International Journal of Mathematics and Mathematical Sciences Volume 2008, Article ID 753632,8pages
doi:10.1155/2008/753632
Research Article
On Faster Convergent Infinite Series
Du ˇsan Hol ´y,1Ladislav Matej´ı ˇcka,1and L’udov´ıt Pinda2
1Department of Physical Engineering of Materials, Faculty of Industrial Technologies in P ´uchov,
Trenˇc´ın University of Alexander Dubˇcek in Trenˇc´ıin, I. Krasku 491/30, P ´uchov 02001, Slovakia
2Faculty of Informatics, Univerzity of Economics, Dolnozemsk´a 1, Bratislava 852 35, Slovakia
Correspondence should be addressed to Ladislav Matej´ıˇcka,[email protected]
Received 12 July 2007; Revised 20 November 2007; Accepted 8 January 2008
Recommended by Laszlo Toth
Suffcient conditions, necessary conditions for faster convergent infinite series, fasterτ-convergent infinite series are studied. The faster convergence of infinite series of Kummer’s type is proved.
Copyrightq2008 Duˇsan Hol ´y 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.
1. Introduction
There have been papers devoted to the study of faster convergence of sequences. Certain methods concerning acceleration of convergence of sequences of partial sums of fixed series using linear or nonlinear transformations of partial sums of series are studied in 1. The
acceleration field of subsequence matrix transformations with respect to the convergence rate of the sequence being accelerated are studied in2. In3, it is discussed a class of methods
for summing sequences which are generalizations of a method due to Salzer 4, which
accelerate some convergent sequences especially monotone sequences. In5, is characterized
the summability field of a matrixAby showingAis convergence preserving over the set of all sequences which converge faster than some fixed sequencex,Ais convergence preserving over the set of all sequences, orAonly preserves the limit of a set of constant sequences. Statistical acceleration convergence of sequences was discussed in6. The notion of faster convergent
series with positive terms is defined in7and the notion ofτ-convergent series is defined in 8–10. The statistical convergence of infinite series as a special case ofτ-convergence of infinite series is discussed in11–14.
In this paper, some questions related to sufficient conditions and necessary conditions for faster convergent infinite series are studied, faster τ-convergent series are defined and studied, and faster convergence of series of Kummer’s type is proved. In7, page 146, it is
c, and limn→∞an/cn p > 0, then the Kummer series ∞n1bn has the same sum as∞n1an and is a faster convergent series than∞n1an, if all terms of∞n1bnare positive. Hence we can calculate the unknown sumafaster by summation terms of the Kummer series∞n1bn, which is constructed by∞n1an,
∞
n1cn, andp. InLemma 4.1, we proved a faster convergence of the Kummer series∞n1bnto the unknown sumaof∞n1anwithout conditions of positivity ofp and terms of∞n1an,∞n1bn,∞n1cn.
We denote byNthe set of all positive integers and byRthe set of all real numbers.
Definition 1.1see2. Let∞n1an,∞n1bnbe convergent real series with the same sum, with nonzero terms and such that bn bn 1 · · ·/0, n ∈ N. The series
∞
n1an is called faster convergent than∞n1bnif limn→∞an an 1 · · ·/bn bn 1 · · · 0.
Lemma 1.2see7. Let∞n1an,∞n1bnbe convergent real series with positive terms and with the
same sum. If limn→∞an/bn 0, then limn→∞an an 1 · · ·/bn bn 1 · · · 0.
In what follows, we do not assume equality of sum of convergent series∞n1an,∞n1bn because we can construct series∞n1a∗n,
∞
n1b∗n, wherea∗n anandb∗nbnforn≥2 with the same sum.
2. Faster convergent series Lemma 2.1. Let∞n1an,
∞
n1bnbe convergent real series with positive terms. If
lim n→∞
an an 1 · · ·
bn bn 1 · · · 0, then lim infn→∞
an
bn 0. 2.1
Proof. By the way of contradiction, we suppose that lim infn→∞an/bn q > 0, then there existsn0 ∈ Nsuch that for every n > n0 we have 0 < q < a
n/bn,where 0 < q < q.From this followsqb
n bn 1 · · · < an an 1 · · ·forn > n0,which is a contradiction with
limn→∞an an 1 · · ·/bn bn 1 · · · 0,similarly forq∞.
