On the c₀-equivalence and permutations of series

2020/08/09 by Bartoszewicz, Artur, Fechner, Włodzimierz, Świątczak, Aleksandra +1
#40A05 #40A35 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2008.03785

Abstract

Assume that a convergent series of real numbers ∑_n=1^∞ a_n has the property that there exists a set A⊆ \N such that the series ∑_{n ∈ A} a_n is conditionally convergent. We prove that for a given arbitrary sequence (b_n) of real numbers there exists a permutation σ\colon \N → \N such that σ(n) = n for every n ∉ A and (b_n) is c₀-equivalent to a subsequence of the sequence of partial sums of the series ∑_n=1^∞ a_σ(n). Moreover, we discuss a connection between our main result with the classical Riemann series theorem.

Related

• Constructive aspects of Riemann's permutation theorem for series
• On the set of limit points of conditionally convergent series
• On Riemann's Theorem About Conditionally Convergent Series
• Type R λ-Permutation Approach to Velleman's Open Problem
• Achievement sets of conditionally convergent series