# Homework Help: From an integral to a gamma to a series

1. Sep 14, 2012

### deathquasar

1. The problem statement, all variables and given/known data
Using Euler's Gamma and a proper substitution prove the relation above:
$\int_0^1 1/x^x=\sum_{n=1}^\infty 1/n^n$

2. Relevant equations
How to resolve this XD?

3. The attempt at a solution

2. Sep 14, 2012

### 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.

3. Sep 14, 2012

### 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!

4. Sep 14, 2012

### 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).

5. Sep 14, 2012

### 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?

6. Sep 14, 2012

### clamtrox

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

7. Sep 14, 2012

### deathquasar

ok done :D thank you very much ^^