Convergence of sub-series' and sub-signed series' in terms of the asymptotic $\psi$-density

TL;DR

This paper investigates the convergence properties of sub-series and sub-signed series of divergent series using the novel concept of asymptotic ψ-density to measure the size of subsets of natural numbers.

Contribution

It introduces the use of asymptotic ψ-density to analyze convergence and divergence of sub-series and sub-signed series, providing a more precise measure than linear density.

Findings

Characterization of convergence based on asymptotic ψ-density

Extension to sub-signed series with restricted sign sequences

Insights into the size of subsets influencing series convergence

Abstract

Given a non-negative real sequence $\{c_n\}_n$ such that the series $\sum_{n=1}^{\infty}c_n$ diverges, it is known that the size of an infinite subset $A\subset\mathbb{N}$ can be measured in terms of the linear density such that the sub-series $\sum_{n\in A}c_n$ 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 $A\subset\mathbb{N}$ in a more precise way in terms of the recently introduced asymptotic $\psi$-density. The convergence of the associated sub-signed series $\sum_{n=1 }^{\infty}m_nc_n$ is also discussed, where $\{m_n\}_n$ is a real sequence with values restricted to the set $\{-1, 0, 1\}$.