Preface
Contents
1 Integrals
1.1 A Beautiful Integral by the English Mathematician James Joseph Sylvester
1.2 Strange Limits with Trigonometric Integrals
1.3 Two Curious Logarithmic Integrals with Parameter
1.4 Exploring More Appealing Logarithmic Integrals with Parameter: The First Part
1.5 Exploring More Appealing Logarithmic Integrals with Parameter: The Second Part
1.6 More Good-Looking Logarithmic Integrals: The First Part
1.7 More Good-Looking Logarithmic Integrals: The Second Part
1.8 More Good-Looking Logarithmic Integrals: The Third Part
1.9 Special and Challenging Integrals with Parameter Involving the Inverse Hyperbolic Tangent: The First Part
1.10 Special and Challenging Integrals with Parameter Involving the Inverse Hyperbolic Tangent: The Second Part
1.11 Some Startling Generalizations Involving Logarithmic Integrals with Trigonometric Parameters
1.12 More Startling and Enjoyable Logarithmic Integrals Involving Trigonometric Parameters
1.13 Surprisingly Bewitching Trigonometric Integrals with Appealing Closed Forms: The First Act
1.14 Surprisingly Bewitching Trigonometric Integrals with Appealing Closed Forms: The Second Act
1.15 Surprisingly Bewitching Trigonometric Integrals with Appealing Closed Forms: The Third Act
1.16 Surprisingly Bewitching Trigonometric Integrals with Appealing Closed Forms: The Fourth Act
1.17 Playing with a Hand of Fabulous Integrals: The First Part
1.18 Playing with a Hand of Fabulous Integrals: The Second Part
1.19 Another Fabulous Integral Related to a Curious Binoharmonic Series Representation of ζ(3)
1.20 Pairs of Appealing Integrals with the Logarithm Function, the Dilogarithm Function, and Trigonometric Functions
1.21 Valuable Logarithmic Integrals Involving Skew-Harmonic Numbers: A First Partition
1.22 Valuable Logarithmic Integrals Involving Skew-Harmonic Numbers: A Second Partition
1.23 Valuable Logarithmic Integrals Involving Skew-Harmonic Numbers: A Third Partition
1.24 A Pair of Precious Generalized Logarithmic Integrals
1.25 Two Neat and Useful Generalizations with Logarithmic Integrals Involving Skew-Harmonic Numbers
1.26 A Special Logarithmic Integral and a Generalization of It
1.27 Three Useful Generalized Logarithmic Integrals Connected to Useful Generalized Alternating Harmonic Series
1.28 Three Atypical Logarithmic Integrals Involving log(1+x3), Related to a Special Form of Symmetry in Double Integrals
1.29 Four Curious Integrals with Cosine Function Which Lead to Beautiful Closed Forms, Plus Two Exotic Integrals
1.30 Aesthetic Integrals That Must Be Understood as Cauchy Principal Values, Generated by a Beautiful Generalization
1.31 A Special Integral Generalization, Attacked with Strategies Involving the Cauchy Principal Value Integrals, That Also Leads to Two Famous Results by Ramanujan
1.32 Four Magical Integrals Beautifully Calculated, Which Generate the Closed Forms of Four Classical Integrals
1.33 A Bouquet of Captivating Integrals Involving Trigonometric and Hyperbolic Functions
1.34 Interesting Integrals to Evaluate, One of Them Coming from Lord Kelvin's Work
1.35 A Surprisingly Awesome Fractional Part Integral with Forms Involving tan(x) and cot(x)
1.36 A Superb Integral with Logarithms and the Inverse Tangent Function and a Wonderful Generalization of It
1.37 More Wonderful Results Involving Integrals with Logarithms and the Inverse Tangent Function
1.38 Powerful and Useful Sums of Integrals with Logarithms and the Inverse Tangent Function
1.39 Two Beautiful Sums of Integrals, Each One Involving Three Integrals, Leading to a Possible Unexpected Result
1.40 Tackling Curious Logarithmic Integrals with a Radical in the Denominator
