微积分下 第二版 课后习题答案 同济大学(14)

?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)

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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)

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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???)