Some generalizations of Olivier's theorem

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