How can I prove this elegant equation I discovered using Wolfram Alpha?

In summary, A user discovered an elegant equation on Wolfram Alpha and asked for help proving it. They were directed to a Wikipedia article and a MathWorld proof, but also encouraged to not feel bad for not being able to prove it themselves.
  • #1
superadvanced
3
0
Playing around with Wolfram Alpha I discovered an elegant looking little equation. Judging by the decimal approximation of both sides, there seems to be an extremely high probability that it is true. A picture of the equation is attached but ill try to type it too:

sum(1/n^n,n,1,inf)=integral(1/x^x,x,0,1)

My question is does anyone know how to go about showing this? Wolfram doesn't have much to say about either side of the equation other than decimal approximations. Obviously there is no known anti-derivative for 1/x^x. Thoughts?
 

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  • #2
You may wish to look at the "[URL Dream[/url] article in Wikipedia.
 
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  • #3
wow that's really simple. i feel stupid.
 
  • #4
superadvanced said:
wow that's really simple. i feel stupid.

Why feel stupid? It's not an obvious result by any means. The derivation in the linked wikipedia article isn't trivial either. Straightforward if you know what to do, maybe, but not simple enough to feel stupid for not thinking of it.

You found something neat. Don't feel bad that you couldn't prove it. Maybe next time you find something neat you will be able to prove it (even if it is still the case that someone else has proved it previously).
 
  • #6
Mute said:
Why feel stupid? It's not an obvious result by any means. The derivation in the linked wikipedia article isn't trivial either. Straightforward if you know what to do, maybe, but not simple enough to feel stupid for not thinking of it.

You found something neat. Don't feel bad that you couldn't prove it. Maybe next time you find something neat you will be able to prove it (even if it is still the case that someone else has proved it previously).

lol I don't understand the wiki proof but there was a much simpler proof I found when googling sophomore's dream. Its on wolfram math world.
 

Related to How can I prove this elegant equation I discovered using Wolfram Alpha?

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