A Classical Olivier’s Theorem and Statistical Convergence
Šalát, Tibor ; Toma, Vladimír
Annales mathématiques Blaise Pascal, Tome 10 (2003), p. 305-313 / Harvested from Numdam
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L. Olivier démontrait en 1827 un résultat classique sur la vitesse de convergence vers zéro d’une série convergente à termes positifs décroissants. Nous démontrons que ce résultat reste valable si nous omettons la monotonie des termes de la série, en remplaçant l’opération limite par la limite statistique ou encore par des généralisations de ce concept.

L. Olivier proved in 1827 the classical result about the speed of convergence to zero of the terms of a convergent series with positive and decreasing terms. We prove that this result remains true if we omit the monotonicity of the terms of the series when the limit operation is replaced by the statistical limit, or some generalizations of this concept.

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     title = {A Classical Olivier's Theorem and Statistical Convergence},
     journal = {Annales math\'ematiques Blaise Pascal},
     volume = {10},
     year = {2003},
     pages = {305-313},
     doi = {10.5802/ambp.179},
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     mrnumber = {2031274},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AMBP_2003__10_2_305_0}
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Šalát, Tibor; Toma, Vladimír. A Classical Olivier’s Theorem and Statistical Convergence. Annales mathématiques Blaise Pascal, Tome 10 (2003) pp. 305-313. doi : 10.5802/ambp.179. http://gdmltest.u-ga.fr/item/AMBP_2003__10_2_305_0/

[1] Brown, T. C.; Freedman, A. R. The uniform density of sets of integers and Fermat’s last theorem, C. R. Math. Rep. Acad. Sci. Canada, Tome XII (1990), pp. 1-6 | MR 1043085 | Zbl 0701.11011

[2] Fast, H. Sur la convergence statistique, Coll. Math., Tome 2 (1951), pp. 241-244 | MR 48548 | Zbl 0044.33605

[3] Fridy, J. A. On statistical convergence, Analysis, Tome 5 (1985), pp. 301-313 | MR 816582 | Zbl 0588.40001

[4] Halberstam, H.; Roth, K. F. Sequences I, Oxford University Press, Oxford (1966) | MR 210679 | Zbl 0141.04405

[5] Knopp, K. Theorie und Anwendung der unendlichen Reihen 3. Aufl., Springer (1931) | MR 28430 | Zbl 0001.39201

[6] Kostyrko, P.; Šalát, T.; Wilczński, W. ℐ-convergence, Real Anal. Exch., Tome 26 (2000-2001), pp. 669-689 | MR 1844385 | Zbl 1021.40001

[7] Olivier, L. Remarques sur les séries infinies et leur convergence, J. reine angew. Math., Tome 2 (1827), pp. 31-44 | Article | MR 1577632 | Zbl 002.0044cj

[8] Powel, B. J.; Šalát, T. Convergence of subseries of the harmonic series and asymptotic densities of sets of positive integers, Publ. Inst. Math.(Beograd), Tome 50(64) (1991), pp. 60-70 | MR 1252159 | Zbl 0745.11008

[9] Schoenberg, I. J. The integrability of certain functions and related summability methods, Amer. Math. Monthly, Tome 66 (1959), pp. 361-375 | Article | MR 104946 | Zbl 0089.04002

[10] Šalát, T. On statistically convergent sequences of real numbers, Math. Slovaca, Tome 30 (1980), pp. 139-150 | MR 587239 | Zbl 0437.40003