Lemma 2.2. Let∞
n1an, ∞
n1bnbe convergent real series with positive terms. Let limn→∞an/bn
exist. Then
lim n→∞
an
bn
0 iff lim n→∞
an an 1 · · ·
bn bn 1 · · · 0. 2.2
The next example shows that under the conditions ofLemma 2.1, for all 0< r≤ ∞, there exist a series∞n1an,∞n1bnsuch that lim supn→∞an/bn r.
Example 2.3. Let{αn;n∈N}be any sequence of real positive numbers which satisfyαn≤1/n4 forn≥1. We define the series∞n1anrα1 α1/12 rα2 α2/22 · · · rαk αk/k2 · · · and ∞
n1bnα1 12α1 α2 22α2 · · · αk k2αk · · ·. Then we have
lim n→∞
a2n−1 a2n · · ·
b2n−1 b2n · · · nlim→∞
rαn αn 1 · · · 1/n2αn αn 1 · · ·
αn αn 1 · · · n2αn αn 1 · · · 0,
lim n→∞
a2n a2n 1 · · · b2n b2n 1 · · · nlim→∞
rαn 1 αn 2 · · · 1/n2αn αn 1 · · ·
Duˇsan Hol ´y et al.
lim n→∞
a2n−1
b2n−1 r,
lim n→∞
a2n
b2n
0. 2.3
If∞n1anα1 α1/12 2α2 α2/22 · · · kαk αk/k2 · · ·, then limn→∞a2n−1/b2n−1 ∞.
Remark 2.4. It is evident that in general from limn→∞an an 1 · · ·/bn bn 1 · · · 0 does not follow lim infn→∞an/bn 0.For example, if we puta2n b2n 1/nn 1, n∈N,and
a2n−1 −b2n−1 −1/nn 1, n∈N, we obtain limn→∞an an 1 · · ·/bn bn 1 · · · 0
and{−1,1}is the set of all cluster points of the sequence{an/bn;n∈N}.
In general, the condition limn→∞an/bn 0 does not imply the condition limn→∞an
an 1 · · ·/bn bn 1 · · · 0 as it follows from the next example.
Example 2.5. Let{αn;n ∈N}be a sequence such thatαn qn, n ∈N,0 < q < 1/2. We define convergent series∞n1an α1 α1 α2/22 α2/22 · · · αk/k2 αk/k2 · · · and∞n1bn
α1−α1/2 α2/3−α2/4 · · · αk/2k−1−αk/2k · · ·. From the definition, we obtain
b2n b2n 1 · · · −αn
2n
αn 1
2n 12n 2
αn 2
2n 32n 4 · · ·
< 1
2n
−α αn 1 αn 2 · · ·
1 2n
−qn qn 1 qn 2 · · · q
n2q−1 2n1−q <0.
2.4
It is obvious thatb2n 1 b2n 2 · · ·>0 and limn→∞an/bn 0.From2k m−12k m>
k m2, k, m∈Nwe have
a2k−1 a2k · · ·
b2k−1 b2k · · ·
2αk/k2 αk 1/k 12 · · ·
αk/2k2k−1 αk 1/
2k 12k 2 · · · >2 fork≥1. 2.5
Thus limn→∞an an 1 · · ·/bn bn 1 · · ·/0.
Lemma 2.6. Let∞n1an,∞n1bnbe convergent real series with nonzero terms. Letbn bn 1 · · ·/0
for alln∈N. Letlia lim infn→∞|1 an 1/an an 2/an · · · |, lsa lim supn→∞|1 an 1/an
an 2/an · · · |, lib lim infn→∞|1 bn 1/bn bn 2/bn · · · |, lsb lim supn→∞|1 bn 1/bn
bn 2/bn · · · |,then
1if lsa< ∞, lib >0,limn→∞an/bn 0,then limn→∞an an 1 · · ·/bn bn 1 · · · 0,
2if an an 1 · · ·/0 for alln∈N, lia>0, lsb<∞,limn→∞an an 1 · · ·/bn
bn 1 · · · 0,then limn→∞an/bn 0,
3 ifan an 1 · · ·/0 for all n ∈ N, 0 < lia, lsa < ∞,0 < lib, lsb < ∞,
Proof. For everyn∈N, we have
an an 1 · · · bn bn 1 · · ·
1 an 1/an an 2/an · · ·
bn/an bn 1/an bn 2/an · · ·
1 an 1/an an 2/an · · ·
bn/an1 bn 1/bn bn 2/bn · · ·
an/bn1 an 1/an an 2/an an 3/an · · · 1 bn 1/bn bn 2/bn bn 3/bn · · ·
.
