# “Closed” form for ∑1nn\sum \frac{1}{n^n}

Earlier today, I was talking with my friend about some “cool” infinite series and the value they converge to like the Basel problem, Madhava-Leibniz formula for $\pi/4, \log 2$ and similar alternating series etc.

One series that popped into our discussion was $\sum\limits_{n=1}^{\infty} \frac{1}{n^n}$.

Proving the convergence of this series is trivial but finding the value to which converges has defied me so far. Mathematica says this series converges to $\approx 1.29129$.

We were joking that it should have something to do with $\pi,e,\phi,\gamma$ or at the least it must be a transcendental number :-).

My questions are:

1. What does this series converge to?
2. Does this series arise in any context and are there interesting trivia to be known about this series?

I am actually slightly puzzled that I have not been able to find much about this series on the Internet. (At least my Google search did not yield any interesting results).

Certainly you need to check out Sophomore’s Dream.