= Solution
The <probability generating function> of a nonnegative <integer-valued random variable> is
$$
\boxed{G(s)=\mathbb E[s^X]=\sum_{j=0}^{\infty}p(j)s^j.}
$$
Use $s^0=1$, including at $s=0$. The <power series> converges absolutely for every real $s\in[-1,1]$, since $\sum_jp(j)|s|^j\le1$. In particular it is always defined on the usual probability-generating interval $[0,1]$. It may also be defined for $|s|>1$, depending on the <radius of convergence> and convergence at its endpoints; the precise real <domain> of finite expectation is $\{s\in\mathbb R:\sum_jp(j)|s|^j<\infty\}$.
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