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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 $|z|<R$. 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 $|z|<r<R$ 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.

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3  
Hint: Both the root test and the ratio test seem applicable here. –  Srivatsan Aug 7 '11 at 2:16
    
Other potentially useful facts: $x \mapsto x^\alpha$ is increasing and continuous on $[0, \infty)$. –  Dylan Moreland Aug 7 '11 at 2:31
10  
There's an explicit formula for the radius of convergence of a power series: $R = \liminf_{n \rightarrow \infty} |a_n|^{-{1 \over n}}$. –  Zarrax Aug 7 '11 at 3:00
4  
Root test yes, ratio test no. –  Robert Israel Aug 7 '11 at 7:43

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