On a new application of almost increasing sequences
© Özarslan and Keten; licensee Springer 2013
Received: 3 September 2012
Accepted: 6 November 2012
Published: 9 January 2013
In (Bor in Int. J. Math. Math. Sci. 17:479-482, 1994), Bor has proved the main theorem dealing with summability factors of an infinite series. In the present paper, we have generalized this theorem on the summability factors under weaker conditions by using an almost increasing sequence instead of a positive non-decreasing sequence.
MSC:40D15, 40F05, 40G99.
If we take , then summability reduces to summability. Also, if we take and , then we get summability. Furthermore, if we take , and for all values of n, reduces to summability. Finally, if we take and , then we get summability (see ).
Before stating the main theorem, we must first introduce some further notations.
2 Known result
then the series is summable , .
3 The main result
The aim of this paper is to generalize Theorem A for summability under weaker conditions. For this, we need the concept of an almost increasing sequence. A positive sequence is said to be almost increasing if there exists a positive increasing sequence and two positive constants A and B such that (see ). Obviously, every increasing sequence is an almost increasing sequence but the converse need not be true as can be seen from the example . Also, one can find some results dealing with absolute almost convergent sequences (see ). So, we are weakening the hypotheses of Theorem A replacing the increasing sequence by an almost increasing sequence. Now, we shall prove the following theorem.
are satisfied, then the series is summable , .
Remark It should be noted that if we take as a positive non-decreasing sequence, and , then we get Theorem A. In this case, conditions (21) and (22) reduce to conditions (16) and (17), respectively. Also, the condition ‘ is a non-increasing sequence’ and the conditions (18)-(20) are automatically satisfied.
This completes the proof of the theorem. If we take , then we get a result concerning the summability factors. If we take , then we have another result dealing with summability. If we take and for all values of n, then we get a result dealing with summability. If we take , and for all values of n, then we get a result for summability. □
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