I have to proof this, but I don't know how to do this.
Let $R$ be the radius of convergence for $\sum_{k=0}^\infty a_k(x-a)^k$ and suppose that $\displaystyle\lim_{n \to +\infty} \left|\frac{a_{n+1}}{a_n}\right|=L$. Then:
(a) if $L$ is a nonzero finite real number, $R = \frac{1}{L}$,
(b) if $L=0, R = \infty$,
(c) if $L=\infty, R=0$.