1.41 Calculating More Logarithmic Integrals with a Radical in the Denominator
1.42 Somewhat Atypical Integrals with Curious Closed Forms
1.43 More Atypical Integrals with Curious Closed Forms
1.44 A Wonderful Trigonometric Integral by Larry Glasser
1.45 Resistant Logarithmic Integrals That Are Good to Know
1.46 Appealing Parameterized Integrals with Logarithms and the Dilogarithm Function, Related to Harmonic Series
1.47 An Encounter with Six Useful Integrals Involving Logarithms and the Dilogarithm
1.48 A Battle with Three Challenging Integrals Involving Logarithms and the Dilogarithm
1.49 Fascinating Polylogarithmic Integrals with Parameter Involving the Cauchy Product of Two Series
1.50 A Titan Involving Alternating Harmonic Series of Weight 7
1.51 A Tough Integral Approached by Clever Transformations
1.52 An Unexpected Closed Form, Involving Catalan's Constant, of a Nice Integral with the Dilogarithm
1.53 A Group of Six Special, Challenging Generalized Integrals Involving Curious Closed Forms
1.54 Amazing and (Very) Useful Integral Beasts Involving log2(sin(x)), log2(cos(x)), log3(sin(x)), and log3(cos(x))
1.55 Four Challenging Integrals with the Logarithm and Trigonometric Functions, Giving Nice Closed Forms
1.56 Advanced Integrals with Trigonometric Functions, Related to Fourier Series and Harmonic Series
1.57 Gems of Integration Involving Splendid Ideas About Symmetry in Two Dimensions
1.58 More Gems of Integration, This Time Involving Splendid Ideas About Symmetry in Three Dimensions
1.59 The Complete Elliptic Integral of the First Kind at Play
1.60 Evaluating An Esoteric-Looking Integral Involving a Triple Series with Factorial Numbers, Leading to ζ(4)
2 Hints
2.1 A Beautiful Integral by the English Mathematician James Joseph Sylvester
2.2 Strange Limits with Trigonometric Integrals
2.3 Two Curious Logarithmic Integrals with Parameter
2.4 Exploring More Appealing Logarithmic Integrals with Parameter: The First Part
2.5 Exploring More Appealing Logarithmic Integrals with Parameter: The Second Part
2.6 More Good-Looking Logarithmic Integrals: The First Part
2.7 More Good-Looking Logarithmic Integrals: The Second Part
2.8 More Good-Looking Logarithmic Integrals: The Third Part
2.9 Special and Challenging Integrals with Parameter Involving the Inverse Hyperbolic Tangent: The First Part
2.10 Special and Challenging Integrals with Parameter Involving the Inverse Hyperbolic Tangent: The Second Part
2.11 Some Startling Generalizations Involving Logarithmic Integrals with Trigonometric Parameters
2.12 More Startling and Enjoyable Logarithmic Integrals Involving Trigonometric Parameters
2.13 Surprisingly Bewitching Trigonometric Integrals with Appealing Closed Forms: The First Act
2.14 Surprisingly Bewitching Trigonometric Integrals with Appealing Closed Forms: The Second Act
2.15 Surprisingly Bewitching Trigonometric Integrals with Appealing Closed Forms: The Third Act
2.16 Surprisingly Bewitching Trigonometric Integrals with Appealing Closed Forms: The Fourth Act
2.17 Playing with a Hand of Fabulous Integrals: The First Part
2.18 Playing with a Hand of Fabulous Integrals: The Second Part
2.19 Another Fabulous Integral Related to a Curious Binoharmonic Series Representation of ζ(3)
2.20 Pairs of Appealing Integrals with the Logarithm Function, the Dilogarithm Function, and Trigonometric Functions
2.21 Valuable Logarithmic Integrals Involving Skew-Harmonic Numbers: A First Partition
2.22 Valuable Logarithmic Integrals Involving Skew-Harmonic Numbers: A Second Partition
2.23 Valuable Logarithmic Integrals Involving Skew-Harmonic Numbers: A Third Partition
2.24 A Pair of Precious Generalized Logarithmic Integrals
