TITLE: Prove that $\sum_{n=1}^\infty \frac{n!}{n^n}$is irrational.
QUESTION [50 upvotes]: Let $$S= \sum_{n=1}^\infty \frac{n!}{n^n}$$
(Does anybody know of a closed form expression for $S$?)
It is easy to show that the series converges.
Prove that $S$ is irrational.
I tried the sort of technique that works to prove $e$ is irrational, but got bogged down.

REPLY [3 votes]: This doesn't answer the question.
But.
Wolfy calculates it as
$1.8798538621752585334863061450709600388198734004892899048296176691222963$
$866612142113617650197389123532397...$
and the inverse symbolic calculator
doesn't know,
so I would be surprised
if there is a
reasonable formula.