Convergent subseries of divergent series

Let X be the set of positive real sequences x= (xn) such that the series ∑ nxn is divergent. For each x∈ X, let Ix be the collection of all A⊆ N such that the subseries ∑ n∈Axn is convergent. Moreover, let A be the set of sequences x∈ X such that lim nxn= 0 and Ix≠ Iy for all sequences y= (yn) ∈ X with lim inf nyn+1/ yn> 0. We show that A is comeager and that contains uncountably many sequences x which generate pairwise nonisomorphic ideals Ix. This answers, in particular, an open question recently posed by M. Filipczak and G. Horbaczewska.