I have a question: What is the value of $\sum _{n=1}^{\infty \:}\frac{n!}{n^n}$? Only I've calculated the following identity: $$\sum _{n=1}^{\infty \:}\frac{n!}{n^n}=\int _0^{1}\left(1+x\cdot \ln \left(x\right)\right)^{-2}dx$$ I found this identity, when I watched The Sophomore's dream in wikipedia, but I would know the value exact of the serie.

Regards,

Ronald.