数学分析试题集锦（一）：13

? E?c a) ? x1 a) ?￥ l ???D l ?u× 1.p/ ò a) ?￥ l ?×. (1) 1P n=1 (2x)n n! ; (2) 1P n=1 ln(x+1) n+1 x n+1; (3) 1P n=1 [(n+1n )nx]n; (4) 1P n=1 xn2 2n ; (5) 1P n=1 [3+(?1)n] n x n; (6) 1P n=1 3n+(?2)n n (x+1) n; (7) 1P n=1 (2n)!! (2n+1)!!x n; (8) 1P n=1 (1+ 1n)?n62xn; (9) 1P n=1 (?1)n n npn x n; (10) 1P n=1 xn 5n+7n; (11) 1P n=1 (n!)2 (2n)!x n; (12) 1P n=1 (1+ 12 +￠￠￠+ 1n)xn; (13) 1P n=1 nxn; 1 (14) 1P n=1 (x?2)2n?1 (2n?1)! ; (15) 1P n=1 an2xn (0 < a < 1); (16) 1P n=1 xn np: 2
 ! a) ?1P n=0 anxn￥ l ???1R
P n=0 1bnxn￥ l ???1Q
) ?/ ) ?￥ l ???
 (1) 1P n=1 anx2n
(2) 1P n=1 (an +bn)xn
(3) 1P n=0 anbnxn . 3
 !1P n=0 akxk1?M (n = 0;1;:::;x1 > 0)
p￡
?0 < x < x1 H
 μ (1) 1P n=0 anxn l ?
(2) flfl flfl 1P n=0 anxn flfl flfl? M. x2 a a a))) ? ? ?￥￥￥???ééé 1
 !f(x) = 1P n=0 anxn?jxj < r H l ?
* 1?1P n=0 an n+1r n+1 l ? Hμ Z r 0 f(x)dx = 1X n=0 an n+1r n+1 ? ?1P n=0 anxn?x = r H ^? l ?. 2. ?¨ 5￡ üR10 1n(1?x)x dx = ? 1P n=1 1 n2. 2 3.¨?[±s?[sp/ ) ?￥?
 (1) 1P n=1 xn n ; (2) 1P n=1 nxn ; (3) 1P n=1 n(n+1)xn ; (4) 1P n=1 (?1)n?1 n(2n?1)x 2n ; (5) 1P n=1 n2+1 n!2n x n ; (6) 1P n=1 (?1)nn3 (n+1)! x n ; (7) 1P n=0 x4n?1 4n+1; (8) 1P n=0 (2n+1 ?1)xn ; (9) 1P n=1 n2xn?1 ; (10) 1P n=1 (2n+1)2 n! x 2n+1 . 4.p/ ) ?￥?
 (1) 1P n=1 2n?1 2n ; (2) 1P n=1 1 n(2n+1) . 5.￡ ü
 (1) 1P n=0 x4n (4n)! ?@Z?y (4) = y
3 (2) 1P n=0 xn (n!)2 ?@Z?xy 00 +y0 ?y = 0 . 6
 !f(x) ^ a) ?1P n=0 anxn(?R;R) ￥?f ?
?f(x)1f ?
5) ???CQ a￥[
?f(x)1}f ?
5) ???C} Q a￥[. 7
 !f(x) = 1P n=1 xn n2 ln(1+n): (1)p￡
f(x)[?1;1] ??
f0(x)(?1;1)= ??
(2)p￡
f(x)?x = ?1 V?
(3)p￡
lim x!1? f0(x) = +1
(4)p￡
f(x)?x = 1? V?. x3fff ? ? ?￥￥￥ a a a))) ? ? ?ZZZ 7 7 7 1. ?¨'?f ?￥Z T
|/ f ?Z 71 ? X ? ) ?
i a ü l ?uW. (1) 1a?x;a 6= 0; (2) 1(1+x)2; (3) 1(1+x)3; (4) cos2 x ; (5) sin3 x ; (6) xp1?3x; (7) (1+x)e?x ; 4 (8) ln(x+p1+x2); (9) 11?3x+2x2; (10) arcsinx; (11) ln(1+x+x2); (12) xarctanx?lnp1+x2; (13) Rx0 sintt dt; (14) Rx0 cost2dt: 2
 ?¨ a) ?Mep/ f ?￥ ? X ? Z 7 T
 (1) ln(1+x2)1+x ; (2) (arctanx)2 ; (3) ln2(1?x): 3
|/ f ?·??x0Z 71 ü à) ?
 (1) 1a?x;x0 = b(6= a); (2) ln 12+2x+x2;x0 = ?1; (3) lnx;x0 = 2; (4) ex;x0 = 1: 4
Z 7ddx(ex?1x )1x￥ a) ?
iw1 = 1P n=1 n (n+1)!: 5
 k|f(x) = lnxZ 7?x?1x+1￥ a) ?. 6
 !f ?f(x)uW(a;b)=￥ò¨? ?Báμ?
'iM > 0
 5 BMx 2 (a;b)
μ jf(n)(x)j? M;n = 1;2;:::; ￡ ü
(a;b)=?i?xDx0
μ f(x) = 1X n=0 f(n)(x0) n! (x?x0) 2: 6