# Article

 Title: Some generalizations of Olivier's theorem (English) Author: Faisant, Alain Author: Grekos, Georges Author: Mišík, Ladislav Language: English Journal: Mathematica Bohemica ISSN: 0862-7959 (print) ISSN: 2464-7136 (online) Volume: 141 Issue: 4 Year: 2016 Pages: 483-494 Summary lang: English . Category: math . Summary: 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: \endgraf (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}. \endgraf (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 $\mathbb N$. (English) Keyword: convergent series Keyword: Olivier's theorem Keyword: ideal Keyword: $\mathcal {I}$-convergence Keyword: $\mathcal {I}$-monotonicity MSC: 11B05 MSC: 40A05 MSC: 40A35 idZBL: Zbl 06674858 idMR: MR3576795 DOI: 10.21136/MB.2016.0057-15 . Date available: 2017-01-03T15:17:08Z Last updated: 2020-07-01 Stable URL: http://hdl.handle.net/10338.dmlcz/145956 . Reference: [1] Bandyopadhyay, S.: Mathematical Analysis: Problems and Solutions.Academic Publishers, Kolkata (2006). Reference: [2] Knopp, K.: Theory and Applications of Infinite Series.Springer, Berlin (1996), German. Zbl 0842.40001 Reference: [3] Kostyrko, P., Šalát, T., Wilczyński, W.: $\scr I$-convergence.Real Anal. Exch. 26 (2001), 669-685. MR 1844385 Reference: [4] Krzyž, J.: Olivier's theorem and its generalizations.Pr. Mat. 2 (1956), Polish, Russian 159-164. Zbl 0075.25802, MR 0084609 Reference: [5] Olivier, L.: Remarks on infinite series and their convergence.J. Reine Angew. Math. 2 (1827), French 31-44. MR 1577632 Reference: [6] Šalát, T., Toma, V.: A classical Olivier's theorem and statistical convergence.Ann. Math. Blaise Pascal 10 (2003), 305-313. Zbl 1061.40001, MR 2031274, 10.5802/ambp.179 .

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