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This question already has an answer here:

Prove that $\sum_{n=1}^\infty \frac{1}{n^n}=\int_0^1 \frac{1}{x^x}dx$. This comes from this question Evaluate $\sum_0^\infty \frac{1}{n^n}$.

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marked as duplicate by 6005, user127.0.0.1, Davide Giraudo, Michael Hoppe, Namaste Mar 8 '14 at 17:43

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ I agree. It's a duplicate. $\endgroup$ – user103828 Mar 8 '14 at 15:53
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This identity is called Sophomore's dream (a "dream" because it seems too good to be true, and at the same time an identity because it is indeed true!).

It's proof can be found in Pedro Tamaroff's answer to this related (actually, its exactly the same) question.

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