Extending Tests for Convergence of Number Series

Extending tests for convergence of number series

Elijah Liﬂyand, Sergey Tikhonov, and Maria Zeltser

September, 2012

Liﬂyand et al Extending monotonicity September, 2012 1 / 28 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, Schl¨omilch are related to monotonicity.

Goals and History

Main Goal: Relax the monotonicity assumption for the sequence of terms of the series.

Liﬂyand et al Extending monotonicity September, 2012 2 / 28 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, Schl¨omilch are related to monotonicity.

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 Tests by Abel, Cauchy, de la Vallee Poussin, Dedekind, Dirichlet, du Bois Reymond, Ermakov, Leibniz, Maclaurin, Olivier, Sapogov, Schl¨omilch are related to monotonicity.

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

Liflyand et al Extending monotonicity September, 2012 2 / 28 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.

References

Books:

Liflyand et al Extending monotonicity September, 2012 3 / 28 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.

References

Books: T.J.I’a Bromwich, Introduction to the Theory of Inﬁnite Series, MacMillan, New York, 1965.

Liflyand et al Extending monotonicity September, 2012 3 / 28 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.

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.

Liﬂyand et al Extending monotonicity September, 2012 3 / 28 D.D. Bonar and M.J. Khoury, Real Inﬁnite Series, MAA, Washington, DC, 2006.

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.

Liﬂyand et al Extending monotonicity September, 2012 3 / 28 References

Liﬂyand et al Extending monotonicity September, 2012 3 / 28 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.

Liﬂyand et al Extending monotonicity September, 2012 4 / 28 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.

Tests

Ermakov’s test.

Liﬂyand et al Extending monotonicity September, 2012 5 / 28 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.

Tests

Ermakov’s test. In its simplest form, it is given as follows.

Liﬂyand et al Extending monotonicity September, 2012 5 / 28 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.

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.

Liﬂyand et al Extending monotonicity September, 2012 5 / 28 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.

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 September, 2012 5 / 28 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.

Tests

Ermakov’s test. In its simplest form, it is given as follows.

Liﬂyand et al Extending monotonicity September, 2012 5 / 28 We will extend such a more general assertion.

Tests

Ermakov’s test. In its simplest form, it is given as follows.

Liﬂyand et al Extending monotonicity September, 2012 5 / 28 Tests

Ermakov’s test. In its simplest form, it is given as follows.

Liﬂyand et al Extending monotonicity September, 2012 5 / 28 If {ak} is a positive monotone decreasing null sequence, then the series (1) and

∞ X k 2 a2k k=1 converge or diverge simultaneously.

This assertion is a partial case of the following classical result due to Schl¨omilch. Let (1) be a series whose terms are positive and non-increasing,

Tests

The well known Cauchy condensation test states that

Liﬂyand et al Extending monotonicity September, 2012 6 / 28 This assertion is a partial case of the following classical result due to Schl¨omilch. Let (1) be a series whose terms are positive and non-increasing,

Tests

The well known Cauchy condensation test states that

If {ak} is a positive monotone decreasing null sequence, then the series (1) and

∞ X k 2 a2k k=1 converge or diverge simultaneously.

Liﬂyand et al Extending monotonicity September, 2012 6 / 28 Let (1) be a series whose terms are positive and non-increasing,

Tests

The well known Cauchy condensation test states that

If {ak} is a positive monotone decreasing null sequence, then the series (1) and

∞ X k 2 a2k k=1 converge or diverge simultaneously.

This assertion is a partial case of the following classical result due to Schl¨omilch.

Liﬂyand et al Extending monotonicity September, 2012 6 / 28 Tests

The well known Cauchy condensation test states that

If {ak} is a positive monotone decreasing null sequence, then the series (1) and

∞ X k 2 a2k k=1 converge or diverge simultaneously.

