Blah blah blah blah
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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