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Some Tauberian theorems for the product method of Borel and Cesàro summability
Computers and Mathematics with Applications 64 (2012) 2871–2876
Contents lists available at SciVerse ScienceDirect
Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa
Some Tauberian theorems for the product method of Borel and Cesàro summability✩ Yılmaz Erdem ∗ , Ümit Totur 1 Adnan Menderes University, Department of Mathematics, 09010 Aydin, Turkey
article
abstract
info
Article history: Received 30 November 2011 Received in revised form 22 April 2012 Accepted 29 April 2012
In this paper we prove several new Tauberian theorems for the product of Borel and Cesàro summability methods which improve classical Tauberian theorems for the Borel summability method. © 2012 Elsevier Ltd. All rights reserved.
Keywords: Borel summability Cesàro summability (B)(C , α) summability Tauberian conditions Tauberian theorems
1. Introduction and preliminaries In 1913, the first Tauberian theorem related to the Borel summability method which gives the relation between Borel and Cesàro summability methods was given by Hardy and Littlewood [1]. Next, these well-known results extended and investigated the conditions needed for a Borel summable sequence to be convergent by Çanak and Totur [2]. In literature, a number of authors; Hardy [3], Jakimovski [4] and Rajagopal [5] investigated several sufficient conditions such that Borel summability implies (C , α). Pati [6] proved some Tauberian theorems for the product of Abel and Cesàro summability methods, and gave some ∞ ∞ suitable conditions so that (A) (C , α) summability of n=0 an to s for some α > 0 implies (C , α) summability of n=0 an to s. These Tauberian theorems in the sense of Pati were generalized by Çanak et al. [7], and Erdem and Çanak [8]. In what ∞ follows, we give some definitions and preliminary results in order to obtain our main theorems. n Let n=0 an be an infinite series of real numbers with sequence of nth partial sums (sn ) = ( k=0 ak ). Definition 1. Let Aαn be defined by the generating function (1 − x)−α−1 = Aα0 = 1,
Aαn =
∞
n =0
Aαn xn (|x| < 1), where
Γ (n + α + 1) α(α + 1) · · · (α + n) = , n! Γ (n + 1)Γ (α + 1)
for α > −1. If sαn =
Snα Aα n
=
n 1 α−1 A sk → s α An k=0 n−k
as n → ∞, we say that the sequence (sn ) is (C , α) summable to s, and it is denoted by sn → s (C , α). ✩ The paper has been evaluated according to old Aims and Scope of the journal. If the paper has not been scanned for the plagiarism, please do it. This is one of the old submissions. ∗ Corresponding author. Tel.: +90 256 218 2000; fax: +90 256 213 5379. E-mail addresses: [email protected], [email protected] (Y. Erdem), [email protected], [email protected] (Ü. Totur). 1 Tel.: +90 256 218 2000; fax: +90 256 213 5379.
0898-1221/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2012.04.022
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Y. Erdem, Ü. Totur / Computers and Mathematics with Applications 64 (2012) 2871–2876
1 Particularly, for α = 1, we observe arithmetic means of (sn ) that s1n = σn (s) = n+ k=0 sk . Denote τn = nan (n = 1 α 0, 1, 2, . . .) and define τn as (C , α) mean of (τn ). It is well known that if β > α > −1, then (C , α) ⊂ (C , β) [9], and also (C , α) summability method is regular [10].
n
Definition 2. If the corresponding series e−x
∞
sn
n! n=0
xn → s,
∞
sn n n=0 n! x
converges for all x and
x → ∞,
we say that the sequence (sn ) is Borel summable to s, and it is denoted by sn → s (B). Here, the Borel summability method is regular [3]. In Definition 2, if we change (sn ) by (sαn ), then an is (B) (C , α) summable to s and represented by sn → s (B) (C , α). One can easily see that the (B) (C , 0) summability method is the Borel summability method. In the next assertion, Szàsz [11] investigated the composition of two summability methods, as follows. Theorem 3. If sn → s (B) as n → ∞, then sn → s (B) (C , α). Particularly for α = 1, if sn → s (B), then σn (s) → s (B). Moreover, since the Borel summability method is regular, the (B) (C , α) summability method is also regular. Throughout this paper, we refer the symbols un = o(1) and un = O(1) by un → 0 as n → ∞ and (un ) is bounded for large enough n, respectively. Now let us give the following classical Tauberian theorems. Theorem 4 ([3, Theorem 143]). If convergent to s.
