A few remarks on divergent sequences: Rates of divergence

Abstract

We continue investigation of certain classes of sequences of positive real numbers which are important in asymptotic analysis of divergent processes. Among others, we define the notion of quotient speed, give representation results for some kinds of sequences and improve several our earlier results concerning selection principles and games.

A few remarks on divergent sequences: Rates of divergence II

“…This note can be viewed as an addendum to the paper [10], because here we extend and improve the following result ( [10,Theorem 3.11]; for the definitions see the next section): Theorem 1.1. The following selection properties are equivalent and all are satisfied:…”

“…Note that for λ 0 = 1 we have the well-known class ARV s of sequences (see [7][8][9][10]), and it holds…”

“…In fact, the above theorem is valid for any fixed k ∈ N. The methodology we used in the proof of this result did not allow us to show that the second coordinate here can be replaced by the smaller class Tr (∞) (R ∞,s ) introduced also in [10]. One of main aims of this note is to show that it is possible.…”

“…In several recent papers, the authors found out nice relations between theory of divergent processes in asymptotic analysis and theory of selection principles and games (see [6,[8][9][10]). This note can be viewed as an addendum to the paper [10], because here we extend and improve the following result ( [10,Theorem 3.11]; for the definitions see the next section): Theorem 1.1.…”

Note on selection principles of Kocinac

“…Well-known Kočinac's selection principles α i , i ∈ {1, 2, 3, 4}, are subject of the research in many papers that appeared in the literature in recent years (see, e.g., [2][3][4][5][6][7][8]12]). These selection principles are based on the famous Arhangel'skiȋ's properties introduced and studied in 1972 (see, e.g., [1] and [7]).…”