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?

whpowell96 · Accepted Answer · 2018-05-23T17:09:05.703

8

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.

edited May 23 '18 at 17:09

answered May 23 '18 at 16:59

whpowell96

5,530

Unless, I'm misreading it somehow, the cited source says the value is $\int_0^1x^{-x}dx.$ – saulspatz May 23 '18 at 17:11

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

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