| Title:
|
Some generalizations of Olivier's theorem (English) |
| Author:
|
Faisant, Alain |
| Author:
|
Grekos, Georges |
| Author:
|
Mišík, Ladislav |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
141 |
| Issue:
|
4 |
| Year:
|
2016 |
| Pages:
|
483-494 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $\lim \limits _{n \to \infty } n a_n = 0$. In the present paper: \endgraf (a) We formulate and prove a necessary and sufficient condition for having $\lim \limits _{n \to \infty } n a_n = 0$; Olivier's theorem is a consequence of our Theorem \ref {import}. \endgraf (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 \ref {Iol}, when one takes as $\mathcal I$ the ideal of all finite subsets of $\mathbb N$. (English) |
| Keyword:
|
convergent series |
| Keyword:
|
Olivier's theorem |
| Keyword:
|
ideal |
| Keyword:
|
$\mathcal {I}$-convergence |
| Keyword:
|
$\mathcal {I}$-monotonicity |
| MSC:
|
11B05 |
| MSC:
|
40A05 |
| MSC:
|
40A35 |
| idZBL:
|
Zbl 06674858 |
| idMR:
|
MR3576795 |
| DOI:
|
10.21136/MB.2016.0057-15 |
| . |
| Date available:
|
2017-01-03T15:17:08Z |
| Last updated:
|
2020-07-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/145956 |
| . |
| Reference:
|
[1] Bandyopadhyay, S.: Mathematical Analysis: Problems and Solutions.Academic Publishers, Kolkata (2006). |
| Reference:
|
[2] Knopp, K.: Theory and Applications of Infinite Series.Springer, Berlin (1996), German. Zbl 0842.40001 |
| Reference:
|
[3] Kostyrko, P., Šalát, T., Wilczyński, W.: $\scr I$-convergence.Real Anal. Exch. 26 (2001), 669-685. MR 1844385 |
| Reference:
|
[4] Krzyž, J.: Olivier's theorem and its generalizations.Pr. Mat. 2 (1956), Polish, Russian 159-164. Zbl 0075.25802, MR 0084609 |
| Reference:
|
[5] Olivier, L.: Remarks on infinite series and their convergence.J. Reine Angew. Math. 2 (1827), French 31-44. MR 1577632 |
| Reference:
|
[6] Šalát, T., Toma, V.: A classical Olivier's theorem and statistical convergence.Ann. Math. Blaise Pascal 10 (2003), 305-313. Zbl 1061.40001, MR 2031274, 10.5802/ambp.179 |
| . |
