The ideal test for the divergence of a series
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Published:2023-04-07
Issue:3
Volume:117
Page:
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ISSN:1578-7303
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Container-title:Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas
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language:en
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Short-container-title:Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.
Author:
Filipów Rafał, Kwela AdamORCID, Tryba Jacek
Abstract
AbstractWe generalize the classical Olivier’s theorem which says that for any convergent series $$\sum _n a_n$$
∑
n
a
n
with positive nonincreasing real terms the sequence $$(n a_n)$$
(
n
a
n
)
tends to zero. Our results encompass many known generalizations of Olivier’s theorem and give some new instances. The generalizations are done in two directions: we either drop the monotonicity assumption completely or we relax it to the monotonicity on a large set of indices. In both cases, the convergence of $$(na_n)$$
(
n
a
n
)
is replaced by ideal convergence. In the second part of the paper, we examine families of sequences for which the assertions of our generalizations of Olivier’s theorem fail. Here, we are interested in finding large linear and algebraic substructures in these families.
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Computational Mathematics,Geometry and Topology,Algebra and Number Theory,Analysis
Reference27 articles.
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