A Note on Olivier's Theorem and Convergence in Erdos-Ulam Density

Bukor, József

doi:10.28924/2291-8639-18-2020-332

Research Article

A Note on Olivier's Theorem and Convergence in Erdos-Ulam Density

József Bukor¹

Department of Mathematics and Informatics, J. Selye University, Komárno, Slovakia

Abstract

Olivier's Theorem says that if Σan is a convergent positive series and (a_n) is monotone decreasing, then na_n→0. Salat and Toma [4] proved that the monotonicity condition can be omitted if the convergence of (nan)n is replaced by the statistical convergence. The aim of this note is to give an alternative proof and generalization of this result.

How to Cite

J. Bukor, A Note on Olivier's Theorem and Convergence in Erdos-Ulam Density, Int. J. Anal. Appl. 18 (2020), 332–336. https://doi.org/10.28924/2291-8639-18-2020-332

BibTeX

@article{Bukor2020,
  author  = {Bukor, József},
  title   = {A Note on Olivier's Theorem and Convergence in Erdos-Ulam Density},
  journal = {Int. J. Anal. Appl.},
  volume  = {18},
  year    = {2020},
  pages   = {332--336},
  doi     = {10.28924/2291-8639-18-2020-332}
}

References

[1] J. Bukor, F. Filip and J. T. Toth, Sets with countably infinitely many prescribed weighted densities, Rocky Mountain J. Math. (to appear).
[2] A. Faisant,G. Grekos and L. Misik, Some generalizations of Olivier's theorem, Math. Bohem. 141 (4) (2016), 483-494.
[3] C. P. Niculescu and G. T. Prajitura, Some open problems concerning the convergence of positive series, Ann. Acad. Rom. Sci. Ser. Math. Appl. 6 (1) (2014), 92-107.
[4] T. Salat and V. Toma, A Classical Olivier's Theorem and Statistical Convergence, Ann. Math. Blaise Pascal, 10 (2) (2003), 305-313.