Subseries Numbers for Convergent Subseries

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.