Let $\sum_0^{\infty} a_n z^n$ have radius of convergence $R$ with $0< R< \infty$. Let $\alpha>0$. Find the radius of convergence of $\sum_0^{\infty} |a_n|^{\alpha} z^n$.
I tried to start with what I am given: so the series $\sum_0^{\infty} a_n z^n$ converges uniformly and absolutely for every $|z|. I tried to do some computations to end up with $|a_n|^{\alpha}$ but the fact that $\alpha$ can be any positive real makes the task difficult. Is there some general method to find the radius of convergence of a power series given some power series $\sum_0^{\infty} a_n z^n$? I've seens this type of questions around but I was never sure where to start. Ratio and Cauchy/Hadamar's tests aside, what does one need to do to find the radius of convergence?
Is this a good start: Say I take some $r$ such that $|z| then $\sum_0^{\infty} a_n r^n$ converges so there is some $N$ past which $ a_n r^n <\epsilon$. But again $\alpha$ being a positive real confuses me. Thx for any answers.
sequences-and-series
complex-analysis
convergence-divergence
power-series