Factorial of a Poisson random variable ($\mathbb{E}[X!]=e^{-\lambda }/(1-\lambda ),0<\lambda <1$)

If $X$ has a Poisson distribution with parameter $\lambda >0$, then the ordinary factorial $X!$ has finite expectation exactly when $\lambda <1$. Indeed, its nonnegative expectation series is $e^{-\lambda }{\sum }_{k\ge 0}{\lambda }^k$, since $k!$ cancels the denominator of the probability mass function. The geometric series gives the formula and proves divergence for $\lambda \ge 1$. This differs from a falling-factorial moment of fixed order, which is finite for every finite Poisson distribution parameter.