Extending tests for convergence of number series
Elijah Liflyand, Sergey Tikhonov, and Maria Zeltser
September, 2012
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Goals and History
Main Goal:
Relax the monotonicity assumption for the sequence of terms of the series.
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Goals and History
Main Goal:
Relax the monotonicity assumption for the sequence of terms of the series.
In various well-known tests for convergence/divergence of number
series
∞
X
ak ,
(1)
k=1
with positive ak , monotonicity of the sequence of these ak is the basic
assumption.
Liflyand et al
Extending monotonicity
September, 2012
2 / 28
Goals and History
Main Goal:
Relax the monotonicity assumption for the sequence of terms of the series.
In various well-known tests for convergence/divergence of number
series
∞
X
ak ,
(1)
k=1
with positive ak , monotonicity of the sequence of these ak is the basic
assumption.
Such series are frequently called monotone series.
Liflyand et al
Extending monotonicity
September, 2012
2 / 28
Goals and History
Main Goal:
Relax the monotonicity assumption for the sequence of terms of the series.
In various well-known tests for convergence/divergence of number
series
∞
X
ak ,
(1)
k=1
with positive ak , monotonicity of the sequence of these ak is the basic
assumption.
Such series are frequently called monotone series.
Tests by Abel, Cauchy, de la Vallee Poussin, Dedekind, Dirichlet, du
Bois Reymond, Ermakov, Leibniz, Maclaurin, Olivier, Sapogov,
Schlömilch are related to monotonicity.
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References
Books:
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References
Books:
T.J.I’a Bromwich, Introduction to the Theory of Infinite Series,
MacMillan, New York, 1965.
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Extending monotonicity
September, 2012
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References
Books:
T.J.I’a Bromwich, Introduction to the Theory of Infinite Series,
MacMillan, New York, 1965.
K. Knopp, Theory and Application of Infinite Series, Blackie& Son
Ltd., London-Glasgow, 1928.
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Extending monotonicity
September, 2012
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References
Books:
T.J.I’a Bromwich, Introduction to the Theory of Infinite Series,
MacMillan, New York, 1965.
K. Knopp, Theory and Application of Infinite Series, Blackie& Son
Ltd., London-Glasgow, 1928.
G.M. Fichtengolz, Infinite Series: Rudiments, Gordon and Breach,
New York, 1970.
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Extending monotonicity
September, 2012
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References
Books:
T.J.I’a Bromwich, Introduction to the Theory of Infinite Series,
MacMillan, New York, 1965.
K. Knopp, Theory and Application of Infinite Series, Blackie& Son
Ltd., London-Glasgow, 1928.
G.M. Fichtengolz, Infinite Series: Rudiments, Gordon and Breach,
New York, 1970.
D.D. Bonar and M.J. Khoury, Real Infinite Series, MAA, Washington,
DC, 2006.
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Tests
In its initial form the Maclaurin-Cauchy integral test reads as follows:
Consider a non-negative monotone decreasing function f defined on
[1, ∞). Then the series
∞
X
f (k)
(2)
k=1
converges if and only if the integral
Z
∞
f (t) dt
1
is finite. In particular, if the integral diverges, then the series diverges.
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Tests
Ermakov’s test.
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Tests
Ermakov’s test.
In its simplest form, it is given as follows.
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Tests
Ermakov’s test.
In its simplest form, it is given as follows.
Let f be a continuous (this is not necessary) non-negative monotone
decreasing function for t > 1.
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Tests
Ermakov’s test.
In its simplest form, it is given as follows.
Let f be a continuous (this is not necessary) non-negative monotone
decreasing function for t > 1.
If for t large enough f (et )et /f (t) ≤ q < 1, then series (2) converges,
Liflyand et al
Extending monotonicity
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Tests
Ermakov’s test.
In its simplest form, it is given as follows.
Let f be a continuous (this is not necessary) non-negative monotone
decreasing function for t > 1.
If for t large enough f (et )et /f (t) ≤ q < 1, then series (2) converges,
while if f (et )et /f (t) ≥ 1, then series (2) diverges.
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Extending monotonicity
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Tests
Ermakov’s test.
In its simplest form, it is given as follows.
Let f be a continuous (this is not necessary) non-negative monotone
decreasing function for t > 1.
