Closed form of $\sum \frac{x^n}{n^n}$
Solution 1:
$\sum_{n=1}^\infty \frac{x^n}{n^n} = x $ Sphd$(-x;1)$
But, before saying "that's a joke", read the preamble of the paper : "The Sophomore's Dream Function", http://fr.scribd.com/doc/34977341/Sophomore-s-Dream-Function
By the way, this leads to :
$\sum_{n=1}^\infty \frac{x^n}{n^n} = x\int_{0}^1 {t^{-xt}}dt$
(From Eq.6:1 and Eq.1:2)