This assertion is a partial case of the following classical result due to Schl¨omilch. Let (1) be a series whose terms are positive and non-increasing,

Liﬂyand et al Extending monotonicity September, 2012 6 / 28 u0 < u1 < u2 < ... be a sequence of positive integers such that

∆u k ≤ C. ∆uk−1 Then series (1) converges if and only if the series

∞ ∞ X X ∆ukauk = (uk+1 − uk)auk k=1 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.

Tests

and let

Liﬂyand et al Extending monotonicity September, 2012 7 / 28 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.

Tests

and let

u0 < u1 < u2 < ... be a sequence of positive integers such that

∆u k ≤ C. ∆uk−1 Then series (1) converges if and only if the series

∞ ∞ X X ∆ukauk = (uk+1 − uk)auk k=1 k=1 converges.

Liﬂyand et al Extending monotonicity September, 2012 7 / 28 Let {ak} be a positive monotone null sequence. If series (1) is convergent, then kak is a null sequence.

Tests

and let

u0 < u1 < u2 < ... be a sequence of positive integers such that

∆u k ≤ C. ∆uk−1 Then series (1) converges if and only if the series

∞ ∞ X X ∆ukauk = (uk+1 − uk)auk k=1 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:

Liﬂyand et al Extending monotonicity September, 2012 7 / 28 Tests

and let

u0 < u1 < u2 < ... be a sequence of positive integers such that

∆u k ≤ C. ∆uk−1 Then series (1) converges if and only if the series

∞ ∞ X X ∆ukauk = (uk+1 − uk)auk k=1 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.

Liﬂyand et al Extending monotonicity September, 2012 7 / 28 Tests

Sapogov’s test reads as follows.

If {bk} is a positive monotone increasing sequence, then the series

∞ X bk (1 − ) bk+1 k=1 as well as

∞ X bk+1 ( − 1) bk k=1

converges if the sequence {bk} is bounded and diverges otherwise.

Liﬂyand et al Extending monotonicity September, 2012 8 / 28 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 sums of bk is bounded, then the series akbk is convergent. k k=1 ∞ P P (ii) If {ak} ∈ BV and bk is convergent, then the series akbk is k k=1 convergent. Two well-known and widely used corollaries of this test - Dirichlet’s and Abel’s tests - involve monotone sequences.

Tests

The tests of Dedekind and of du Bois Reymond are united as the next assertion.

Liﬂyand et al Extending monotonicity September, 2012 9 / 28 Let {ak} and {bk} be two sequences.

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| < ∞.

Liﬂyand et al Extending monotonicity September, 2012 9 / 28 Two well-known and widely used corollaries of this test - Dirichlet’s and Abel’s tests - involve monotone sequences.

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.

Liﬂyand et al Extending monotonicity September, 2012 9 / 28 Dirichlet’s and Abel’s tests - involve monotone sequences.

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.

Liﬂyand et al Extending monotonicity September, 2012 9 / 28 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.

Liﬂyand et al Extending monotonicity September, 2012 9 / 28 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 akbk is convergent. k=1 One of corollaries of this test is the celebrated Leibniz test: ∞ P k Let {ak} be a monotone null sequence. Then the series (−1) ak k=1 is convergent. Further, Abel’s test reads as follows. P Let {ak} be a bounded monotone sequence and bk a convergent k ∞ P series. Then the series akbk is convergent. k=1

Tests

The ﬁrst one is as follows.

Liﬂyand et al Extending monotonicity September, 2012 10 / 28 One of corollaries of this test is the celebrated Leibniz test: ∞ P k Let {ak} be a monotone null sequence. Then the series (−1) ak k=1 is convergent. Further, Abel’s test reads as follows. P Let {ak} be a bounded monotone sequence and bk a convergent k ∞ P series. Then the series akbk is convergent. k=1

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 akbk is convergent. k=1

Liﬂyand et al Extending monotonicity September, 2012 10 / 28 Further, Abel’s test reads as follows. P Let {ak} be a bounded monotone sequence and bk a convergent k ∞ P series. Then the series akbk is convergent. k=1

Tests

The ﬁrst one is as follows.

