 Solution

Using the probability mass function of the Poisson distribution directly givesFor each fixed positive integer $n$, this series converges: the ratio of successive nonzero terms is $(1+1/j{)}^n/(j+1)$, which tends to zero. Equivalently, using the polynomial identity in (ii) and the factorial moments from (a)(iv) gives $\mathbb{E}[X^n]={\sum }_{k=1}^{n}A(n,k)$; the extension of that identity to $X=0,1$ is justified in (iv).