Skip to main content

Notes on Summability Factors of Infinite Series


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




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 [1]) that for , (1.5) holds if and only if


For , the series is said to be -summable, , (Sulaiman [2]), 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 [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


is that


2. Lemmas

Lemma 2.1.

Let be nonincreasing, . Then for , ,



Since is nonincreasing, then .




Lemma 2.2.

For ,



Since ,  then is nonincreasing and hence


Lemma 2.3 (see [3]).

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


one has


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 , .


We have


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


Applying HÖlder's inequality, we have


This completes the proof of the theorem.


  1. Das G: On some methods of summability. The Quarterly Journal of Mathematics Oxford Series 1966,17(2):244–256.

    MATH  Article  Google Scholar 

  2. Sulaiman WT: Notes on two summability methods. Pure and Applied Mathematika Sciences 1990,31(1–2):59–69.

    MATH  MathSciNet  Google 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.2005.09.019

    MATH  MathSciNet  Article  Google Scholar 

  4. Singh N, Sharma N: On summability factors of infinite series. Proceedings of Mathematical Sciences 2000,110(1):61–68. 10.1007/BF02829481

    MATH  MathSciNet  Article  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to W T Sulaiman.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Sulaiman, W.T. Notes on Summability Factors of Infinite Series. J Inequal Appl 2011, 365453 (2011).

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI:


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