I apologise if this is off topic.


Consider the quantity
$$
F(m,n,k)=\frac{(m)\_k}{k!n^{k-1} }
$$
where $m,n \in \mathbb{N}.$ For moderately large $n$, it seems that the approximation
$$
\sum\_{k=1}^{K} k F(n,n,k) \approx en,
$$
where $K$ is the largest integer $k$ for which $k!\leq n,$
is excellent with the quality of approximation essentially independent of $n\in [10^2,10^6].$


There must be a (deep?) mathematical reason for this, I feel. Any comments, references, pointers welcome.



A (surprising?) expression for $e$
Just typing in the answer from the comments so the question does not stay "unanswered".


The expression on the left can be rewritten
$$
\sum\_{k} k \binom{n}{k} n^{-k}=\left(1+\frac{1}{n}\right)^{n-1},
$$ and the contribution to the sum comes from the very few first terms, those with $k!\leq n.$


