1. Ряды (853737), страница 9

Просмтор этого файла доступен только зарегистрированным пользователям. Но у нас супер быстрая регистрация: достаточно только электронной почты!

Текст из файла (страница 9)

&(11 jxj < 1,limf(x)=lim=nn!1n!1 1 + xn1=2 x = 1.; nlim!1 fn(x) 6= 0 (4.4) x,  jxj 1 x 6= ;1:; , . + (;1 ;1) (1 +1): > 4.6. @ . 1 (;1)n 1 ; x !nX1+x :n=1 n< !. + D = IR n f;1g:!, x = 1 . x 6= 1 x 2 D . . jfn+1(x)j :F. 8 0 . d = nlim!1 jfn(x)jn+1 j1 + xjnjfnj1;xjn+1 (x)jd = nlim!1 jfn(x)j = nlim!1 n + 1 j1 + xjn+1 j1 ; xjn =n11;x1;x1;x :== nlim!1 1 + x !1 1 + x 1 ! = nlimn 1+n1 + n1 1 + x F x,  d< 1 . x,  d > 1, ( ). x,  d = 1, + . . .

@ 0 . &21;xx = 1 + x () x = 0:d = 1 () 1 + x = 1 () 4 11 ;; x = ;1 ; x132 x = 0 1X(;1)n n1 :(4:5)n=1! N , .,  (. 3.5). , (4.5), , 0 , (4.5) . , ., , x = 0 . # d < 1: &88>1 ; x < 1>;2x < 0>>>>>>< 1+x< 1+x1;x() >()d < 1 () 1 + x < 1 () >>>1;x2>>>>: 1 + x > ;1: 1+x > 08<() : x1 +> x0> 0() x 2 (0 +1):; , (0 +1) . . x = 0 . >-, $ !!#$#+6 5$6 .1 (3 ; 4)n1nXX1)2)2n 43n 4n=1n=1 1 +1 n1XX1 42)45)ln(1+4)n( ; 2)n=1n=111 n;1XX8)47)2 sin 3n;1 4nn=1n=1nxxxxn xnxx1 tgnX3)4xn=1n1 1X4n nn=1 41 ( + ) !nX9)6)xx xn=11 7 1) 3 3 { 0 42) (;1 ;1) (;1 1) (1 +1) { 0 4+1 3); 4 + 4 + { 0 ,n=;1 = ; 4 +2 ZZ " 4xnnn n133nn:4) (;1 1) (3 +1) { 0 4 = 1 " 4pp5) (; ; 1 ; 1) { 0 4 116) ;1 ; 4 4 +1 { 0 47) (;1 +1) { 0 48) (;1 ;1) (1 +1) { 0 49) (;1 1) { 0 .xee||||| -( 3Sn(x) = f1(x) + f2(x) + : : : + fn(x)1X n- ".

fn(x).n=13S (x) = nlim!1 Sn(x) ". , 0  .C x, " fn(x)  nlim!1 Sn(x). 4.7. @ 1 nXx:n=0< @ Sn(x): &Sn(x) = f0(x) + f1(x) + ::: + fn;1(x) =8n>< 1 + x + x2 + : : : + xn;1 = x ; 1 x 6= 1=>x;1: n x = 1:; S (x) = nlim!1 Sn(x) nlim!1 Sn(x): &8>1 jxj < 1>>< 1;xlimS(x)=>n!1 n1 jxj > 1 x = 1>>: 9n x = ;1:1348., S (x) = 1 ;1 x jxj < 1: . x 1 nXx .>n=0 4.8. @ 1X1:n=1 (n + x)(n + x + 1)< @ Sn(x) = f1(x) + f2(x) + ::: + fn (x): ; fn(x) = n +1 x ; n + 1x + 1 ! 1! 111Sn(x) = x + 1 ; x + 2 + x + 2 ; x + 3 + : : : +! 1!111+ x + n ; 1 ; x + n + x + n ; x + n + 1 = x +1 1 ; x + n1 + 1 x 62 f;n ; 1 ;n ::: ;1g:8.,! 111 ;=limS(x)=limnn!1n!1 x + 1 x + n + 1x+1 x 6= ;n n 2 N:; , S (x) = x +1 1 x 6= ;n n 2 NII: >-, $ !!#$#+6 5$3  "! %"< , !:1 11 n n;11XXX143)1) ( ;)42)nn ctg xn=0n=1n=0 ln (2 ; 1)xxxx ;1 2 (;1 1)1) 0 = 141 + 1 + 1ln(2;1)2) ln(2 ; 1) ; 1 , 2 2 2 2 +1 4ctg x3) ctg x ; 1 , 12 2 + 2 NIIxxxxxxx< x <eeek < x <k|||||135k:: 50/2/0 12><8/*=/0>/@ 0 -( .