If for t large enough f (et )et /f (t) ≤ q < 1, then series (2) converges,
while if f (et )et /f (t) ≥ 1, then series (2) diverges.
In fact, it is known that one can get a family of tests by replacing et
with a positive increasing function ϕ(t) satisfying certain properties.
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Extending monotonicity
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Tests
Ermakov’s test.
In its simplest form, it is given as follows.
Let f be a continuous (this is not necessary) non-negative monotone
decreasing function for t > 1.
If for t large enough f (et )et /f (t) ≤ q < 1, then series (2) converges,
while if f (et )et /f (t) ≥ 1, then series (2) diverges.
In fact, it is known that one can get a family of tests by replacing et
with a positive increasing function ϕ(t) satisfying certain properties.
We will extend such a more general assertion.
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Tests
The well known Cauchy condensation test states that
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Tests
The well known Cauchy condensation test states that
If {ak } is a positive monotone decreasing null sequence, then the
series (1) and
∞
X
2k a2k
k=1
converge or diverge simultaneously.
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Tests
The well known Cauchy condensation test states that
If {ak } is a positive monotone decreasing null sequence, then the
series (1) and
∞
X
2k a2k
k=1
converge or diverge simultaneously.
This assertion is a partial case of the following classical result due to
Schlömilch.
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Tests
The well known Cauchy condensation test states that
If {ak } is a positive monotone decreasing null sequence, then the
series (1) and
∞
X
2k a2k
k=1
converge or diverge simultaneously.
This assertion is a partial case of the following classical result due to
Schlömilch.
Let (1) be a series whose terms are positive and non-increasing,
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Tests
and let
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Tests
and let
u0 < u1 < u2 < ... be a sequence of positive integers such that
∆uk
≤ C.
∆uk−1
Then series (1) converges if and only if the series
∞
X
k=1
∆uk auk =
∞
X
(uk+1 − uk )auk
k=1
converges.
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Tests
and let
u0 < u1 < u2 < ... be a sequence of positive integers such that
∆uk
≤ C.
∆uk−1
Then series (1) converges if and only if the series
∞
X
k=1
∆uk auk =
∞
X
(uk+1 − uk )auk
k=1
converges.
In the theory of monotone series there are statements on the behavior
of its terms. Such is Abel–Olivier’s kth term test:
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Tests
and let
u0 < u1 < u2 < ... be a sequence of positive integers such that
∆uk
≤ C.
∆uk−1
Then series (1) converges if and only if the series
∞
X
k=1
∆uk auk =
∞
X
(uk+1 − uk )auk
k=1
converges.
In the theory of monotone series there are statements on the behavior
of its terms. Such is Abel–Olivier’s kth term test:
Let {ak } be a positive monotone null sequence. If series (1) is
convergent, then kak is a null sequence.
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Tests
Sapogov’s test reads as follows.
If {bk } is a positive monotone increasing sequence, then the series
∞
X
k=1
(1 −
bk
bk+1
)
as well as
∞
X
bk+1
(
− 1)
bk
k=1
converges if the sequence {bk } is bounded and diverges otherwise.
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Tests
The tests of Dedekind and of du Bois Reymond are united as the next
assertion.
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Tests
The tests of Dedekind and of du Bois Reymond are united as the next
assertion.
P
Sequence of bounded variation – {ak } ∈ BV :
|ak+1 − ak | < ∞.
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Tests
The tests of Dedekind and of du Bois Reymond are united as the next
assertion.
P
Sequence of bounded variation – {ak } ∈ BV :
|ak+1 − ak | < ∞.
Let {ak } and {bk } be two sequences.
(i) If {ak } ∈ BV , {ak } is a null sequence, and the sequence of partial
∞
P
P
ak bk is convergent.
sums of
bk is bounded, then the series
k
(ii) If {ak } ∈ BV and
convergent.
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k=1
P
bk is convergent, then the series
k
∞
P
ak bk is
k=1
Extending monotonicity
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Tests
The tests of Dedekind and of du Bois Reymond are united as the next
assertion.
P
Sequence of bounded variation – {ak } ∈ BV :
|ak+1 − ak | < ∞.
Let {ak } and {bk } be two sequences.
