# What is the exact value of $\sum\limits_{x=1}^∞\frac1{x^x}$?

On the internet I found an evaluation of the integral $\displaystyle \int_0^1 x^{-x} \,\mathrm{d}x$ which results in $$\sum_{x=1}^\infty \frac{1}{x^x}.$$ Seeing this graphically I found that the sum does seem to converge to approximately $1.291$. So as it converges shouldn't we be able to find its exact value? Can someone please tell me what this value is or how I may be able to find it?

Edit:

While searching about this I came across Sophomore's Dream and this question which is quite interesting but none of them can quite give a closed value (in terms of known constants like $\pi$, $\phi$ or e). So does $\displaystyle \sum_{x=1}^\infty \frac{1}{x^x}$ or $\displaystyle \int_0^1 x^{-x} \,\mathrm{d}x$ not have such a value? Is it irrational? So many questions.

• The series you wrote seems to be missing an $n$ somewhere. – Marra May 15 '18 at 10:41
• Is this meant to be $\sum_{n=1}^{\infty}{\frac{1}{n^n}}$? – Rhys Hughes May 15 '18 at 10:45
• Yes, extremely sorry about that. – Raheeb Hassan May 15 '18 at 10:59
• There is no known "closed form" of the type you ask about. – GEdgar May 15 '18 at 11:23
• – cgiovanardi May 15 '18 at 13:10