Liﬂyand et al Extending monotonicity September, 2012 10 / 28 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 akbk is convergent. k=1 One of corollaries of this test is the celebrated Leibniz test: ∞ P k Let {ak} be a monotone null sequence. Then the series (−1) ak k=1 is convergent. Further, Abel’s test reads as follows. P Let {ak} be a bounded monotone sequence and bk a convergent k ∞ P series. Then the series akbk is convergent. k=1

Liflyand et al Extending monotonicity September, 2012 10 / 28 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.

Classes of sequences and functions

Let us introduce certain classes of functions and sequences more general than monotone.

Liflyand et al Extending monotonicity September, 2012 11 / 28 To introduce a counterpart for functions, we will assume all functions to be defined on (0, ∞), locally of bounded variation, and vanishing at infinity.

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].

Liﬂyand et al Extending monotonicity September, 2012 11 / 28 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.

Liﬂyand et al Extending monotonicity September, 2012 11 / 28 Setting ak = f(k) in this case, we obtain {ak} ∈ W MS. 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 WMS.

Classes of sequences and functions

Deﬁnition We say that a non-negative function f deﬁned on (0, ∞), is weak monotone, writtenWM, if

f(t) ≤ Cf(x) for any t ∈ [x, 2x].

Liﬂyand et al Extending monotonicity September, 2012 12 / 28 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 WMS.

Classes of sequences and functions

Deﬁnition We say that a non-negative function f deﬁned on (0, ∞), is weak monotone, writtenWM, if

f(t) ≤ Cf(x) for any t ∈ [x, 2x].

Setting ak = f(k) in this case, we obtain {ak} ∈ W MS.

Liﬂyand et al Extending monotonicity September, 2012 12 / 28 In some problems one should consider a smaller class than WMS.

Classes of sequences and functions

Deﬁnition We say that a non-negative function f deﬁned on (0, ∞), is weak monotone, writtenWM, if

f(t) ≤ Cf(x) for any t ∈ [x, 2x].

Setting ak = f(k) in this case, we obtain {ak} ∈ W MS. 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.

Liﬂyand et al Extending monotonicity September, 2012 12 / 28 Classes of sequences and functions

Deﬁnition We say that a non-negative function f deﬁned on (0, ∞), is weak monotone, writtenWM, if

f(t) ≤ Cf(x) for any t ∈ [x, 2x].

Liflyand et al Extending monotonicity September, 2012 12 / 28 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, −τ that is, {ak} such that there exists τ > 0 so that k ak ↓, is a proper subclass of GMS. One of the simple basic properties of GMS is GMS ( WMS.

Classes of sequences and functions

Denote ∆ak = ak+1 − ak.

Liﬂyand et al Extending monotonicity September, 2012 13 / 28 The class of quasi-monotone sequences, −τ that is, {ak} such that there exists τ > 0 so that k ak ↓, is a proper subclass of GMS. One of the simple basic properties of GMS is GMS ( WMS.

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.

Liﬂyand et al Extending monotonicity September, 2012 13 / 28 −τ that is, {ak} such that there exists τ > 0 so that k ak ↓, is a proper subclass of GMS. One of the simple basic properties of GMS is GMS ( WMS.

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,

Liﬂyand et al Extending monotonicity September, 2012 13 / 28 is a proper subclass of GMS. One of the simple basic properties of GMS is GMS ( WMS.

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, −τ that is, {ak} such that there exists τ > 0 so that k ak ↓,

Liﬂyand et al Extending monotonicity September, 2012 13 / 28 One of the simple basic properties of GMS is GMS ( WMS.

Classes of sequences and functions

Denote ∆ak = ak+1 − ak.