(i) If {ak } ∈ BV , {ak } is a null sequence, and the sequence of partial
∞
P
P
ak bk is convergent.
sums of
bk is bounded, then the series
k
(ii) If {ak } ∈ BV and
convergent.
k=1
P
bk is convergent, then the series
k
∞
P
ak bk is
k=1
Two well-known and widely used corollaries of this test -
Liflyand et al
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Tests
The tests of Dedekind and of du Bois Reymond are united as the next
assertion.
P
Sequence of bounded variation – {ak } ∈ BV :
|ak+1 − ak | < ∞.
Let {ak } and {bk } be two sequences.
(i) If {ak } ∈ BV , {ak } is a null sequence, and the sequence of partial
∞
P
P
ak bk is convergent.
sums of
bk is bounded, then the series
k
(ii) If {ak } ∈ BV and
convergent.
k=1
P
bk is convergent, then the series
k
∞
P
ak bk is
k=1
Two well-known and widely used corollaries of this test Dirichlet’s and Abel’s tests - involve monotone sequences.
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Tests
The first one is as follows.
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Tests
The first one is as follows.
Let {ak } be a monotone null sequence and {bk } be a sequence such
that the sequence of its partial sums is bounded. Then the series
∞
P
ak bk is convergent.
k=1
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Tests
The first one is as follows.
Let {ak } be a monotone null sequence and {bk } be a sequence such
that the sequence of its partial sums is bounded. Then the series
∞
P
ak bk is convergent.
k=1
One of corollaries of this test is the celebrated Leibniz test:
∞
P
(−1)k ak
Let {ak } be a monotone null sequence. Then the series
k=1
is convergent.
Liflyand et al
Extending monotonicity
September, 2012
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Tests
The first one is as follows.
Let {ak } be a monotone null sequence and {bk } be a sequence such
that the sequence of its partial sums is bounded. Then the series
∞
P
ak bk is convergent.
k=1
One of corollaries of this test is the celebrated Leibniz test:
∞
P
(−1)k ak
Let {ak } be a monotone null sequence. Then the series
k=1
is convergent.
Further, Abel’s test reads as follows.
P
Let {ak } be a bounded monotone sequence and
bk a convergent
series. Then the series
∞
P
k
ak bk is convergent.
k=1
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Classes of sequences and functions
Let us introduce certain classes of functions and sequences more general
than monotone.
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Classes of sequences and functions
Let us introduce certain classes of functions and sequences more general
than monotone.
Definition
We call a non-negative null (that is, tending to zero at infinity) sequence
{ak } weak monotone, written WMS, if for some positive absolute constant
C it satisfies
ak ≤ Can
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for any
k ∈ [n, 2n].
Extending monotonicity
September, 2012
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Classes of sequences and functions
Let us introduce certain classes of functions and sequences more general
than monotone.
Definition
We call a non-negative null (that is, tending to zero at infinity) sequence
{ak } weak monotone, written WMS, if for some positive absolute constant
C it satisfies
ak ≤ Can
for any
k ∈ [n, 2n].
To introduce a counterpart for functions, we will assume all functions
to be defined on (0, ∞), locally of bounded variation, and vanishing
at infinity.
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Classes of sequences and functions
Definition
We say that a non-negative function f defined on (0, ∞), is weak
monotone, written WM, if
f (t) ≤ Cf (x)
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for any
t ∈ [x, 2x].
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Classes of sequences and functions
Definition
We say that a non-negative function f defined on (0, ∞), is weak
monotone, written WM, if
f (t) ≤ Cf (x)
for any
t ∈ [x, 2x].
Setting ak = f (k) in this case, we obtain {ak } ∈ W M S.
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Classes of sequences and functions
Definition
We say that a non-negative function f defined on (0, ∞), is weak
monotone, written WM, if
f (t) ≤ Cf (x)
for any
t ∈ [x, 2x].
Setting ak = f (k) in this case, we obtain {ak } ∈ W M S.
Clearly, in these definitions 2n and 2x can be replaced by [cn] (where
[a] denotes the integer part of a) and cx, respectively, with some
c > 1 and another constant C.
