From an integral to a gamma to a series

  • #1

Homework Statement


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

[itex]\sum_1^\infty \frac{1}{n!}\int_0^\infty e^{-y(n+1)}y^n[/itex]

but now?
 
  • #6
938
9
Now do a small change of variables so you can write the integral as a gamma function
 
  • #7
ok done :D thank you very much ^^
 

Related Threads on From an integral to a gamma to a series

Replies
0
Views
5K
  • Last Post
Replies
8
Views
1K
Replies
6
Views
7K
Replies
5
Views
3K
Replies
2
Views
4K
  • Last Post
Replies
0
Views
1K
  • Last Post
Replies
3
Views
938
Replies
3
Views
2K
  • Last Post
Replies
3
Views
507
Replies
14
Views
1K
Top