Think I might need to refresh myself in power series before doing this one. Anyway, how should I proceed with this question?

Consider the Branching Process { } where is the population size of the nth generation. Assume and that the pgf of the common offspring distribution N is

(i) Express as a power series and hence find .

(ii) If for n = 0,1, ..., write down an equation relating and . Hence, or otherwise, evaluate for n = 0,1,2.

(iii) Find the extinction probability .

Thank youu!