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Notes on
Summability Factors of Infinite Series
Journal of Inequalities and Applications volume 2011, Article number: 365453 (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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F365453/MediaObjects/13660_2010_Article_2335_Equ1_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F365453/MediaObjects/13660_2010_Article_2335_Equ2_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F365453/MediaObjects/13660_2010_Article_2335_Equ3_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F365453/MediaObjects/13660_2010_Article_2335_Equ4_HTML.gif)
where as
.
The series is said to be summable
, if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F365453/MediaObjects/13660_2010_Article_2335_Equ5_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F365453/MediaObjects/13660_2010_Article_2335_Equ6_HTML.gif)
and it is said to be summable ,
, if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F365453/MediaObjects/13660_2010_Article_2335_Equ7_HTML.gif)
where is as defined by (1.1).
For ,
summability reduces to
summability.
The series is said to be
bounded or
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F365453/MediaObjects/13660_2010_Article_2335_Equ8_HTML.gif)
By , we denote the set of sequences
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F365453/MediaObjects/13660_2010_Article_2335_Equ9_HTML.gif)
It is known (Das [1]) that for , (1.5) holds if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F365453/MediaObjects/13660_2010_Article_2335_Equ10_HTML.gif)
For , the series
is said to be
-summable,
, (Sulaiman [2]), if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F365453/MediaObjects/13660_2010_Article_2335_Equ11_HTML.gif)
where as
.
It is quite reasonable to give the following definition.
For , the series
is said to be
-summable,
, if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F365453/MediaObjects/13660_2010_Article_2335_Equ12_HTML.gif)
where as
.
We also assume that ,
are positive sequences of numbers such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F365453/MediaObjects/13660_2010_Article_2335_Equ13_HTML.gif)
A positive sequence is said to be a quasi-
-power increasing sequence,
, if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F365453/MediaObjects/13660_2010_Article_2335_Equ14_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F365453/MediaObjects/13660_2010_Article_2335_Equ15_HTML.gif)
is that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F365453/MediaObjects/13660_2010_Article_2335_Equ16_HTML.gif)
2. Lemmas
Lemma 2.1.
Let be nonincreasing,
. Then for
,
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F365453/MediaObjects/13660_2010_Article_2335_Equ17_HTML.gif)
Proof.
Since is nonincreasing, then
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F365453/MediaObjects/13660_2010_Article_2335_Equ18_HTML.gif)
Therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F365453/MediaObjects/13660_2010_Article_2335_Equ19_HTML.gif)
Lemma 2.2.
For ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F365453/MediaObjects/13660_2010_Article_2335_Equ20_HTML.gif)
Proof.
Since ,  then
is nonincreasing and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F365453/MediaObjects/13660_2010_Article_2335_Equ21_HTML.gif)
Lemma 2.3 (see [3]).
If is a quasi-f-increasing sequence, where
,
,
, then under the conditions
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F365453/MediaObjects/13660_2010_Article_2335_Equ22_HTML.gif)
one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F365453/MediaObjects/13660_2010_Article_2335_Equ23_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F365453/MediaObjects/13660_2010_Article_2335_Equ24_HTML.gif)
are all satisfied, then the series is summable
,
.
Proof.
We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F365453/MediaObjects/13660_2010_Article_2335_Equ25_HTML.gif)
In order to prove the result, it is sufficient, by Minkowski's inequality, to show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F365453/MediaObjects/13660_2010_Article_2335_Equ26_HTML.gif)
Applying HÖlder's inequality, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F365453/MediaObjects/13660_2010_Article_2335_Equ27_HTML.gif)
This completes the proof of the theorem.
References
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
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DOI: https://doi.org/10.1155/2011/365453