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How can we show that $$\sum_{n=1}^{+\infty} \frac{1}{n^n} = \int_0^1 \frac{1}{x^x}dx$$

Could you give me some hints??


marked as duplicate by Thomas Andrews, Jack D'Aurizio, Aaron Maroja, user99914, André Nicolas real-analysis Feb 9 '15 at 22:53

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    $\begingroup$ Sometimes called the Sophomore's dream $\endgroup$ – Thomas Andrews Feb 9 '15 at 21:53
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    $\begingroup$ Now that is interesting. Thanks for the link, @ThomasAndrews. I never would have imagined that is true. Fascinating! +1 for both of you. $\endgroup$ – MPW Feb 9 '15 at 22:02

Hint: Write $x^{x} = e^{x\ln x}$, use $$e^{x} = \sum_{k=0}^{\infty}\frac{x^k}{k!}$$

and integrate term by term.

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    $\begingroup$ I'm glad I could help! $\endgroup$ – Aaron Maroja Feb 13 '15 at 23:06

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