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There is an peculiar algebraic equality: $$\int_0^1\frac 1{x^x}\mathrm dx=\sum_{n=1}^{+\infty}\frac 1{n^n}.$$

Some context.

The proof is relatively straightforward, one can write $\frac 1{x^x}=e^{-x\ln(x)}$ and use the exponential series $$e^{-x\ln(x)}=\displaystyle\sum_{n=0}^{+\infty}\frac 1{n!}(-x\ln(x))^n,$$ then apply a famous theorem to invert $\sum$ and $\int$...

The question.

My question is the following: is there a geometric interpretation of this equality?

Any ideas would be greatly appreciated.

Edit. This question seems to have been already asked here but no answer was provided yet.

E. Joseph
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    This has been asked (but not answered) before: https://math.stackexchange.com/q/1298750/42969 – Martin R Oct 18 '24 at 07:46
  • @MartinR Thanks for the link, i will include it in the question. Maybe this new question will revive the interest for this seemingly open problem. – E. Joseph Oct 18 '24 at 08:03
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    I think some application of here could be used to generate a geometric description - As $n$ goes to infinity, probably the error term for Euler-Maclaurin goes to $0$ which could be should shown geometrically – Kraken Oct 18 '24 at 08:23

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