= Total progeny with critical geometric offspring
{title2=$G_Z(s)=1-\sqrt{1-s}$}
Start a <branching process> with one ancestor and <offspring distribution> $\mathbb P(N=k)=2^{-k-1}$ for $k\ge0$. Its <probability generating function> is $G_N(s)=1/(2-s)$ and its mean offspring count is $1$, making it a <critical branching process>. Splitting the family at the ancestor gives the <total progeny> recursion $Z=1+\sum_{i=1}^N Z_i$. The <probability generating function of an independent random sum> then gives $G_Z=s/(2-G_Z)$, whose root satisfying $G_Z(0)=0$ is $1-\sqrt{1-s}$. Its limit at $s\uparrow1$ equals $1$, consistent with finite <total progeny> almost surely; its <derivative> diverges there, so the <expectation> of $Z$ is nevertheless infinite.
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