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This is obviously the sort of problem that fascinates a kid, as it did me when I was younger. I became reminded of it when looking for creative ways to bound the sum below from Series involving factorial:

$\displaystyle \sum_{n=1}^{\infty}\frac{1!+...+n!}{(n+2)!}$

So, is there a closed form for $A=\displaystyle \sum_{n=0}^{\infty}\frac{1}{n^n}=\frac{1}{1}+\frac{1}{2^2}+\frac{1}{3^3}+...?$

If you're wondering, $A$ is a poor upper bound for each term in the sum.

ShakesBeer
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    Search for Sophomore's dream. – Hakim Oct 09 '14 at 21:37
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    http://en.wikipedia.org/wiki/Sophomore's_dream – Seth Oct 09 '14 at 21:38
  • Thank you, I knew it would be something obvious but I didn't know what to google (I suppose there is no closed form then?) – ShakesBeer Oct 09 '14 at 21:38
  • @Shakespeare No, there is no way to express the integral formula in closed form in terms of elementary functions (which is non-trivial to prove, but it is possible), although the expression is a simple integral of a closed form expression so that's pretty close. – user2566092 Oct 09 '14 at 21:44
  • @user2566092 well what about non-elementary functions? and what about the proof? (links would be great) – ShakesBeer Oct 09 '14 at 21:45
  • @Shakespeare The Risch algorithm http://en.wikipedia.org/wiki/Risch_algorithm can be used to show that the function being integrated has no anti-derivative expressible in closed form in terms of elementary functions. I'm not sure about non-elementary functions, but often people do define exotic ones and then they show identities between them. Maybe that's the case here, I'm just not sure. – user2566092 Oct 09 '14 at 21:48
  • @user2566092 Thank you :). Post that as an answer and I'd happily accept and unpvote – ShakesBeer Oct 09 '14 at 22:01

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