J. Comp. & Math. Sci. Vol. 1(4), 501-513 (2010).
A note on index summability of an infinite series MAHENDRA MISRA1, U.K.MISRA2 and S.BUXI3 1
Principal Government Science CollegeMlkangiri, Orissa, India e-mail:mahendramisra@2007.gmail.com 2 Department of Mathematics Berhampur University Berhampur – 760007, Orissa, India e-mail:umakanta_misra@yahoo.com 3 Department of Mathematics Gunupur Science College Gunupur, Odisha ABSTRACT In this paper we have established a theorem on
N , pn , n ; f
k
summability of an infinite series..
Key words:
N , pn , n
k
summability, N , p n , n ; f
k
summability . AMS Classification No: 40D25
1. INTRODUCTION Let
a
n
(1.2)
be an infinite series
with sequence of partial sums
s n .
Let p n be a sequence of non-negative e numbers such that n
(1.1) Pn
p , as n , ( Pi 0
p i 0 ; i 1). The sequence-to-sequence transformation
tn
1 Pn
n
p s , ( P
n
0)
0
defines the sequence
t n of
mean of the sequence
(N , pn )
sn generated
by the sequence of co-efficients The series
a
n
p n .
is said to be summable
N , p n , k 1 if (Bor[1]) k
Pn (1.3) n 1 p n
k 1 k
t n t n 1 .
Journal of Computer and Mathematical Sciences Vol. 1 Issue 4, 30 June, 2010 Pages (403-527)
502
Mahendra Misra et al., J.Comp.&Math.Sci. Vol.1(4),501-513 (2010)
Similarly, the sequence-to-sequence transformation (1.4) Tn
1 Pn
n
pn s , ( Pn 0)
0
defines the sequence Tn of
N, p n
s n egenerated by the sequence of co-efficients p n . mean of the sequenc
The series
a
n
is said to be N , p n k ,
, k 1 if
Let
Let f be a function of n , if
{ f (
k
n
)}k ( n ) k 1 Tn Tn1 ,
n 1
then the series
a
Pn n 1 p n
k 1 k
Tn Tn 1
.
n
is said to be
N , p n , n ; f k , k 1 summable . Clearly for
f ( n ) n , 0 ,
N , pn , n ; f
(1.5)
N , pn , n ; k , k 1, 0 summable..
k
N , pn , n ; k .
and for 0
N , pn , n ; f
k
N , pn , n k .
We consider n , n and q n as seq-
n
be any sequence of positive e
numbers. The series
a
n
is said to o
uences of positive numbers such that n
Qn q , as n . 0
be summable N , p n , n k , k 1, if
k 1 n
Tn Tn 1
k
2. Known Theorems:- On dealing with
.
n 1
Clearly N , p n ,
Pn pn
N , pn
k
and
Sulaiman[4] proved the following theorem:
k
Theorem-A:-Let t n denote the
N , p n ,1 1 N , p n .
N, p -mean of the series a .
Further, if
n
n
k
n k k 1 Tn Tn1 , n 1
then the series
N , qn , n k , k 1 summability,,
Letbe n , n and q n sequence off positive numbers such that n
a
n
is said to be
Qn q , as n . 0
Journal of Computer and Mathematical Sciences Vol. 1 Issue 4, 30 June, 2010 Pages (403-527)
Mahendra Misra et al., J.Comp.&Math.Sci. Vol.1(4), 501-513 (2010) 503 Let us write
Tn
1 1 k n
n n 1 n
t n 1 .
If k 1 k n n k n n 1
k 1 k 1 k
q q O n 1 Q Q Q
n n 1 n
n n 1 n
k 1
n n 1 n
,
k
n Tn
k 1
k 1
qn Qn
k
,
k
k n Tn
k
,
k
Pn1 k n Tn pn
then the series
k
Pn q n k n Tn p n Qn
a
n
k
n is summable
N , q n , n k , k 1.
k
,
Theoram-B for N , q n , n ; k , k 1, 0 , Summability methods.
n n 1 n
k 1
k
Pn 1 k n Tn pn
k
.
