A classical Olivier’s theorem and statistical convergence. (English) Zbl 1061.40001

L. Olivier [J. Reine Angew. Math. 2, 31–44 (1827)] proved the classical result about the speed of convergence to zero of the terms of a convergent series with positive and decreasing terms. In this paper, the authors 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. For details, we refer the reader to the paper.


40A05 Convergence and divergence of series and sequences
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[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, XII, 1-6 (1990) · Zbl 0701.11011
[2] Fast, H., Sur la convergence statistique, Coll. Math., 2, 241-244 (1951) · Zbl 0044.33605
[3] Fridy, J. A., On statistical convergence, Analysis, 5, 301-313 (1985) · Zbl 0588.40001
[4] Halberstam, H.; Roth, K. F., Sequences I (1966) · Zbl 0141.04405
[5] Knopp, K., Theorie und Anwendung der unendlichen Reihen 3. Aufl. (1931) · Zbl 0001.39201
[6] Kostyrko, P.; Šalát, T.; Wilczński, W., \( \mathcal{I} \)-convergence, Real Anal. Exch., 26, 669-689 (20002001) · Zbl 1021.40001
[7] Olivier, L., Remarques sur les séries infinies et leur convergence, J. reine angew. Math., 2, 31-44 (1827) · ERAM 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), 50(64), 60-70 (1991) · Zbl 0745.11008
[9] Schoenberg, I. J., The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66, 361-375 (1959) · Zbl 0089.04002
[10] Šalát, T., On statistically convergent sequences of real numbers, Math. Slovaca, 30, 139-150 (1980) · Zbl 0437.40003
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