 Total progeny with critical geometric offspring

Start a branching process with one ancestor and offspring distribution $\mathbb{P}(N=k)=2^{-k-1}$ for $k\ge 0$. 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 $s\uparrow 1$ equals $1$, consistent with finite total progeny almost surely; its derivative diverges there, so the expectation of $Z$ is nevertheless infinite.