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Classes of sequences and functions
Definition
We say that a non-negative function f defined on (0, ∞), is weak
monotone, written WM, if
f (t) ≤ Cf (x)
for any
t ∈ [x, 2x].
Setting ak = f (k) in this case, we obtain {ak } ∈ W M S.
Clearly, in these definitions 2n and 2x can be replaced by [cn] (where
[a] denotes the integer part of a) and cx, respectively, with some
c > 1 and another constant C.
In some problems one should consider a smaller class than W M S.
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Classes of sequences and functions
Denote ∆ak = ak+1 − ak .
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Classes of sequences and functions
Denote ∆ak = ak+1 − ak .
Definition. A positive null sequence {ak } is called general monotone
if it satisfies
2n
X
|∆ak | ≤ Can ,
k=n
for any n and some absolute constant C.
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Classes of sequences and functions
Denote ∆ak = ak+1 − ak .
Definition. A positive null sequence {ak } is called general monotone
if it satisfies
2n
X
|∆ak | ≤ Can ,
k=n
for any n and some absolute constant C.
The class of quasi-monotone sequences,
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Classes of sequences and functions
Denote ∆ak = ak+1 − ak .
Definition. A positive null sequence {ak } is called general monotone
if it satisfies
2n
X
|∆ak | ≤ Can ,
k=n
for any n and some absolute constant C.
The class of quasi-monotone sequences,
that is, {ak } such that there exists τ > 0 so that k −τ ak ↓,
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Classes of sequences and functions
Denote ∆ak = ak+1 − ak .
Definition. A positive null sequence {ak } is called general monotone
if it satisfies
2n
X
|∆ak | ≤ Can ,
k=n
for any n and some absolute constant C.
The class of quasi-monotone sequences,
that is, {ak } such that there exists τ > 0 so that k −τ ak ↓,
is a proper subclass of GM S.
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Extending monotonicity
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Classes of sequences and functions
Denote ∆ak = ak+1 − ak .
Definition. A positive null sequence {ak } is called general monotone
if it satisfies
2n
X
|∆ak | ≤ Can ,
k=n
for any n and some absolute constant C.
The class of quasi-monotone sequences,
that is, {ak } such that there exists τ > 0 so that k −τ ak ↓,
is a proper subclass of GM S.
One of the simple basic properties of GM S is GM S ( W M S.
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Classes of sequences and functions
A similar function class:
Definition
We say that a non-negative function f is general monotone, GM, if for all
x ∈ (0, ∞)
Z
2x
|df (t)| ≤ Cf (x).
x
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Classes of sequences and functions
A similar function class:
Definition
We say that a non-negative function f is general monotone, GM, if for all
x ∈ (0, ∞)
Z
2x
|df (t)| ≤ Cf (x).
x
We just remark that if f (·) is a GM function,
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Classes of sequences and functions
A similar function class:
Definition
We say that a non-negative function f is general monotone, GM, if for all
x ∈ (0, ∞)
Z
2x
|df (t)| ≤ Cf (x).
x
We just remark that if f (·) is a GM function,
then for ak = f (k) there holds {ak } ∈ GM S.
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Extending monotonicity
September, 2012
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Classes of sequences and functions
A similar function class:
Definition
We say that a non-negative function f is general monotone, GM, if for all
x ∈ (0, ∞)
Z
2x
|df (t)| ≤ Cf (x).
x
We just remark that if f (·) is a GM function,
then for ak = f (k) there holds {ak } ∈ GM S.
And of course GM ( W M .
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September, 2012
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Extension of ”monotone” tests
An extension of the Maclaurin-Cauchy integral test reads as follows:
Theorem
Let f be a WM function. Then the series
∞
X
f (k)
k=1
and integral
Z
∞
f (t) dt
1
converge or diverge simultaneously.
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Extension of ”monotone” tests
An extension of Ermakov’s test:
Theorem
Let f be a WM function and let ϕ(t) be a monotone increasing, positive
function having a continuous derivative and satisfying ϕ(t) > t for all t
large enough. If for t large enough
f (ϕ(t))ϕ0 (t)
≤ q < 1,
f (t)
then series (2) converges, while if
f (ϕ(t))ϕ0 (t)
≥ 1,
f (t)
then series (2) diverges.
