New Results on the Order of Functions at Infinity

Abstract

Recently, new classes of positive and measurable functions, M(ρ) and M(±∞), have been defined in terms of their asymptotic behaviour at infinity, when normalized by a logarithm (Cadena et al., , 2017. Looking for other suitable normalizing functions than logarithm seems quite natural. It is what is developed in this paper, studying new classes of functions of the type lim x→∞ log U (x)/H(x) = ρ < ∞ for a large class of normalizing functions H. It provides subclasses of M(0) and M(±∞).

A new test for convergence of positive series

“…As in Lemma 2.2, from Cadena, Kratz and Omey [5], we obtain A(x) f (w i (x)) B(x) with A, B ∈ RV θ , and (3.2) follows. Starting from (3.2) we use a property of regular variation: If U (x) ∈ RV α , then ln U (x)/ ln x → α, see [2], to obtain (3.1).…”

“…Cadena, Kratz and Omey [5] showed that (2.4) (with w(n) = n) holds if and only if f (x) = a [x] satisfies the following property.…”

“…This test about convergence/divergence of the series a i is sometimes called Cauchy's second test [5]. It was re-invented, for example, in Rao [12].…”

“…Bingham, Goldie and Teugels [2]). Recently, Cadena, Kratz and Omey [5] described a new class of functions that covers the class of functions regularly varying at infinity, and in the other recent paper of these authors [6] that new class of functions was used for characterization of the tail probability distribution functions under general settings. Taking that new class into consideration enables us to further reconsider and develop the earlier tests on convergence/divergence of positive series.…”