√
∞
n=0
an is Borel summable to s and n∆sn = n(sn − sn−1 ) = o( n), then
∞
n =0
an is
Hardy and Littlewood [12] obtained the following strong result by using Theorem 4. We refer the reader to Hardy and Littlewood [13] for another proof. Theorem 5 ([3, Theorem 156]). If
√
∞
n =0
an is Borel summable to s and n∆sn = O( n), then
∞
n =0
an is convergent to s.
The main purpose of the present paper is to prove some new Tauberian theorems for the (B) (C , α) summability method which improve classical Tauberian theorems mentioned above, and to give the short proof of some classical Tauberian theorems as special cases of our results. 2. Lemmas We begin with quoting some lemmas which are needed in proving our theorems. Lemma 6 ([14,15]). For α > −1, τnα = n∆sαn = n(sαn − sαn−1 ). Lemma 7 ([15,3]). For α > −1, τnα+1 = (α + 1) (sαn − snα+1 ). Lemma 8 ([7]). For α > −1, n∆τnα+1 = (α + 1) (τnα − τnα+1 ). 1 1 sα+1 + (1 − α+ )σn (sα+1 ). Lemma 9 ([5]). σn (sα ) = α+ 1 n 1 The following lemma is an auxiliary result of the present paper.
Lemma 10. If sn → s(B) (C , α), then sn → s(B) (C , α + 1). Proof. From the definition of the (B) (C , α) summability method, sαn → s(B), we get σn (sα ) → s(B). One can obtain from Lemma 9 that
σn (sα+1 ) =
Aα0 σ0 (sα ) + Aα1 σ1 (sα ) + · · · + Aαn σn (sα ) Aα0 + Aα1 + · · · + Aαn
,
(1)
n where Aα0 + Aα1 + · · · + Aαn = ( α+ + 1)Aαn [3, Theorem 51]. Next, by Definition 2, we find 1
B0 (x, σn (sα )) = e−x
∞
σ n ( sα )
n =0
α
Br (x, σn (s )) = e
−x
∞
xn n!
α
σn−r (s )
n =0
xn n!
=e
−x
y
ey Br −1 (y)dy 0
From the last equations, we get lim inf Br −1 (x) ≥ lim Br (x) ≥ lim sup Br −1 (x), x→∞
x→∞
x→∞
(r ≥ 1) (see [16]).
Y. Erdem, Ü. Totur / Computers and Mathematics with Applications 64 (2012) 2871–2876
2873
and lim Br (x) = lim B0 (x).
x→∞
x→∞
Thus, we find out lim sup |Br (x)| ≥ lim sup |B0 (x)|. x→∞
x→∞
α
Since (σn (s )) is Borel summable to s then the sequences (σ0 (sα )), (σ1 (sα )), . . ., (σn−1 (sα )) are Borel summable to s. By the 1 equality (1), we obtain that (σn (sα+1 )) is Borel summable to s. Hence, from Lemma 9, (sα+ ) is Borel summable to s. n Remark 11. It is clear that if α = 0 in Lemma 10, then we conclude that if (sn ) is Borel summable to s, then (σn (s)) is Borel summable to s. 3. Main results In this section, we give the proof of the following Tauberian theorems for the (B) (C , α) summability method. Theorem 12. If
∞
n =0
an is (B) (C , α) summable to s for some α ≥ 0, and
√
α
τn = O( n) ∞ then n=0 an is (C , α) summable to s. Proof. From the hypothesis of the theorem, sαn → s (B).
(2)
By Lemma 6, we have
√ τnα = n1sαn = O( n).
(3)
It follows from the equalities (2), (3) and Theorem 5 that sαn → s and therefore we obtain sn → s (C , α). Thus, one can conclude that
∞
n=0
an is (C , α) summable to s.
n=0 an is (B) (C , α) summable to s for some α ≥ 0, and √ τnα = o( n) ∞ then n=0 an is (C , α) summable to s.
