= Solution
By hypothesis the <power series> converges at every positive real argument, so its <radius of convergence> is infinite. Termwise <differentiation of a power series> is therefore valid in a neighborhood of $s=1$. For $k\ge1$,
$$
G^{(k)}(s)=\sum_{j=k}^{\infty}
j(j-1)\cdots(j-k+1)p(j)s^{j-k}.
$$
At $s=1$, the right side is the <expectation> of the <falling factorial> of $X$:
$$
\boxed{G^{(k)}(1)=\mathbb E[X(X-1)\cdots(X-k+1)].}
$$
The special case $k=1$ gives
$$
\boxed{G'(1)=\mathbb E[X].}
$$
For $k=0$, take the empty product as $1$; the equality becomes $G(1)=1$. These are the <factorial moments>.
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