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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) = [itex]\sum[/itex][itex]^{\infty}_{n=0}[/itex] a[itex]_{n}[/itex] x[itex]^{n}[/itex]

???

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) = [itex]\sum[/itex][itex]^{\infty}_{n=0}[/itex] a[itex]_{n}[/itex] x[itex]^{n}[/itex]

???

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