Ih-convergence and convergence of positive series

Vladimír Baláž, Alexander Maťašovský, Tomáš Visnyai


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 ΣnNh(n)=+∞ and Ih={AN : ΣnAh(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 hR+R+ such that modified Olivier's Theorem remains valid for ideal Ih.


<em>I</em>-convergence; convergence of positive series; Olivier's theorem; admissible ideals

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ISSN: 1331-0623 (Print), 1848-8013 (Online)