Theorem-C:-Let
t n denote the
N, p -mean of the series a n
Then the series
a
n
n is summable
1
Subsequently, Jena has proved an analogue theorem. He proved: Theorem-B:- Let
t n denote the
N, p -mean of the series a n
n
and
Let 1 k
q n Qn
k 1 q O Q
k k 1 n
n 1
n n 1 n
n n 1 n
If k 1 n
1 k
If
Tn n t n 1 .
and
Tn n t n 1 . 2
1
n
let
N , q n , n , k 1. k
n 1
,
Later on Samanta 3 have generalized
k 1
and
k
,
n n 1 n
q n Qn
k k 1
Pn pn
k k 1
k k 1
k k 1 q O , Q k
qn Qn
qn Qn
k n Tn
k
,
k
k n Tn
k
,
k
Pn1 k n Tn pn
Journal of Computer and Mathematical Sciences Vol. 1 Issue 4, 30 June, 2010 Pages (403-527)
k
504
Mahendra Misra et al., J.Comp.&Math.Sci. Vol.1(4), 501-513 (2010)
then the series
a
k
n is summable
n
n Tn
N , qn , n ; k , k 1, 0 . In this paper, we generalize TheoremC for N , q n , n ; f k , k 1, summability methods. 3. Main Theorem:- Let
t n denote the
N, p -mean of the series a n
n
and
let 1
Tn n t n 1 .
f
k
n
n 1
q nk 1 n Qn
k
f k 1 q O Q n f n 1 n
(3.2)
qn Qn
n f n 1 n
(3.3)
k
n n
k
k
(3.4)
k 1
k n Tn
n n
n Tn n f n 1 n
f n n , =f f n n
then the series
a
n
n is summable
N , qn , n ; f k , k 1, 0.
Let n be the n-th N, q n mean
If (3.1)
,
Proof of the Theorem:
1 k
(3.5)
k
k
k
of the series
n n 1
,
k 1
k
k 1
n 1 Qn qn Qn Qn1 1
Qn q n P11
Pn1 p n
1 n1 P a r 1 r Qn qn QnQn1 1 r 1
Qn qn P11
,
n n
n . Then
n 1 = P `1a Qn qn Q n Q n 1 1
,
qn Qn
n
Qn q n a
Pn pn k
a
k
n Pr 1 a r Pn11 Qn q 0 Q0 q n n r 1 (By Abel’s partial summation formula)
Journal of Computer and Mathematical Sciences Vol. 1 Issue 4, 30 June, 2010 Pages (403-527)
Mahendra Misra et al., J.Comp.&Math.Sci. Vol.1(4), 501-513 (2010) 505 1 1 n 1 P P 1 k 1 T Qn 1 q n Qn q n 1 P11 Qn Qn 1 1 p
Qn q n 1 Q n 1 q n
p Q n q n 1 Qn 1 q n P1 P P 1 Pn Pn 1 1k 1 Tn Pn11 q 0 Qn 1 n . pn
1 1 n 1 P k 1 Qn 1q n Qn q n 1 T Qn Qn 1 1 p
Qn q n 1 Qn 1 q n
1 1 k
1 1 P 1 Qn q n 1 Qn 1 q n k T T p
1 1 Pn q 0 n k Tn p n Q n
Tn ,1 Tn , 2 Tn ,3 Tn , 4 Tn,5 Tn ,6 Tn ,7
, say..
To prove t he t heorem, by Minkowski’s inequality it is sufficient to show that
f n k n k 1 Tn ,r
k
, r 1, 2,3, 4,5,6,7 .
