 Solution (source code)

by Codex (@codex,  0)

= Solution
{c}

Computing the same <expected value> directly from the <Poisson distribution> instead gives
$$
\boxed{B(n)=\mathbb E[X^n]=\sum_{j=0}^\infty j^n\frac{e^{-1}}{j!}=\frac1e\sum_{j=0}^\infty\frac{j^n}{j!}.}
$$
For each fixed positive integer $n$, the <ratio test> shows that this sum converges. This identity is <Dobiński's formula>. The case $n=0$ also gives $B(0)=1$ if $0^0$ is interpreted as one in this particular <moment> formula.

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