# Power series

A power series in $x$ is an infinite series of the form $a_0 + a_1 x + a_2 x^2 + \dotsb$, a sum of powers of $x$. It can also be written in the shorthand form $\sum_{n=0}^\infty a_n x^n.$

Such series can be used to represent many functions such as $1/(1+x)$, $\sin x$, $\cos x$ and $e^x$. They may only be valid for some values of $x$. For example, $\frac{1}{1+x}=1-x+x^2-x^3+\dotsb$ is only valid when $|x|<1$, but the series for $\sin x$, $\cos x$ and $e^x$ are valid for all values of $x$.