I'm trying to answer the following question, having a hard time figuring out where to begin. Any help appreciated:
Let f be analytic everywhere it is defined, and for $|z| < 2$ the following holds:
$$ f(z) = \sum_{n=0}^{\infty}\frac{n+1}{2^n}z^n $$
Let $(a_n)$ be the sequence of co-efficients for the taylor series of f around $z = \pi$, that is $f(z) = \sum_{n=0}^{\infty}a_n(z-\pi)^n$. find the radius of convergence for the series $\sum_{n=0}^{\infty}a_nz^n$ (no need to calculate the $a_n$'s).
I can calculate the various derivatives of f at zero based on the formula given for the co-efficients around zero above, but having a hard time figuring out how to go from there. Its supposed to be related to the textbook material to do with the uniqueness principle for complex analytic functions