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I need to show that $$\int_0^1 x^{-x} \mathrm dx = \sum\limits_{n = 1}^\infty n^{-n}$$

I tried using that $x^{-x} = e^{-x\ln(x)}$, but I don't know what to do about when $x = 0$ since the equality does not hold. I also tried using the power series expansion of $e^x$, but I don't really know where to go from there. Any help would be appreciated.

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marked as duplicate by mrtaurho, Eevee Trainer, Lord Shark the Unknown, カカロット, Gerry Myerson Feb 28 at 5:29

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