Publication result detail

Kummer test and regular variation

ŘEHÁK, P.

Original Title

Kummer test and regular variation

English Title

Kummer test and regular variation

Type

WoS Article

Original Abstract

We establish relations among the Kummer test, certain generalization of discrete regular variation, and regular variation on time scales. More precisely, we give a new interpretation (including new proof) of the Kummer test which detects convergence of series. We show that the limit relation from the Kummer test can be rewritten in terms of recently introduced concept of refined regularly varying sequences with respect to an auxiliary sequence tau. The theory of such sequences can be developed by transforming them into the new time scale T = tau(N), which then enables us to utilize the existing results for regularly varying functions on time scales. Replace this sentence by "In particular, the Karamata type theorem and the representation for refined regularly varying sequences yield not only the Kummer test, but provide also asymptotic formulae for the partial sums of series and the representation for the sequences satisfying the Kummer test.

English abstract

We establish relations among the Kummer test, certain generalization of discrete regular variation, and regular variation on time scales. More precisely, we give a new interpretation (including new proof) of the Kummer test which detects convergence of series. We show that the limit relation from the Kummer test can be rewritten in terms of recently introduced concept of refined regularly varying sequences with respect to an auxiliary sequence tau. The theory of such sequences can be developed by transforming them into the new time scale T = tau(N), which then enables us to utilize the existing results for regularly varying functions on time scales. Replace this sentence by "In particular, the Karamata type theorem and the representation for refined regularly varying sequences yield not only the Kummer test, but provide also asymptotic formulae for the partial sums of series and the representation for the sequences satisfying the Kummer test.

Keywords

Kummer test; Regularly varying sequence; Karamata theorem; Time scale

Key words in English

Kummer test; Regularly varying sequence; Karamata theorem; Time scale

Authors

ŘEHÁK, P.

RIV year

2021

Released

28.02.2020

Publisher

SPRINGER WIEN

Location

WIEN

ISBN

0026-9255

Periodical

MONATSHEFTE FUR MATHEMATIK

Volume

192

Number

2

State

Republic of Austria

Pages from

419

Pages to

426

Pages count

8

URL

https://link.springer.com/article/10.1007/s00605-019-01361-y

BibTex

@article{BUT162150,
  author="Pavel {Řehák}",
  title="Kummer test and regular variation",
  journal="MONATSHEFTE FUR MATHEMATIK",
  year="2020",
  volume="192",
  number="2",
  pages="419--426",
  doi="10.1007/s00605-019-01361-y",
  issn="0026-9255",
  url="https://link.springer.com/article/10.1007/s00605-019-01361-y"
}