Deﬁnition. A positive null sequence {ak} is called general monotone if it satisﬁes

Liﬂyand et al Extending monotonicity September, 2012 13 / 28 Classes of sequences and functions

Denote ∆ak = ak+1 − ak.

Deﬁnition. A positive null sequence {ak} is called general monotone if it satisﬁes

Liﬂyand et al Extending monotonicity September, 2012 13 / 28 We just remark that if f(·) is a GM function,

then for ak = f(k) there holds {ak} ∈ GMS. And of course GM ( WM.

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

Liﬂyand et al Extending monotonicity September, 2012 14 / 28 then for ak = f(k) there holds {ak} ∈ GMS. And of course GM ( WM.

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,

Liﬂyand et al Extending monotonicity September, 2012 14 / 28 And of course GM ( WM.

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} ∈ GMS.

Liﬂyand et al Extending monotonicity September, 2012 14 / 28 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} ∈ GMS. And of course GM ( WM.

Liﬂyand et al Extending monotonicity September, 2012 14 / 28 Extension of ”monotone” tests

An extension of the Maclaurin-Cauchy integral test reads as follows: Theorem Let f be aWM function. Then the series

∞ X f(k) k=1 and integral Z ∞ f(t) dt 1 converge or diverge simultaneously.

Liﬂyand et al Extending monotonicity September, 2012 15 / 28 Extension of ”monotone” tests

An extension of Ermakov’s test: Theorem Let f be aWM 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.

Liﬂyand et al Extending monotonicity September, 2012 16 / 28 Let f be aWM function. Then both series

∞ ∞ X X f(uk)∆uk and f(uk+1)∆uk k=1 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 X ∆ukauk = (uk+1 − uk)auk k=1 k=1 converges.

Extension of ”monotone” tests

Theorem

Let {uk} be an increasing sequence of positive numbers such that uk+1 = O(uk) and uk → ∞.

Liﬂyand et al Extending monotonicity September, 2012 17 / 28 Let {ak} be a WMS. Then (1) converges if and only if the series

∞ ∞ X X ∆ukauk = (uk+1 − uk)auk k=1 k=1 converges.

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 aWM function. Then both series

∞ ∞ X X f(uk)∆uk and f(uk+1)∆uk k=1 k=1 R ∞ converge (or diverge) with 1 f(t) dt.

Liﬂyand et al Extending monotonicity September, 2012 17 / 28 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 aWM function. Then both series

∞ ∞ X X f(uk)∆uk and f(uk+1)∆uk k=1 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 X ∆ukauk = (uk+1 − uk)auk k=1 k=1 converges.

Liﬂyand et al Extending monotonicity September, 2012 17 / 28 Analyzing one Dvoretzky’s result and its proof, we obtain the following statement. Theorem

If {ak}, {bk} ∈ W MS, 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 ≤ MRj, we have ak < bk.

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.

Liﬂyand et al Extending monotonicity September, 2012 18 / 28 we obtain the following statement. Theorem

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,

Liﬂyand et al Extending monotonicity September, 2012 18 / 28 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

Liﬂyand et al Extending monotonicity September, 2012 18 / 28 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, k ∈ N. uk

Extension of ”monotone” tests

Our next result is ”dual” to the above Schl¨omilch-type extensions.

Liﬂyand et al Extending monotonicity September, 2012 19 / 28 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, k ∈ N. uk

Extension of ”monotone” tests

Our next result is ”dual” to the above Schl¨omilch-type extensions.

We recall that the increasing sequence {uk} is called lacunary if uk+1/uk ≥ q > 1.

Liﬂyand et al Extending monotonicity September, 2012 19 / 28 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, k ∈ N. uk

Extension of ”monotone” tests

Our next result is ”dual” to the above Schl¨omilch-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.

