• Research Article
• 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:

## Abstract

New result concerning summability of the infinite series is presented.

## Keywords

• Positive Constant
• General Result
• Infinite Series
• Positive Sequence
• Nondecreasing Sequence

## 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

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

where as .

The series is said to be summable , if
and it is said to be summable , , if

where is as defined by (1.1).

For , summability reduces to summability.

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

where as .

It is quite reasonable to give the following definition.

For , the series is said to be -summable, , if

where as .

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

holds for (see ).

Das , in 1966, proved the following result.

Theorem 1.1.

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

Recently Singh and Sharma  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

## 2. Lemmas

Lemma 2.1.

Let be nonincreasing, . Then for , ,

Proof.

Since is nonincreasing, then .

Lemma 2.2.

For ,

Proof.

Since ,  then is nonincreasing and hence

Lemma 2.3 (see ).

If is a quasi-f-increasing sequence, where , , , then under the conditions

## 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

are all satisfied, then the series is summable , .

Proof.

In order to prove the result, it is sufficient, by Minkowski's inequality, to show that

This completes the proof of the theorem.

## Authors’ Affiliations

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

## References 