Extending tests for convergence of number series

Abstract

Keywords:Convergence/divergence of number series Monotone/weak monotone/general monotone functions and sequences Analyzing several classical tests for convergence/divergence of number series, we relax the monotonicity assumption for the sequence of terms of the series. We verify the sharpness of the obtained results on corresponding classes of sequences and functions.

Double cosine-sine series and Nikol'skii classes in uniform metric

“…The following theorem is an extension of the Lorentz result and results from [23] to the two-dimensional mixed case. Applications of onedimensional weak monotonicty can be found in [12]. Also, we note the paper by Dyachenko and Tikhonov [7], where the estimates of Fourier coefficients satisfying another definition of weak monotonicity are given.…”

“…Theorem 4. (i) Assume that r, s ∈ Z + , {a jk } j, k∈N satisfies the condition (11) and h(x, y) is the sum of (12). If m, n ∈ N, ω ∈ BB ∩ ∆ 2 and the condition…”

Summability characterizations of positive sequences

“…For more information on this asymptotic behavior of summable sequences of positive numbers we refer to [7,5,8] and references therein.…”

Uniform convergence of Hankel transforms

“…The latter is a generalization of the well known Abel-Olivier's test that deals with nonnegative monotone functions (see also [16], where the monotonicity assumption is relaxed). We emphasize that Theorem 4.7 g only needs to be real-valued, instead of nonnegative.…”