2.6
From this follows our assertion.
Remark 2.7. The condition lim infn→∞|1 bn 1/bn bn 2/bn · · · |>0 can be satisfied, for example, if lim supn→∞|bn 1/bn|<1/2.In fact, if lim supn→∞|bn 1/bn|< r <1/2.then there existsn0∈N such that for everyn > n0, we have
bn 1
bn
bn 2 bn
bn 3 bn · · ·
≤bn 1
bn bn 2
bn 1
bn 1
bn bn 3
bn 2
bn 2
bn 1
bn 1
bn · · ·
≤r r2 r3 · · · r
1−r,
2.7
and thus|1 bn 1/bn bn 2/bn · · · | ≥ 1−2r/1−r > 0.The condition lim supn→∞|1
an 1/an an 2/an · · · |<∞can be satisfied, for example, if lim supn→∞|an 1/an|<1.Indeed, if lim supn→∞|an 1/an|α <1,then there existsn0∈Nsuch that for everyn > n0, we have
1 an 1
an
an 2 an · · ·
1 an 1
an
an 2 an 1
an 1 an
an 3 an 2
an 2 an 1
an 1 an · · ·
≤1 β β2 β3 · · · 1
1−β <∞, where α < β <1.
2.8
Conversely, from the condition lim infn→∞|1 bn 1/bn bn 2/bn · · · | > 0 need not follow the condition lim supn→∞|bn 1/bn| < 1/2. For example, if we put a1/0, a2n 1/2n,
a2n 1 −1/2n,n 1,2, . . . , then∞n1an is convergent series and lim supn→∞|an 1/an| 1,
lim supn→∞|1 an 1/an an 2/an · · · |1.
3.τ-convergent series
Definition 3.1see10. We say that a sequence{an;n∈ N}hasτ-limit a real numberLand we writeτ-limn→∞anL, if for eachε >0 the setAε {n;|an−L| ≥ε}belongs to the ideal
τ, whereτis an admissible ideal of subsets ofNwhich is additiveifA, B∈τ, thenA∪B∈τ,
hereditaryifB⊂A∈τ, thenB∈τ, containing all singletons and not containingN.
We denote byτf the ideal of all finite subsets ofN.
Definition 3.2 see 10. We say that ∞n1anτ-converges to a real number L and we write
Duˇsan Hol ´y et al.
Definition 3.3. Let∞n1an,n∞1bnbeτ-convergent real series with nonzero terms such thatbn
bn 1 · · ·/0, n∈N. A series∞n1anis calledτ-faster convergent than∞n1bnifτ-limn→∞an 1 an 2 · · ·/bn 1 bn 2 · · · 0.
Definition 3.4see9. Letτbe an admissible ideal of subsets ofN. A numberx∈Ris said to be aτ-cluster point of∞n1xnif for eachε >0 the set{n∈N;|
n
k1xk−x|< ε}is not fromτ.
Remark 3.5. Of course, ifτ is an admissible ideal of subsets ofN, byτ-cluster point of a real sequence{xn;n∈N}, we mean a numberx∈R, where for eachε >0 the set{n∈N;|xn−x|< ε} is not fromτ.Moreover, we say that∞−∞is theτ-cluster point of a real sequence{xn;n∈N}
if for eachc >0c <0the set{n∈N;xn> c}{n∈N;xn< c}is not fromτ.
Remark 3.6. If {xn;n ∈ N} is a real sequence, τ is an admissible ideal, and X {x;
xis aτ-cluster point of{xn;n ∈ N}}, then X /∅., Indeed, if{xn;n ∈ N} is bounded, then by8there exists aτ-cluster point of{xn;n∈N}.If{xn;n∈N}is not bounded, then either
∞or−∞isτ-cluster point of{xn;n∈N}according toRemark 3.5or there existsl∈Rsuch
that{n∈N;|xn|> l} ∈τ.If for somel∈R{n∈N;|xn|> l} ∈τ, thenKl{n∈N;|xn| ≤l}/∈τ.
Consider the idealυ {Kl∩A;A ∈τ}.Because{xn;n ∈ Kl}is a bounded set, there exists
x∈R,|x| ≤lsuch thatxis aυ-cluster point of{xn;n∈K
l}.Letε >0. Sincexis aυ-cluster
point of{xn;n∈Kl}, then the set{n∈Kl;|xn−x|< ε}/∈υand then also is not fromτ. Sinceτ has a hereditary property, then{n∈N;|xn−x|< ε}∈/τ. Soxis a cluster point of{xn;n∈N}.
