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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 \(\mathbb{N}\).
40A05 Convergence and divergence of series and sequences
40A35 Ideal and statistical convergence
11B05 Density, gaps, topology
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