Original scientific paper

\(I^h\)-convergence and convergence of positive series

Vladimír Baláž ; Faculty of Chemical and Food Technology, Slovak University of Technology in Bratislava, Bratislava, Slovakia
Alexander Maťašovský ; Faculty of Chemical and Food Technology, Slovak University of Technology in Bratislava, Bratislava, Slovakia
Tomáš Visnyai ; Faculty of Chemical and Food Technology, Slovak University of Technology in Bratislava, Bratislava, Slovakia

Abstract

In 1827 L. Olivier proved result about the speed of convergence to zero of the terms of convergent positive series with non-increasing terms so-called Olivier's Theorem. T. Šalát and V. Toma made remark that the monotonicity condition in Olivier's Theorem can be dropped if the convergence of the sequence (nan) is weakened by means of the notion of I-convergence for an appropriate ideal I. Results of this type are called a modified Olivier's Theorem.

In connection with this we will study the properties of summable ideals Ih where h: R+→R+ is a function such that Σn∈Nh(n)=+∞ and Ih={A⊊N : Σn∈Ah(n)<+∞}. We show that Ih-convergence and Ih*-convergence are equivalent. What does not valid in general.
Further we also show that the modified Olivier's Theorem is not valid for summable ideals Ih in generally. We find sufficient conditions for real function h: R+→R+ such that modified Olivier's Theorem remains valid for ideal Ih.

Keywords

\( I^{h}\)-convergence; convergence of positive series; Olivier's theorem; admissible ideals

Hrčak ID:

303374

URI

https://hrcak.srce.hr/303374

Publication date:

2.6.2023.

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