Infinite continued fractions and convergents

Question

I've got some question about infinite continued fractions and convergents:

Given:

$n = 205320043521075746592613$

$e = 70760135995620281241019$

$\frac{e}{n}=[0, 2, 1, 9, 6, 54, 5911, 1, 5, 1, 1, . . .]$

Why $\frac{e}{n}=[0,2,1,9,6,54]$ is a good convergent?

Why don't we choose for $\frac{e}{n}=[0, 2, 1, 9, 6, 54, 5911]$ or $\frac{e}{n}=[0, 2, 1, 9, 6, 54, 5911,1]$?

The book says you always have to stop before a big number. In this case 5911. Why?

Thnx for your help!

Answer 1

The goal is to get a good rational approximation with a relatively small denominator. If we didn't care about the size of the numerator and denominator, we'd just use the original fraction. Also, I have read that one of the applications of continued fractions is in approximating desired gear ratios by combining gears with a small number of teeth. A gear with 5911 teeth would be hard to machine, and fragile.

Answer 2

A large coefficient means, that it needs a large new partial denominator to improve the previous approximation $\implies$ the previous approximation was (already) very good.