Corollary 13. If
∞
Here, we emphasize that if we take α = 0 in Corollary 13, then we have Theorem 4. Corollary 14. If
∞
n =0
an is (B) (C , α) summable to s for some α ≥ 0, and
τnα = O(1) ∞ then n=0 an is (C , α) summable to s. Corollary 15. If
∞
n =0
an is (B) (C , α) summable to s for some α ≥ 0, and
α
τn = o(1) ∞ then n=0 an is (C , α) summable to s. Taking α = 0 in Corollary 15, we obtain the following corollary. Corollary 16. If
then
∞
n =0
τn = nan = o(1) ∞ n =0
an is Borel summable to s and
an is convergent to s.
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Y. Erdem, Ü. Totur / Computers and Mathematics with Applications 64 (2012) 2871–2876
Theorem 17. If
then
n∆τnα+1 ∞
∞
n=0
an is (B) (C , α) summable to s for some α ≥ 0, and
√ = O( n)
(4)
an is (C , α) summable to s.
n =0
Proof. We can find by hypothesis and Lemma 10 that sαn → s (B) implies 1 sα+ → s (B). n
(5)
Then, we have
(sαn − snα+1 ) → 0 (B). By Lemma 7, we get
τnα+1 → 0 (B).
(6)
It follows from (6) and the hypothesis (4), we obtain
τnα+1 = o(1)
(7)
by Theorem 5. Thus, we have by Lemma 6 that 1 n1sα+ = o(1). n
(8) α+1
Then, we can find n1sn n1snα+1
√
n
= O(1) from the identity (8) and hence
= o(1).
Equivalently,
√
1 n1sα+ = o( n). n
(9)
By (5) and (9), we get 1 sα+ → s, n
(10)
from Theorem 4. Substituting (7) and (10) into Lemma 7, we obtain sαn → s. Consequently, we conclude that sn → s (C , α). That is,
∞
n =0
an is (C , α) summable to s.
Theorem 18. If
∞
n=0
an is (B) (C , α + 1) summable to s for some α ≥ 0, and
τnα+1 = o(1) ∞ then n=0 an is (C , α) summable to s. Proof. By hypothesis, we have 1 sα+ → s (B). n
(11)
Replacing α by α + 1 in Lemma 6, we obtain
τnα+1 = n1snα+1 = o(1).
(12) α+1
From the identity (12), we have n1sn α+1
n1sn
√
n
= O(1), it follows from
= o(1)
that
√
1 n1sα+ = o( n). n
(13)
Y. Erdem, Ü. Totur / Computers and Mathematics with Applications 64 (2012) 2871–2876
2875
By (11) and (13) and Theorem 4, we find out snα+1 → s.
(14)
Next, substituting (12) and (14) into Lemma 7, we obtain sαn → s, equivalently, sn → s (C , α). Then, this completes that
∞
n =0
an is (C , α) summable to s.
By taking α = 0 in Theorem 18, we obtain the following corollary.
τn1 = then
∞
an is (B) (C , 1) summable to s and
∞
Corollary 19. If
n =0
n
1
n + 1 k=0
kak = o(1)
an is convergent to s.
n =0
Here, we note that Corollary 16 implies Corollary 19. Theorem 20. If
∞
n =0
an is (B) (C , α + 1) summable to s for some α ≥ 0, and
n∆τnα+2 = o(1) then
(15)
an is (C , α) summable to s.
∞
n =0
1 Proof. By using the hypothesis of the assertion and Lemma 10, we obtain that sα+ → s (B) implies n 2 sα+ → s (B). n
(16)
Replacing α by α + 1 in Lemma 7, we obtain 1 2 (α + 2) (sα+ − sα+ ) = τnα+2 → 0 (B). n n
(17)
From the hypothesis (15), we have n∆τnα+2 = O(1), and it follows n1τnα+2
√
n
= o(1)
which is equivalent to
√
n1τnα+2 = o( n).
(18)
By (17) and (18), we get
τnα+2 → 0
(19)
from Theorem 4. Replacing α by α + 1 in Lemma 6, we get
τnα+2 = n1snα+2 = o(1).
(20) α+2
From the identity (20), we have n1sn 2 n1sα+ n
√
n
= O(1), and thus
= o(1).
