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Notes on Summability Factors of Infinite Series
Journal of Inequalities and Applications volume 2011, Article number: 365453 (2011)
New result concerning summability of the infinite series is presented.
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.
Let , . Then if is -summable, it is -summable.
Recently Singh and Sharma  proved the following theorem.
Let , and let be a monotonic nondecreasing sequence for . The necessary and sufficient condition that is -summable whenever
Let be nonincreasing, . Then for , ,
Since is nonincreasing, then .
Since , then is nonincreasing and hence
Lemma 2.3 (see ).
If is a quasi-f-increasing sequence, where , , , then under the conditions
Our aim is to present the following new general result.
Let , and let be a quasi-f-increasing sequence, where , , and (2.6), and
are all satisfied, then the series is summable , .
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.
Das G: On some methods of summability. The Quarterly Journal of Mathematics Oxford Series 1966,17(2):244–256.
Sulaiman WT: Notes on two summability methods. Pure and Applied Mathematika Sciences 1990,31(1–2):59–69.
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
Singh N, Sharma N: On summability factors of infinite series. Proceedings of Mathematical Sciences 2000,110(1):61–68. 10.1007/BF02829481
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Sulaiman, W.T. Notes on Summability Factors of Infinite Series. J Inequal Appl 2011, 365453 (2011). https://doi.org/10.1155/2011/365453
- Positive Constant
- General Result
- Infinite Series
- Positive Sequence
- Nondecreasing Sequence