2.25 Two Neat and Useful Generalizations with Logarithmic Integrals Involving Skew-Harmonic Numbers
2.26 A Special Logarithmic Integral and a Generalization of It
2.27 Three Useful Generalized Logarithmic Integrals Connected to Useful Generalized Alternating Harmonic Series
2.28 Three Atypical Logarithmic Integrals Involving log(1+x3), Related to a Special Form of Symmetry in Double Integrals
2.29 Four Curious Integrals with Cosine Function Which Lead to Beautiful Closed Forms, plus Two Exotic Integrals
2.30 Aesthetic Integrals That Must Be Understood as Cauchy Principal Values, Generated by a Beautiful Generalization
2.31 A Special Integral Generalization, Attacked with Strategies Involving the Cauchy Principal Value Integrals, That Also Leads to Two Famous Results by Ramanujan
2.32 Four Magical Integrals Beautifully Calculated, Which Generate the Closed Forms of Four Classical Integrals
2.33 A Bouquet of Captivating Integrals Involving Trigonometric and Hyperbolic Functions
2.34 Interesting Integrals to Evaluate, One of Them Coming from Lord Kelvin's Work
2.35 A Surprisingly Awesome Fractional Part Integral with Forms Involving tan(x) and cot(x)
2.36 A Superb Integral with Logarithms and the Inverse Tangent Function and a Wonderful Generalization of It
2.37 More Wonderful Results Involving Integrals with Logarithms and the Inverse Tangent Function
2.38 Powerful and Useful Sums of Integrals with Logarithms and the Inverse Tangent Function
2.39 Two Beautiful Sums of Integrals, Each One Involving Three Integrals, Leading to a Possible Unexpected Result
2.40 Tackling Curious Logarithmic Integrals with a Radical in the Denominator
2.41 Calculating More Logarithmic Integrals with a Radical in the Denominator
2.42 Somewhat Atypical Integrals with Curious Closed Forms
2.43 More Atypical Integrals with Curious Closed Forms
2.44 A Wonderful Trigonometric Integral by Larry Glasser
2.45 Resistant Logarithmic Integrals That Are Good to Know
2.46 Appealing Parameterized Integrals with Logarithms and the Dilogarithm Function, Related to Harmonic Series
2.47 An Encounter with Six Useful Integrals Involving Logarithms and the Dilogarithm
2.48 A Battle with Three Challenging Integrals Involving Logarithms and the Dilogarithm
2.49 Fascinating Polylogarithmic Integrals with Parameter Involving the Cauchy Product of Two Series
2.50 A Titan Involving Alternating Harmonic Series of Weight 7
2.51 A Tough Integral Approached by Clever Transformations
2.52 An Unexpected Closed Form, Involving Catalan's Constant, of a Nice Integral with the Dilogarithm
2.53 A Group of Six Special, Challenging Generalized Integrals Involving Curious Closed Forms
2.54 Amazing and (Very) Useful Integral Beasts Involving log2(sin(x)), log2(cos(x)), log3(sin(x)), and log3(cos(x))
2.55 Four Challenging Integrals with the Logarithm and Trigonometric Functions, Giving Nice Closed Forms
2.56 Advanced Integrals with Trigonometric Functions, Related to Fourier Series and Harmonic Series
2.57 Gems of Integration Involving Splendid Ideas About Symmetry in Two Dimensions
2.58 More Gems of Integration, This Time Involving Splendid Ideas About Symmetry in Three Dimensions
2.59 The Complete Elliptic Integral of the First Kind at Play
2.60 Evaluating An Esoteric-Looking Integral Involving a Triple Series with Factorial Numbers, Leading to ζ(4)
3 Solutions
3.1 A Beautiful Integral by the English Mathematician James Joseph Sylvester
3.2 Strange Limits with Trigonometric Integrals
3.3 Two Curious Logarithmic Integrals with Parameter
3.4 Exploring More Appealing Logarithmic Integrals with Parameter: The First Part
