Power Series

Form

$$\sum _{n=0}^{\infty }{a}_n{x}^n$$

Can also be centered around a value c

$$\sum _{n=0}^{\infty }{a}_n(x-c{)}^n$$

Intervals of Convergence

Elementary Function Series

$f(x)$	Maclaurin Series	IOC
$\frac{1}{1-x}$	${\sum }_{n=0}^{\infty }{x}^n$	$(-1,1)$
$\ln (1-x)$	$-{\sum }_{n=1}^{\infty }\frac{x^n}{n}$	$[-1,1)$
$\ln (1+x)$	${\sum }_{n=1}^{\infty }(-1{)}^{n+1}\frac{x^n}{n}$	$(-1,1]$
$\arctan x$	${\sum }_{n=0}^{\infty }(-1{)}^n\frac{x^{2n+1}}{2n+1}$	$[-1,1]$
$e^x$	${\sum }_{n=0}^{\infty }\frac{x^n}{n!}$	$(-\infty ,\infty )$
$\cos x$	${\sum }_{n=0}^{\infty }(-1{)}^n\frac{x^{2n}}{(2n)!}$	$(-\infty ,\infty )$
$\sin x$	${\sum }_{n=0}^{\infty }(-1{)}^n\frac{x^{2n+1}}{(2n+1)!}$	$(-\infty ,\infty )$