Power series

A power series is a series in the form

$\sum_{n=0}^\infty c_n(x-a)^n$

Power series can be used to represent functions in the form

$\frac{a}{1-f(x)}$

since this is the formula for an infinite geometric series. For example,

$\frac{x}{1+5x}=x\sum_{n=0}^\infty(-5x)^n$

over the interval of convergence, which is $|x|<\frac{1}{5}$

Functions can sometimes be differentiated or integrated to put them in a form where they can represented by a power series. For example,

$\frac{d}{dx}\arctan(x)=\frac{1}{1+x^2}=\sum_{n=0}^\infty(-1)^n x^2n$

so

$\arctan(x)=\int\sum_{n=0}^\infty(-x^2)^ndx=\sum_{n=0}^\infty(-1)^n\frac{x^{2n+1}}{2n+1}$

An important application of power series is in Taylor and Maclaurin series.