So, we obtain
√
n1snα+2 = o( n).
(21)
By (16) and (21), we get snα+2 → s,
(22)
from Theorem 4. Next, changing α by α + 1 in Lemma 7, we have 1 2 (α + 2) (sα+ − sα+ ) = τnα+2 . n n
(23)
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Y. Erdem, Ü. Totur / Computers and Mathematics with Applications 64 (2012) 2871–2876
By (19), (22) and (23), we obtain 1 sα+ → s. n
(24)
Then, replacing α by α + 1 in Lemma 8, we have
(α + 2) (τnα+1 − τnα+2 ) = n∆τnα+2 .
(25)
By (15) and (19) and from the identity (25), we find
τnα+1 → 0.
(26)
Next from the identities (24), (26) and Lemma 7, we obtain sαn → s. Hence, we get sn → s (C , α). That is to say,
∞
n =0
an is (C , α) summable to s.
Acknowledgment The authors are grateful to the anonymous referee who made a number of useful suggestions, which improved the quality of this paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
G.H. Hardy, J.E. Littlewood, The relations between Borel’s and Cesàro’s method of summation, Proc. London Math. Soc. 11 (1913) 1–16. İ. Çanak, Ü. Totur, Some Tauberian theorems for Borel summability methods, Appl. Math. Lett. 23 (2010) 302–305. G.H. Hardy, Divergent Series, second ed., Chelsea, New York, NY, 1991. A. Jakimovski, Some relations between the methods of summability of Abel, Borel, Cesàro, Hölder and Hausdorff, J. Anal. Math. 3 (1954) 346–381. C.T. Rajagopal, On a theorem connecting Borel and Cesàro summabilities, J. Indian Math. Soc. (N.S.) 24 (1961) 433–442. T. Pati, On Tauberian theorems, in: D. Rath, S. Nanda (Eds.), Sequences, Summability and Fourier Analysis, Narosa Publishing House, 2005, pp. 84–96. İ. Çanak, Y. Erdem, Ü. Totur, Some Tauberian theorems for (A)(C , α) summability method, Math. Comput. Modelling 52 (5–6) (2010) 738–743. Y. Erdem, İ. Çanak, A Tauberian theorem for (A)(C , α) summability, Comput. Math. Appl. 60 (11) (2010) 2920–2925. D. Borwein, Theorems on some methods of summability, Q. J. Math. 9 (1) (1958) 310–316. J. Boos, Classical and Modern Methods in Summability, Oxford University Press, New York, 2000. O. Szász, On products of summability methods, Proc. Amer. Math. Soc. 3 (1952) 257–262. G.H. Hardy, J.E. Littlewood, Theorems concerning the summability of series by Borel’s exponential method, Rend. Circ. Mat. Palermo 41 (1916) 36–53. G.H. Hardy, J.E. Littlewood, On the Tauberian theorems for Borel summability, J. London Math. Soc. 18 (1943) 194–200. E. Kogbetliantz, Sur le séries absolument sommables par la méthode des moyennes arihtmétiques, Bull. Sci. Math. 49 (2) (1925) 234–251. E. Kogbetliantz, Sommation des séries et intégrals divergents par les moyennes arithmétiques et typiques, Memorial Sci. Math. 51 (1931) 1–84. M.R. Parameswaran, Some product theorems in summability, Math. Z. Bd. 68 (1957) 19–26.
Contents lists available at SciVerse ScienceDirect
Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa
Some Tauberian theorems for the product method of Borel and Cesàro summability✩ Yılmaz Erdem ∗ , Ümit Totur 1 Adnan Menderes University, Department of Mathematics, 09010 Aydin, Turkey
article
abstract
info
Article history: Received 30 November 2011 Received in revised form 22 April 2012 Accepted 29 April 2012
In this paper we prove several new Tauberian theorems for the product of Borel and Cesàro summability methods which improve classical Tauberian theorems for the Borel summability method. © 2012 Elsevier Ltd. All rights reserved.
