value of $\sum_{k=1}^{\infty} k^{-k}$.

Question

I want to compute the value of $\sum_{k=1}^{\infty} k^{-k}$.

First i try to check whether this series converges,

\begin{align} \lim_{k \rightarrow \infty} k^{-k} = \lim_{k\rightarrow \infty} e^{-k \log(k)} =0 \end{align} Thus i see that series converges.

But how do we know the exact value of this series?

Answer 1

This is called the Sophomore's Dream.