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