??3n?2n1n1n1n?1n1?1n1?1n??[()?()]??()??()??()??() （2）因为?n?n?16n?123n?12n?132n?023n?03? 而由
?(1n知r?12，其和为11?r?1?2
?02)n1?12?由
?(1n113)知r?3，其和为11???3n?0r1?12 3?故?3n?2nn?1?133n?1622?3?2?2 7．设排球每一次下落后的高度依次为:
h1?34h,h2?34h1?(34)2h,
h3?34h2?(34)3h,h4,
4?34h3?(34)h,??hn?34hn?1?(34)nh??反弹的总距离s??hn3?n31n?h?3hn?1?(34)?n?14h?(34)?h? n?041?348．由已知可得：
CD?bsin?,DE?CDcos(90o??)?CDsin??b(sin?)2,EF?DEsin??b(sin?)3,FG?EFcos(90o??)?EFsin??b(sin?)4,??L=|CD|+|DE|+|EF|+|FG|+?=
?b(sin?)n??bsin??(sin?)n?bsin??1bn?1n?01?sin??sin?1?sin?
习题4-3
1． （1）R?limann??a?lim2n(n2?1)1n?1?n?1n??2((n?1)2?1)2 46
…
当x??111时，级数收敛，所以该级数的收敛域为[?,] 222
（2）R?liman1n?lim?1
n??an??1n?1n?1当x?4时，级数收敛，当x?6时，级数发散， 所以该级数的收敛域为[4,6)
nx2n?1（3）该幂级数只含有奇次幂项，记un?n，则有 n2?(?3)un?1(n?1)x2n?1(2n?1?(?3)n?1)12lim?lim?x 2n?1nnn??un??3nx(2?(?3))n当x?3时，级数收敛，当x?3时，级数发散，于是收敛半径R?当x??3时，级数发散，所以该级数的收敛域为(?3,3
3)
n2n（4）该幂级数只含有偶次幂项，记un?2(x?a)，则有
un?12n?1(x?a)2n?22 lim?lim?2x?an2nn??un??2(x?a)n当x?a?222时，级数收敛，当x?a?时，级数发散，于是收敛半径R? 222222,?a?) 时，级数发散，所以收敛域为(?a?222当x??a?2． （1）设s(x)??nxn?1x??n?1(?1?x?1)
?
?x0s(x)dx??(?nx0n?1n?1)dx??xn?n?1x1?x(?1?x?1)
?1?x?故 s(x)????(1?x)2?1?x??(?1?x?1)
x2n?1(?1?x?1) （2）设s(x)??2n?1n?1
s?(x)??x2n?2?n?1?11?x2(?1?x?1)
47
xx111s(x)?s(0)??dx??dx???0x2?1?0x2?1dx01?x2x11x11???dx?[??dx]0(x?1)(x?1)20x?1x?1
xx1111?[?dx??dx]?[ln(1?x)?ln(1?x)]01?x20x?1211?x?ln(?1?x?1)21?xx?
（3）设s(x)???(2n?1)xn?1n?n(?1?x?1)
则s(x)?2?(n?1)xn?1n??xnn?1(?1?x?1)
令u(x)??(n?1)xn?1?n?1?(?1?x?1)
?x0u(x)dx??xn?1x2?1?x(?1?x?1)
??x2?2x?x2u(x)???1?x???(1?x)2??(?1?x?1)
2x?x2x3x?x2故s(x)?2???21?x(1?x)(1?x)2xn（4）设s(x)??n?2n(n?1)??(?1?x?1)
(?1?x?1)
xn?1s?(x)??(?1?x?1)
n?1n?2s??(x)??xn?2?n?2?11?x(?1?x?1)
s?(x)?s?(0)??x0x01dx??ln(1?x)(?1?x?1) 1?xs(x)?s(0)??ln(1?x)dx?(1?x)ln(1?x)?x(?1?x?1)
习题4-4 1． (1)
48
1??4?x2211x21?41?(2n?1)!!2n???x2n?1n!23n?12?1?x?1?(?)?2?4??2?12111???1)(??n?1)2n?(?)(??x??1?22??1??2???2?n?1n!?4????(?2?x?2)2n?12n1?cos2x1?xn?12???(?1)（2）sinx?22n?1(2n)!(???x???)
（3）设f(x)?ln(x?1?x2)
1?22f?(x)?11?x?n?12?(1?x)111(?)(??1)(??n?1)22?1??2(x2)nn!n?1
??1??(?1)nx(2n?1)!!2nx(2n)!!?n(?1?x?1)(2n?1)!!2nxdx(2n)!!(?1?x?1)f(x)?f(0)??dx??(?1)0n?1??x0(2n?1)!!?x??(?1)nx2n?1(2n)!!(2n?1)n?1（4）a?exxlna
(lna)nn??xn!n?0?(???x???)
（5）设f(x)?(1?x)ln(1?x)
f?(x)?1?ln(1?x)?1??(?1)n?1?n?1xnn(?1?x?1)
f(x)?f(0)??dx??(?1)0n?1x?n?1?x0xndxn?x??(?1)n?1n?1?xn(n?1)?n?1
(?1?x?1)（6）
x1?x2?x[1??(?1)nn?1(2n?1)!!2nx]
(2n)!!(?1?x?1)
?x??(?1)nn?1?(2n?1)!!2n?1x(2n)!! 49
2． f(x)?1111?????3x2?3x?2x?1x?2111?? x?42x?41?31?2 ??13?? 1n?1??1nn(x?4)n?02n(x?4)
n?032
???(11n?02n?1?3n?1)(x?4)n(?6?x??2)
??1?x?4注：收敛域：???3?1?????7?x??1?6?x???6?x??2 ??1?x?4?1??2?23． （1）18o??10
cos?1?110?1?2!(10)2?4!(?10)4?1?6!(10)6??
|r?16!(?2|10)6?10?4
cos??1?1(?)21?102!10?4!(10)4?0.9511
?（2）
11?x4?1??(?1)n(2n?1)!!4nx?1)
n?1(2n)!!x(?1?1?2101?x4dx?1?2??(?1)n(2n?1)!!2n)!!(4n?1)(14n?12)
n?1(|r5!!2|?6!!?113?(1132)?10?4 1?2101?x4dx?12?110?(12)5?124(12)9?0.4969 ?4． 设s(x)??n2xn???x???
n?1n!???s(x)??nxn??1n1n?2)!x??xn?x2ex(n?1)!?xexn?1n?2(n?1(n?1)!?则?n2?s(1)?2e
n?1n! 50
(???x???)