Definition 3.7. Letτbe an admissible ideal of subsets ofN. Let∞n1xnbe an infinite series of real numbers and let X {x ∈ R;x is aτ-cluster point of ∞n1xn}.IfX is bounded, then
ssupXsinfXis said to be aτ-lim supn→∞n∞1xnτ-lim infn→∞∞n1xn. If supX ∞
or∞isτ-cluster pointinfX −∞or−∞isτ-cluster point, thenτ-lim supn→∞∞n1xn ∞ τ-lim infn→∞∞n1xn−∞.
Definition 3.8. We say that a sequence{xn;n∈R}isτ-bounded aboveτ-bounded bellow, if
there existm∈Rsuch that{n∈N;xn> m} ∈τ{n∈N;xn< m} ∈τand isτ-bounded if it is
τ-bounded above and below simultaneously.
It is obvious that if ∞n1an is faster convergent than
∞
n1bn, then ∞
n1an is τ-faster convergent, the ∞n1bn. Generally, from the fact that ∞n1an isτ-faster convergent,∞n1bn does not hold that∞n1anis faster convergent than∞n1bn.
It is obvious that forτ-convergent series whereτhas the property p1:
ifM∈τ, thenM 1{n 1;n∈M∩N} ∈τ, M−1{n−1>0;n∈M∩N} ∈τ
3.1 τf orτst {A ⊂ N; asymptotic density A 0} have p1, we obtain similar lemma as
Lemma 2.6.
Lemma 3.9. Letτ be an admissible ideal of subsets ofNwith property p1. Let∞n1an,∞n1bn be
τ-convergent real series with nonzero terms. Letbn bn 1 · · ·/0, for alln∈N. Let
tia τ-lim inf
n→∞
1 an 1
an
an 2
an · · ·
, tsa τ-lim sup
n→∞
1 an 1
an
an 2
an · · · ,
tib τ-lim inf
n→∞
1 bn 1
bn
bn 2 bn · · ·
, tsb τ-lim sup
n→∞
1 bn 1
bn
bn 2 bn · · ·
.
Then
1 iftsa<∞, tib>0, τ-limn→∞an/bn 0, thenτ-limn→∞an an 1 · · ·/bn bn 1 · · · 0,
2if an an 1 · · ·/0 for all n∈N, tia>0, tsb<∞, τ-limn→∞an an 1 · · ·/bn
bn 1 · · · 0,thenτ-limn→∞an/bn 0,
3if an an 1 · · ·/0 for all n ∈ N,0 < tia, tsa < ∞,0 < tib, tsb < ∞,then
τ-limn→∞an/bn 0,if and only if τ-limn→∞an an 1 · · ·/bn bn 1 · · · 0.
Proof. First, we note that ifτ-lim supn→∞xn < ∞, then sequence {xn;n ∈ N} isτ-bounded abovesimilarly forτ- lim inf. Next, we show that if{an;n∈N}is aτ-convergent sequence to 0 and{bn;n∈N}is aτ-bounded sequence, then{anbn : n∈N}is aτ-convergent sequence to 0. There exist k ∈ Rsuch that the set Nb {n;|bn| ≥ k} is fromτ. Let ε > 0.The set
Naε {n;|an| ≥ε/k}is fromτ. Ifn∈N\Naε∪Nb, then|an|< ε/Kand|bn|< kand so
|anbn|< ε. From this and from properties of idealτ, follows that{anbn: n∈N}τ-converges to
0. Then the proof follows fromLemma 2.6.
The following examples show that we cannot replace lim byτ-lim in Lemmas1.2,2.1, and2.2.
Example 3.10. Let∞n1bnbe a convergent series with positive terms such that ∞
n1bn bn 1
bn 2 · · ·is convergente.g.,n∞1bn ∞n1αn−1, 0< α < 1. LetM{ij; 1< ij< ij 1, j∈N},
M ∈ τ be an infinite subset ofN, whereτ is an arbitrary admissible ideal with property p1 different fromτf. We define a real series∞n1anin the following way:
an
⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ 1
nbn, n >1, n /ij, ij∈M, j2,3, . . .
bij−1 bij−1 1 · · · n∈M\ {i1},
c, n1,
3.3
wherecbe a real number such that∞n1an ∞n1bn. We obtainτ-limn→∞an/bn 0,an
an 1 · · ·/bn bn 1 · · ·>1 for alln≥i2.
