Solution
= Solution
For the <geometric distribution> on $\{1,2,\ldots\}$,
$$
\mathbb P(X=m)=p(1-p)^{m-1}.
$$
Summing the <geometric series>,
$$
G_X(s)
=\sum_{m=1}^{\infty}p(1-p)^{m-1}s^m
=\boxed{\frac{ps}{1-(1-p)s}}.
$$
= Solution
For the <geometric distribution> on $\{1,2,\ldots\}$,
$$
\mathbb P(X=m)=p(1-p)^{m-1}.
$$
Summing the <geometric series>,
$$
G_X(s)
=\sum_{m=1}^{\infty}p(1-p)^{m-1}s^m
=\boxed{\frac{ps}{1-(1-p)s}}.
$$