Subseries Numbers for Convergent Subseries

TL;DR

This paper explores the minimal number of index sets needed to ensure divergence or convergence of subseries of conditionally convergent series, linking these to known continuum cardinal characteristics.

Contribution

It introduces the concept of subseries numbers for convergent and divergent subseries, connecting these to established cardinal characteristics of the continuum.

Findings

Cardinal characteristics related to subseries divergence and convergence are identified.

Dual results show the minimal number of series needed to prevent simultaneous convergence.

Connections are established between subseries properties and well-known continuum cardinal characteristics.

Abstract

Every conditionally convergent series of real numbers has a subseries that diverges. The subseries numbers, previously studied in arXiv:1801.06206 , answer the question how many subsets of the natural numbers are necessary, such that every conditionally convergent series has a subseries that diverges, with the index set being one of our chosen sets. By restricting our attention to subseries generated by an index set that is both infinite and coinfinite, we may ask the question where the subseries have to be convergent. The answer to this question is a cardinal characteristic of the continuum. We consider several closely related variations to this question, and show that our cardinal characteristics are related to several well-known cardinal characteristics of the continuum. In our investigation, we simultaneously will produce dual results, answering the question how many conditionally convergent series one needs such that no single infinite coinfinite set of indices makes all of the series converge.