A note on the irrationality of certain infinite series
type: Item
instance of: scholarly article
title: A note on the irrationality of certain infinite series
zbMATH Open document ID: 0772.11020
author: Masao Toyoizumi
author: Takeshi Okano
published in: Proceedings of the Japan Academy. Series A
publication date: 1993-06-29T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.pja/1195511710
review text: Les auteurs montrent le résultat suivant: Soit \(\{a_ n\}_{n\geq 1}\) une suite de nombres entiers vérifiant \(a_ 1\geq 2\), \(a_{n+1}\geq a_ n\) pour \(n\) suffisamment grand et \(\lim_{n\to\infty}a_ n=+\infty\). Alors les nombres \(1, \alpha=\sum_{k\geq 1} \prod_{1\leq n\leq k} a_ n^{-1}\) et \(\beta=\sum_{k\geq 1} (-1)^{k-1} \prod_{1\leq n\leq k} a_ n^{-1}\) sont linéairement indépendants sur \(\mathbb Q\). La preuve est la même que celle donnée par \textit{C. L. Siegel} [Transcendental numbers. Princeton Univ. Press (1949; Zbl 0039.04402)] pour montrer que \(e\) n'est pas un nombre quadratique.
Mathematics Subject Classification ID: 11J72
zbMATH DE Number: 221217
zbMATH Keywords: irrationality
zbMATH Keywords: infinite series
reviewed by: François Gramain
MaRDI profile type: Publication
OpenAlex ID: W1967243572
cites work: Q3207052
cites work: Transcendental Numbers. (AM-16)
DOI: 10.3792/PJAA.68.212
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description: scientific article; zbMATH DE number 221217