I have a question about this proof:
Prove that the two power series
$$\sum_{i=0}^\infty a_nz^n\quad \text{ and }\quad \sum_{i=0}^\infty na_nz^{n-1}$$
have the same radius of convergence, by using only Abel’s Lemma.
I tried showing that if $\|a_nz_0^n\|$ is bounded then $\|na_nz_0^{n-1}\|$ is also bounded and therefore both series will have the same radius of convergence. Is this a reasonable way to approach the problem?
(Abel's Lemma: Take $z_0 \ne 0$. If $\|a_nz_0^n\|$ is bounded ($\leqslant M$) for all natural $n$ then $\forall r$ s.t. $0\leqslant r \leqslant |z_0|$, $\sum_{i=0}^\infty \|a_n\|z^n$ is convergent.)