Keywords: Borel summability Cesàro summability (B)(C , α) summability Tauberian conditions Tauberian theorems
1. Introduction and preliminaries In 1913, the first Tauberian theorem related to the Borel summability method which gives the relation between Borel and Cesàro summability methods was given by Hardy and Littlewood [1]. Next, these well-known results extended and investigated the conditions needed for a Borel summable sequence to be convergent by Çanak and Totur [2]. In literature, a number of authors; Hardy [3], Jakimovski [4] and Rajagopal [5] investigated several sufficient conditions such that Borel summability implies (C , α). Pati [6] proved some Tauberian theorems for the product of Abel and Cesàro summability methods, and gave some ∞ ∞ suitable conditions so that (A) (C , α) summability of n=0 an to s for some α > 0 implies (C , α) summability of n=0 an to s. These Tauberian theorems in the sense of Pati were generalized by Çanak et al. [7], and Erdem and Çanak [8]. In what ∞ follows, we give some definitions and preliminary results in order to obtain our main theorems. n Let n=0 an be an infinite series of real numbers with sequence of nth partial sums (sn ) = ( k=0 ak ). Definition 1. Let Aαn be defined by the generating function (1 − x)−α−1 = Aα0 = 1,
Aαn =
∞
n =0
Aαn xn (|x| < 1), where
Γ (n + α + 1) α(α + 1) · · · (α + n) = , n! Γ (n + 1)Γ (α + 1)
for α > −1. If sαn =
Snα Aα n
=
n 1 α−1 A sk → s α An k=0 n−k
as n → ∞, we say that the sequence (sn ) is (C , α) summable to s, and it is denoted by sn → s (C , α). ✩ The paper has been evaluated according to old Aims and Scope of the journal. If the paper has not been scanned for the plagiarism, please do it. This is one of the old submissions. ∗ Corresponding author. Tel.: +90 256 218 2000; fax: +90 256 213 5379. E-mail addresses: [email protected], [email protected] (Y. Erdem), [email protected], [email protected] (Ü. Totur). 1 Tel.: +90 256 218 2000; fax: +90 256 213 5379.
0898-1221/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2012.04.022
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Y. Erdem, Ü. Totur / Computers and Mathematics with Applications 64 (2012) 2871–2876
1 Particularly, for α = 1, we observe arithmetic means of (sn ) that s1n = σn (s) = n+ k=0 sk . Denote τn = nan (n = 1 α 0, 1, 2, . . .) and define τn as (C , α) mean of (τn ). It is well known that if β > α > −1, then (C , α) ⊂ (C , β) [9], and also (C , α) summability method is regular [10].
n
Definition 2. If the corresponding series e−x
∞
sn
n! n=0
xn → s,
∞
sn n n=0 n! x
converges for all x and
x → ∞,
we say that the sequence (sn ) is Borel summable to s, and it is denoted by sn → s (B). Here, the Borel summability method is regular [3]. In Definition 2, if we change (sn ) by (sαn ), then an is (B) (C , α) summable to s and represented by sn → s (B) (C , α). One can easily see that the (B) (C , 0) summability method is the Borel summability method. In the next assertion, Szàsz [11] investigated the composition of two summability methods, as follows. Theorem 3. If sn → s (B) as n → ∞, then sn → s (B) (C , α). Particularly for α = 1, if sn → s (B), then σn (s) → s (B). Moreover, since the Borel summability method is regular, the (B) (C , α) summability method is also regular. Throughout this paper, we refer the symbols un = o(1) and un = O(1) by un → 0 as n → ∞ and (un ) is bounded for large enough n, respectively. Now let us give the following classical Tauberian theorems. Theorem 4 ([3, Theorem 143]). If convergent to s.
√
∞
n=0
an is Borel summable to s and n∆sn = n(sn − sn−1 ) = o( n), then
∞
n =0
an is
Hardy and Littlewood [12] obtained the following strong result by using Theorem 4. We refer the reader to Hardy and Littlewood [13] for another proof. Theorem 5 ([3, Theorem 156]). If
√
∞
n =0
an is Borel summable to s and n∆sn = O( n), then
∞
n =0
an is convergent to s.
