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.
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.
@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, Volume 10 (2003) no. 2, pp. 305-313. doi : 10.5802/ambp.179. http://archive.numdam.org/articles/10.5802/ambp.179/
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