Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 4 / 12H / b / iii

Table of contents
- Solution iii

Solution

Consecutive convergents obey the standard error estimate

0 < α - \frac{p _{n}}{q _{n}} < \frac{1}{q _{n} q _{n + 1}} \leq \frac{1}{a _{n + 1} q _{n}^{2}} .

If the partial quotients are unbounded, then for every

c > 0

there are infinitely many indices with

a_{n + 1} > 1/ c

; otherwise the whole sequence would be bounded after adjoining its finite initial segment. At each such index,

α - \frac{p _{n}}{q _{n}} < \frac{c}{q _{n}^{2}} .

The denominators of the infinite continued fraction tend to infinity, so these give infinitely many distinct rational numbers of the required kind.

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