Example 3.11. LetM{nj;n11, nj< nj 1, nj 1−nj≥2, j∈N},M∈τbe an infinite subset of N, whereτis an arbitrary admissible ideal with property p1 different fromτf. Let{Bnj}
∞
j1be a sequence of positive real numbers such thatBnj1 < Bnj < mBnj1,limj→∞Bnj0,wherem >1.
PutAk Bk/2k, k ∈ M. LetAk, Bkfork /∈Mbe defined as follows: ifnj < k < nj 1, where j ∈Nwe putεnj Anj−Anj1/nj 1−nj, Ak Anj−k−njεnj, Bk Bnj −k−njεnj.For k /∈M, we haveAk/Bk< Anj/Bnj1Bnj/2
njB
nj1 < m/2
nj, wheren
j < k < nj 1.From this and
from the definitionAk, k ∈M, it follows limk→∞Ak/Bk 0,henceτ-limk→∞Ak/Bk 0.It is obvious thatAk−Ak 1/Bk−Bk 1 1 fork /nj−1, j >1.LetA0> A1andB0> B1be real numbers. Putan An−1−An, bnBn−1−Bn, n∈N.The series∞n1an,∞n1bnare convergent with positive terms andτ-limn→∞an an 1 · · ·/bn bn 1 · · · 0, τ-limn→∞an/bn 1.
Remark 3.12. Ifτhave not the property p1, we get similar lemma asLemma 3.9, but with shift indices of given series.In general, it does not hold that ifτ-limn→∞ana, thenτ-limn→∞an 1
Duˇsan Hol ´y et al.
4. Kummer series
In7, is showed: if∞n1anis a convergent real series with positive terms and with an unknown sumaand∞n1cnis a convergent real series with positive terms and with a known sumc, then if limn→∞an/cn p >0 the Kummer series∞n1bn, whereb1a1 pc−c1andbnan−pcn forn≥2, has the same sum as∞n1anand ifbn>0 is a faster convergent series then∞n1an.
In the following lemma, we prove by usingLemma 2.6 the faster convergence of the Kummer series∞n1bn to the unknown sum of
∞
n1an without conditions of positivity ofp and terms of∞n1an,∞n1bn,∞n1cn.
Lemma 4.1. Let ∞n1an be a convergent real series with nonzero terms and with the sum a. Let
lim infn→∞|1 an 1/an an 2/an · · · | > 0. Letan an 1 · · ·/0 for alln ∈ N. Let ∞n1cn
be a convergent series with nonzero terms and with the sumc. Let limn→∞an/cn p /0. If for the
series∞n1bn, whereb1 a1 pc−c1andbn an−pcn, forn ≥ 2, is bn/0, forn ≥ 2, and lim supn→∞|1 bn 1/bn bn 2/bn · · · | <∞, then∞n1bn aand∞n1bn is a faster convergent
series than∞n1an.
Proof. Since limn→∞bn/an 0, the proof follows fromLemma 2.6.
In the next example, we construct, by using ofRemark 2.7andLemma 4.1, the Kummer series∞n1bnfaster convergent than the series∞n1ansuch that seriesn∞1an,∞n1cn,∞n1bn does not have positive terms.
Example 4.2. Let∞n1an∞n1−1n/4n3n2 √n, the sum is unknown. It is evident that
lim supn→∞|an 1/an| 1/4 < 1/2.Let∞n1cn ∞n1−1n/4nn2, then limn→∞an/cn 1/3 and from ∞n1xn/n2 −x
0ln1 −t/tdt, x ∈ −1,1, it follows
∞
n1−1n/4nn2
0
−1/4ln1−t/tdtthe sum is known. The Kummer series is
∞
n1bnb1 ∞n2an−1/3cn, where ∞n2bn ∞n2−1n 1/4n3n√n3n2 √n,b1 1/48 1/3
0
−1/4ln1−t/tdt. It
is obvious that lim supn→∞|bn 1/bn| 1/4 < 1. Because ∞n1an is series with alternating signs such that for alln ∈ Nwe have that |an 1| < |an|, we get for n ∈ Na2n a2n 1 · · · >
0, a2n 1 a2n 2 · · ·<0.Hence
∞
n1bnis a faster convergent series than
∞
n1anand it has the same sum.
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