n 1
Now, m 1
k 1
k
f n
n
T n ,1
k
n2 m 1
f n n n 2
k
k 1
1 Qnk
n 1
P 1 p
1
1 k q n T
k
Journal of Computer and Mathematical Sciences Vol. 1 Issue 4, 30 June, 2010 Pages (403-527)
506
Mahendra Misra et al., J.Comp.&Math.Sci. Vol.1(4), 501-513 (2010)
m 1
k 1
k
f n n n2
1 Qn
n 1 P k q n p 1
k
1 k
T
1 Qn
n 1
k
q n 1
k 1
,
(using Holder’s inequality) k
m
P O (1) 1 p
k
P O (1) 1 p
f
k 1
k
k k 1 f O (1) k k 1 f 1 m
k
f O (1) 1 f O (1) f 1 m
k
n
1
k
m
k 1
f n
n 1
k
m
m 1
k
1 k T
k
k 1
T
P p
k
f k k 1 k
k
T
q , by(3.1) Q
q Q
k
T
k
k
P q p Q
k
T
k
, by (3.5)
O ( 1 ) , as m ,by(3.2). Next, m 1
k
P q p Q
k 1
k
q n Qn
f n k n k 1 Tn , 2
k
n2
Journal of Computer and Mathematical Sciences Vol. 1 Issue 4, 30 June, 2010 Pages (403-527)
Mahendra Misra et al., J.Comp.&Math.Sci. Vol.1(4), 501-513 (2010) 507
m1
k
f n n
k 1
n2
P qn 1 1 p
1 Qnk1
m 1
1 Q n 1
k k 1 f n n n2
1 1 k
n1
k
T
n 1 P k q n 1 p 1 n 1
1 Q n 1
q n
1
1
k
1 k k
T
k
k 1
,
(using Holder’s inequality) k
m
O (1 )
1
P p
k
P O (1) 1 p
k
T
1
k
f k k 1
k k 1 f O (1 ) k k 1 1 f
k
f O (1) 1 f m
m
O (1 )
1
f
k 1
f k
n
m
m 1
n
n 1
k
m
1 k k
k
k 1
k 1
T
k
q n 1 Q n 1
, Q
f k k 1 q k
P q p Q
k
by(3.1)
T
k
k
P q p Q
k
k
T
k
P q p Q
k
T
k
, by(3.5)
O (1 ) , as m , by(3.2) Journal of Computer and Mathematical Sciences Vol. 1 Issue 4, 30 June, 2010 Pages (403-527)
505
Mahendra Misra et al., J.Comp.&Math.Sci. Vol.1(4), 501-513 (2010)
Next, m 1
k
f n k n k 1 T n , 3 n2
m 1
f
k 1
k
n
n
n2
m 1
q
f n n
Q n 1
n2
n 1
k
1 1 k
T
1
1
k 1
k
n 1
1 Q nk 1
n 1 q n 1 1 1 Q n 1
k
n 1
1 k k
q n
k
k 1
1
1
T
,
(using Holder’s inequality) m
O (1 )
k
1 k k
T
1
m
O (1 )
1
m 1
k
f
n
1 k 1
f k
k
T
k
O (1 )
1 m
O (1 )
1
k
f f
f
k
k 1
q n 1 Q n 1
q Q
q Q
q Q
k 1
q Q
k
T
k
k
, by(3.1)
f k k 1
k k 1 f (1 ) k k 1 1 f m
n
1
m
O
k 1
k
n
T
k
k
T
k
, by(3.5)
O ( 1 ) , as m , by(3.3). Journal of Computer and Mathematical Sciences Vol. 1 Issue 4, 30 June, 2010 Pages (403-527)
Mahendra Misra et al., J.Comp.&Math.Sci. Vol.1(4), 501-513 (2010) 509 Further, m 1
k 1
f k
n
n
k
Tn,4
n2
m 1
f
k 1
k
n
n
n2
m 1
k 1
f n n k
n2
q nk Q nk Q nk1
q nk Q nk Q n 1
n 1
1
Q q n n 1 q n
n 1 q n 1
Q n 1 q n
k
1
1 k T
k
k
1 Q n 1
1 k k
T
n 1
q
n
1
k
k 1
,
(using Holder’s inequality) m
O (1 )
k
1 k k
T
k
1
k k m1 q q Q k k 1 n n n 1 f n n Q Q q n 1 n n1 n m
O (1 )
k
1 k k
T
k
1
m
O (1)
1
1
f k k 1
k
n 1
q n k k 1 f n n Q
T
k
k k 1 f O (1 ) k k 1 1 f m
m 1
q Q
f k k 1 q Q
k
T
n 1
,
k
Journal of Computer and Mathematical Sciences Vol. 1 Issue 4, 30 June, 2010 Pages (403-527)
by(3.1)
510
Mahendra Misra et al., J.Comp.&Math.Sci. Vol.1(4), 501-513 (2010) k
f O (1) 1 f m
m
O (1 )
1
f
O ( 1 ) , as m
k
k 1
q Q
k 1
q Q
k
k
T
k
k
T
, by( 3 . 5 )
, by(3.3).
