Solution
= Solution
Put $q=1-p$. Summing the <geometric series> in the definition of the <probability generating function> gives
$$
\boxed{G_X(s)=\mathbb E[s^X]
=p\sum_{k=0}^{\infty}(qs)^k
=\frac{p}{1-qs},\qquad|s|<\frac1q.}
$$
This is the zero-based <geometric distribution>, counting failures before the first success.