Convergent subseries of divergent series

TL;DR

This paper investigates the structure of subseries of divergent positive sequences, showing that a large set of such sequences generate many nonisomorphic ideals, answering an open question in the field.

Contribution

It demonstrates that the set of sequences with specific convergence properties is comeager and contains uncountably many that produce pairwise nonisomorphic ideals, advancing understanding of divergent series.

Findings

The set  is comeager.

Contains uncountably many sequences with nonisomorphic ideals.

Answers an open question by Filipczak and Horbaczewska.

Abstract

Let $\mathscr{X}$ be the set of positive real sequences $x=(x_n)$ such that the series $\sum_n x_n$ is divergent. For each $x \in \mathscr{X}$, let $\mathcal{I}_x$ be the collection of all $A\subseteq \mathbf{N}$ such that the subseries $\sum_{n \in A}x_n$ is convergent. Moreover, let $\mathscr{A}$ be the set of sequences $x \in \mathscr{X}$ such that $\lim_n x_n=0$ and $\mathcal{I}_x\neq \mathcal{I}_y$ for all sequences $y=(y_n) \in \mathscr{X}$ with $\liminf_n y_{n+1}/y_n>0$. We show that $\mathscr{A}$ is comeager and that contains uncountably many sequences $x$ which generate pairwise nonisomorphic ideals $\mathcal{I}_x$. This answers, in particular, an open question recently posed by M. Filipczak and G. Horbaczewska.