= Solution
The geometric family law on the nonnegative integers has <probability generating function>
$$
\boxed{G(s)=\sum_{k\geq0}2^{-k-1}s^k=\frac1{2-s}},\qquad |s|<2.
$$
Its <expected value> is $G'(1)=1$, so this is a <geometric-offspring critical branching process>. Starting from $G_0(s)=s$, verify the rational iterate by induction. If $G_n(s)=[n-(n-1)s]/[n+1-ns]$, then
$$
G(G_n(s))=\frac1{2-G_n(s)}=\frac{n+1-ns}{n+2-(n+1)s},
$$
which has the same form with $n$ replaced by $n+1$. Thus, for $n\geq1$,
$$
\boxed{G_n(s)=\frac{n-(n-1)s}{n+1-ns},\qquad |s|<1+\frac1n}.
$$
The stated disk is the actual power-series convergence domain, not merely a place where the rational continuation can be evaluated. Its simple pole is at $s=1+1/n$.
The extinction <probability> at generation $n$ is the constant coefficient:
$$
\boxed{P(X_n=0)=G_n(0)=\frac n{n+1}}.
$$
For later use, expanding the remaining <geometric series> gives
$$
P(X_n=j)=\frac1{(n+1)^2}\left(\frac n{n+1}\right)^{j-1},\qquad j\geq1.
$$
In particular $P(X_n>0)=1/(n+1)$. At $n=0$ the initial population is one deterministically; its <PGF> is $s$, and the displayed radius involving $1/n$ is used only for $n\geq1$.
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