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.
Best Answer
Using $$ n! = \int_0^{ + \infty } {\mathrm{e}^{ - t} t^n \mathrm{d}t} = n^{n + 1} \int_0^{ + \infty } {\mathrm{e}^{ - ns} s^n \mathrm{d}s} $$ and the fact that $0<\mathrm{e}^{-s}s<\mathrm{e}^{-1}<1$ for $s>0$, we obtain $$ \sum\limits_{n = 1}^\infty {\frac{{n!}}{{n^n }}} = \sum\limits_{n = 1}^\infty {n\int_0^{ + \infty } {\mathrm{e}^{ - ns} s^n \mathrm{d}s} } = \int_0^{ + \infty } {\sum\limits_{n = 1}^\infty {n(\mathrm{e}^{ - s} s)^n } \mathrm{d}s} = \int_0^{ + \infty } {\frac{{\mathrm{e}^{ - s} s}}{{(1 - \mathrm{e}^{ - s} s)^2 }}\mathrm{d}s} . $$ The numerical value is $1.879853862\ldots$.