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International Journal of Scientific & Engineering Research, Volume 3, Issue 6, June-2012 1 ISSN 2229-5518
IJSER © 2012
http://www.ijser.org
On nk
N,q ; nSummability Factors of Infinite
Series (III) Aradhana Dutt Jauhari
ABSTRACT: A theorem concerning some new absolute summability method is proved. Many other results some known and unknown are derived.
KEY WORDS AND PHRASES: Absolute Summability, Summability.
—————————— ——————————
1- INTRODUCTION :
Let na be a given infinite series with partial sumsns and n nu na . Let
n and
nt be the thn Cesàro mean of
order ( >-1)of the sequencens and
nu respectevily.The series na is said to be summable k
C, ,k 1 ,if
kk 1
n n 1
n 1
n (1.1)
Or equivalently,
k1
n
n 1
n t
where,
n n n 1t n
A series na is summable k
C, ; n , if the series
kk k 1
n n 1
n 1
n n ; (1.2)
(n) , a positive non-decreasing sequence such that nm n m .
where n
is the thn Cesàro mean of order of na ,It follows that above definition reduces to that of FLETT [4].
Let np be a sequence of positive numbers such that,
n
n v
v 0
P p
as n
.
The sequence -to -sequence transformation
International Journal of Scientific & Engineering Research, Volume 3, Issue 6, June-2012 2 ISSN 2229-5518
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n
n v v
v 0n
1T p s
P
nP 0(n 0) (1.3) defines the sequence of weighted means of the
sequencens generated by the sequence of coefficients
np . The series na is said to be summable nk
N,p ,k 1 if
Email id- aditya_jauhari@ rediffmail.com, Deptt. Of Applied Sciences (Mathematics), N.I.E.T., Greater Noida ,U.P. (INDIA)
kk 1
nn n 1
n 1 n
Pu u
p (1.4)
and it is said to be summable nk
N,p ; n ,k 1 ,if
kk k 1
n nn n 1
n 1 n n
P PT T
p p ; (1.5)
In 1995, SULAIMAN, W.T. [8] proved the theorem.The main objective of this paper is to generalize the theorem of SULAIMAN[8]. However our theorem
is as follows. 2-THEOREM:
Let { np } and { nq } be two sequences of positive real numbers such that-
n 1 np (p ) (2.1)
n n 1q (q ) (2.2)
n n n np Q (P q ) (2.3)
The series na is summable n
kN,p , n then n na is summable n
kN,q , n if –
1/k
n n n n nn
n n n n n
Q p Q P q
q P q p Q (2.4)
1/k
n n n nn
n n n n
p Q P q
P q p Q (2.5)
3-PROOF OF THE THEOREM:
In order to prove the theorem it is sufficient to prove that
by LEINDLER [5]
kk k 1
n nn n 1
n 1 n n
P PT T
p p
where ,
n
n v v
v 0n
1T p s
P
International Journal of Scientific & Engineering Research, Volume 3, Issue 6, June-2012 3 ISSN 2229-5518
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Let n n n 1x T T , n 1 0 0,s a
nn
n v 1 v
v 1n n 1
px P a
P P
nn n 1
n v 1 v
v 1n
P Px P a
p
n 1 n 2
n n 1
n 1 n 1
1 P Pa x
P p
Considering-
n v
n v
v 0 0n
1t q a
Q
n 1 n
n v v 1 v 1 n v v
v 0 v 0n
1t Q ( a ) Q a
Q
n 1 n
v v 1 v 1 v v
v 0 v 0n
1Q (a ) a
Q (3.1)
so,
n 1
n n 1 v v 1 v 1
v 0n
n 2 n 1 n
v v 1 v 1 v v v v
v 0 v 0 v 0n 1
1t t Q (a )
Q
1 Q (a ) a a
Q
n 1 n 2
v v 1 v 1 v v 1 v 1
