# Article

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Keywords:
convergent series; Olivier's theorem; ideal; $\mathcal {I}$-convergence; $\mathcal {I}$-monotonicity
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 $\mathbb N$.
References:
[1] Bandyopadhyay, S.: Mathematical Analysis: Problems and Solutions. Academic Publishers, Kolkata (2006).
[2] Knopp, K.: Theory and Applications of Infinite Series. Springer, Berlin (1996), German. Zbl 0842.40001
[3] Kostyrko, P., Šalát, T., Wilczyński, W.: $\scr I$-convergence. Real Anal. Exch. 26 (2001), 669-685. MR 1844385
[4] Krzyž, J.: Olivier's theorem and its generalizations. Pr. Mat. 2 (1956), Polish, Russian 159-164. MR 0084609 | Zbl 0075.25802
[5] Olivier, L.: Remarks on infinite series and their convergence. J. Reine Angew. Math. 2 (1827), French 31-44. MR 1577632
[6] Šalát, T., Toma, V.: A classical Olivier's theorem and statistical convergence. Ann. Math. Blaise Pascal 10 (2003), 305-313. DOI 10.5802/ambp.179 | MR 2031274 | Zbl 1061.40001

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