Can there be multiple power series representations for a function?

[dumbQuestion]

I guess this is a simple question. Say I am tasked with finding the Taylor series for a given function. Well say that the function is analytic and so we know there's a taylor series representation for it. Am I gauranteed that this representation is the only power series representation for it? Or could there be others? What if the function is not analytic but we can create a power series representation on some restricted interval? Is that power series unique for it? Also, what if I have a function that I can construct a power series on more than one interval in the domain. How do those series relate to each other?

 

[Erland]

If a function is analytic in a symmetric neighbourhood of a point a, then the Taylor series about a is the only power series about a which represents the function in this neighbourhood. Because if we differentiate the power series repeated times and evaluate at a, we find that the coefficients of the series are determined by the derivatives of the function at a.

And if there is a power series representation about a in such a neighbourhood, then the function is analytic there (it could be taken as a definition of "analytic") and the series is the Taylor series about a.