Liﬂyand et al Extending monotonicity September, 2012 19 / 28 In diﬀerent terms, {uk} ∈ Λ is true if and only if there exists r ∈ N such that

uk+r ≥ q > 1, k ∈ N. uk

Extension of ”monotone” tests

Our next result is ”dual” to the above Schl¨omilch-type extensions.

k X uj ≤ Cuk. j=1

Liﬂyand et al Extending monotonicity September, 2012 19 / 28 Extension of ”monotone” tests

Our next result is ”dual” to the above Schl¨omilch-type extensions.

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, k ∈ N. uk

Liﬂyand et al Extending monotonicity September, 2012 19 / 28 Extension of ”monotone” tests

¯ We denote ∆auk := auk − auk+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 ¯ uk|∆auk |, k=1 and ∞ X ukauk k=1 converge or diverge simultaneously.

Liﬂyand et al Extending monotonicity September, 2012 20 / 28 We proceed to a smaller class than W MS. 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 f be a GM function. Then the integrals R ∞ R ∞ 1 f(t) dt and 1 t|df(t)| converge or diverge simultaneously.

General monotonicity

It is also possible to get equiconvergence results for an important case un = n, where the lacunarity can no more help.

Liﬂyand et al Extending monotonicity September, 2012 21 / 28 Proposition. Let {ak} be a GMS. Then series (1) and

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 MS. Indeed, assuming general monotonicity of the sequences, we prove the following result.

Liﬂyand et al Extending monotonicity September, 2012 21 / 28 A similar result for functions: Proposition. Let f be a GM function. Then the integrals R ∞ R ∞ 1 f(t) dt and 1 t|df(t)| converge or diverge simultaneously.

General monotonicity

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 Proposition. Let f be a GM function. Then the integrals R ∞ R ∞ 1 f(t) dt and 1 t|df(t)| converge or diverge simultaneously.

General monotonicity

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

Proposition. Let {ak} be a GMS. Then series (1) and

Liﬂyand et al Extending monotonicity September, 2012 21 / 28 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: k just take ak = 1/ ln k everywhere except n = 2 where an = 2/ ln n.

Negative type results

We now proceed to a group of tests where

Liﬂyand et al Extending monotonicity September, 2012 22 / 28 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: k just take ak = 1/ ln k everywhere except n = 2 where an = 2/ ln n.

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 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: k just take ak = 1/ ln k everywhere except n = 2 where an = 2/ ln n.

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 As for extending the Leibniz test, it cannot hold without additional assumption of the boundedness of variation: k just take ak = 1/ ln k everywhere except n = 2 where an = 2/ ln n.

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 k just take ak = 1/ ln k everywhere except n = 2 where an = 2/ ln n.

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.

Liﬂyand et al Extending monotonicity September, 2012 22 / 28 An immediate natural extension of WMS is the class deﬁned by

k X an a ≤ C , k n n=k/2

or a bit more general

[ck] X an a ≤ C k n n=[k/c]

for some c > 1.

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 or a bit more general

[ck] X an a ≤ C k n n=[k/c]

for some c > 1.

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 WMS is the class deﬁned by

k X an a ≤ C , k 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 WMS is the class deﬁned by

k X an a ≤ C , k n n=k/2

or a bit more general

[ck] X an a ≤ C k n n=[k/c]

for some c > 1.

Liﬂyand et al Extending monotonicity September, 2012 23 / 28 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 WMS is a subclass of the broadly used ∆2-class,

Wider classes

The principle diﬀerence between WMS and these classes is that the latter two allow certain amount of zero members, unlike WMS 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 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 WMS is a subclass of the broadly used ∆2-class,

Wider classes

The principle diﬀerence between WMS and these classes is that the latter two allow certain amount of zero members, unlike WMS that forbid even

Liﬂyand et al Extending monotonicity September, 2012 24 / 28 In conclusion, note that WMS is a subclass of the broadly used ∆2-class,

Wider classes

The principle diﬀerence between WMS and these classes is that the latter two allow certain amount of zero members, unlike WMS 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 WMS and these classes is that the latter two allow certain amount of zero members, unlike WMS that forbid even

