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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. $ $ –  Did Dec 7 '12 at 6:13
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up vote 2 down vote accepted

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 $|z|\lt1$? –  Gerry Myerson Dec 7 '12 at 4:56
    
Hadamard is overkill. But the question indicates he/she clearly does not know Hadamard. –  Bombyx mori Dec 7 '12 at 5:06
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Which seems to be a reason (apart from the obvious conceptual one) not to use Hadamard, no? –  Did Dec 7 '12 at 6:11
    
This is a good point. Thank you. –  Bombyx mori Dec 7 '12 at 6:12
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