3.5 Exploring More Appealing Logarithmic Integrals with Parameter: The Second Part
3.6 More Good-Looking Logarithmic Integrals: The First Part
3.7 More Good-Looking Logarithmic Integrals: The Second Part
3.8 More Good-Looking Logarithmic Integrals: The Third Part
3.9 Special and Challenging Integrals with Parameter Involving the Inverse Hyperbolic Tangent: The First Part
3.10 Special and Challenging Integrals with Parameter Involving the Inverse Hyperbolic Tangent: The Second Part
3.11 Some Startling Generalizations Involving Logarithmic Integrals with Trigonometric Parameters
3.12 More Startling and Enjoyable Logarithmic Integrals Involving Trigonometric Parameters
3.13 Surprisingly Bewitching Trigonometric Integrals with Appealing Closed Forms: The First Act
3.14 Surprisingly Bewitching Trigonometric Integrals with Appealing Closed Forms: The Second Act
3.15 Surprisingly Bewitching Trigonometric Integrals with Appealing Closed Forms: The Third Act
3.16 Surprisingly Bewitching Trigonometric Integrals with Appealing Closed Forms: The Fourth Act
3.17 Playing with a Hand of Fabulous Integrals: The First Part
3.18 Playing with a Hand of Fabulous Integrals: The Second Part
3.19 Another Fabulous Integral Related to a Curious Binoharmonic Series Representation of ζ(3)
3.20 Pairs of Appealing Integrals with the Logarithm Function, the Dilogarithm function, and Trigonometric Functions
3.21 Pairs of Appealing Integrals with the Logarithm Function, the Dilogarithm function, and Trigonometric Functions
3.22 Valuable Logarithmic Integrals Involving Skew-Harmonic Numbers: A Second Partition
3.23 Valuable Logarithmic Integrals Involving Skew-Harmonic Numbers: A Third Partition
3.24 A Pair of Precious Generalized Logarithmic Integrals
3.25 Two Neat and Useful Generalizations with Logarithmic Integrals Involving Skew-Harmonic Numbers
3.26 A Special Logarithmic Integral and a Generalization of It
3.27 Three Useful Generalized Logarithmic Integrals Connected to Useful Generalized Alternating Harmonic Series
3.28 Three Atypical Logarithmic Integrals Involving log(1+x3), Related to a Special Form of Symmetry in Double Integrals
3.29 Four Curious Integrals with Cosine Function Which Lead to Beautiful Closed Forms, Plus Two Exotic Integrals
3.30 Aesthetic Integrals That Must Be Understood as Cauchy Principal Values, Generated by a Beautiful Generalization
3.31 A Special Integral Generalization, Attacked with Strategies Involving the Cauchy Principal Value Integrals, That Also Leads to Two Famous Results by Ramanujan
3.32 Four Magical Integrals Beautifully Calculated, Which Generate the Closed Forms of Four Classical Integrals
3.33 A Bouquet of Captivating Integrals Involving Trigonometric and Hyperbolic Functions
3.34 Interesting Integrals to Evaluate, One of Them Coming from Lord Kelvin's Work
3.35 A Surprisingly Awesome Fractional Part Integral with Forms Involving tan(x), cot(x)
3.36 A Superb Integral with Logarithms and the Inverse Tangent Function, and a Wonderful Generalization of It
3.37 More Wonderful Results Involving Integrals with Logarithms and the Inverse Tangent Function
3.38 Powerful and Useful Sums of Integrals with Logarithms and the Inverse Tangent Function
3.39 Two Beautiful Sums of Integrals, Each One Involving Three Integrals, Leading to a Possible Unexpected Result
3.40 Tackling Curious Logarithmic Integrals with a Radical in the Denominator
3.41 Calculating More Logarithmic Integrals with a Radical in the Denominator
3.42 Somewhat Atypical Integrals with Curious Closed Forms
3.43 More Atypical Integrals with Curious Closed Forms
3.44 A Wonderful Trigonometric Integral by Larry Glasser
