高数下册知识点罗列(7)

xln?1?x?????x?1??1??ln???x?1??2????x?1????x?1?1ln1??ln2?????????2????n???n?1?x?1??????1?x?3?n??x?1??1???ln2????1?2nn?1????n?1n????1?1????1n???ln2??ln2???x?1?????x?1???nn?12?n?1?2???n?2?n?2???

例3：将函数f(x)? [ 解]：

x展开成x?1的幂级数。

2x2?7x?4???1x1411?4111??(?)????2?x?1??2x2?7x?4(x?4)(2x?1)9x?42x?19?31?x?131??33???n4?1?2n1?x?1nn(x?1)nnn???(?1)?()(x?1)?4(?1)?2() ???n??27n?0327n?0327n?03x?12(x?1)11?1且-1??1，故收敛区间??x? 3322x例4：将函数f(x)?展开成x的幂级数，并确定其收敛域。

1?2x又，?1??x12x1?n [ 解]：???(2x)??2n?1xn

1?2x21?2x2n?1n?12x?1??11111?x?，x??时，幂级数发散，收敛域为(?,) 222221?x展开成x的幂级数，并确定其收敛域。 1?x2）利用求导或积分进行展开 例1、将函数将函数f(x)?arctan[ 解]：1）f??x??1?1?x?1????1?x?n?2?1?x???1?x????1??1?x?2??1?x?22?1?x???1?x??1?x??222?1 21?x ????1?n?0?x2n ?1?x?1

?? 2）

?x0f??x?dx?????1?x2ndx??n?00n?0xn??1?n2n?1?x2n?1?f?x??f?0???n?0???1?n2n?1?x2n?1

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?f?0??arctan ?f?x??1?x1?xx?0?arctan1??4

?4???n?0???1?nn2n?1?x2n?1 收敛区间为??1,1? ??n?0?? x??1??2n?1?xn?0??1?n2n?1??1?nn2n?1???1?2n?12n?1???n?0????1?nn2n?1收敛

x?1???2n?1?xn?0n2n?1???1?2n?1??n?0??1?2n?1??1???n?0??1?2n?1收敛

?2n?1?xn?0??1?收敛域??1,1?

3）?f?x???4??n?0???1?n2n?1?x2n?1 收敛域为??1,1?

例2、将函数将函数f(x)?11?x11?xln?arctan?x展开成x的幂级数，并确定其收敛域。 41?x21?x??11111114n[ 解]：1）f??x????????1??1??x?1??x4n 2441?x41?x21?x1?xn?0n?1 ?1?x?1 2）

?x0f??x?dx???n?1?x0?1x4n?14n?1 xdx???x?f?x??f?0???n?14n?1n?14n?14n? ?f?0???1?x?11?x1?ln?arctan?x?1?x?41?x2??x?0?0

?x4n?1x4n?1 ?f?x??f?0??? 收敛区间为??1,1? ??n?14n?1n?14n?1?x1 x??1??发散 ?????n?14n?1n?14n?1n?14n?1??1??x4n?11 x?1??发散 ????n?14n?1n?14n?1n?14n?1?4n?1?4n?1???1?4n?1x4n?1 ?收敛域??1,1?

4n?1n?1?x4n?1 3）?f?x??? 收敛域为??1,1?

n?14n?1? 32

6、利用已知幂函数展开式求和：见上课例题

?x??????xxnxn?12??x??例1、 1） ?n??n?1????ln?1??，?1??1??2?x?2

2??n?1?n?0?n?1??2?n?12?nn?02?x??1nn?1???????1?x??5? ?n?5??n?1?n?05??n?1?n?0nn?1n?1x?x??5ln?1??，?1??1??5?x?5

5?5?4n?1 ???1??ln 5n?1n?0?n?x??1n2n?1???????1?x??3? 2）?2n?1????2n?1?!n?0?2n?1?!n?03n2n?1x?sin，???x???

3??1?32n?1?sin3 ?

n?0?2n?1?!?n??xnxn?1xn 3）????x??xex，???x???

n?1?n?1?!n?0n!n?0n!?3n ??e2

n?0n!?6、富里叶级数：1) 三角函数系正交性: 见教材

a0?2) 富里叶级数: 形如???ancosnx?bnsinnx?,其中

2n?1 an? bn?f?x?cosnxdx, ?n?0,1,2,3,...? ????1?1???f?x?cosnxdx, ?n?1,2,3,...?

??3) 函数展开成富里叶级数: 利用狄利克雷定理与上述公式展开.

例1、将下列周期为2?的函数展开成富里叶级数 f?x????x???x?0;

00?x???

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