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2. On weighted densities
- Creator:
- Giuliani-Antonini, Rita, Grekos, Georges, and Mišík, Ladislav
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- asymptotic density, logarithmic density, $f$-density, and continuity
- Language:
- English
- Description:
- The continuity of densities given by the weight functions $n^{\alpha }$, $\alpha \in [-1,\infty [$, with respect to the parameter $\alpha $ is investigated.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
3. Some generalizations of Olivier's theorem
- Creator:
- Faisant, Alain, Grekos, Georges, and Mišík, Ladislav
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- convergent series, Olivier’s theorem, ideal, I-convergence, and I-monotonicity
- Language:
- English
- Description:
- Let ∑ ∞ n=1 an be a convergent series of positive real numbers. L. Olivier proved that if the sequence (an) is non-increasing, then lim n→∞ nan = 0. In the present paper: (a) We formulate and prove a necessary and sufficient condition for having lim n→∞ nan = 0; Olivier’s theorem is a consequence of our Theorem 2.1. (b) We prove properties analogous to Olivier’s property when the usual convergence is replaced by the I-convergence, that is a convergence according to an ideal I of subsets of ℕ. Again, Olivier’s theorem is a consequence of our Theorem 3.1, when one takes as I the ideal of all finite subsets of ℕ.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public