v 0 v 0n n 1
1 1Q (a ) Q (a )
Q Q
n 1v
v v 1 v 1
v 0n n 1
qQ (a )
Q Q
n 1v
v 1 v v
v 0n n 1
qQ a
Q Q
nv n 1 n 2
v 1 v n 1
v 0n n 1 n 1 n 1
q 1 P PQ x
Q Q P p
(3.2)
Using Abel’s transformation, we have-
n 1 v1 2v 1 vn
n n 1 1
v 0 0n n 1 v 1 1
P PQqt t x
Q Q P p+
International Journal of Scientific & Engineering Research, Volume 3, Issue 6, June-2012 4 ISSN 2229-5518
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nn 1 n v 1 v 2
v 1
v 0n 1 v 1
Q P Px
P p
n 1n v 1 v v v 1
v
v 0n n 1 v 1 v
n 1 n n n 1n
n 1 n
q Q P Px
Q Q P p
Q P P x
P p
Clearly,
v 1 v v v 1 v v 1
v 1 v v
v 1 v v 1 v v
Q p Q qQ
P P P P P
so,
n 1 n 1n n v 1
n n 1 v 1 v v v v 1 v
v 0 v 0n n 1 n n 1 v
q q Pt t Q x x Q
Q Q Q Q p
n 1n v 1 n n
v v v 1 n n
v 0n n 1 v n n
q P q Px q x
Q Q p Q p
=1 2 3 4
(say)
Now for n na to be summable nk
N,q , n if-
kk k 1
n nn n 1
n 1 n n
Q QT T
q q (3.3)
Now by Minköwaski’s inequality, we have-
kk k 1
n n
1 2 3 4n 1 n n
Q Q
q q
k k 1k
n n
1n 1 n n
k k 1 k k 1kk
n n n n
2 3n 1 n 1n n n n
Q QM
q q
Q Q Q Q +
q q q q
k k 1k
n n
4n 1 n n
Q Q
q q
where M is some positive constant.
=11 12 13 14
(say)
Now,
International Journal of Scientific & Engineering Research, Volume 3, Issue 6, June-2012 5 ISSN 2229-5518
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k k 1k
v v
11 1v 1 v v
Q QM
q q
k k 1 kn 1
v v nv 1 v v
v 1 v 0v v n n 1
Q Q qM Q x
q q Q Q
kk k 1k n 1
v 1 vn n n nv vk 1 k k
n 1 v 0n n n n n 1 v
Q qQ Q q 1 qx
q q q Q Q q
kk n 1k 1 k n 1 n 1k kv 1n n n
v v v vk 1 k kn 1 v 0 v 0n n n n 1 v
QQ Q qx q q
q q Q Q q
Using Hölder’s inequality-
k k 1 k 1kn 1
k kv v vn nv v v
n 1 v 0n n n 1 v v v
Q P pQ qx q
q Q Q q p P
k k 1 kn 1
k kn v v v v vv v v
n 1 v 0n n 1 v v v v v
q Q P Q p Px q
Q Q q p q P p
k k 1n 1
k kv v v nv v v
v 1 n v 1v v v n n 1
Q P P qx q
q p p Q Q v v v vp Q P q
k k 1n 1
k kv v v vv v
v 0 v v v v
Q P P qx
q p Q p
k k 1 kn 1
kv v v v v v v vv
v 0 v v v v v v v v
Q P P q Q p P qx
q p Q p P q Q p
k k 1n 1
kv vv
v 0 v v
P Px
p p
1
Again
k k 1k
v v
12 2v 1 v v
Q QM
q q
kk k 1n 1
n n n v 1v 1 v v
v 1 v 0n n n n 1 v
Q Q q PM Q x
q q Q Q p
International Journal of Scientific & Engineering Research, Volume 3, Issue 6, June-2012 6 ISSN 2229-5518
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kk k 1k n 1
n n n v 1v 1 v vk k
n 1 v 0n n n n 1 v
Q Q q PQ x
q q Q Q p
kk
n 1n n v 1 v 1 v
v vkn 1 v 0n n n 1 v v
Q q P Q qx
q Q Q p q
k k kn 1
k kv v 1 v 1nv v vk
n 1 v 0n n 1 v v v
k 1n 1
v
v 0
Q P Qqq x *
Q Q q p q
* q
Using Hölder’s inequality-
k k kk 1 n 1
k kv v vn n 1v v vk
n 1 v 0n n 1 v v v
Q Q Pq Qq x
Q Q q q p
k 1 k kn 1
k kv v v vnv v v
n 1 v 0n n 1 v v v v
P Q Q Pqx q
Q Q p q q p
k 1 k k
k kv v v v nv v v
v 1 n v 1v v v v n n 1
P P Q Q qx q
p p q q Q Q
k 1 k k
kv v v v v v v vv
v 1 v v v v v v v v
P P q Q p Q P qx
p p Q q q P p Q
{using condition (2.4) of theorem }
k k 1
kv vv
v 1 v v
P Px
p p
(1)
Further,
k k 1k
v v
13 3v 1 v v
Q QM
q q
kk k 1n 1
v v v 1nv v v 1
v 1 v 0v v n n 1 v
