A little analysis

1.

Show that quadratic convergence implies that the number of correct digits doubles at each iteration (see the top of page 45). When I say ``show that,'' I mean write a sketch of a proof, keeping in mind that as a non-mathematician I have a fairly lax definition of proof.

2.

What approximation did you have to make in order to derive the doubling-digit heuristic? What is the implication of that approximation.

Note: this problem is necessarily vague. Do your best.