S11(x) S2(x) : : : Sn(x) : : : { ".X fn(x) S (x) { .n=1@ , " S (x)  . " Sn(x), " fn (x): @, " Sn(x) . , " S (x) , .. " Sn(x) + . " S (x) .. 5.1. < . Sn(x) = xn n 2 NII x 2 G0 1]: !, " Sn(x) G0 1] . n. !,"(0 x 2 G0 1)S (x) = nlimS(x)=n!11 x = 1 x = 1: > 5.2. < .!8>< n3x 1 ; x x 2 G0 1=n]Sn(x) = >:n0 x 2 (1=n 1] n 2 NII: ;1Z=n1!Z1Z13Sn(x)dx = n x n ; x dx + 0dx =001=n10 23 1=nxx= @n2 2 ; n3 3 A = 12 ; 13 = 16 :0; S (x) = nlim!1 Sn(x) = 0 x 2 G0 1] ( x = 0 0 , x 2"(0#1] 0 . fSn(x)g n0 = x1 + 1  .), Z10Z1S (x)dx = 0dx = 0:0136819>><Z=; , , .. >: Sn(x)dx>! 0Z1 S (x)dx: >0 , #..

. + ("") , ., 0 ". , , 1.X. D { . ". fn(x) ..n=11X x 2 D fn(x) .n=11X8 + D fn(x) n=1 0 +, " > 0  n0  x , n > n0 x 2 D jSn(x) ; S (x)j < ": 5.3. . "Sn(x) = 1 +nxn2x2 n 2 NII x 2 IR n- ". .&. 0 ..< @ . .nxnxS (x) = nlim!1 2 1 2! =!1 Sn(x) = nlim!1 1 + n2x2 = nlimn n2 + xx !=0= nlim!1 1 2n n2 + x x 2 IR:8., ". x IR.137,, 0 . IR. 80 . .

" Sn(x) IR. nx !0 n(1 + n2x2) ; nx2n2x3 2n;nx C0=Sn(x) = 1 + n2x2 =(1 + n2x2)2(1 + n2x2)2Sn0 (x) = 0 () n ; n3x2 = 0 () x = n1 C 1! 1! n n = 1:Sn n =21 + n2 n128., n 1 1! ,1 x = n , jSn(x) ; S (x)j = Sn n ; 0 = 2 > " ( ". .#, 1=2). % ,  IR: > 5.4.

. "Sn(x) = xe;n x n 2 NII x 2 IR n- ". .&. 0 ..< @ . .;n x = 0S (x) = nlim!1 Sn(x) = nlim!1 xe x 2 IR: 8., ". x IR.,, 0 . IR. 80 . . " Sn(x) IR.Sn0 (x) = (xe;n x )0 = e;n x + xe;n x (;2n2x) = e;n x (1 ; 2x2n2)C2 22 22 22 22 22 2pSn0 (x) = 0 () 1 ; 2x2n2 = 0 () x2 = 21n2 () x = 22 n1 Cp0 p121Sn @ 2 n A = 22 n1 e;1=2:1388., ."p10 p n1 01122 Sn(x) + @; 2 e;1=2 n A @ 2 e;1=2 n A . x 2 IR pjSn(x)j 22 e;1=2 n1 :', x IR: S (x) = 0, pjSn(x) ; S (x)j 22 e;1=2 n1 (5:1); " > 0 + n0 , p2 e;1=2 1 < "2n0 n n0 #pp2 e;1=2 1 2 e;1=2 1 2n 2n0 " (5.1) , " > 0 n0 ( x) , n n0 x IR jSn(x) ; S (x)j < ": % , IR: >-, $ !!#$#+6 5$" %"< n( ) 2 INI 0 -! " %"< . 3 - "0 .1) n( ) = sinn2 @0 ]42) n( ) = ;nx2 2 @0 +1)4!122 @;1 1]43) n( ) = (1 + ) 1 ; (1 + 2)n!124) n( ) = (1 + ) 1 ; (1 + 2)n2 @1 2]5) n( ) = ;nx 2 @0 +1)SSxxSxxSxxSxxexxnnSxxx1) 43) 45) .xxxnxe::2) 44) 4|||||139x ," ,# { ".

.1XE ". fn (x) x 2 D  n=11X +.  an , x 2 Dn=1 jfn(x)j an ". + D . 5.5. &. . 1 sin nxX:n=1 n3< ; x IR sin nx 1 n3 n31 1X n3 +. , n=1 ,# ". IR . > 5.6. &. . 1X1p x 2 G0 +1):n=0 2n 1 + nx< ; x 0 p11 1 () p 11 + nx 1 () p1 +nx2n 1 + nx 2n1X 21n +.

( n=0  ), ,# ". G0 +1) . >140 5.7. F., !n1 ln 1 + xXnxnn=1 + G1 + " +1), " { +. .< F x 2 G1 + " +1) !!nnln 1 + xln 1 + 1 + "nn 0!nln 1 + 1 + "= 0:nlim!1n8., +. C , !nln 1 + 1 + "0<< C:n; x 1 + " " { +. , !nln 1 + x0 < nxn (1 +C ")n :1X! , (1 +C ")n +.n=1 (  ), ,# , ". G1 + " +1), " { +.