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Extension of ”monotone” tests
Theorem
Let {uk } be an increasing sequence of positive numbers such that
uk+1 = O(uk ) and uk → ∞.
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Extension of ”monotone” tests
Theorem
Let {uk } be an increasing sequence of positive numbers such that
uk+1 = O(uk ) and uk → ∞.
Let f be a WM function. Then both series
∞
X
f (uk )∆uk
and
k=1
f (uk+1 )∆uk
k=1
converge (or diverge) with
Liflyand et al
∞
X
R∞
1
f (t) dt.
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Extension of ”monotone” tests
Theorem
Let {uk } be an increasing sequence of positive numbers such that
uk+1 = O(uk ) and uk → ∞.
Let f be a WM function. Then both series
∞
X
f (uk )∆uk
and
k=1
∞
X
f (uk+1 )∆uk
k=1
R∞
converge (or diverge) with 1 f (t) dt.
Let {ak } be a WMS. Then (1) converges if and only if the series
∞
X
k=1
∆uk auk =
∞
X
(uk+1 − uk )auk
k=1
converges.
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Extension of ”monotone” tests
An extension of Abel–Olivier’s kth term test:
Theorem
Let {ak } be a WMS. If series (1) converges, then kak is a null sequence.
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Extension of ”monotone” tests
An extension of Abel–Olivier’s kth term test:
Theorem
Let {ak } be a WMS. If series (1) converges, then kak is a null sequence.
Analyzing one Dvoretzky’s result and its proof,
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Extension of ”monotone” tests
An extension of Abel–Olivier’s kth term test:
Theorem
Let {ak } be a WMS. If series (1) converges, then kak is a null sequence.
Analyzing one Dvoretzky’s result and its proof,
we obtain the following statement.
Theorem
If {ak }, {bk } ∈ W M S, and the series are convergent and divergent,
respectively, then for every M > 1 there exist infinitely many Rj ,
Rj → ∞, such that for all k with Rj ≤ k ≤ M Rj , we have ak < bk .
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Extension of ”monotone” tests
Our next result is ”dual” to the above Schlömilch-type extensions.
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Extension of ”monotone” tests
Our next result is ”dual” to the above Schlömilch-type extensions.
We recall that the increasing sequence {uk } is called lacunary if
uk+1 /uk ≥ q > 1.
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Extending monotonicity
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Extension of ”monotone” tests
Our next result is ”dual” to the above Schlömilch-type extensions.
We recall that the increasing sequence {uk } is called lacunary if
uk+1 /uk ≥ q > 1.
A more general class of sequences is the one in which each sequence
can be split into finitely-many lacunary sequences. In the latter case
we will write {uk } ∈ Λ.
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Extending monotonicity
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Extension of ”monotone” tests
Our next result is ”dual” to the above Schlömilch-type extensions.
We recall that the increasing sequence {uk } is called lacunary if
uk+1 /uk ≥ q > 1.
A more general class of sequences is the one in which each sequence
can be split into finitely-many lacunary sequences. In the latter case
we will write {uk } ∈ Λ.
This is true if and only if
k
X
uj ≤ Cuk .
j=1
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Extending monotonicity
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Extension of ”monotone” tests
Our next result is ”dual” to the above Schlömilch-type extensions.
We recall that the increasing sequence {uk } is called lacunary if
uk+1 /uk ≥ q > 1.
A more general class of sequences is the one in which each sequence
can be split into finitely-many lacunary sequences. In the latter case
we will write {uk } ∈ Λ.
This is true if and only if
k
X
uj ≤ Cuk .
j=1
In different terms, {uk } ∈ Λ is true if and only if there exists r ∈ N
such that
uk+r
≥ q > 1,
uk
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k ∈ N.
Extending monotonicity
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Extension of ”monotone” tests
¯ u := au − au .
We denote ∆a
k
k
k+1
Proposition. Let {ak } be a non-negative WMS, and let a sequence {uk }
be such that {uk } ∈ Λ and uk+1 = O(uk ).
Then the series (1),
∞
X
¯ u |,
uk |∆a
k
k=1
and
∞
X
uk auk
k=1
converge or diverge simultaneously.
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Extending monotonicity
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General monotonicity
It is also possible to get equiconvergence results for an important case
un = n, where the lacunarity can no more help.
