Convergence of series on large set of indices

Year, pages: 2015, 1095 - 1106

We prove that if $∑_n=1^∞a_n=∞$ and $(a_n)$ is non-decreasing, then $∑_{n\in A}a_n=∞$ for any set $A\subset\mathbb{N}$ of positive lower density. We introduce a Cauchy-like definition of $\mathcal I$-convergence of series. We prove that the $\mathcal I$-convergence of series coincides with the convergence on large set of indexes if and only if $\mathcal I$ is a $P$-ideal. It turns out that $\mathcal I$-convergence of series $∑_n=1^∞ a_n$ implies $\mathcal I$-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 $\mathcal I$ ideal of naturals for which $\mathcal I$-convergence of $(a_n)$ to zero implies $\mathcal I$-convergence of series $∑_n=1^∞ a_n=∞$ for every sequence $(a_n)$.

Głąb, S., Olczyk, M. 2015. Convergence of series on large set of indices. In Mathematica Slovaca, vol. 65, no.5, pp. 1095-1106. 0139-9918. DOI: https://doi.org/10.1515/ms-2015-0075

Głąb, S., Olczyk, M. (2015). Convergence of series on large set of indices. Mathematica Slovaca, 65(5), 1095-1106. 0139-9918. DOI: https://doi.org/10.1515/ms-2015-0075