The Continued Fraction of $e$

The continued fraction expansion of $e$ begins $[2, 1, 2, 1, 1, 4, 1, 1, 6, \ldots]$ . The obvious pattern in fact does continue, as Euler proved in 1737 (see [#!euler:contfrac!#]), and we will prove in this section. As an application, Euler gave a proof that $e$ is irrational by noting that its continued fraction is infinite.

The proof we give below draws heavily on the proof in [#!cohn:contfrac!#], which describes a slight variant of a proof of Hermite (see [#!olds:contfrac!#]). The continued fraction representation of $e$ is also treated in the German book [#!perron!#], but the proof requires substantial background from elsewhere in that text.