Suppose that $Z$ is a random variable that tells us the number of offspring in a branching process $X(n)_{n \geq 0}$. Assume that $\mathbb{E}(Z)=\mu$ exists. Then $$W(n)=\sum_{k=0}^n X(k)$$ is the total number of individuals who have ever lived until time $n\geq 0$.
I want to evaluate $\mathbb{E}(W(n))$ and I want to understand $\displaystyle \lim_{n \to \infty} \mathbb{E}(W(n))$ with regards to its dependence on $\mu$.
Without loss of generality, assume that $X(0)=1$. By linearity of expectation, I know that $$\mathbb{E}(W(n))=1+\sum_{k=1}^n \mathbb{E}(X(k))=1+\sum_{k=1}^n \mathbb{E}\left(\sum_{j=1}^{X(k-1)} Z_{k-1, j} \right)=1+\sum_{k=1}^n\sum_{j=0}^{X(k-1)} \mathbb{E}(Z_{k-1, j}) $$ $$=1+\sum_{k=1}^n\sum_{j=0}^{X(k-1)} \mu=1+\sum_{k=1}^n (X(k-1)+1)\mu=1+\mu\left(\sum_{k=1}^n X(k-1)\right)+n\mu $$ $$ =1+\mu(W(n)-1)+n\mu, $$ where $Z_{k-1}, j$ is the total number of offspring produced by person $j$ in the $(k-1)$th generation (meaning that $Z_{k-1}, j$ has the same distribution as $Z$).
I don't think that the expectation I computed is correct. Also, I wonder if it is possible for $\displaystyle \lim_{n \to \infty} \mathbb{E}(W_n)=\infty$ even though the population becomes extinct with probability 1.
I will use two standard facts about the process $\{X_n\}$. First, extinction probability is 1 if $\mu \leq 1$. Second $EX_k=\mu ^{k}$ for all $k$. Hence $EW_n=1+\mu +\mu^{2} +...+\mu ^{n}$. It follows that $EW_n \to \infty$ if $\mu =1$ even though exttinction is certain in this case.