Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 11F / Solution

Setand recursivelyThen $p_n/q_n=[a_0,ldots,a_n]$. The determinant identityfollows by induction. Writing the remaining complete quotient as ${\theta }_{n+1}>1$ givesso $\theta$ lies strictly between consecutive convergents and their order alternates. For odd $n$,Moreover $|\theta -p_n/q_n|<1/(q_n{q}_{n+1})$, and $q_n\to \infty$, proving convergence.

The continued-fraction algorithm gives