# Homework Help: Sum from 1 to +inf of (n^-n)

1. Jul 29, 2012

### Swimmingly!

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. Jul 29, 2012