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

2010/05/17 by Bakir Farhi, Farhi, Bakir
Computer Science · Mathematics · #11A05 #FOS: Mathematics #Iterative Methods for Nonlinear Equations #Mathematical and Theoretical Analysis #Number Theory (math.NT) #Numerical Methods and Algorithms

paper · pdf · doi:10.48550/arxiv.1005.2951

openalex publication_date 2010/05/17 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

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} := ∫₀¹ xⁿ (1 - x)^m e^x d x (n, m ∈ ℕ) by 0. In addition, we present a new way for finding again the well-known regular continued fraction expansion of e.

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