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

TL;DR

This paper presents a novel method to derive continued fraction convergents of the number e by approximating specific integrals as zero, offering a new approach to obtaining e's continued fraction expansion.

Contribution

It introduces a new integral approximation technique for deriving continued fraction convergents of e, providing an alternative to traditional methods.

Findings

Continued fraction convergents of e can be obtained via integral approximations.

A new method for deriving e's continued fraction expansion is proposed.

The approach links integral approximations to continued fraction theory.

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 $e$.