The main purpose of the present paper is to prove some new Tauberian theorems for the (B) (C , α) summability method which improve classical Tauberian theorems mentioned above, and to give the short proof of some classical Tauberian theorems as special cases of our results. 2. Lemmas We begin with quoting some lemmas which are needed in proving our theorems. Lemma 6 ([14,15]). For α > −1, τnα = n∆sαn = n(sαn − sαn−1 ). Lemma 7 ([15,3]). For α > −1, τnα+1 = (α + 1) (sαn − snα+1 ). Lemma 8 ([7]). For α > −1, n∆τnα+1 = (α + 1) (τnα − τnα+1 ). 1 1 sα+1 + (1 − α+ )σn (sα+1 ). Lemma 9 ([5]). σn (sα ) = α+ 1 n 1 The following lemma is an auxiliary result of the present paper.
Lemma 10. If sn → s(B) (C , α), then sn → s(B) (C , α + 1). Proof. From the definition of the (B) (C , α) summability method, sαn → s(B), we get σn (sα ) → s(B). One can obtain from Lemma 9 that
σn (sα+1 ) =
Aα0 σ0 (sα ) + Aα1 σ1 (sα ) + · · · + Aαn σn (sα ) Aα0 + Aα1 + · · · + Aαn
,
(1)
n where Aα0 + Aα1 + · · · + Aαn = ( α+ + 1)Aαn [3, Theorem 51]. Next, by Definition 2, we find 1
B0 (x, σn (sα )) = e−x
∞
σ n ( sα )
n =0
α
Br (x, σn (s )) = e
−x
∞
xn n!
α
σn−r (s )
n =0
xn n!
=e
−x
y
ey Br −1 (y)dy 0
From the last equations, we get lim inf Br −1 (x) ≥ lim Br (x) ≥ lim sup Br −1 (x), x→∞
x→∞
x→∞
(r ≥ 1) (see [16]).
Y. Erdem, Ü. Totur / Computers and Mathematics with Applications 64 (2012) 2871–2876
2873
and lim Br (x) = lim B0 (x).
x→∞
x→∞
Thus, we find out lim sup |Br (x)| ≥ lim sup |B0 (x)|. x→∞
x→∞
α
Since (σn (s )) is Borel summable to s then the sequences (σ0 (sα )), (σ1 (sα )), . . ., (σn−1 (sα )) are Borel summable to s. By the 1 equality (1), we obtain that (σn (sα+1 )) is Borel summable to s. Hence, from Lemma 9, (sα+ ) is Borel summable to s. n Remark 11. It is clear that if α = 0 in Lemma 10, then we conclude that if (sn ) is Borel summable to s, then (σn (s)) is Borel summable to s. 3. Main results In this section, we give the proof of the following Tauberian theorems for the (B) (C , α) summability method. Theorem 12. If
∞
n =0
an is (B) (C , α) summable to s for some α ≥ 0, and
√
α
τn = O( n) ∞ then n=0 an is (C , α) summable to s. Proof. From the hypothesis of the theorem, sαn → s (B).
(2)
By Lemma 6, we have
√ τnα = n1sαn = O( n).
(3)
It follows from the equalities (2), (3) and Theorem 5 that sαn → s and therefore we obtain sn → s (C , α). Thus, one can conclude that
∞
n=0
an is (C , α) summable to s.
n=0 an is (B) (C , α) summable to s for some α ≥ 0, and √ τnα = o( n) ∞ then n=0 an is (C , α) summable to s.
Corollary 13. If
∞
Here, we emphasize that if we take α = 0 in Corollary 13, then we have Theorem 4. Corollary 14. If
∞
n =0
an is (B) (C , α) summable to s for some α ≥ 0, and
τnα = O(1) ∞ then n=0 an is (C , α) summable to s. Corollary 15. If
∞
n =0
an is (B) (C , α) summable to s for some α ≥ 0, and
α
τn = o(1) ∞ then n=0 an is (C , α) summable to s. Taking α = 0 in Corollary 15, we obtain the following corollary. Corollary 16. If
then
∞
n =0
τn = nan = o(1) ∞ n =0
an is Borel summable to s and
an is convergent to s.
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Y. Erdem, Ü. Totur / Computers and Mathematics with Applications 64 (2012) 2871–2876
Theorem 17. If
then
n∆τnα+1 ∞
∞
n=0
an is (B) (C , α) summable to s for some α ≥ 0, and
√ = O( n)
(4)
an is (C , α) summable to s.
n =0
Proof. We can find by hypothesis and Lemma 10 that sαn → s (B) implies 1 sα+ → s (B). n
(5)
Then, we have
(sαn − snα+1 ) → 0 (B). By Lemma 7, we get
τnα+1 → 0 (B).
