sequences and series - What is exact value of $sum_{n=1}^{infty} frac{1}{n^n}$ .

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

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.