3.45 Resistant Logarithmic Integrals That Are Good to Know
3.46 Appealing Parameterized Integrals with Logarithms and the Dilogarithm Function, Related to Harmonic Series
3.47 An Encounter with Six Useful Integrals Involving Logarithms and the Dilogarithm
3.48 A Battle with Three Challenging Integrals Involving Logarithms and the Dilogarithm
3.49 Fascinating Polylogarithmic Integrals with Parameter Involving the Cauchy Product of Two Series
3.50 A Titan Involving Alternating Harmonic Series of Weight 7
3.51 A Tough Integral Approached by Clever Transformations
3.52 An Unexpected Closed Form, Involving Catalan's Constant, of a Nice Integral with the Dilogarithm
3.53 A Group of Six Special, Challenging Generalized Integrals Involving Curious Closed Forms
3.54 Amazing and (Very) Useful Integral Beasts Involving log2(sin(x)), log2(cos(x)), log3(sin(x)), and log3(cos(x))
3.55 Four Challenging Integrals with the Logarithm and Trigonometric Functions, Giving Nice Closed Forms
3.56 Advanced Integrals with Trigonometric Functions, Related to Fourier Series and Harmonic Series
3.57 Gems of Integration Involving Splendid Ideas About Symmetry in Two Dimensions
3.58 More Gems of Integration, This Time Involving Splendid Ideas About Symmetry in Three Dimensions
3.59 The Complete Elliptic Integral of the First Kind at Play
3.60 Evaluating an Esoteric-Looking Integral Involving a Triple Series with Factorial Numbers, Leading to ζ(4)
4 Sums and Series
4.1 A Remarkable IMC Limit Problem Involving a Curious Sum with the Reciprocal of a Product with Two Logarithms
4.2 Two Series with Tail Involving the Double Factorial, Their Generalizations, and a ζ(2) Representation
4.3 Six Enjoyable Sums Involving the Reciprocal of the Central Binomial Coefficient and Two Series Derived from Them
4.4 A Great Time with a Special Binoharmonic Series
4.5 A Panel of (Very) Useful Cauchy Products of Two Series: From Known Cauchy Products to Less Known Ones
4.6 Good-to-Know Generating Functions: The First Part
4.7 Good-to-Know Generating Functions: The Second Part
4.8 Good-to-Know Generating Functions: The Third Part
4.9 Good-to-Know Generating Functions: The Fourth Part
4.10 Good-to-Know Generating Functions: The Fifth Part
4.11 Good-to-Know Generating Functions: The Sixth Part
4.12 Good-to-Know Generating Functions: The Seventh Part
4.13 Two Nice Sums Related to the Generalized Harmonic Numbers, an Asymptotic Expansion Extraction, a Neat Representation of log2(2), and a Curious Power Series
4.14 Opening the World of Harmonic Series with Beautiful Series That Require Athletic Movements During Their Calculations: The First (Enjoyable) Part
4.15 Opening the World of Harmonic Series with Beautiful Series That Require Athletic Movements During Their Calculations: The Second (Enjoyable) Part
4.16 A Special Harmonic Series in Disguise Involving Nice Tricks
4.17 A Few Nice Generalized Series: Most of Them May Be Seen as Applications of The Master Theorem of Series
4.18 Useful Relations Involving Polygamma with the Argument n/2 and the Generalized Skew-Harmonic Numbers
4.19 A Key Classical Generalized Harmonic Series
4.20 Revisiting Two Classical Challenging Alternating Harmonic Series, Calculated by Exploiting a Beta Function Form
4.21 A Famous Classical Generalization with Alternating Harmonic Series, Derived by a New Special Way
4.22 Seven Useful Generalized Harmonic Series
4.23 A Special Challenging Harmonic Series of Weight 4, Involving Harmonic Numbers of the Type H2n
4.24 Two Useful Atypical Harmonic Series of Weight 4 with Denominators of the Type (2n+1)2
4.25 Another Special Challenging Harmonic Series of Weight 4, Involving Harmonic Numbers of the Type H2n
