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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:
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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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