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
First we show that this series converges when $n \in \Z_{\ge 0}$.

Consider the coefficients:
 * $1, \paren {1 - \dfrac 1 {1!} }, \paren {1 - \dfrac 1 {1!} + \dfrac 1 {2!} }, \ldots$

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

Starting from the $\paren {n + 1}$th term, there is a factor of $\paren {n - n}$.

In this case, all subsequent terms of the expansion equal $0$.

Therefore the series converges.

Then we show that this series diverges for $n \notin \Z_{\ge 0}$.

We do this via Divergence Test.

That is, we show that $\displaystyle \lim_{k \mathop \to \infty} \frac { {!k} \, n^{\underline k} } {k!} \ne 0$.

$\displaystyle \lim_{k \mathop \to \infty} \frac { {!k} \, n^{\underline k} } {k!}$ exists.

By Power Series Expansion for Exponential Function, we have that $\displaystyle \lim_{k \mathop \to \infty} \frac {k!} {!k} = e$.

Hence $\displaystyle \lim_{k \mathop \to \infty} n^{\underline k}$ would exist as well.

For $k > \floor {n + 2}$:

Since $n - k \ne 0$, $\size {n^{\underline k} }$ increases without bound.

Hence its limit does not exist, a contradiction.

Therefore $\displaystyle \lim_{k \mathop \to \infty} \frac { {!k} \, n^{\underline k} } {k!}$ does not exist for $n \notin \Z_{\ge 0}$.

By Divergence Test, our series diverges.