How to evaluate $\sum_{n=1}^{\infty }\frac{1}{n^{n}}$?

Question

$$\sum_{n=1}^{\infty }\frac{1}{n^{n}}$$

I have no idea how to even start computing this series. I do know, however, that this series definitely converges. Solving it numerically results in a solution close to 1.29. But, how would one compute this series analytically?

Answer 1

This sum has no known closed form, but the following relation is true: $$\sum_{k=1}^{\infty}k^{-k} = \int_0^1x^{-x}dx.$$ See here.