# A simple continued fraction expansion for $e^n$

24 Apr 2020 Reyes-Bustos Cid

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... (read more)

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• NUMBER THEORY