Exercise 6, § 8

For each real number $x > 1$, there is a unique infinite sequence $(t_n)$ of integers $\geq 1$ such that $t_{n+1} \geq t_n^2$ for all $n \geq 1$ and such that $x = \prod_{n=1}^{\infty} \left( 1 + \frac{1}{t_n} \right)$ (use Exercise 19 of § 7). The number $x$ is rational if and only if $t_{n+1} = t_n^2$ form a certain value of $n$ onwards. [To show that the condition is necessary, show that if we put $x_n = \prod_{k=n+1}^{\infty} \left( 1 + \frac{1}{t_k} \right)$ then the denominators of the fractions $\frac{x_n}{x_n - 1}$ (in their irreducible form) form a decreasing sequence].