Some generalizations of Olivier's theorem

• Published in: Mathematica bohemica Vol. 141; no. 4; pp. 483 - 494
• Main Authors: Grekos, Georges, Faisant, Alain, Misik, Ladislav
• Format: Journal Article
• Language: English
• Published: Institute of Mathematics, Czech Academy of Sciences 01.01.2016; Institute of Mathematics of the Czech Academy of Science
• Subjects: convergent series; ideal; mathcal{I}$-convergence; mathcal{I}$-monotonicity; Mathematics; Number Theory; Olivier's theorem
• ISSN: 0862-7959; 2464-7136
• DOI: 10.21136/MB.2016.0057-15

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 $\mathbb N$.