Abstract
A series $\sum x_k$ is $\mathcal{F}$-convergent to $s$ if the sequence $(\sum_{k=1}^n x_k)$ of its partial sums is $\mathcal{F}$-convergent to $s$. We describe filters $\mathcal{F}$ for which $\mathcal{F}$-convergence of a series $\sum x_k$ implies $\mathcal{F}$-convergence to $0$ of the series terms $x_k$. If $(x_k)$ is small enough with respect to a given filter $\mathcal{F}$, then there is an $\mathcal{F}$-subseries $\sum_{k\in I} x_k$ which is absolutely convergent in the usual sense. Filters corresponding to summable ideals, Erdős-Ulam ideals, matrix summability ideals, lacunary ideals and Louveau-Veličković ideals are considered.
Citation
Alexander Leonov. Cihan Orhan. "On Filter Convergence of Series." Real Anal. Exchange 40 (2) 459 - 474, 2015.
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