# From an integral to a gamma to a series

## Homework Statement

Using Euler's Gamma and a proper substitution prove the relation above:
$\int_0^1 1/x^x=\sum_{n=1}^\infty 1/n^n$

## Homework Equations

How to resolve this XD?

## The Attempt at a Solution

Maybe you should start by expanding x-x = e -x ln x as a series and then trying to think of a nice substitution.

but I've to pass through the euler's gamma representation, and expanding this function is quite horrible using taylor. I'll try anyway with your idea!,Ty!

Don't expand it into a full Taylor series, just the exponential. That should give you a clue what the substitution should be (remembering that you want a gamma function in the end).

ok, I expanded just the exp, and now I've this, (with some transformations) and quite looks like what I want.

$\sum_1^\infty \frac{1}{n!}\int_0^\infty e^{-y(n+1)}y^n$

but now?

Now do a small change of variables so you can write the integral as a gamma function

ok done :D thank you very much ^^