Convergence of sub-series' and sub-signed series' in terms of the asymptotic ψ-density

2024/08/07 by Heittokangas, Janne, Latreuch, Zinelaabidine
#11B05 #40A05 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2408.03973

Abstract

Given a non-negative real sequence \c_n\_n such that the series ∑_n=1^∞c_n diverges, it is known that the size of an infinite subset A⊂ℕ can be measured in terms of the linear density such that the sub-series ∑_{n∈ A}c_n either (a) converges or (b) still diverges. The purpose of this research is to study these convergence/divergence questions by measuring the size of the set A⊂ℕ in a more precise way in terms of the recently introduced asymptotic ψ-density. The convergence of the associated sub-signed series ∑_n=1 ^∞m_nc_n is also discussed, where \m_n\_n is a real sequence with values restricted to the set \-1, 0, 1\.

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