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Abstract and Applied Analysis
Volume 2011, Article ID 213816, 8 pages
doi:10.1155/2011/213816
Research Article
Summability of Sequences and Selection Properties
Dragan Djurčić,1 Ljubiša D. R. Kočinac,2 and Mališa R. Žižović3
1
Technical Faculty Čačak, Svetog Save 65, 32000 Čačak, Serbia
Faculty of Sciences and Mathematics, University of Niš, 18000 Niš, Serbia
3
Singidunum University, Belgrade, Serbia
2
Correspondence should be addressed to Dragan Djurčić, dragandj@tfc.kg.ac.rs
Received 9 February 2011; Accepted 10 April 2011
Academic Editor: Jean Pierre Gossez
Copyright q 2011 Dragan Djurčić et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We prove that some classes of summable sequences of positive real numbers satisfy several
selection principles related to a special kind of convergence.
1. Introduction
By Æ , Ê, and Ê we denote the set of natural numbers, real numbers, and the extended real
line Ê ∪ {−∞, ∞}, respectively.
Let Ë denote the set of sequences a an n∈ of positive real numbers.
We begin with the following definitions of selection principles.
Let A and B be nonempty subsets of Ë. Then the symbol S1 A, B denotes the selection
principle.
For each sequence an : n ∈ Æ of elements of A there is a sequence b bn n∈ ∈ B
such that bn ∈ an for each n ∈ Æ .
The following infinitely long game G1 A, B is naturally associated to the previous
selection principle.
Two players, ONE and TWO, play a round for each positive integer. In the n-th
round ONE chooses a sequence an ∈ A, and TWO responds by choosing an element
bn ∈ an . TWO wins a play a1 , b1 ; . . . ; an , bn ; . . . if b bn n∈ ∈ B; otherwise, ONE
wins.
Another selection principle
fin A, B
is defined as follows.
For each sequence an : n ∈ Æ of elements of A there is a sequence b ∈ B such that
b ∩ an is finite for each n ∈ Æ .
It is clear how the corresponding game Gfin A, B is defined.
2
Abstract and Applied Analysis
A strategy of a player is a function σ from the set of all finite sequences of moves of
the other player into the set of admissible moves of the strategy owner.
A strategy σ for the player TWO is a coding strategy if TWO remembers only the most
recent move by ONE and by TWO before his next move. More precisely the moves of TWO
are b1 σa1 , ∅; bn σan , bn−1 , n ≥ 2.
In this paper we introduce also the following game. Let i ∈ Æ be a fixed but arbitrary
wi
natural number. We define the game 1 A, B for two players, ONE and TWO, who play
a round for each n ∈ Æ . In the i-th round ONE plays a sequence ai ai,m m∈ ∈ A, and TWO
responds by choosing a finite set Fi {ai,mi1 , . . . , ai,mik }. In the n-th round, n / i, ONE plays a
sequence an an,m m∈ ∈ A, and TWO responds by choosing an element an,mn ∈ an . TWO
wins a play if the sequence b a1,m1 , . . . , ai−1,mi−1 ; ai,mi1 , . . . , ai,mik ; ai1,mi1 , . . . belongs to B;
otherwise, ONE wins.
For more information on selection principles and games see the survey papers in 1, 2
and references therein.
In a number of papers by the authors published in the last few years it was
demonstrated that some subclasses A and B of Ë satisfy certain selection principles and
game theoretical statements for A and B classes of divergent sequences related to celebrated
Karamata’s theory of regular variation 3–6 see 7–12, and for A and B classes of sequences
converging to 0 see 13. For other results concerning sequences and sequence spaces see
14–16.
In this paper our selections are related to special kinds of convergence of series. More
precisely, we start by a sequence of summable sequences and during the selection process we
control not only the convergence of series, but also the nature of that convergence.
2. Results
We use the following notations for the classes of sequences we deal with:
1
1,S
a ∈ Ë:
a ∈ Ë:
∞
∞
an < ∞ ,
n1
an S ,
for S ∈ 0, ∞,
n1
1,α,β a ∈ 1,S : S ∈ α, β ,
1,α,β 1,α,β
1,β ,
2.1
for α, β ∈ 0, ∞,
for α, β ∈ 0, ∞.
Notice that the sequence x xn n∈ , xn S/2n , belongs to the class 1,S , so that all
the classes above are nonempty.
Theorem 2.1. For each S ∈ 0, ∞ and each ε εS ∈ 0, S TWO has a winning coding strategy
w1 1,S 1,S−ε,S
, .
in the game 1
Proof. Let σ denote a strategy of TWO, and let S > 0 and ε εS ∈ 0, S be fixed. Suppose
that in the first round ONE chooses a sequence x1 x1,m m∈ from 1,S . There is k ∈ Æ
Abstract and Applied Analysis
3
k
such that ∞
mk1 x1,m < ε/2, and thus M S −
m1 x1,m ∈ 0, ε/2. Player TWO plays
σx1 {x1,1 , . . . , x1,k }—a finite subset of x1 .
