- Research Article
- Open Access

# Notes on Summability Factors of Infinite Series

- W T Sulaiman
^{1}Email author

**2011**:365453

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

© W. T. Sulaiman. 2011

**Received:**5 November 2010**Accepted:**19 January 2011**Published:**26 January 2011

## Abstract

New result concerning summability of the infinite series is presented.

## Keywords

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

## 1. Introduction

Necessary and sufficient conditions for the method to be regular are

(i)
for each
*,*

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

where as .

where is as defined by (1.1).

For , summability reduces to summability.

where as .

It is quite reasonable to give the following definition.

where as .

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.

## 2. Lemmas

## 3. Result

Our aim is to present the following new general result.

Theorem 3.1.

are all satisfied, then the series is summable , .

Proof.

This completes the proof of the theorem.

## Authors’ Affiliations

## References

- Das G:
**On some methods of summability.***The Quarterly Journal of Mathematics Oxford Series*1966,**17**(2):244–256.MATHView ArticleGoogle Scholar - Sulaiman WT:
**Notes on two summability methods.***Pure and Applied Mathematika Sciences*1990,**31**(1–2):59–69.MATHMathSciNetGoogle Scholar - 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 - 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

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.