Convergent subseries of divergent series

Abstract

Let $\mathcal{X}$ be the set of positive real sequences $x=(x_n)$ such that the series ${\sum }_n{x}_n$ is divergent. For each $x\in \mathcal{X}$, 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 $\mathcal{A}$ be the set of sequences $x\in \mathcal{X}$ such that ${\lim }_n{x}_n=0$ and ${\mathcal{I}}_x\ne {\mathcal{I}}_y$ for all sequences $y=(y_n)\in \mathcal{X}$ with ${\mathrm{liminf}}_n{y}_{n+1}/y_n>0$. We show that $\mathcal{A}$ 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.

6 pp; comments are welcome