Factorial-moment inversion ($\mathrm{Pr}(X=k)=\frac{1}{k!}{\sum }_{r\ge 0}\frac{(-1{)}^r\mathbb{E}(X{)}_{k+r}}{r!}$)

For a nonnegative integer-valued random variable, the displayed inversion is valid when $\mathbb{E}(X{)}_s{s}^k/s!\to 0$. Indeed the gap between consecutive Jordan-Bonferroni exact-occurrence inequalities is $\mathbb{E}(X{)}_{k+m+1}/[k!(m+1)!]$, which tends to zero under this hypothesis. No exchange of an infinite signed sum with an expectation is needed.