考研数学之幂级数展开与求和

考研数学之幂级数展开与求和

来源：文都图书

级数在考研数学中属于数一和数三要考查的内容，其核心内容为幂级数展开与求和，今天我们就来详细学习一下幂级数的展开与求和步骤。

幂级数展开与求和在考试中常以解答题形式出现。要学好展开与求和，首先，我们需要两大工具：1、常见泰勒级数及收敛域;2、逐项展开与逐项求导。其次，要掌握常用方法。

展开常用方法，一是直接展开，这种考法较少，二是间接展开，以这种考法居多。间接展开解题的要点如下：

(1)转化，将函数f(x)在某非零点处展开，转化到在x=0处展开。

(2)拆项，将函数拆成两项之和或差，然后利用常见函数的幂级数展开将两个展开式求和或者求差便可。

(3)因式分解，将函数分解成两项之积，一般其中一个因式为低次(至多为二次)多项式，另一个用常见幂级数展开式展开。

(4)求导法，先对函数求导，再用常见幂级数展开式展开，最后逐项积分。

(5)积分法，先对函数积分，再用常见幂级数展开式展开，最后逐项求导。

幂级数求和是展开的逆问题，比展开要难，考研中常用到的方法如下。

(1)直接套用已知的基本展开式，后者拆后套用。

(2)系数的分母中含有n的阶乘的，考虑用指数函数,或者正弦函数与余弦函数的某种组合。

(3)系数的分母中含有n、n+1、n+2的可以先逐项求导。系数的分子中含有n、n+1、n+2的可以先逐项积分。

除此之外，展开与求和部分还会考一些综合性题目，如跟微分方程结合在一起考查。总之主要方法还是如上综述的方法。望考生们多

联系，以体会上述方法。此外建议考生找一些类似的题目，强化练习。学会利用其方法和技巧，考研数学会涉及很多题目考察很多知识点，对待这些题目，我们要从运用的基本知识，及其解题方法，从理论到实践系统性的掌握，建议参考一下汤家凤的2017《考研数学复习大全》认真备考吧，预祝考试顺利。

When you are old and grey and full of sleep,

And nodding by the fire, take down this book,

And slowly read, and dream of the soft look

Your eyes had once, and of their shadows deep;

How many loved your moments of glad grace,

And loved your beauty with love false or true,

But one man loved the pilgrim soul in you,

And loved the sorrows of your changing face;

And bending down beside the glowing bars,

Murmur, a little sadly, how love fled

And paced upon the mountains overhead

And hid his face amid a crowd of stars.

The furthest distance in the world

Is not between life and death

But when I stand in front of you

Yet you don't know that

I love you.

The furthest distance in the world

Is not when I stand in front of you

Yet you can't see my love

But when undoubtedly knowing the love from both Yet cannot be together.

The furthest distance in the world

Is not being apart while being in love

But when I plainly cannot resist the yearning

Yet pretending you have never been in my heart. The furthest distance in the world

Is not struggling against the tides

But using one's indifferent heart

To dig an uncrossable river

For the one who loves you.