# Criteria for a power series representation?

1. Sep 12, 2012

### Master J

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: Sep 12, 2012
2. Sep 12, 2012

### mathman

Assuming the power series is correct for small enough x, then a radius of convergence (r) can be defined where f(x) = power series for |x| < r.

3. Sep 12, 2012

### Stephen Tashi

As mathman suggests, the question of whether f(x) can be expanded as a power series is not a yes-or-no question. Some power series converge for all values of x; some power series only converge for particualr values of x.

In the wikipedia article on analytic functions (http://en.wikipedia.org/wiki/Analytic_function) under the section "alternate characterizations", conditions are given for a real valued function to be expandable in a power series for all points x in an open set D.