Branching probabilities maths question

Let a, b in R and consider a branching process with branching probabilities p0=a/4 , p1= 1/4 and pn = b/2^n for n ≥ 2


(a) What relations must be satisfied by a and b for these probabilities to be valid branching probabilities?

(b) What is the radius of convergence of the probability generating function
G(x) =∑(n=0 to infinity)Pn x^n

Thanks =)

(a) The probabilities should sum to 1.
a/4 + 1/4 + (b/4 + b/8 + b/16 + ...) = 1
==> a/4 + 1/4 + (b/4)(1 + 1/2 + 1/4 + ...) = 1
==> a/4 + 1/4 + (b/4) * 1/(1 - 1/2) = 1, via infinite geometric series
==> a/4 + 1/4 + 2b/4 = 1
==> (a + 2b + 1)/4 = 1
==> a + 2b = 3.

b) Note that
G(x) = a/4 + (1/4)x + ∑(n = 2 to infinity) (b/2^n) x^n
........= a/4 + (1/4)x + ∑(n = 2 to infinity) b(x/2)^n.

Since this latter series is geometric, it converges iff |x/2| < 1 <==> |x| < 2.
So, the radius of convergence equals 2.

I hope this helps!