Next, m 1
f n k n k 1 T n , 5
k
n2
m 1
k 1
k
f n
n
n2
1 Q n 1
n 1 P 1 1 p
k
q n 1
1 Q n 1
k
1 k k
n 1
q n 1 1
T
k
k 1
,
(using Holder’s inequality)
m
P O (1) 1 1 p m
k
1 k k k T
k
m 1
k
k k 1 f O (1 ) k k 1 1 f m
Q n 1
n 1
P 1 k O (1) T k k 1 p f 1 1
q f n k n k 1 n 1 k
q Q
, by(3.1) Q
f k k 1 q P 1 p
k
Journal of Computer and Mathematical Sciences Vol. 1 Issue 4, 30 June, 2010 Pages (403-527)
k
T
k
Mahendra Misra et al., J.Comp.&Math.Sci. Vol.1(4), 501-513 (2010) 511 k
f O (1) 1 f m
O (1) f 1 m
k
O (1 ) , as m
k 1
k 1
k
q Q
P 1 p
q Q
P 1 k T p
k
k
T
k
k
, by(3.5)
,by(3.4).
Next, m 1
k
k k 1 f n n T n , 6 n2
m 1
k 1
f n n k
n2
q nk Q nk Q nk1
n 1
Q n 1 q n m 1
k 1
f n n k
n 2
1
P q n 1 p
1
1 k T
k
k k P 1 Qn 1 1 k k q nk n1 k q T n p qn Qnk Qn1 1
1 Q n 1
n 1
q 1
n
k 1
,
(using Holder’s inequality) m
O (1 )
1
P 1 p
k
k
1 k k
T
k
Journal of Computer and Mathematical Sciences Vol. 1 Issue 4, 30 June, 2010 Pages (403-527)
k
512
Mahendra Misra et al., J.Comp.&Math.Sci. Vol.1(4), 501-513 (2010)
m 1
qn Qn
k 1
f k
n
n
n 1
m
O (1 )
1
P 1 p
k
k
q n Q n 1
1 k 1
k
f
Q n 1 q n
k
k
k
T
, Q
f k k 1 q k k 1 f O (1 ) k k 1 1 f m
m
O (1)
1
O (1) 1 m
k
f f
f
k
O (1 ) , as m
k 1
k 1
P 1 p
P 1 p
q Q
k
k
q Q
q Q
P 1 p
by(3.1)
k
k
k
k
T
k
, by(3.4).
k k 1 f n n Tn , 7
k
n 1
m
k
k
T
Finally, m
T
1
1 k k k 1 Pn q 0 n n T n f n n p Q n 1 n n
k
Journal of Computer and Mathematical Sciences Vol. 1 Issue 4, 30 June, 2010 Pages (403-527)
,by(3.5)
Mahendra Misra et al., J.Comp.&Math.Sci. Vol.1(4), 501-513 (2010) 513
k
m
k
k k 1 P q f n n n 0 n n 1 pn Qn k
m
O (1) f n n k
k 1
n 1
m
O (1 )
n 1
f
n n
k
n n
1 k k
n
k
Pn q n n p Q n n
k
k 1
Tn
1
k
f n n
k 1
k
k
Pn q n p n Qn
k
k
Tn
k
n
k
Tn
k
,
by(3.5)
O (1 ) , as m , by(3.2). This completes the proof of the Theorem. REFERENCES
summability
of an infinite series,, Ph.D thesis submitted to Berhampur 1. Bor, H., On two summability methods, University (2008). Math. Proc. Cambridge. Philos., Vol. 4. Sulaiman, W.T., On a New absolute 97, 147-149 (1985). summability method., International journal Math.& Math. Science, Vol. 2. Jena.K., On N , p n , n k summabilit 21, No. 3, 603-606 (1998). summability method , Ph.D thesis submitted to Berhampur University (2009). 3. Samanta, P., On
N , pn , n ; k
Journal of Computer and Mathematical Sciences Vol. 1 Issue 4, 30 June, 2010 Pages (403-527)