, . > 5.8. &. . 1 xX x 2 G0 +1):n=1 1 + n4 x2141< 8 . ". . & x !0 1 + n4x2 ; 2xn4x4 21;n0fn(x) = 1 + n4x2 = (1 + n4x2)2 = (1 + n4xx2)2 Cfn0 (x) = 0 () 1 ; n4x2 = 0 () x2 = n14 () x = n12 C 1!fn n2 = 21n2 :! , x 2 G0 +1) 0 fn(x) 21n2 1 1X . 2n2 +. ,n=1 ,# ". G0 +1) . >-, $ !!#$#+6 5$3"  B;,  "0 %"< " ".1 p1 ; 4n1 cosXX1)2@;11]42)2 (;1 +1)423nn=1n=11 2 ;nx1 5 + sinXX3)2 @0 +1)44)2 (;1 +1)4nnxn=1n=1 3 +1 5 ; cos1XX1p2@0+1)46)2 (;1 +1)45)n2nxn 1+n=1 2 +n=0 21 ;n2 x2X7)2 (;1 +1)2xx exnxen=1nxxnxnxxexxe:|||||142xx 6-/ .

/0 0812 -//@ 04" + 1Xan(x ; x0)nn=0 x0 an { . , x { . .8 { "..1X& 6.5 , an(x ; x0)n n=0 . x = x0, x IR  +. R, 0 (x0 ; R x0 + R) Gx0 ; R x0 + R].7 + x = x0 R + ., .. + 0 + . ..7 R , (x0 ; R x0 + R) { .1XE an(x ; x0)n . x = x0n=0 R = 0. E x IR R = +1. 6.1. @ .

1 nXn!x :n=0< , , F. &:jfn(x)j = n!jxjnC jfn+1(x)j = (n + 1)!jxjn+1Cjfn+1(x)j = lim (n + 1)!jxjn+1 = lim n!(n + 1)jxjnjxj =limn!1n!1n!1 jfn (x)jn!jxjnn!jxjn(1 x 6= 0= nlim(n+1)jxj=!10 x = 0:8., x = 0 . . !. + f0g. R = 0: >143 6.2. @ . 1 xnX:(2n)!n=0< , , F.

&:n+1njfn(x)j = (2jxnj )! C jfn+1(x)j = (2(jnxj+ 1))! Cn+1(2n)!jfjxjn+1 (x)jnlim!1 jfn(x)j = nlim!1 (2n + 2)!jxjn =njxj(2n)!jxjjxj=lim= nlim!1 (2n)!(2n + 1)(2n + 2)jxjn n!1 (2n + 1)(2n + 2) = 0 < 1 x IR:8., x IR. . . (;1 +1): R = +1: > 6.3. @ . 1 (x ; 1)2nX:n=1 n9n< (.) -#.&:2nfn(x) = (x n;91)n Cvvu2 u2nqnu(x ; 1) nunt (x ; 1)t1 =f(x)=lim=limlimnn!1n!1n!1n9n9nvu22u(x;1)(x;1)nt 1= 9 nlim!1 n = 9 :2(x;1)! , , 9 < 1 ., jx ; 1j < 3 , jx ; 1j > 3: 8., R = 3, (1 ; 3 1 + 3) = (;2 4):144& .

./, x = ;2, x = 4, 1 32n X1 1X=:nn9nn=1n=1% , , . 8.,. (;2 4): > 6.4. @ . 1 (n!)2 nXx:n=1 (2n)!< &. F. , .22(n!)((n+1)!)njfn(x)j = (2n)! jxj C jfn+1(x)j = (2(n + 1))! jxjn+1C2n+1 (2n)!jf((n+1)!)jxjn+1 (x)jnlim!1 jfn(x)j = nlim!1 (2n + 2)!(n!)2jxjn =222njxj(2n)!(n+1)jxj(n!)(n+1)jxj= nlim!1 (2n + 1)(2n + 2) =!1 (2n)!(2n + 1)(2n + 2)(n!)2jxjn = nlim!212n 1+ n1 jxj:!!= jxj nlim=!1 244n 1 + 21n 1 + 22n! , , 14 jxj < 1 ., jxj < 4 , jxj > 4: 8., R = 4 (;4C 4):& . .

x = 4 1 (n!)2 nX4:n=1 (2n)!145@ # aan+1 : &:n2 4n2 n+1(n!)((n+1)!)4 Can = (2n)! C an+1 = (2(n + 1))!an+1 = ((n + 1)!)24n+1(2n)! =an(2n + 2)!(n!)24n(n!)2(n + 1)24n4(2n)! = (n +!1)24n+1 > 1= (2n)!(2=n + 1)(2n + 2)(n!)24n 4 n + 1 (n + 1) n + 122 n 2 NI .% , fang , 0, .,, nlim !1 an 6= 0: ! 1X (n!)2 n4 . : (2n=1 n)! x = ;4 1X(n!)2 4n:(;1)n (2n)!n=1/, , 0 , 1 (n!)2 nX(2n)! 4 n=1 #. J ,  0 , 0 ,  1X(n!)2 4n(;1)n (2n)!n=1 , , ., , 0 .&, , . (;4 4): > 6.5. @ . 1 xnXp :n=1 n146< , , F.

Характеристики

Список файлов лекций