= Solution
{c}
Under the extra hypothesis that $G(s)$ exists for every real $s$, its <power series> has infinite <radius of convergence>. <Termwise differentiation of a power series> is therefore valid to every finite order. For $k\geq1$,
$$
G^{(k)}(s)=\sum_{j=k}^\infty j(j-1)\cdots(j-k+1)p(j)s^{j-k}.
$$
At $s=1$ this is the <expected value> of the <falling factorial> of $X$:
$$
\boxed{G^{(k)}(1)=\mathbb E[(X)_k]=\mathbb E[X(X-1)\cdots(X-k+1)].}
$$
In particular $\boxed{G'(1)=\mathbb E[X]}$. The terms with $j<k$ vanish because their <falling factorials> contain a zero factor. These <factorial moments> are finite under the stated hypothesis.
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