= Solution
This is the ordinary <factorial> of $X$, rather than a falling-factorial <moment>. Substituting the <probability mass function> of the <Poisson distribution> gives the nonnegative <series>
$$
\mathbb E[X!]=\sum_{k=0}^\infty k!e^{-\lambda}\frac{\lambda^k}{k!}=e^{-\lambda}\sum_{k=0}^\infty\lambda^k.
$$
The <geometric series> converges exactly when $0<\lambda<1$. Therefore \b[$\displaystyle\mathbb E[X!]=\frac{e^{-\lambda}}{1-\lambda}$ for $0<\lambda<1$]. For $\lambda\geq1$, its terms fail to tend to zero and the <expectation> is infinite. This is the <factorial of a Poisson random variable>.
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