We assume $m>1$. We study a slightly different problem: $k$ being a fixed strictly positive integer, we assume that there are $k$ individuals in generation 0. We denote by $W_n$ the number of individuals in the $n$-th generation and define $u_n$ as the probability that the sequence $(W_n)_{n\in\mathbb{N}^*}$ takes the value $k$ for the first time at rank $n$: $$u_n=P\left((W_n=k)\cap\left(\bigcap_{i=1}^{n-1}(W_i\neq k)\right)\right)$$ For $n$ and $r$ non-zero natural integers, $u_n^{(r)}$ is the probability that the sequence $(W_n)_{n\in\mathbb{N}^*}$ takes the value $k$ for the $r$-th time at rank $n$. Verify that the series $\sum_{n\geqslant 1}u_n s^n$ and $\sum_{n\geqslant 1}u_n^{(r)}s^n$ converge when $s\in[-1,1]$.
We assume $m>1$. We study a slightly different problem: $k$ being a fixed strictly positive integer, we assume that there are $k$ individuals in generation 0. We denote by $W_n$ the number of individuals in the $n$-th generation and define $u_n$ as the probability that the sequence $(W_n)_{n\in\mathbb{N}^*}$ takes the value $k$ for the first time at rank $n$:
$$u_n=P\left((W_n=k)\cap\left(\bigcap_{i=1}^{n-1}(W_i\neq k)\right)\right)$$
For $n$ and $r$ non-zero natural integers, $u_n^{(r)}$ is the probability that the sequence $(W_n)_{n\in\mathbb{N}^*}$ takes the value $k$ for the $r$-th time at rank $n$.
Verify that the series $\sum_{n\geqslant 1}u_n s^n$ and $\sum_{n\geqslant 1}u_n^{(r)}s^n$ converge when $s\in[-1,1]$.