Some generalizations of Olivier's theorem

Faisant, Alain, Grekos, Georges, and Mišík, Ladislav. "Some generalizations of Olivier's theorem." Mathematica Bohemica 141.4 (2016): 483-494. <http://eudml.org/doc/287530>.

@article{Faisant2016,
abstract = {Let $\sum \limits _\{n=1\}^\infty a_n$ be a convergent series of positive real numbers. L. Olivier proved that if the sequence $(a_n)$ is non-increasing, then $\lim \limits _\{n \rightarrow \infty \} n a_n = 0$. In the present paper: (a) We formulate and prove a necessary and sufficient condition for having $\lim \limits _\{n \rightarrow \infty \} n a_n = 0$; Olivier’s theorem is a consequence of our Theorem . (b) We prove properties analogous to Olivier’s property when the usual convergence is replaced by the $\mathcal \{I\}$-convergence, that is a convergence according to an ideal $\mathcal \{I\}$ of subsets of $\mathbb \{N\}$. Again, Olivier’s theorem is a consequence of our Theorem , when one takes as $\mathcal \{I\}$ the ideal of all finite subsets of $\mathbb \{N\}$.},
author = {Faisant, Alain, Grekos, Georges, Mišík, Ladislav},
journal = {Mathematica Bohemica},
keywords = {convergent series; Olivier’s theorem; ideal; $\mathcal \{I\}$-convergence; $\mathcal \{I\}$-monotonicity},
language = {eng},
number = {4},
pages = {483-494},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Some generalizations of Olivier's theorem},
url = {http://eudml.org/doc/287530},
volume = {141},
year = {2016},
}

TY - JOUR
AU - Faisant, Alain
AU - Grekos, Georges
AU - Mišík, Ladislav
TI - Some generalizations of Olivier's theorem
JO - Mathematica Bohemica
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 141
IS - 4
SP - 483
EP - 494
AB - Let $\sum \limits _{n=1}^\infty a_n$ be a convergent series of positive real numbers. L. Olivier proved that if the sequence $(a_n)$ is non-increasing, then $\lim \limits _{n \rightarrow \infty } n a_n = 0$. In the present paper: (a) We formulate and prove a necessary and sufficient condition for having $\lim \limits _{n \rightarrow \infty } n a_n = 0$; Olivier’s theorem is a consequence of our Theorem . (b) We prove properties analogous to Olivier’s property when the usual convergence is replaced by the $\mathcal {I}$-convergence, that is a convergence according to an ideal $\mathcal {I}$ of subsets of $\mathbb {N}$. Again, Olivier’s theorem is a consequence of our Theorem , when one takes as $\mathcal {I}$ the ideal of all finite subsets of $\mathbb {N}$.
LA - eng
KW - convergent series; Olivier’s theorem; ideal; $\mathcal {I}$-convergence; $\mathcal {I}$-monotonicity
UR - http://eudml.org/doc/287530
ER -