TITLE: Closed form expression for the series $\sum n!/n^n$
QUESTION [5 upvotes]: It is quite well known that the series $\sum n!/n^n$ converges. For instance, the question of convergence was addressed in this thread. However, I was wondering if there is a closed form expression for this sum.
By closed form, I mean an expression where the answer is in the form of a function of a well known kind, or an expression which could even be a definite integral of some function.
I tried to use generating functions and find an expression for this sum but I did not go very far. If someone can derive an expression for this sum starting from some known power series, that would be great.

REPLY [6 votes]: Using
$$
n! = \int_0^{ + \infty } {e^{ - t} t^n dt}  = n^{n + 1} \int_0^{ + \infty } {e^{ - ns} s^n ds} 
$$
and the fact that $0<e^{-s}s<e^{-1}<1$ for $s>0$, we obtain
$$
\sum\limits_{n = 1}^\infty  {\frac{{n!}}{{n^n }}}  = \sum\limits_{n = 1}^\infty  {n\int_0^{ + \infty } {e^{ - ns} s^n ds} }  = \int_0^{ + \infty } {\sum\limits_{n = 1}^\infty  {n(e^{ - s} s)^n } ds}  = \int_0^{ + \infty } {\frac{{e^{ - s} s}}{{(1 - e^{ - s} s)^2 }}ds} .
$$
The numerical value is $1.879853862\ldots$.