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From an integral to a gamma to a series

  1. Sep 14, 2012 #1
    1. The problem statement, all variables and given/known data
    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]


    2. Relevant equations
    How to resolve this XD?


    3. The attempt at a solution
     
  2. jcsd
  3. Sep 14, 2012 #2
    Maybe you should start by expanding x-x = e -x ln x as a series and then trying to think of a nice substitution.
     
  4. Sep 14, 2012 #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!
     
  5. Sep 14, 2012 #4
    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).
     
  6. Sep 14, 2012 #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?
     
  7. Sep 14, 2012 #6
    Now do a small change of variables so you can write the integral as a gamma function
     
  8. Sep 14, 2012 #7
    ok done :D thank you very much ^^
     
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