A closed form for $\sum_{k = 1}^{\infty} k^{-k}$?

Question

Is there any closed form for this expression? $$\displaystyle\sum_{k = 1}^{\infty} k^{-k}$$ I got this while playing with my scientific calculator, so I really have no idea about how I could find one.

the overwhelming majority of things you can wonder about by playing with your calculator have no closed forms. — mercio, Oct 08 '13 at 14:11
That deserves to be the most upvoted Math.SE comment in history. — Soham Chowdhury, Oct 08 '13 at 14:14

Abhimanyu Pallavi Sudhir · Accepted Answer · 2018-04-28T12:15:56.950

0

There isn't a known closed form in terms of elementary functions. I've seen it being called the "Sophomore's dream function" $\mathrm{Sphd}(x) = \sum_{k = 1}^{\infty} k^{-k-x}$ at least once, due to its significance in the Sophomore's Dream.

edited Apr 28 '18 at 12:15

answered Oct 08 '13 at 14:07

Abhimanyu Pallavi Sudhir

3,306

How do you know there isn't one? It is very unlikely, but your claim is unfounded. – Pedro Oct 08 '13 at 14:14

A closed form for $\sum_{k = 1}^{\infty} k^{-k}$?

1 Answers1