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

Solution 1:

Two different power series (around the same point) cannot converge to the same function.

If the power series both have positive radius of convergence, and their $n$th coefficients differ, then the $n$th derivatives of the functions they define also differs, so they cannot be the same function.

Solution 2:

Yes and no, as Henning mentioned, two power series around the same point of the form $$f(x)=\sum_{n=0}^{n=\infty}a_n(x-c)^n$$ cannot converge to the same function if the coefficients $a_n$ are different. This is the case encountered in most calculus classes.

But, definitely don't take that to mean that the Laurent series centered at the same point will always have the same coefficients $a_n$ as the Taylor series at that point. For proof of that consider the Laurent and Taylor expansions for $f(z)=\frac{1}{1-z}$ around $z=1$.