Liﬂyand et al Extending monotonicity September, 2012 24 / 28 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

2−k, k 6= 2n; a = k n−2, k = 2n. This sequence is doubling but not W MS. Obviously,

∞ ∞ n X n X 2 2 a n = 2 n2 n=1 n=1 diverges

Wider classes

that is, the one

Liﬂyand et al Extending monotonicity September, 2012 25 / 28 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

2−k, k 6= 2n; a = k n−2, k = 2n. This sequence is doubling but not W MS. Obviously,

∞ ∞ n X n X 2 2 a n = 2 n2 n=1 n=1 diverges

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 To illustrate this, the easiest way is to consider Cauchy’s condensation test. Take {ak} such that

2−k, k 6= 2n; a = k n−2, k = 2n. This sequence is doubling but not W MS. Obviously,

∞ ∞ n X n X 2 2 a n = 2 n2 n=1 n=1 diverges

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 Take {ak} such that

2−k, k 6= 2n; a = k n−2, k = 2n. This sequence is doubling but not W MS. Obviously,

∞ ∞ n X n X 2 2 a n = 2 n2 n=1 n=1 diverges

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 This sequence is doubling but not W MS. Obviously,

∞ ∞ n X n X 2 2 a n = 2 n2 n=1 n=1 diverges

Wider classes

that is, the one

2−k, k 6= 2n; a = k n−2, k = 2n.

Liﬂyand et al Extending monotonicity September, 2012 25 / 28 Obviously,

∞ ∞ n X n X 2 2 a n = 2 n2 n=1 n=1 diverges

Wider classes

that is, the one

2−k, k 6= 2n; a = k n−2, k = 2n. This sequence is doubling but not W MS.

Liﬂyand et al Extending monotonicity September, 2012 25 / 28 Wider classes

that is, the one

2−k, k 6= 2n; a = k n−2, k = 2n. This sequence is doubling but not W MS. Obviously,

∞ ∞ n X n X 2 2 a n = 2 n2 n=1 n=1 diverges Liﬂyand et al Extending monotonicity September, 2012 25 / 28 Theorem. Let f(·) ∈ WM. Then the following series and integrals converge or diverge simultaneously:

Z ∞ X f(t) dt; f(k); 1 k

X (uk+1 − uk)f(uk), provided uk ↑, uk+1 = O(uk); k

X X ukf(uk), uk|f(uk+1) − f(uk)|, with uk lacunary, uk+1 = O(uk); k k

X Z ∞ k|f(k + 1) − f(k)|, t |df(t)|, provided f(·) is GM. k 1

Overview

The main results may be summarized as the following statement.

Liﬂyand et al Extending monotonicity September, 2012 26 / 28 Z ∞ X f(t) dt; f(k); 1 k

X (uk+1 − uk)f(uk), provided uk ↑, uk+1 = O(uk); k

X X ukf(uk), uk|f(uk+1) − f(uk)|, with uk lacunary, uk+1 = O(uk); k k

X Z ∞ k|f(k + 1) − f(k)|, t |df(t)|, provided f(·) is GM. k 1

Overview

The main results may be summarized as the following statement. Theorem. Let f(·) ∈ WM. Then the following series and integrals converge or diverge simultaneously:

Liﬂyand et al Extending monotonicity September, 2012 26 / 28 X (uk+1 − uk)f(uk), provided uk ↑, uk+1 = O(uk); k

X X ukf(uk), uk|f(uk+1) − f(uk)|, with uk lacunary, uk+1 = O(uk); k k

X Z ∞ k|f(k + 1) − f(k)|, t |df(t)|, provided f(·) is GM. k 1

Overview

The main results may be summarized as the following statement. Theorem. Let f(·) ∈ WM. Then the following series and integrals converge or diverge simultaneously:

Z ∞ X f(t) dt; f(k); 1 k

Liﬂyand et al Extending monotonicity September, 2012 26 / 28 X X ukf(uk), uk|f(uk+1) − f(uk)|, with uk lacunary, uk+1 = O(uk); k k

