a series from continued fraction expansion of an irrational number

Question

if $\theta $ is a irrational number, it has a continued fraction expansion $$[a_{0}, \dots a_{n}, \dots ] = a_{0}+\frac{1}{a_{1}+\frac{1}{\dots}} $$ the associated rational approximations is $p_{n}/q_{n}= [a_{0}, \dots a_{n} ] $ I want to show that $\sum {\frac{1}{q_{n} }} $is finite from the recursion formular I obtain that $$p_{n}q_{n-1}-p_{n-1}q_{n}=(-1)^{n-1}$$ it follows that $p_{n}/q_{n}$ is an alternating series, and I have no clue what to do next.

Answer 1

From the recurrence relations $$ p_n = a_n p_{n-1} + p_{n-2} \\ q_n = a_n q_{n-1} + q_{n-2} $$ and the fact that all $a_n \ne 0$ for an irrational number it follows that in particular $$ q_n \ge q_{n-1} + q_{n-2} $$ and therefore $q_n \ge F_n$ ($n$-th Fibonacci number). The conclusion follows because $\sum \frac{1}{F_n}$ is convergent.