The ideal test for the divergence of a series

TL;DR

This paper extends Olivier's theorem by exploring ideal convergence of the sequence n*a_n under relaxed monotonicity conditions, and investigates large algebraic structures where the theorem's assertions fail.

Contribution

It introduces new generalizations of Olivier's theorem using ideal convergence and analyzes the algebraic structure of sequences where the theorem does not hold.

Findings

Generalizations include dropping monotonicity or relaxing it to large index sets.

Established conditions under which n*a_n converges in the ideal sense.

Identified large linear and algebraic substructures in sequences failing the theorem.

Abstract

We generalize the classical Olivier's theorem which says that for any convergent series $\sum_n a_n$ with positive nonincreasing real terms the sequence $(n a_n)$ tends to zero. Our results encompass many known generalizations of Olivier's theorem and give some new instances. The generalizations are done in two directions: we either drop the monotonicity assumption completely or we relax it to the monotonicity on a large set of indices. In both cases, the convergence of $(na_n)$ is replaced by ideal convergence. In the second part of the paper, we examine families of sequences for which the assertions of our generalizations of Olivier's theorem fail. Here, we are interested in finding large linear and algebraic substructures in these families.