
> 4.4. @ . 1 n xX2 tg 3n :n=1< @ . , .. + x, .! 0 + D:() x D = x 2 IR 3n 6= 2 (2k + 1) n 2 NII k 2 ZZ :130 x = 0 { . . x 6= 0 x 2 D . . F.jfn+1(x)j : 8 0 . d = nlim!1 jfn(x)jx2n+1 tg x tg 3n+1 n+1 jfn+1 (x)j3d = nlim!1 n x = 2 nlim!1 tg x =!1 jfn(x)j = nlim2 tg 3n 3n x 3nx+1 = 2 < 1:= 2 nlim !1 n 338., F x + D, .; x = 0, . +)(nD = x 2 IR x 6= 2 (2k + 1)3 n 2 NII k 2 ZZ : > 4.5. @ .
1 1X:(4:4)n1+xn=1< @ . . &D = x 2 IR x 6= ;1 = IR n f;1g:& . . jfn+1(x)j F. 8 0 . d = nlim!1 jfn(x)j :njjfj1+xn+1(x)jd = nlim!1 jfn(x)j = nlim!1 j1 + xn+1j :E jxj > 1, 1njxj 1 + xn 1 < 1:d = nlim=!1 n+1 jxj 1 + xn1+1 jxj1318., F (4.4) x, jxj > 1. E + jxj 1x 6= ;1 d = 1, 0 , F (4.4). . 80 . nlim!1 fn(x) , jxj 1 x 6= ;1 . 0 .
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
