Consider a branching process where the offspring distribution is given by $$P(X = k) = \frac{1}{2^{k+1}}$$
what is the probability that the process becomes extinct at exactly at the $n$th generation?
The answer is supposed to be $\frac{1}{n(n+1)}$ but I'm not sure how to get there.
Wouldn't it be the generating function $\phi_n(0) - \phi_{n-1}(0)$?
The generating function is $$ \phi(t)=\sum_{k\ge 0} {\mathbb{P}}(X=k) t^k=\frac{1}{2-t} $$ and it's easy to prove by induction that the $n$-fold composition of $\phi(t)$ with itself is $$ \phi^{(n)}(t)=\frac{n-(n-1)t}{(n+1)-nt} $$ so the probability that the process is extinct by the $n$th generation is $$ \phi^{(n)}(0)=\frac{n}{n+1}$$ and as you say the probability that it becomes extinct exactly at generation $n$ is $$ \phi^{(n)}(0)-\phi^{(n-1)}(0)=\frac{n}{n+1}-\frac{n-1}{n}=\frac{1}{n(n+1)}. $$
