LAUSR

Abstract Let x = {xn}n=1∞ $\begin{array}{} \displaystyle \{x_n\}_{n=1}^{\infty} \end{array}$ be a sequence of positive numbers, and ????x be the collection of all subsets A ⊆ ℕ such that ∑k∈A $\begin{array}{} \displaystyle \sum_{k\in A} \end{array}$ xk < +∞. The aim of this article is to study how large the summable subsequence could be. We define the upper density of summable subsequences of x as the supremum of the upper asymptotic densities over ????x, SUD in brief, and we denote it by D*(x). Similarly, the lower density of summable subsequences of x is defined as the supremum of the lower asymptotic densities over ????x, SLD in brief, and we denote it by D*(x). We study the properties of SUD and SLD, and also give some examples. One of our main results is that the SUD of a non-increasing sequence of positive numbers tending to zero is either 0 or 1. Furthermore, we obtain that for a non-increasing sequence, D*(x) = 1 if and only if lim infk→∞nxn=0, $\begin{array}{} \displaystyle \liminf_{k\to\infty}nx_n=0, \end{array}$ which is an analogue of Cauchy condensation test. In particular, we prove that the SUD of the sequence of the reciprocals of all prime numbers is 1 and its SLD is 0. Moreover, we apply the results in this topic to improve some results for distributionally chaotic linear operators.