Stochastic Processes-branching process

Let {Xn : n 1} be a branching process with X0 = 1 whose offspring ?ξ? distribution has mean μ =E(?ξ?) ?∈? (0, ?∞? ). Recall that Xn denotes the number of particles in the nth generation of the process. Let Nn = ?∑?i=0?n????∑?i=0?∞??? Xi denote the number of particles born by time n, and N = Xi the number of particles ever born. Let ???, ?ψ?n and ψ be the probability generating functions for the random variables ?ξ? , Nn and N, respectively.

(a) Show that ψn(s)=s?(ψn-1(s)) and then take n -> ?∞? to conclude that ψ(s)=s?(ψ (s))

(b) Use this formula to show that if μ < 1 then E(N) = 1/(1-μ).