# Criteria for a power series representation?

I've used many different power series representations of functions and seem to always take it for granted that functions which are "nice" and continuous have such a representation.

But what is the criteria for a function to have a power series representation? I know of some that don't, but how can one tell if a function can be represented as such?

EDIT:

I may as well ask, is there a proof or derivation for the power series identity

f(x) = $\sum$$^{\infty}_{n=0}$ a$_{n}$ x$^{n}$

???

Last edited: