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Keywords:
convergent series; Olivier's theorem; ideal; $\mathcal {I}$-convergence; $\mathcal {I}$-monotonicity
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:
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