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

[Swimmingly!]

Homework Statement

http://latex.codecogs.com/examples/00a93bb6c6c645f9802b88f4c1c986fc.gif
Let's call this L.
Find the exact value of L.

Homework Equations

http://latex.codecogs.com/examples/6835d744da9ce19b352158cd01b91e91.gif
1+1/2=1,5

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?

 

[awkward]

According to some fragmentary information on this web page

http://forums.wolfram.com/mathgroup/archive/2006/Apr/msg00062.html

the series has been investigated, but not much is known about it.