Factorial as Sum of Series of Subfactorial by Falling Factorial over Factorial/Condition for Convergence

Theorem
Consider the series:

This converges only when $n \in \Z_{\ge 0}$, that is, when $n$ is a non-negative integer.

Proof
Consider the coefficients:
 * $1, \left({1 - \dfrac 1 {1!} }\right), \left({1 - \dfrac 1 {1!} + \dfrac 1 {2!} }\right), \ldots$

By Power Series Expansion for Exponential Function, they converge to $\dfrac 1 e$.

Thus none of the terms ever reaches $0$ except when there is a factor of $\left({n - n}\right)$.

In this case, all subsequent terms of the expansion equal $0$ and indeed, the sequence converges.