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

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An elegant equation was discovered involving the sum and integral of 1/n^n and 1/x^x, respectively, which shows high probability of being true based on decimal approximations. The original poster seeks guidance on proving this equation, noting the lack of information from Wolfram Alpha and the absence of a known anti-derivative for 1/x^x. Responses emphasize that the result is not trivial and that it's common to struggle with such proofs. A linked article on "Sophomore's Dream" provides additional insights, with one user mentioning a simpler proof found on Wolfram MathWorld. The discussion highlights the challenges of mathematical proofs while encouraging exploration and learning.
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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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You may wish to look at the "[URL Dream[/url] article in Wikipedia.
 
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wow that's really simple. i feel stupid.
 
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).
 
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.
 
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