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Extending monotonicity
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General monotonicity
It is also possible to get equiconvergence results for an important case
un = n, where the lacunarity can no more help.
We proceed to a smaller class than W M S. Indeed, assuming general
monotonicity of the sequences, we prove the following result.
Liflyand et al
Extending monotonicity
September, 2012
21 / 28
General monotonicity
It is also possible to get equiconvergence results for an important case
un = n, where the lacunarity can no more help.
We proceed to a smaller class than W M S. Indeed, assuming general
monotonicity of the sequences, we prove the following result.
Proposition. Let {ak } be a GMS. Then series (1) and
X
k|∆ak |
k
converge or diverge simultaneously.
Liflyand et al
Extending monotonicity
September, 2012
21 / 28
General monotonicity
It is also possible to get equiconvergence results for an important case
un = n, where the lacunarity can no more help.
We proceed to a smaller class than W M S. Indeed, assuming general
monotonicity of the sequences, we prove the following result.
Proposition. Let {ak } be a GMS. Then series (1) and
X
k|∆ak |
k
converge or diverge simultaneously.
A similar result for functions:
Liflyand et al
Extending monotonicity
September, 2012
21 / 28
General monotonicity
It is also possible to get equiconvergence results for an important case
un = n, where the lacunarity can no more help.
We proceed to a smaller class than W M S. Indeed, assuming general
monotonicity of the sequences, we prove the following result.
Proposition. Let {ak } be a GMS. Then series (1) and
X
k|∆ak |
k
converge or diverge simultaneously.
A similar result for functions:
Proposition.
Let
R∞
R ∞f be a GM function. Then the integrals
f
(t)
dt
and
1
1 t|df (t)| converge or diverge simultaneously.
Liflyand et al
Extending monotonicity
September, 2012
21 / 28
Negative type results
We now proceed to a group of tests where
Liflyand et al
Extending monotonicity
September, 2012
22 / 28
Negative type results
We now proceed to a group of tests where
extending monotonicity to weak monotonicity fails in that or another
sense.
Liflyand et al
Extending monotonicity
September, 2012
22 / 28
Negative type results
We now proceed to a group of tests where
extending monotonicity to weak monotonicity fails in that or another
sense.
Sapogov type test cannot be true if {bk } (as well as {1/bk }) is WMS.
Liflyand et al
Extending monotonicity
September, 2012
22 / 28
Negative type results
We now proceed to a group of tests where
extending monotonicity to weak monotonicity fails in that or another
sense.
Sapogov type test cannot be true if {bk } (as well as {1/bk }) is WMS.
A counterexample can be constructed against generalization of Abel’s
test.
Liflyand et al
Extending monotonicity
September, 2012
22 / 28
Negative type results
We now proceed to a group of tests where
extending monotonicity to weak monotonicity fails in that or another
sense.
Sapogov type test cannot be true if {bk } (as well as {1/bk }) is WMS.
A counterexample can be constructed against generalization of Abel’s
test.
As for extending the Leibniz test, it cannot hold without additional
assumption of the boundedness of variation:
Liflyand et al
Extending monotonicity
September, 2012
22 / 28
Negative type results
We now proceed to a group of tests where
extending monotonicity to weak monotonicity fails in that or another
sense.
Sapogov type test cannot be true if {bk } (as well as {1/bk }) is WMS.
A counterexample can be constructed against generalization of Abel’s
test.
As for extending the Leibniz test, it cannot hold without additional
assumption of the boundedness of variation:
just take ak = 1/ ln k everywhere except n = 2k where an = 2/ ln n.
Liflyand et al
Extending monotonicity
September, 2012
22 / 28
Wider classes
We shall show that WMS is, in a sense, the widest class for which such
tests are still valid.
Liflyand et al
Extending monotonicity
September, 2012
23 / 28
Wider classes
We shall show that WMS is, in a sense, the widest class for which such
tests are still valid.
An immediate natural extension of W M S is the class defined by
ak ≤ C
k
X
an
,
n
n=k/2
Liflyand et al
Extending monotonicity
September, 2012
23 / 28
Wider classes
We shall show that WMS is, in a sense, the widest class for which such
tests are still valid.
