Convergence of Series on Large Set of Indices

Abstract

Abstract We prove that if an = ∞ and (an) is non-decreasing, then an = ∞ for any set A ⊂ ℕ of positive lower density. We introduce a Cauchy-like definition of I-convergence of series. We prove that the I-convergence of series coincides with the convergence on large set of indexes if and only if I is a P-ideal. It turns out that I-convergence of series an implies I-convergence of (an) to zero. The converse implication does not hold for analytic P-ideals and it is independent of ZFC that there is I ideal of naturals for which I-convergence of (an) to zero implies I-convergence of series an = ∞ for every sequence (an).