Ih-convergence and convergence of positive series

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 (na_n) 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 I^h where h: R⁺→R⁺ is a function such that Σ_n∈Nh(n)=+∞ and I^h={A⊊N : Σ_n∈Ah(n)<+∞}. We show that I^h-convergence and I^h*-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 I^h in generally. We find sufficient conditions for real function h: R⁺→R⁺ such that modified Olivier's Theorem remains valid for ideal I^h.