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Sum from 1 to +inf of (n^-n)

  1. Jul 29, 2012 #1
    1. The problem statement, all variables and given/known data
    http://latex.codecogs.com/examples/00a93bb6c6c645f9802b88f4c1c986fc.gif [Broken]
    Let's call this L.
    Find the exact value of L.

    2. Relevant equations
    http://latex.codecogs.com/examples/6835d744da9ce19b352158cd01b91e91.gif [Broken]
    1+1/2=1,5


    3. The attempt at a solution
    • L>0.
    • 1,5>L
    • The function is always growing
    Therefore there must be an definite answer in ]1,5 ; 0[
    Wolfram gives 1,291286...


    Spam of possible methods:
    -Trying to find the value relating it to integration. Impossible. The integral is not defined.
    -Try to find an adequate Taylor Expansion. Can't find a function that fits AND I don't know any pretty method to relation expansions to functions.
    -Try to find a relatable sum such as that of the differences or quocients of the next term. Couldn't do much with it.
    -Use n^-n=e^ln(n^-n)=e^(-n*ln n). Can't do much with it.
    -Turn into a product problem n^-n=ln(e^(n^-n)). Can't do much with it.
    -Use a general formula for the sum of a^(k) to infinity. Haven't tried but it doesn't seem to simplify.


    Main question:
    Is it representable using already known constants?
     
    Last edited by a moderator: May 6, 2017
  2. jcsd
  3. Jul 29, 2012 #2
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