On the third edition of Ahlfors' Complex Analysis, page 39 Theorem 2 it states: The derived series $\sum_{1}^{\infty}na_n z^{n-1}$ has the same radius of convergence, because $\sqrt[n]n \rightarrow 1$.
I'm confused about it.
The radius of convergence of $\sum_{0}^{\infty}a_nz^n$ is $\lim_{n\rightarrow \infty} \sup \sqrt[n]{|a_n|}$
We know that \begin{equation*} \sum_{1}^{\infty}na_nz^{n-1}=\sum_{0}^{\infty}(n+1)a_{n+1}z^n=\sum_{0}^{\infty} b_nz^n \end{equation*} So, the radius of convergence of $\sum_{1}^{\infty}na_nz^{n-1}$ should be $\lim_{n\rightarrow \infty} \sup \sqrt[n]{|b_n|}=\lim_{n\rightarrow \infty} \sup \sqrt[n]{|(n+1)a_{n+1}|}$
If we want to prove these two power series have the same radius of convergence, we should prove:
(i) $\sqrt[n]{n+1} \rightarrow 1$. But I'm confused that Ahlfors says he wants $\sqrt[n]n \rightarrow 1$.
(ii) $\lim_{n\rightarrow \infty} \sup \sqrt[n]{|a_n|} = \lim_{n\rightarrow \infty} \sup \sqrt[n]{|a_{n+1}|}$. But I don't know how to write a rigorous proof for this.