Open Access

Notes on Summability Factors of Infinite Series

Journal of Inequalities and Applications20112011:365453

https://doi.org/10.1155/2011/365453

Received: 5 November 2010

Accepted: 19 January 2011

Published: 26 January 2011

Abstract

New result concerning summability of the infinite series is presented.

1. Introduction

Let be a given infinite series with sequence of partial sums . Let denote the sequence of means of   . The transform of is defined by
(1.1)
where
(1.2)

Necessary and sufficient conditions for the method to be regular are

(i) for each ,

(ii) , where is a positive constant independent of .

The series is said to be summable , , if
(1.3)
where
(1.4)

where as .

The series is said to be summable , if
(1.5)
where
(1.6)
and it is said to be summable , , if
(1.7)

where is as defined by (1.1).

For , summability reduces to summability.

The series is said to be bounded or if
(1.8)
By , we denote the set of sequences satisfying
(1.9)
It is known (Das [1]) that for , (1.5) holds if and only if
(1.10)
For , the series is said to be -summable, , (Sulaiman [2]), if
(1.11)

where as .

It is quite reasonable to give the following definition.

For , the series is said to be -summable, , if
(1.12)

where as .

We also assume that , are positive sequences of numbers such that
(1.13)
A positive sequence is said to be a quasi- -power increasing sequence, , if there exists a constant such that
(1.14)

holds for (see [3]).

Das [1], in 1966, proved the following result.

Theorem 1.1.

Let , . Then if is -summable, it is -summable.

Recently Singh and Sharma [4] proved the following theorem.

Theorem 1.2.

Let , and let be a monotonic nondecreasing sequence for . The necessary and sufficient condition that is -summable whenever
(1.15)
is that
(1.16)

2. Lemmas

Lemma 2.1.

Let be nonincreasing, . Then for , ,
(2.1)

Proof.

Since is nonincreasing, then .
(2.2)
Therefore
(2.3)

Lemma 2.2.

For ,
(2.4)

Proof.

Since ,  then is nonincreasing and hence
(2.5)

Lemma 2.3 (see [3]).

If is a quasi-f-increasing sequence, where , , , then under the conditions
(2.6)
one has
(2.7)

3. Result

Our aim is to present the following new general result.

Theorem 3.1.

Let , and let be a quasi-f-increasing sequence, where , , and (2.6), and
(3.1)

are all satisfied, then the series is summable , .

Proof.

We have
(3.2)
In order to prove the result, it is sufficient, by Minkowski's inequality, to show that
(3.3)
Applying HÖlder's inequality, we have
(3.4)

This completes the proof of the theorem.

Authors’ Affiliations

(1)
Department of Computer Engineering, College of Engineering, University of Mosul

References

  1. Das G: On some methods of summability. The Quarterly Journal of Mathematics Oxford Series 1966,17(2):244–256.MATHView ArticleGoogle Scholar
  2. Sulaiman WT: Notes on two summability methods. Pure and Applied Mathematika Sciences 1990,31(1–2):59–69.MATHMathSciNetGoogle Scholar
  3. Sulaiman WT: Extension on absolute summability factors of infinite series. Journal of Mathematical Analysis and Applications 2006,322(2):1224–1230. 10.1016/j.jmaa.2005.09.019MATHMathSciNetView ArticleGoogle Scholar
  4. Singh N, Sharma N: On summability factors of infinite series. Proceedings of Mathematical Sciences 2000,110(1):61–68. 10.1007/BF02829481MATHMathSciNetView ArticleGoogle Scholar

Copyright

© W. T. Sulaiman. 2011

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.