In the second round ONE chooses a sequence x2 x2,m m∈ ∈ 1,S , and then TWO
responds by choosing σx2 , σx1 x2,m2 such that x2,m2 < M/2 which is possible because
limm → ∞ x2,m 0.
In the n-th round, n ≥ 3, ONE chooses xn xn,m m∈ ∈ 1,S , and TWO’s response is
σxn , xn−1,mn−1 xn,mn such that xn,mn < xn−1,mn−1 /2n−1 < M/2n−1 , and so on.
Set yn x1,n for n ≤ k and yn xn−k1,mn−k1 for n > k. Let us prove y yn n∈ ∈
1,S−ε,S
. We have
∞
k
∞
k
∞
yn yn yn x1,m yn
n1
n1
m1
nk1
S−M
∞
nk1
yn < S − M M
∞
1
i1
nk1
2i
2.2
S.
On the other hand,
∞
n1
yn >
k
ε
x1,m S − M > S − .
2
m1
2.3
That is, y ∈ 1,S−ε,S .
Corollary 2.2. For each S ∈ 0, ∞ and each ε εS ∈ 0, S the selection principle
1,S 1,S−ε,S
,
is true.
fin Notice that one can prove a refinement of Theorem 2.1 and Corollary 2.2 in the sense
that it is possible to have additional control of selections giving the sequence y. For this we
need the following definitions and notation.
Definition 2.3 see 13. A sequence xn n∈ ∈
each λ ≥ 1 it satisfies
lim
n→∞
Ë is said to belong to the class Tr−∞,× if for
xnλ
0,
xn
2.4
where r denotes the integer part of r ∈ Ê.
Definition 2.4 see 9. For a sequence x xn n∈ ∈ Ë, the Landau-Hurwicz sequence wx wn xn∈ of x is defined by
wn x : sup{|xm − xk | : m ≥ n, k ≥ n},
n ∈ Æ.
2.5
4
by
Abstract and Applied Analysis
Given a sequence x xn n∈ ∈
Ë we denote by Sx Sn xn∈ the sequence defined
Sn x n
xi ,
i1
n ∈ Æ.
2.6
1,α,β
Let TrÊ−∞, be the set of all sequences a an n∈ ∈ 1,α,β such that wSa ∈ Tr−∞,× .
Theorem 2.5. For each S ∈ 0, ∞ and each ε εS ∈ 0, S TWO has a winning coding strategy
w1 1,S 1,S−ε,S
, TrÊ−∞, .
in the game 1
Proof. The strategy σ of player TWO and the sequence y yn n∈ are actually from the proof
of Theorem 2.1. Therefore, y ∈ 1,S−ε,S . Besides, since, by construction, the series
∞
yn1
n1
2.7
yn
is convergent, we have
lim
n→∞
∞
yk1
kn
yk
0.
2.8
Consider now the sequence Sy Sn yn∈ . This sequence is convergent by the
d’Alembert criterion, and let Sy be its limit. It remains to prove wSy wn Sy n∈ ∈
Tr−∞,× . It is enough to prove
wn1 Sy
lim
0.
n → ∞ w n Sy
2.9
First, notice that
w n Sy S y − Sn y ,
n ∈ Æ.
2.10
Thus we get
wn1 Sy
S y − Sn1 y
yn1
0.
lim
lim 1 − lim
n → ∞ w n Sy
n → ∞ S y − Sn y
n → ∞ yn1 yn2 · · ·
2.11
That is 2.9, since by 2.8 and the fact that for n sufficiently large it holds
yn2 yn3 yn2
yn2 yn3
yn2 yn3
··· ·
··· ≤
··· ,
yn1 yn1
yn1 yn2 yn1
yn1 yn2
2.12
Abstract and Applied Analysis
5
we have
yn1
1
1.
lim
n → ∞ yn1 yn2 · · ·
n → ∞ 1 yn2 /yn1 yn3 /yn1 · · ·
lim
2.13
The theorem is proved.
Corollary 2.6. The selection principle
fin 1,S
1,S−ε,S
, TrÊ−∞, is true.
The following two theorems give other selection results for defined classes of
sequences: one of the fin -type and the other of the 1 -type.
Theorem 2.7. For each S ∈ 0, ∞ the selection principle
fin 1,S
, 1,∞ is satisfied.
Proof. Consider first the case S ∈ 0, ∞. Let xn : n ∈ Æ , xn xn,m m∈ , be a sequence of
elements of 1,S . For each n ∈ Æ let zni xn,i , i ≤ k kn, be a finite subset of xn such that
S/2 < ki1 zni < S. Arrange now znp , n ∈ Æ , p ∈ {1, 2, . . . , kn}, in the sequence y yj j∈
in which the position of an element is determined first by n and then by p, that is,
y z11 , . . . , z1k1 ; . . . ; zn1 , . . . , znkn ; . . . .
