Some generalizations of Olivier’s theorem. (English) Zbl 1389.40004

Faisant, Alain; Grekos, Georges; Mišík, Ladislav

Math. Bohem. 141, No. 4, 483-494 (2016).

Summary: Let \(\sum\limits_{n=1}^\infty a_n\) be a convergent series of positive real numbers. L. Olivier [J. Reine Angew. Math. 2, 31–44 (1827; ERAM 002.0044cj)] proved that if the sequence \((a_n)\) is non-increasing, then \(\lim\limits_{n\to\infty}na_n=0\). In the present paper:
(a) We formulate and prove a necessary and sufficient condition for having \(\lim\limits_{n\to\infty}na_n=0\); Olivier’s theorem is a consequence of our Theorem 2.1.
(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 3.1, when one takes as \(\mathcal{I}\) the ideal of all finite subsets of \(\mathbb{N}\).

MSC:

40A05	Convergence and divergence of series and sequences
40A35	Ideal and statistical convergence
11B05	Density, gaps, topology

Keywords:

convergent series; Olivier’s theorem; ideal; \(\mathcal{I}\)-convergence; \(\mathcal{I}\)-monotonicity