Publication detail
Kummer test and regular variation
ŘEHÁK, P.
English title
Kummer test and regular variation
Type
journal article in Web of Science
Language
en
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 in English
Kummer test; Regularly varying sequence; Karamata theorem; Time scale
Released
28.02.2020
Publisher
SPRINGER WIEN
Location
WIEN
ISSN
0026-9255
Volume
192
Number
2
Pages from–to
419–426
Pages count
8
BIBTEX
@article{BUT162150,
author="Pavel {Řehák},
title="Kummer test and regular variation",
year="2020",
volume="192",
number="2",
month="February",
pages="419--426",
publisher="SPRINGER WIEN",
address="WIEN",
issn="0026-9255"
}