An immediate natural extension of W M S is the class defined by
ak ≤ C
k
X
an
,
n
n=k/2
or a bit more general
[ck]
X
an
ak ≤ C
n
n=[k/c]
for some c > 1.
Liflyand et al
Extending monotonicity
September, 2012
23 / 28
Wider classes
The principle difference between W M S and these classes is that the latter
two allow certain amount of zero members, unlike W M S that forbid even
a single zero, i.e., an0 = 0 implies an = 0 for n ≥ n0 .
Liflyand et al
Extending monotonicity
September, 2012
24 / 28
Wider classes
The principle difference between W M S and these classes is that the latter
two allow certain amount of zero members, unlike W M S that forbid even
a single zero, i.e., an0 = 0 implies an = 0 for n ≥ n0 .
Putting zeros on certain positions, say k = 2n , we easily construct a
counterexample to show that the Cauchy condensation test cannot be
valid for these classes nor its extensions.
Liflyand et al
Extending monotonicity
September, 2012
24 / 28
Wider classes
The principle difference between W M S and these classes is that the latter
two allow certain amount of zero members, unlike W M S that forbid even
a single zero, i.e., an0 = 0 implies an = 0 for n ≥ n0 .
Putting zeros on certain positions, say k = 2n , we easily construct a
counterexample to show that the Cauchy condensation test cannot be
valid for these classes nor its extensions.
In a similar way, just letting a function f to take non-zero values only
close to integer points, one sees that the Maclaurin-Cauchy integral
test may fail as well.
Liflyand et al
Extending monotonicity
September, 2012
24 / 28
Wider classes
The principle difference between W M S and these classes is that the latter
two allow certain amount of zero members, unlike W M S that forbid even
a single zero, i.e., an0 = 0 implies an = 0 for n ≥ n0 .
Putting zeros on certain positions, say k = 2n , we easily construct a
counterexample to show that the Cauchy condensation test cannot be
valid for these classes nor its extensions.
In a similar way, just letting a function f to take non-zero values only
close to integer points, one sees that the Maclaurin-Cauchy integral
test may fail as well.
In conclusion, note that W M S is a subclass of the broadly used
∆2 -class,
Liflyand et al
Extending monotonicity
September, 2012
24 / 28
Wider classes
that is, the one
Liflyand et al
Extending monotonicity
September, 2012
25 / 28
Wider classes
that is, the one
for which the doubling condition a2k ≤ Cak holds for each k ∈ N.
Liflyand et al
Extending monotonicity
September, 2012
25 / 28
Wider classes
that is, the one
for which the doubling condition a2k ≤ Cak holds for each k ∈ N.
We observe that assuming the doubling condition by no means can
guarantee the above tests to be extended.
Liflyand et al
Extending monotonicity
September, 2012
25 / 28
Wider classes
that is, the one
for which the doubling condition a2k ≤ Cak holds for each k ∈ N.
We observe that assuming the doubling condition by no means can
guarantee the above tests to be extended.
To illustrate this, the easiest way is to consider Cauchy’s
condensation test.
Liflyand et al
Extending monotonicity
September, 2012
25 / 28
Wider classes
that is, the one
for which the doubling condition a2k ≤ Cak holds for each k ∈ N.
We observe that assuming the doubling condition by no means can
guarantee the above tests to be extended.
To illustrate this, the easiest way is to consider Cauchy’s
condensation test.
Take {ak } such that
−k
2 ,
k 6= 2n ;
ak =
n−2 ,
k = 2n .
Liflyand et al
Extending monotonicity
September, 2012
25 / 28
Wider classes
that is, the one
for which the doubling condition a2k ≤ Cak holds for each k ∈ N.
We observe that assuming the doubling condition by no means can
guarantee the above tests to be extended.
To illustrate this, the easiest way is to consider Cauchy’s
condensation test.
Take {ak } such that
−k
2 ,
k 6= 2n ;
ak =
n−2 ,
k = 2n .
This sequence is doubling but not W M S.
Liflyand et al
Extending monotonicity
September, 2012
25 / 28
Wider classes
that is, the one
for which the doubling condition a2k ≤ Cak holds for each k ∈ N.
We observe that assuming the doubling condition by no means can
guarantee the above tests to be extended.
