THOMAS’ CALCULUS 学习笔记（Chapter 10-7)

KlavierP

于 2020-07-23 16:40:54 发布

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本文介绍了幂级数的概念、收敛性及其半径，包括如何使用比值测试确定收敛区间。此外，还阐述了幂级数的乘法、对连续函数的幂级数展开及逐项求导和积分的定理。

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文章目录

Power Series

Power Series

Power Series and Convergence

DEFINITIONS
A power series about $x = 0$ is a series of the form
$\sum\limits_{n=0}^\infin c_nx^n=c_0+c_1x+c_2x^2+\dots+c_nx^n+\dots.\tag{1}$ A power series about $x = a$ is a series of the form
$\sum\limits_{n=0}^\infin c_n(x-a)^n=c_0+c_1(x-a)+c_2(x-a)^2+\dots+c_n(x-a)^n+\dots\tag{2}$ in which the center $a$ and the coefficients $c_0,c_1,c_2,\dots,c_n,\dots$ are constants.

Using the Ratio Test to see where it converges and diverges.
e.g.
$\text{(a)}\quad\displaystyle\sum\limits_{n=1}^\infin(-1)^{n-1}\frac{x^n}{n}=x-\frac{x^2}{2}+\frac{x^3}{3}-\cdots$
Solution
Apply the Ratio Test to the series $\Sigma|u_n|$ , where is the $n$ th term of the power series in question.

$\text{(a)}\quad\displaystyle\left|\frac{u_{n+1}}{u_n}\right|=\left|\frac{x^{n+1}}{n+1}\cdot\frac{n}{x}\right|=\frac{n}{n+1}|x|\to|x|$

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