Exercise 5, § 8

For each real number $x \in ]0, 1]$, there is a unique infinite increasing sequence $(q_n)$ of integers > 0 such that $x = \sum_{n=1}^{\infty} 1/(q_1 q_2 \cdots q_n)$. The number $x$ is rational if and only if $q_{n+1} = q_n$ from a certain value of $n$ onwards.