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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 $n$.

The geometric family law on the nonnegative integers has probability generating functionIts expected value is $G^{\prime }(1)=1$, so this is a geometric-offspring critical branching process. Starting from $G_0(s)=s$, verify the rational iterate by induction. If $G_n(s)=[n-(n-1)s]/[n+1-ns]$, thenwhich has the same form with $n$ replaced by $n+1$. Thus, for $n\ge 1$,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:For later use, expanding the remaining geometric series givesIn 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 $n\ge 1$.

For a Galton-Watson process, condition on the number $K$ of children of the initial individual. The descendants of these children over the next $n$ generations are independent copies of $X_n$. The probability generating function of their total, conditional on $K=k$, is $G_n(s{)}^k$. Averaging over the family-size law givesThis is the branching-process generating-function iteration. Conditioning instead on the last generation gives $G_n(G(s))$; both orders agree because $G_n$ is the $n$th iterate of the same function $G$. The argument holds on the usual PGF domain $|s|\le 1$, with larger domains available when the corresponding expectations converge.

For a nonnegative integer-valued random variable, the probability generating function iswith $0\le s\le 1$ always allowed. Its derivative at $1$ from below gives the expectation when finite.

The waiting time $N$ has the positive-integer geometric distribution $P(N=j)=p{q}^{j-1}$. For the capped geometric waiting time, $X=j<k$ exactly when $N=j$, while $X=k$ occurs whenever $N\ge k$. Thus the terminal mass is $q^{k-1}$, givingThe finite sum is a polynomial; any apparent singularity of the rational expression is removable. This includes $k=1$, where $G_X(s)=s$.

Differentiating the rational form at $s=1$ yieldsAs a second interpretation, the expectation is the sum of the tail probabilities $P(X\ge j)=q^{j-1}$ for $1\le j\le k$. Their finite geometric sum gives the same answer.