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.

@article{AMBP_2003__10_2_305_0,
     author = {Tibor \v{S}al\'at and Vladim{\'\i}r Toma},
     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},
     mrnumber = {2031274},
     zbl = {1061.40001},
     language = {en},
     url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.179/}
}
Tibor Šalát; Vladimír Toma. 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. https://ambp.centre-mersenne.org/articles/10.5802/ambp.179/

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