On filter convergence of series

Leonov A., ORHAN C.

Real Analysis Exchange, vol.40, no.2, pp.459-474, 2015 (Scopus) identifier

  • Publication Type: Article / Article
  • Volume: 40 Issue: 2
  • Publication Date: 2015
  • Doi Number: 10.14321/realanalexch.40.2.0459
  • Journal Name: Real Analysis Exchange
  • Journal Indexes: Scopus
  • Page Numbers: pp.459-474
  • Keywords: F-convergence, Filter, I-convergence, Ideal, Series, Statistical convergence
  • Ankara University Affiliated: Yes

Abstract

A series Σ xk is F-convergent to s if the sequence (Σk=1 n xk) of its partial sums is F-convergent to s. We describe filters F for which F- convergence of a series Σ xk implies F-convergence to 0 of the series terms xk. If (xk) is small enough with respect to a given filter F, then there is an F-subseries Σ k∈I xk that is absolutely convergent in the usual sense. Filters corresponding to summable ideals, Erdos-Ulam ideals, matrix summability ideals, lacunary ideals and Louveau-Veličković ideals are considered.