What is exact value of $\sum_{n=1}^{\infty} \frac{1}{n^n}$ .

Question

I would like to ask if anyone would help me with solving the following infinite series. \begin{equation} \sum_{n=1}^{\infty} \frac{1}{n^n} = \,? \end{equation} Thank you in advance.

Answer 1

Although it has no closed form, it does satisfy the remarkable identity $$ \sum_{n=1}^\infty \frac{1}{n^n} = \int_0^1\frac{dx}{x^x}. $$ It is also equal to $\ln(3)^e$ to five decimal places.