Convergent subseries of divergent series

Abstract

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