Abstract Let x = {xn}n=1∞ $\begin{array}{} \displaystyle \{x_n\}_{n=1}^{\infty} \end{array}$ be a sequence of positive numbers, and 𝓙x 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 𝓙x, 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 𝓙x, 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.

中文翻译：

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序列的可求子序列的密度及其应用

Abstract 让 x = {xn}n=1∞ $\begin{array}{} \displaystyle \{x_n\}_{n=1}^{\infty} \end{array}$ 是一个正数序列，并且𝓙x 是所有子集 A ⊆ ℕ 的集合，使得 ∑k∈A $\begin{array}{} \displaystyle \sum_{k\in A} \end{array}$ xk < +∞。本文的目的是研究可求和子序列的大小。我们将 x 的可求子序列的上密度定义为 𝓙x 上的上渐近密度的上限值，简称 SUD，我们用 D*(x) 表示它。类似地，x 的可求子序列的较低密度被定义为 𝓙x 上的较低渐近密度的上限值，简称 SLD，我们用 D*(x) 表示它。我们研究了 SUD 和 SLD 的特性，并给出了一些例子。我们的主要结果之一是趋于零的非递增正数序列的 SUD 是 0 或 1。此外，对于非递增序列，我们得到 D*(x) = 1 当且仅当lim infk→∞nxn=0, $\begin{array}{} \displaystyle \liminf_{k\to\infty}nx_n=0, \end{array}$ 这是柯西凝聚测试的类似物。特别地，我们证明了所有素数的倒数序列的 SUD 为 1，其 SLD 为 0。此外，我们应用本主题的结果来改进分布混沌线性算子的一些结果。