Using Euler's Gamma and a proper substitution prove the relation above:
[itex]\int_0^1 1/x^x=\sum_{n=1}^\infty 1/n^n[/itex]
Homework Equations
How to resolve this XD?
The Attempt at a Solution
Answers and Replies
#2
clamtrox
938
9
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
deathquasar
4
0
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
clamtrox
938
9
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
deathquasar
4
0
ok, I expanded just the exp, and now I've this, (with some transformations) and quite looks like what I want.