divergent series

[də′vər·jənt ′sir·ēz]

(mathematics)

An infinite series whose sequence of partial sums does not converge.

The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Divergent Series

a series in which the sequence of partial sums does not have a finite limit. If the general term of the series does not tend to zero, the series diverges, for example, 1 - 1 + 1 - 1 + … + (– 1)^n-1. The harmonic series 1 + 1/2 + … + 1/2 + … is an example of a divergent series whose general term tends to zero. There exist numerous classes of divergent series that converge in some generalized sense, since to each such divergent series some “generalized sum” may be assigned that possesses the most important properties of the sum of a convergent series.

Mentioned in

References in periodicals archive

The study of theory of matrix transformations has always been of great interest to mathematicians in the study of sequence spaces, which is motivated by special results in summability theory.

Duals and Matrix Classes Involving Cesaro Type Classes of Sequences of Fuzzy Numbers

Boos, Classical and Modern Methods in Summability, Oxford University Press, Oxford, UK, 2000.

On the Characterization of Some Classes of Four-Dimensional Matrices and Almost B-Summable Double Sequences

Mohiuddine, "Statistical (C, 1)(E, 1) summability and Korovkin's theorem," Filomat, vol.

Some Seminormed Difference Sequence Spaces over n-Normed Spaces Defined by a Musielak-Orlicz Function of Order ([alpha], [beta])

The case of r = -1, on the other hand, has been studied in [4] for v = 1 and the summability of the formal series can be read from the properties of the initial data of (5).

Singular Perturbation of Nonlinear Systems with Regular Singularity

We also introduce a new concept of Wijsman strong Cesaro summability with respect to a modulus and show that if a sequence is Wijsman strongly Cesaro summable, then it is Wijsman strongly Cesaro summable with respect to all moduli f.

Density by moduli and Wijsman statistical convergence

Even when Borel summability does not apply, Borel analysis can still be very useful through the use of Borel dispersion relations.

Asymptotic Behaviour of the QED Perturbation Series

Let ([a.sub.nk]) be a nonnegative regular infinite summability matrix and x [member of] [0, [infinity]).

The Approximation Szasz-Chlodowsky Type Operators Involving Gould-Hopper Type Polynomials

This is possible as long as we have absolute summability, i.e., if

Dirichlet-Neumann and Neumann-Neumann waveform relaxation algorithms for parabolic problems

The theory of sequence spaces is the fundamental of summability. Summability is a wide field of mathematics, mainly in analysis and functional analysis, and has many applications, for instance, in numerical analysis to speed up the rate of convergence, in operator theory, the theory of orthogonal series, and approximation theory.

Cesaro summable sequence spaces over the non-Newtonian complex field

If A = (C, 1), the Cesaro matrix, then statistical A-summability is reduced to statistical summability (C, 1) due to Moricz (MORICZ, 2002).

Korovkin type approximation theorem for functions of two variables via statistical summability (C, 1)/O teorema do tipo aproximacao de Korovkin para funcoes de duas variabilidades pela sumabilidade estatistica

Canak and Totur [8] proved a Tauberian theorem for Cesaro summability methods, and Totur and Dik [20] gave some one-sided Tauberian conditions for a general summability method using the general control modulo of integer order.

Some Tauberian Remainder Theorems for Holder Summability