Can there be more than one power series expansion for a function. I guess the answer is NO, for polynomials.
I know that there are more than one series expansion for every function.
But I am talking about power series here.
All Ideas are appreciated 

If a power serie is converge around a points so the function have a n-th derivative for all
$n\in\mathbb{N}$ and $a_n=\frac{f^{(n)}(c)}{n!}$.
So if two series convergence we get:
$a_n=\frac{f^{(n)}(c)}{n!}=b_n$.
