A simple continued fraction expansion for $e^n$
In this paper we discuss the continued fraction expansion \[ e = 3 - \cfrac{1}{4 - \cfrac{2}{5 - \cfrac{3}{6 - \cfrac{4}{7 - \cdots}}} }, \] and its convergence properties. We show that this expansion is a particular case of a continued fraction expansion of $e^n$, for positive integer power \(n\) with denominators given by arithmetic progressions, and more generally, it is a special case of a continued fraction expansion of the incomplete gamma function, or equivalently, of the confluent hypergeometric function. In addition, we give a related family of continued fraction expansions of $e^{\ell/n}$ for positive intengers $1\leq\ell< n $.