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I was thinking about the problem that says: What is the radius of convergence of $\sum_{0}^{\infty}z^{n!}$ ?

Please help. Thanks everyone in advance for your time.

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    22 minutes. $ $2012-12-07

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You mean the radius of convergence of $$\sum_{0}^{\infty}z^{n!}$$

We have $a_{n!}=1$ for all $n!$. Then by Hadamard's theorem we have $$\lim_{n=m!,n\rightarrow \infty}\sup (1)^{1/n}=1$$

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    Hadamard is overkill, no? Isn't it more-or-less trivial that it diverges at $z=1$, and converges (by comparison to $\sum z^n$) for $|z|\lt1$?2012-12-07
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    Hadamard is overkill. But the question indicates he/she clearly does not know Hadamard.2012-12-07
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    Which seems to be a reason (apart from the obvious conceptual one) **not** to use Hadamard, no?2012-12-07
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    This is a good point. Thank you.2012-12-07