On the \(c_0\)-equivalence and permutations of series

Assume that a convergent series of real numbers \(\sum\limits_{n=1}^\infty a_n\) has the property that there exists a set \(A\subseteq \N\) such that the series \(\sum\limits_{n \in A} a_n\) is conditionally convergent. We prove that for a given arbitrary sequence \((b_n)\) of real numbers there exi...

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Bibliographic Details
Published in:arXiv.org 2020-08
Main Authors: Bartoszewicz, Artur, Fechner, Włodzimierz, Świątczak, Aleksandra, Widz, Agnieszka
Format: Article
Language:English
Subjects:
Colon
Convergence
Equivalence
Permutations
Real numbers
Series (mathematics)
Online Access:Get full text
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Summary:Assume that a convergent series of real numbers \(\sum\limits_{n=1}^\infty a_n\) has the property that there exists a set \(A\subseteq \N\) such that the series \(\sum\limits_{n \in A} a_n\) is conditionally convergent. We prove that for a given arbitrary sequence \((b_n)\) of real numbers there exists a permutation \(\sigma\colon \N \to \N\) such that \(\sigma(n) = n\) for every \(n \notin A\) and \((b_n)\) is \(c_0\)-equivalent to a subsequence of the sequence of partial sums of the series \(\sum\limits_{n=1}^\infty a_{\sigma(n)}\). Moreover, we discuss a connection between our main result with the classical Riemann series theorem.
ISSN:2331-8422