*I*^{h}-convergence and convergence of positive series

^{h}

#### 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∈N}

*h(n)*=+∞ and

*I*={

^{h}*A*⊊

*N*: Σ

_{n∈A}

*h(n)*<+∞}. We show that

*I*-convergence and

^{h}*I*-convergence are equivalent. What does not valid in general.

^{h*}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}#### Keywords

#### Full Text:

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