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.