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 $e = 2.71828...$ can be derived by approximating some integral $I_{n, m} := \int_{0}^{1} x^n (1 - x)^m e^x d x$ $(n, m \in \mathbb{N})$ by 0. In addition, we present a new way for finding again the well-known regular continued fraction expansion of $e$.