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The offspring probabilities $P(K=j)=2^{-j-1}$ for $j\ge 0$ have expected value one. For one initial individual, the branching-process generating-function iteration givesFor $n\ge 1$ the PGF disk is $|s|<1+1/n$; $G_0(s)=s$ separately. Conditional on survival, the population is positive geometric with success probability $1/(n+1)$, leading to the exponential limit of surviving geometric branching.

For $n\ge 1$, substitute $s=e^{t/n}$ in the probability generating function. The moment-generating function of $X_n/n$ isThe domain follows from convergence of the generating series; at or above that positive threshold the expectation diverges, even though a rational continuation can be written.

Conditioning on survival, the probability mass function from part (ii) becomes a positive-integer geometric distribution:Its conditional moment-generating function is $e^{t/n}/[n+1-n{e}^{t/n}]$, tending to $1/(1-t)$ for fixed $t<1$. We can prove the required limit directly, without invoking a continuity theorem. Set $r_n=n/(n+1)$. For every $x\ge 0$, the strict upper tail is exactlySince $\log r_n=-1/n+O(n^{-2})$ and $\lfloor nx\rfloor /n\to x$,At $x=0$ both tails equal one. This is the exponential limit of surviving geometric branching. Conditioning is essential: the unconditional variable has probability $n/(n+1)$ of being zero, whereas the rare surviving population has size of order $n$.