From an integral to a gamma to a series

deathquasar

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

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

deathquasar
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!

clamtrox
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).

deathquasar
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?

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

deathquasar
ok done :D thank you very much ^^