To illustrate this, the easiest way is to consider Cauchy’s
condensation test.
Take {ak } such that
−k
2 ,
k 6= 2n ;
ak =
n−2 ,
k = 2n .
This sequence is doubling but not W M S.
Obviously,
∞
X
n=1
n
2 a
2n
=
∞
X
2n
n=1
n2
diverges
Liflyand et al
Extending monotonicity
September, 2012
25 / 28
Overview
The main results may be summarized as the following statement.
Liflyand et al
Extending monotonicity
September, 2012
26 / 28
Overview
The main results may be summarized as the following statement.
Theorem. Let f (·) ∈ W M . Then the following series and integrals
converge or diverge simultaneously:
Liflyand et al
Extending monotonicity
September, 2012
26 / 28
Overview
The main results may be summarized as the following statement.
Theorem. Let f (·) ∈ W M . Then the following series and integrals
converge or diverge simultaneously:
Z
∞
f (t) dt;
1
Liflyand et al
X
f (k);
k
Extending monotonicity
September, 2012
26 / 28
Overview
The main results may be summarized as the following statement.
Theorem. Let f (·) ∈ W M . Then the following series and integrals
converge or diverge simultaneously:
Z
∞
X
f (t) dt;
1
X
(uk+1 − uk )f (uk ),
f (k);
k
provided
uk ↑, uk+1 = O(uk );
Extending monotonicity
September, 2012
k
Liflyand et al
26 / 28
Overview
The main results may be summarized as the following statement.
Theorem. Let f (·) ∈ W M . Then the following series and integrals
converge or diverge simultaneously:
∞
Z
X
f (t) dt;
1
X
(uk+1 − uk )f (uk ),
f (k);
k
provided
uk ↑, uk+1 = O(uk );
k
X
uk f (uk ),
k
X
uk |f (uk+1 ) − f (uk )|, with uk lacunary, uk+1 = O(uk );
k
Liflyand et al
Extending monotonicity
September, 2012
26 / 28
Overview
The main results may be summarized as the following statement.
Theorem. Let f (·) ∈ W M . Then the following series and integrals
converge or diverge simultaneously:
∞
Z
X
f (t) dt;
1
X
f (k);
k
(uk+1 − uk )f (uk ),
provided
uk ↑, uk+1 = O(uk );
k
X
uk f (uk ),
k
X
uk |f (uk+1 ) − f (uk )|, with uk lacunary, uk+1 = O(uk );
k
X
Z
k|f (k + 1) − f (k)|,
∞
t |df (t)|,
provided
f (·) is GM .
1
k
Liflyand et al
Extending monotonicity
September, 2012
26 / 28
Example
Besov space:
Liflyand et al
Extending monotonicity
September, 2012
27 / 28
Example
Besov space:
For k > r, θ, r > 0 and p ≥ 1
r
kf kBp,θ
 1
1/θ
Z
dt
=  t−rθ−1 ωkθ (f ; t)p  .
t
0
Liflyand et al
Extending monotonicity
September, 2012
27 / 28
Example
Besov space:
For k > r, θ, r > 0 and p ≥ 1
r
kf kBp,θ
 1
1/θ
Z
dt
=  t−rθ−1 ωkθ (f ; t)p  .
t
0
Equivalent form:

Z∞
r
kf kBp,θ
=
1/θ
dt
trθ+1 ωkθ (f ; 1/t)p  .
t
1
Liflyand et al
Extending monotonicity
September, 2012
27 / 28
Example
Since ωk (f ; t)p ↑, the function under the integral sign is W M .
Liflyand et al
Extending monotonicity
September, 2012
28 / 28
Example
Since ωk (f ; t)p ↑, the function under the integral sign is W M .
Applying generalized Maclaurin-Cauchy and Cauchy condensation
tests, we obtain
Liflyand et al
Extending monotonicity
September, 2012
28 / 28
Example
Since ωk (f ; t)p ↑, the function under the integral sign is W M .
Applying generalized Maclaurin-Cauchy and Cauchy condensation
tests, we obtain
convenient equivalent form:
∞
X
!1/θ
2rν ωk (f ; 2−ν )p
θ
.
ν=0
Liflyand et al
Extending monotonicity
September, 2012
28 / 28