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We prove that ifa n  = ∞ and (a n ) is non-decreasing, thena n  = ∞ 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 seriesa n  implies I-convergence of (a n ) 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 (a n ) to zero implies I-convergence of seriesa n  = ∞ for every sequence (a n ).

Convergence of Series on Large Set of Indices