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$$\sum_{k=1}^{\infty} \frac{k^k}{(k!)^2}$$

The series converges by the ratio test but how would I find out the exact sum of the series.

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Do you want the value to which the sum converges? or the limit of the sequence being summed? –  amWhy Feb 5 '13 at 20:43
    
Sum, sorry I meant the sum. –  MJBoa Feb 5 '13 at 20:44
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no reason to think there is a nice closed-form expression for the sum. Why do you ask? –  Will Jagy Feb 5 '13 at 20:47
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Well, its on an old test from one of my classes, and the sum was asked for. I'm correct in saying it converges, at least? –  MJBoa Feb 5 '13 at 20:49

1 Answer 1

The sum seems to converge, since $k^k \approx \frac{k!e^k}{\sqrt{2 \pi k}}$ using Stirling's approximation. Cancelling out $k!$ the summand becomes $ \frac{e^k}{k!\sqrt{2 \pi k}} < \frac{e^k}{k!} $ for $k \geq 1$. since the latter sum clearly converges to $e^{e}-1$, by comparison test the former sum converges too.

EDIT: it does seem to have a numerical solution too:

http://www.wolframalpha.com/input/?i=sum%28k^k%2F%28%28k!%29^2%29%2Ck%3D1..ifinity%29

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Not really answering the question, the OP already knows it converges... –  L. F. Feb 5 '13 at 21:02
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And you made a typo in the Wolfram Alpha link (a factorial sign is missing). –  Martin Argerami Feb 5 '13 at 21:06
    
@L.F.: in the discussion above the OP asks if the sum is convergent –  Alex Feb 5 '13 at 21:07
    
@MartinArgerami: no –  Alex Feb 5 '13 at 21:08
    
Right, SEVERAL typos. –  Did Feb 5 '13 at 21:21

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