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 \(x\).
