Francis Galton's surname problem

Question

I am reading a little bit about this problem and am somewhat confused in some of the justifications provided in my readings.

So the problem is that we want to find the probability of the extinction of a surname after $r$ generations.

We denote this probability by $d_r$.

We denote $p_0$ as the percent of adult males who have no male children which reach adult life,

$p_1$ who have one such male child, and so on...

First question,

If $p_0>0$ then why is it true that $d_1 \leq d_2$ ?

Second question

If we define $G(x) = p_0 + p_1x + p_2x^2 + ...$

Why is it safe to assume that the sum terminates after five or six terms?

Furthermore, does this mean the probabilities $p_m$ eventually converge to $0$ after the first 5-6 terms?

Thanks.

This question is an introduction from Stephen Abbott's - Understanding Analysis to the sequences and series of functions chapter.

Answer 1

For the first question, if the name is extinct after one generation, it is not available for the second, so the chance a name is gone after two generations is at least as great as the chance it is gone after one.

For your second question, very few males have more than five or six male children.