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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|

Is this a good start: Say I take some $r$ such that $|z|

  • 3
    Hint: Both the root test and the ratio test seem applicable here.2011-08-07
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    Other potentially useful facts: $x \mapsto x^\alpha$ is increasing and continuous on $[0, \infty)$.2011-08-07
  • 10
    There's an explicit formula for the radius of convergence of a power series: $R = \liminf_{n \rightarrow \infty} |a_n|^{-{1 \over n}}$.2011-08-07
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    Root test yes, ratio test no.2011-08-07

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