Let x = be a sequence of positive numbers, and 𝓙x be the collection of all subsets A ⊆ ℕ such that 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 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.
Powell, B. J.—Šalát, T.: Convergence of subseries of the harmonic series and asymptotic densities of sets of positive integers, Publications de l’Institut Mathematique, Nouvelle serie tome 50(64) (1991), 60–70.
Šalát, T.: On subseries, Math. Z. 85 (1964), 209–225.
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