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 ^^