X Z ∞ k|f(k + 1) − f(k)|, t |df(t)|, provided f(·) is GM. k 1

Overview

The main results may be summarized as the following statement. Theorem. Let f(·) ∈ WM. Then the following series and integrals converge or diverge simultaneously:

Z ∞ X f(t) dt; f(k); 1 k

X (uk+1 − uk)f(uk), provided uk ↑, uk+1 = O(uk); k

Liﬂyand et al Extending monotonicity September, 2012 26 / 28 X Z ∞ k|f(k + 1) − f(k)|, t |df(t)|, provided f(·) is GM. k 1

Overview

The main results may be summarized as the following statement. Theorem. Let f(·) ∈ WM. Then the following series and integrals converge or diverge simultaneously:

Z ∞ X f(t) dt; f(k); 1 k

X (uk+1 − uk)f(uk), provided uk ↑, uk+1 = O(uk); k

X X ukf(uk), uk|f(uk+1) − f(uk)|, with uk lacunary, uk+1 = O(uk); k k

Liﬂyand et al Extending monotonicity September, 2012 26 / 28 Overview

The main results may be summarized as the following statement. Theorem. Let f(·) ∈ WM. Then the following series and integrals converge or diverge simultaneously:

Z ∞ X f(t) dt; f(k); 1 k

X (uk+1 − uk)f(uk), provided uk ↑, uk+1 = O(uk); k

X X ukf(uk), uk|f(uk+1) − f(uk)|, with uk lacunary, uk+1 = O(uk); k k

X Z ∞ k|f(k + 1) − f(k)|, t |df(t)|, provided f(·) is GM. k 1

Liﬂyand et al Extending monotonicity September, 2012 26 / 28 For k > r, θ, r > 0 and p ≥ 1

 1 1/θ Z −rθ−1 θ dt kfkBr =  t ω (f; t)p  . p,θ k t 0 Equivalent form:

 ∞ 1/θ Z rθ+1 θ dt kfkBr =  t ω (f; 1/t)p  . p,θ k t 1

Example

Besov space:

Liﬂyand et al Extending monotonicity September, 2012 27 / 28 Equivalent form:

 ∞ 1/θ Z rθ+1 θ dt kfkBr =  t ω (f; 1/t)p  . p,θ k t 1

Example

Besov space: For k > r, θ, r > 0 and p ≥ 1

 1 1/θ Z −rθ−1 θ dt kfkBr =  t ω (f; t)p  . p,θ k t 0

Liﬂyand et al Extending monotonicity September, 2012 27 / 28 Example

Besov space: For k > r, θ, r > 0 and p ≥ 1

 1 1/θ Z −rθ−1 θ dt kfkBr =  t ω (f; t)p  . p,θ k t 0 Equivalent form:

 ∞ 1/θ Z rθ+1 θ dt kfkBr =  t ω (f; 1/t)p  . p,θ k t 1

Liﬂyand et al Extending monotonicity September, 2012 27 / 28 Applying generalized Maclaurin-Cauchy and Cauchy condensation tests, we obtain convenient equivalent form:

∞ !1/θ X rν −ν θ 2 ωk(f; 2 )p . ν=0

Example

Since ωk(f; t)p ↑, the function under the integral sign is WM.

Liﬂyand et al Extending monotonicity September, 2012 28 / 28 convenient equivalent form:

∞ !1/θ X rν −ν θ 2 ωk(f; 2 )p . ν=0

Example

Since ωk(f; t)p ↑, the function under the integral sign is WM. 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 WM. Applying generalized Maclaurin-Cauchy and Cauchy condensation tests, we obtain convenient equivalent form:

∞ !1/θ X rν −ν θ 2 ωk(f; 2 )p . ν=0

Liﬂyand et al Extending monotonicity September, 2012 28 / 28