4.4 関数項級数

1.

(a)

$\displaystyle \log(1-x)$	$\displaystyle =$	$\displaystyle -\int_{0}^{x}\frac{1}{1-t}dt$
	$\displaystyle =$	$\displaystyle -\int_{0}^{x}\sum_{n=0}^{\infty}t^{n}dt$
	$\displaystyle =$	$\displaystyle -\sum_{n=0}^{\infty}\int_{0}^{x}t^{n}dt$
	$\displaystyle =$	$\displaystyle -\sum_{n=0}^{\infty}\frac{x^{n+1}}{n+1}$

(b)

$\displaystyle \tan^{-1}{x}$	$\displaystyle =$	$\displaystyle \int_{0}^{x}\frac{1}{1+t^{2}}dt$
	$\displaystyle =$	$\displaystyle \int_{0}^{x}\sum_{n=0}^{\infty}(-1)^{n}t^{2n}dt$
	$\displaystyle =$	$\displaystyle \sum_{n=0}^{\infty}(-1)^{n}\int_{0}^{x}t^{2n}dt$
	$\displaystyle =$	$\displaystyle \sum_{n=0}^{\infty}(-1)^{n}\frac{x^{2n+1}}{2n+1}$