Prove if $\lim_{k \rightarrow \infty} (c_k)^{1/k}=0$, the the radius of convergence of the power series, $\sum_{k \geq 0}c_k(z-z_0)^k$, $R=\infty$.
It's already proved in our textbook that $$R=\frac{1}{\lim_{k \rightarrow \infty} (c_k)^{1/k}}$$ Can we say simply that if $\lim_{k \rightarrow \infty} (c_k)^{1/k}=0$, then $R=\infty$? Do we need a more detailed proof here? And if so, how to prove it? Thanks.