2.14
We have
n·
kn
n km
S <
zmi yj ,
2 m1 i1
j1
2.15
where ykn is the last element of xn belonging to the sequence y. Therefore,
S
yj lim
yj > lim n ·
∞.
n→∞
n→∞
2
j1
j1
∞
kn
2.16
That is, y ∈ 1,∞ .
Suppose now that S ∞. This case is treated similarly to the previous case, but here
we require ki1 zni > 1 for each n ∈ Æ ; the sequence y yj j∈ is formed in a similar way as
kn
in the first case. So we have n · 1 < j1 yj for each n ∈ Æ , hence
∞
j1
yj lim
n→∞
kn
yj > lim n · 1 ∞.
j1
2.17
n→∞
That is, y ∈ 1,∞ . The theorem is proved.
Theorem 2.8. For each S ∈ 0, ∞ and each α > 0 the selection principle
1 1,S
, 1,0,α is true.
Proof. Let xn : n ∈ Æ , xn xn,m m∈ , be a sequence of elements in 1,S . For each n ∈ Æ
take yn xn,mn ∈ xn so that y1 ∈ 0, α which is possible since x1,m → 0 as m → ∞ and
6
Abstract and Applied Analysis
yn < α − y1 /2n−1 for n ≥ 2. Then the sequence y yn n∈ witnesses that the statement is true,
because
∞
yn y1 ∞
yn < y1 α − y1 α.
2.18
n2
n1
That is, y ∈ 1,0,α .
For the next result we have to define the following selection principles 2, 17. Notice
that in 18 we developed an interesting technique for proving results concerning these
selection principles and certain classes of sequences from Ë. In 19 we proposed the use
of this technique and these selection principles in other fields of mathematics and its
applications.
Let, as before, A and B be certain nonempty subfamilies of Ë. Then the symbol
αi A, B, i 2, 3, 4, denotes the selection hypothesis that for each sequence an : n ∈ Æ of elements of A there is an element b ∈ B such that:
α2 A,B: for each n ∈ Æ the set an ∩ b is infinite;
α3 A, B: for infinitely many n ∈ Æ the set an ∩ b is infinite;
α4 A, B: for infinitely many n ∈ Æ the set an ∩ b is nonempty.
1,0,α
Theorem 2.9. For each S ∈ 0, ∞ and each α > 0 the selection principles αi 1,S , TrÊ−∞, , i 2, 3, 4, are satisfied.
Proof. We prove that the principle α2 is true hence also α3 and α4 . Let xn : n ∈ Æ , xn xn,m m∈ , be a sequence of sequences from 1,S . Let m1 ∈ Æ be such that ∞
mm1 1 x1,m < α/2.
∞
For k ≤ 2 let mk be a natural number such that mmk 1 xk,m < α/2k . Consider the sequence
y yj j∈ defined in this way:
y x1,m1 1 , x1,m2 2 , . . . ; x2,m2 1 , x2,m2 2 , . . . ; xk,mk 1 , xk,mk 2 , . . ..
2.19
Then y ∩ xn is infinite for each n ∈ Æ . Further, y ∈ 1,0,α because
0<
∞
j1
yj ∞
∞
k1 mmk 1
xk,m <
∞
α
α.
k
2
k1
2.20
We construct now a new sequence z zi in the way described in Table 1 .
∞
Evidently, z ∩ xn is infinite for each n ∈ Æ . Also, 0 < ∞
i1 zi ≤
j1 yj < α, that is,
1,0,α
. By a minor modification of the proof of Theorem 2.5 we obtain wSz ∈ Tr−∞,× .
z∈
1,0,α
This means z ∈ TrÊ−∞, .
Abstract and Applied Analysis
7
Table 1
z1
z2
z3
z4
z5
z6
z7
z8
z9
x1
z1 ∈ y ∩ x1
—
z3 ∈ y ∩ x1
—
—
z6 ∈ y ∩ x1
—
—
···
x2
—
z2 ∈ y ∩ x2
—
—
z5 ∈ y ∩ x2
—
—
—
···
x3
—
—
—
z4 ∈ y ∩ x3
—
—
—
z8 ∈ y ∩ x3
···
x4
—
—
—
—
—
—
z7 ∈ y ∩ x4
—
···
How
any
z2 /z1 < 1/2
z3 /z2 < 1/22
z4 /z3 < 1/23
z5 /z4 < 1/24
z6 /z5 < 1/25
z7 /z6 < 1/26
z8 /z7 < 1/27
···
Acknowledgments
The authors are supported by the Ministry of Science and Technological Development of the
Republic of Serbia. They thank the referees for their several useful comments.
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8
Abstract and Applied Analysis
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