Q Q PqM x q
q q Q Q p
k k 1 kk n 1
v v v 1nv v v 1k k
v 1 v 0v v n n 1 v
Q Q PqM x q
q q Q Q p
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kk k 1n 1 n 1
k kv 1n nv v v 1 vk
n 1 v 0 v 0n n n 1 v
PQ qx q q
q Q Q p
kkk 1 n 1
k kv 1n n n 1v v v 1k
n 1 v 0n n n 1 v
PQ q Qx q
q Q Q p
kkn 1
k kv 1n nv v v 1
n 1 v 0n n n 1 v
PQ qx q
q Q Q p
k 1k
k kv vn nv v v 1
v 1 n v 1n v v n n 1
P PQ qx q
q p p Q Q
k k 1
k kv v v vv v 1
v 1 v v v v
Q P P qx
q p Q p
k k 1 k
kv v v v v 1 v 1 v 1 v 1v
v 1 v v v v v 1 v 1 v 1 v 1
Q P P q p Q p Qx
q p p Q P q P q
{using condition (2.5) of theorem}
k k 1 k k
kv v v 1 v 1v
v 1 v v v 1 v 1
Q P P qx
q p p Q
k k 1
kv vv
v 1 v v
P Px
p p
1
Lastly,
k k 1k
v v
14 4v 1 v v
Q QM
q q
k k 1 k
v v v vv v
v 1 v v v v
Q Q q PM x
q q Q p
k k 1k k
k kn n n nn nk k
n 1 n n n n
Q Q q Px
q q Q p
k kk
kn n n n n n nnk
n 1 n n n n n n n
Q q P p Q P qx
q Q p q P Q p
k k 1
kn nn
n 1 n n
P Px
p p
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1
This completes the proof of theorem.
4 -COROLLARIES :
The following corollaries can be derived from the theorem- Cor.1:
In the special case when n n our theorem reduces to the theorem of SINHA and KUMAR [10].
Cor.2:
If n 0 for every n , then our theorem reduces to theorem of SHARMA[8].
Cor.3:
For n 0 and k=1 our theorem reduces to the following theorem –
Let { np } and { nq } be two sequences of positive real numbers such that-
n 1 np (p ) (4.1)
n n 1q (q ) (4.2)
n n n np Q (P q ) (4.3)
If na is summable n
kN,p then n na is summable n
kN,q if –
n n n
n
n n n
Q p Q
q P q (4.4)
n n
n
n n
p Q
P q (4.5)
Acknowledgement: I am very thankful to Dr. Rajiv Sinha (Associate Professor, S.M.P.G. College ,Chandausi ,U.P., India ) ,whose great inspiration lead me to complete this paper.
REFERENCES
[1] BOR,H.- On two summability methods ;Math. Proc.Cambridge Phil.Soc.97,(1985),147-149.
[2] BOR,H. and THORPE,B.-On some absolute summability methods;Analysis 7,(1987),145-152.
[3] BORWEIN D. and CASS,F.P.-Strong Nölund summability Math Zeith,103,(1968) 91-111.
[4] FLETT,T.M.-Proc.London Math. Soc.7,(1957),113-141.
[5] LEINDLER,L.-Acta math. Hungar 64 (1994) 269-281.
[6] MAZHAR S.M. – On k
C, summability factors of infinite series , Extract du Bulletin de 1`Academic royale de Belgeque 9classe des sciences
)senance du Samedi 6 mars (1971).
[7] MOHAPTRA, R.N.-A note on summability factors, J. Indian Math. Soc. (1967), 213-224 .
[8] SHARMA,N. –Some aspect of weighted mean matrices,Ph.D. Theses,Kurushetra University (2000).
[9] SINHA,P. and KUMAR,H.– A note on nk
N,p ; summability factors of infinite series factors of infinite series , Tamkang J.of Mathematics ,
vol.39, Number 3,193-198 Autumn 2008.
[10]__________-International Journal of Math., (Communicated).
[11] SULAIMAN, W.T.-Inclusion theorem for absolute matrix summability method of infinite series ; Int. Math. Forum,4,no.24 (2009),1181-1189.
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