Show that the power series $\sum_1^\infty a_nz^n$ and $\sum_1^\infty na_nz^n$ have the same radius of convergence

Question

Show that the power series $\sum_1^\infty a_nz^n$ and $\sum_1^\infty na_nz^n$ have the same radius of convergence.

So I solved this problem using a method that I'm not sure is valid, and I just wanted to verify if this is rigorous enough and if not, try to understand the reason. Suppose $\sum_1^\infty a_nz^n$ has radius of convergence $R$, this implies $$\lim_{n\rightarrow \infty} |\frac{a_{n+1}}{a_n}|=\frac{1}{R}$$

Now using the ratio test for the second power series $$\lim_{n\rightarrow \infty} \left|\frac{(n+1)a_{n+1}}{na_n}\right|=\lim_{n\rightarrow \infty} \left|\left(1+\frac{1}{n}\right)\frac{a_{n+1}}{a_n}\right|=\frac{1}{R}$$

Answer 1

Use the formula for radius of convergence and the fact that $\lim n^{1/n}=1$.

Answer 2

Compare reciprocals of radii of convergence: note that$$\lim_{n\to\infty}n^{1/n}=1\implies\limsup_{n\to\infty}|na_n|^{1/n}=\limsup_{n\to\infty}|a_n|^{1/n}.$$