(6)
It follows from (6) and the hypothesis (4), we obtain
τnα+1 = o(1)
(7)
by Theorem 5. Thus, we have by Lemma 6 that 1 n1sα+ = o(1). n
(8) α+1
Then, we can find n1sn n1snα+1
√
n
= O(1) from the identity (8) and hence
= o(1).
Equivalently,
√
1 n1sα+ = o( n). n
(9)
By (5) and (9), we get 1 sα+ → s, n
(10)
from Theorem 4. Substituting (7) and (10) into Lemma 7, we obtain sαn → s. Consequently, we conclude that sn → s (C , α). That is,
∞
n =0
an is (C , α) summable to s.
Theorem 18. If
∞
n=0
an is (B) (C , α + 1) summable to s for some α ≥ 0, and
τnα+1 = o(1) ∞ then n=0 an is (C , α) summable to s. Proof. By hypothesis, we have 1 sα+ → s (B). n
(11)
Replacing α by α + 1 in Lemma 6, we obtain
τnα+1 = n1snα+1 = o(1).
(12) α+1
From the identity (12), we have n1sn α+1
n1sn
√
n
= O(1), it follows from
= o(1)
that
√
1 n1sα+ = o( n). n
(13)
Y. Erdem, Ü. Totur / Computers and Mathematics with Applications 64 (2012) 2871–2876
2875
By (11) and (13) and Theorem 4, we find out snα+1 → s.
(14)
Next, substituting (12) and (14) into Lemma 7, we obtain sαn → s, equivalently, sn → s (C , α). Then, this completes that
∞
n =0
an is (C , α) summable to s.
By taking α = 0 in Theorem 18, we obtain the following corollary.
τn1 = then
∞
an is (B) (C , 1) summable to s and
∞
Corollary 19. If
n =0
n
1
n + 1 k=0
kak = o(1)
an is convergent to s.
n =0
Here, we note that Corollary 16 implies Corollary 19. Theorem 20. If
∞
n =0
an is (B) (C , α + 1) summable to s for some α ≥ 0, and
n∆τnα+2 = o(1) then
(15)
an is (C , α) summable to s.
∞
n =0
1 Proof. By using the hypothesis of the assertion and Lemma 10, we obtain that sα+ → s (B) implies n 2 sα+ → s (B). n
(16)
Replacing α by α + 1 in Lemma 7, we obtain 1 2 (α + 2) (sα+ − sα+ ) = τnα+2 → 0 (B). n n
(17)
From the hypothesis (15), we have n∆τnα+2 = O(1), and it follows n1τnα+2
√
n
= o(1)
which is equivalent to
√
n1τnα+2 = o( n).
(18)
By (17) and (18), we get
τnα+2 → 0
(19)
from Theorem 4. Replacing α by α + 1 in Lemma 6, we get
τnα+2 = n1snα+2 = o(1).
(20) α+2
From the identity (20), we have n1sn 2 n1sα+ n
√
n
= O(1), and thus
= o(1).
So, we obtain
√
n1snα+2 = o( n).
(21)
By (16) and (21), we get snα+2 → s,
(22)
from Theorem 4. Next, changing α by α + 1 in Lemma 7, we have 1 2 (α + 2) (sα+ − sα+ ) = τnα+2 . n n
(23)
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Y. Erdem, Ü. Totur / Computers and Mathematics with Applications 64 (2012) 2871–2876
By (19), (22) and (23), we obtain 1 sα+ → s. n
(24)
Then, replacing α by α + 1 in Lemma 8, we have
(α + 2) (τnα+1 − τnα+2 ) = n∆τnα+2 .
(25)
By (15) and (19) and from the identity (25), we find
τnα+1 → 0.
(26)
Next from the identities (24), (26) and Lemma 7, we obtain sαn → s. Hence, we get sn → s (C , α). That is to say,
∞
n =0
an is (C , α) summable to s.
Acknowledgment The authors are grateful to the anonymous referee who made a number of useful suggestions, which improved the quality of this paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
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