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For a power series defined as

$\begin{aligned}\sum_{n=0}^{+\infty} \end{aligned}a_n\,(z-z_0)^n$

the radius of convergence is equal to

$R = \frac{1}{\limsup\limits_{n\rightarrow+\infty} \sqrt[n]{\left|a_n\right|}}$

that, if it exists, is equivalent to

$R = \frac{1}{\lim\limits_{n\rightarrow+\infty} \sqrt[n]{\left|a_n\right|}}$

My question then is the following: you might propose to me a power series where the latter limit does not exist and therefore must refer to the strict definition?

Thank you!

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    $\begingroup$ $$a_{2n}=2^n\qquad a_{2n+1}=1$$ $\endgroup$ – Did Dec 2 '16 at 20:56
  • $\begingroup$ You could just as well have $a_{2n+1}=0$... $\endgroup$ – Thomas Andrews Dec 2 '16 at 21:01
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    $\begingroup$ That is not recurrence. It is a sequence that alternates between $1$ and powers of $2$. $\endgroup$ – Fimpellizieri Dec 2 '16 at 21:10
  • $\begingroup$ @Manu Please read carefully what is posted to answer your question, before commenting on it. $\endgroup$ – Did Dec 2 '16 at 21:11
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    $\begingroup$ $a_n=(\sin(n))^n$ $\endgroup$ – hamam_Abdallah Dec 2 '16 at 21:28

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