Power Series

Premise

Blah blah blah blah

Motivation

Applications of Power Series

Possibly the most basic reason why power series are useful is that they allow you to approximate any (possibly convoluted, but smooth) function $f(x)$ by its (truncated) power series $f(x) = \sum_{n=1}^{\infty} a_nx^n = \sum_{n \le N} a_nx^n + O(x^{n+1})$.

Power series are easy to manipulate and understand, general functions— not necessarily.

From an answer regarding the utility of Power Series on the [Math StackExchange](http://math.stackexchange.com/questions/564612/what-are-some-practical-uses-of-power-series#answer-564668).


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