The continuity of densities given by the weight functions $n^{\alpha }$, $\alpha \in [-1,\infty [$, with respect to the parameter $\alpha $ is investigated.
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.