1941
DOI: 10.1090/s0002-9904-1941-07477-x
|View full text |Cite
|
Sign up to set email alerts
|

A sequence of limit tests for the convergence of series

Abstract: In this paper, we shall develop a sequence of limit tests for the convergence and divergence of infinite series of positive terms which is similar in form to the De Morgan and Bertrand sequence but involves the ratio of two successive values of the test ratio rather than the test ratio itself. The proof will be based on the following integral test by R. W. Brink: and if R(x) is a function such that R(n) =R nj and such that R(x) *zR(x') when x' >x, a necessary and sufficient condition for the convergence of th… Show more

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...
2
1

Citation Types

0
7
0

Year Published

2018
2018
2022
2022

Publication Types

Select...
3
2

Relationship

0
5

Authors

Journals

citations
Cited by 5 publications
(7 citation statements)
references
References 0 publications
0
7
0
Order By: Relevance
“…We show how the new definitions lead to new convergence/divergence tests. For the undecided cases, we generalize an old result of Martin [11]. In the final remarks, we provide some one-sided results.…”
Section: Introductionmentioning
confidence: 52%
See 1 more Smart Citation
“…We show how the new definitions lead to new convergence/divergence tests. For the undecided cases, we generalize an old result of Martin [11]. In the final remarks, we provide some one-sided results.…”
Section: Introductionmentioning
confidence: 52%
“…c) We prove a generalization of an old result of Martin [11]. Let ln (0) z = z, ln (1) z = ln z and ln (k+1) z = ln ln (k) z for k = 1, 2, .…”
Section: It Follows That N I=a a I W(n ) Bmentioning
confidence: 85%
“…Similarly we have c) We prove a generalization of an old result of Martin [11]. Let ln (0) z = z, ln (1) z = ln z and ln (k+1) z = ln ln (k) z for k = 1, 2, ...…”
Section: It Follows Thatmentioning
confidence: 58%
“…In the present note, we establish necessary and sufficient conditions for convergence of positive series that generalize the original version of the extended Bertrand-De Morgan test [1,9]. The first theorem on a necessary and sufficient condition for convergence of series was obtained by Cauchy [7], widely known as Cauchy's convergence test.…”
mentioning
confidence: 95%
“…The extended Bertrand-De Morgan test is the last test in this hierarchy. It was originally established in [9]. An elementary proof of this test, its connection with Kummer's test, as well as its application to birth-and-death processes is given in [1].…”
mentioning
confidence: 99%