Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 12I / b

Table of contents
- Solution b

Solution

Taking determinants of the convergent matrices gives

q_{n} p_{n - 1} - q_{n - 1} p_{n} = (- 1)^{n}

. The complete-quotient formula then yields

∣ α - p_{n} / q_{n} ∣ < 1/ (q_{n} q_{n + 1}) < 1/ q_{n}^{2}

. Comparing two consecutive errors shows at least one is below

1/ (2 q^{2})

; hence both stated inequalities occur infinitely often (the second is the elementary precursor of Hurwitz's sharper

1/ 5

bound).

 Ancestors (10)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version