MATHEMATICA BOHEMICA, Vol. 141, No. 4, pp. 483-494, 2016

Alain Faisant, Georges Grekos, Département de Mathématiques and Institut Camille Jordan, Université Jean Monnet (Saint-Étienne), 23 Rue du Dr Paul Michelon, 42023 Saint-Étienne Cedex 2, France, e-mail: faisant@univ-st-etienne.fr, grekos@univ-st-etienne.fr; Ladislav Mišík, University of Ostrava, 30. dubna 22, 701 03 Ostrava 1, Czech Republic, and J. Selye University, Bratislavská cesta 3322, 945 01 Komárno, Slovakia, e-mail: ladislav.misik@osu.cz

Abstract: Let $\sum\limits_{n=1}^\infty a_n$ be a convergent series of positive real numbers. L. Olivier proved that if the sequence $(a_n)$ is non-increasing, then $\lim\limits_{n \to\infty} n a_n = 0$. In the present paper: (a) We formulate and prove a necessary and sufficient condition for having $\lim\limits_{n \to\infty} n a_n = 0$; Olivier's theorem is a consequence of our Theorem \ref{import}. (b) We prove properties analogous to Olivier's property when the usual convergence is replaced by the $\mathcal I$-convergence, that is a convergence according to an ideal $\mathcal I$ of subsets of $\mathbb N$. Again, Olivier's theorem is a consequence of our Theorem \ref{Iol}, when one takes as $\mathcal I$ the ideal of all finite subsets of $\mathbb N$.

Keywords: convergent series; Olivier's theorem; ideal; $\mathcal{I}$-convergence; $\mathcal{I}$-monotonicity

References:
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Some generalizations of Olivier's theorem

Alain Faisant, Georges Grekos, Ladislav Mišík

Received September 25, 2015. First published October 8, 2016.