Extensions of the Argument

The method of proof of this section generalizes to show that the continued fraction expansion of $e^{1/n}$ is

$$[1, (n-1), 1, 1, (3n-1), 1, 1, (5n-1), 1, 1, (7n-1),\ldots]$$

for all $n\in\mathbb{N}$ (see Exercise 5.6).