How to obtain the continued fraction convergents of the number $e$ by neglecting integrals

Abstract

In this note, we show that any continued fraction convergent of the number $e=\mathrm{2.71828...}$ can be derived by approximating some integral $I_{n,m}:={\int }_0^1{x}^n(1-x{)}^m{e}^{x}dx$ $(n,m\in \mathbb{N})$ by 0. In addition, we present a new way for finding again the well-known regular continued fraction expansion of $e$.

arXiv comment7 pages, To appear