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

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 ${\sum }_{n=1}^{\infty }{m}_n{c}_n$ is also discussed, where $\{m_n{\}}_n$ is a real sequence with values restricted to the set $\{-1,0,1\}$.

27 pages