A Classical Olivier’s Theorem and Statistical Convergence
Annales mathématiques Blaise Pascal, Tome 10 (2003) no. 2, pp. 305-313.

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.

DOI : 10.5802/ambp.179
Šalát, Tibor 1 ; Toma, Vladimír 1

1 Comenius University Department of Mathematics Mlynská dolina 84248 Bratislava SLOVAK REPUBLIC
@article{AMBP_2003__10_2_305_0,
     author = {\v{S}al\'at, Tibor and Toma, Vladim{\'\i}r},
     title = {A {Classical} {Olivier{\textquoteright}s} {Theorem} and {Statistical} {Convergence}},
     journal = {Annales math\'ematiques Blaise Pascal},
     pages = {305--313},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {10},
     number = {2},
     year = {2003},
     doi = {10.5802/ambp.179},
     zbl = {1061.40001},
     mrnumber = {2031274},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/ambp.179/}
}
TY  - JOUR
AU  - Šalát, Tibor
AU  - Toma, Vladimír
TI  - A Classical Olivier’s Theorem and Statistical Convergence
JO  - Annales mathématiques Blaise Pascal
PY  - 2003
SP  - 305
EP  - 313
VL  - 10
IS  - 2
PB  - Annales mathématiques Blaise Pascal
UR  - http://archive.numdam.org/articles/10.5802/ambp.179/
DO  - 10.5802/ambp.179
LA  - en
ID  - AMBP_2003__10_2_305_0
ER  - 
%0 Journal Article
%A Šalát, Tibor
%A Toma, Vladimír
%T A Classical Olivier’s Theorem and Statistical Convergence
%J Annales mathématiques Blaise Pascal
%D 2003
%P 305-313
%V 10
%N 2
%I Annales mathématiques Blaise Pascal
%U http://archive.numdam.org/articles/10.5802/ambp.179/
%R 10.5802/ambp.179
%G en
%F AMBP_2003__10_2_305_0
Šalát, Tibor; Toma, Vladimír. A Classical Olivier’s Theorem and Statistical Convergence. Annales mathématiques Blaise Pascal, Tome 10 (2003) no. 2, pp. 305-313. doi : 10.5802/ambp.179. http://archive.numdam.org/articles/10.5802/ambp.179/

[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, Volume XII (1990), pp. 1-6 | MR | Zbl

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

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

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

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

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

[7] Olivier, L. Remarques sur les séries infinies et leur convergence, J. reine angew. Math., Volume 2 (1827), pp. 31-44 | DOI | EuDML | MR | Zbl

[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), Volume 50(64) (1991), pp. 60-70 | EuDML | MR | Zbl

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

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

Cité par Sources :