4.26 A First Uncommon Series with the Tail of the Riemann Zeta Function ζ(2)-H2n(2), Related to Weight 4 Harmonic Series
4.27 A Second Uncommon Series with the Tail of the Riemann Zeta Function ζ(2)-Hn(2), Related to Weight 4 Harmonic Series
4.28 A Third Uncommon Series with the Tail of the Riemann Zeta Function ζ(2)-H2n(2), Related to Weight 4 Harmonic Series
4.29 A Fourth Uncommon Series with the Tail of the Riemann Zeta Function ζ(2)-Hn(2), Related to Weight 4 Harmonic Series
4.30 A Fifth Uncommon Series with the Tail of the Riemann Zeta Function ζ(2)-H2n(2), Related to Weight 4 Harmonic Series
4.31 A Sixth Uncommon Series with the Tail of the Riemann Zeta Function ζ(2)-Hn(2), Related to Weight 4 Harmonic Series
4.32 A Seventh Uncommon Series with the Tail of the Riemann Zeta Function ζ(2)-H2n(2), Related to Weight 4 Harmonic Series
4.33 On the Calculation of an Essential Harmonic Series of Weight 5, Involving Harmonic Numbers of the Type H2n
4.34 More Helpful Atypical Harmonic Series of Weight 5 with Denominators of the Type (2n+1)2 and (2n+1)3
4.35 On the Calculation of Another Essential Harmonic Series of Weight 5, Involving Harmonic Numbers of the Type H2n
4.36 A First Unusual Series with the Tail of the Riemann Zeta Function ζ(3)-H2n(3), Related to Weight 5 Harmonic Series
4.37 A Second Unusual Series with the Tail of the Riemann Zeta Function ζ(3)-Hn(3), Related to Weight 5 Harmonic Series
4.38 A Third Unusual Series with the Tail of the Riemann Zeta Function ζ(3)-H2n(3), Related to Weight 5 Harmonic Series
4.39 A Fourth Unusual Series with the Tail of the Riemann Zeta Function ζ(3)-Hn(3), Related to Weight 5 Harmonic Series
4.40 A Fifth Unusual Series with the Tail of the Riemann Zeta Function ζ(3)-H2n(3), Related to Weight 5 Harmonic Series
4.41 A Sixth Unusual Series with the Tail of the Riemann Zeta Function ζ(3)-Hn(3), Related to Weight 5 Harmonic Series
4.42 A Seventh Unusual Series with the Tail of the Riemann Zeta Function ζ(3)-H2n(3), Related to Weight 5 Harmonic Series
4.43 Three More Spectacular Harmonic Series of Weight 5, Involving Harmonic Numbers of the Type H2n and H2n(2)
4.44 Two Atypical Sums of Series, One of Them Involving the Product of the Generalized Harmonic Numbers Hn(3) Hn(6)
4.45 Amazing, Unexpected Relations with Alternating and Non-alternating Harmonic Series of Weights 5 and 7
4.46 A Quintet of Advanced Harmonic Series of Weight 5 Involving Skew-Harmonic Numbers
4.47 Fourier Series Expansions of the Bernoulli Polynomials
4.48 Stunning Fourier Series with log(sin(x)) and log(cos(x)) Raised to Positive Integer Powers, Related to Harmonic Numbers
4.49 More Stunning Fourier Series, Related to Atypical Harmonic Numbers (Skew-Harmonic Numbers)
4.50 And More Stunning Fourier Series, Related to Atypical Harmonic Numbers (Skew-Harmonic Numbers)
4.51 Yet Other Stunning Fourier Series, This Time with the Coefficients Mainly Kept in an Integral Form
4.52 A Pair of (Very) Challenging Alternating Harmonic Series with a Weight 4 Structure, Involving Harmonic Numbers of the Type H2n
4.53 Important Tetralogarithmic Values and More (Curious) Challenging Alternating Harmonic Series with a Weight 4 Structure, Involving Harmonic Numbers H2n
4.54 Two Alternating Euler Sums Involving Special Tails, a Joint Work with Moti Levy, Plus Two Newer Ones
4.55 A (Very) Hard Nut to Crack (An Alternating Harmonic Series with a Weight 4 Structure, Involving Harmonic Numbers of the Type H2n)
4.56 Another (Very) Hard Nut to Crack (An Alternating Harmonic Series with a Weight 4 Structure, Involving Harmonic Numbers of the Type H2n)
4.57 Two Harmonic Series with a Wicked Look, Involving Skew-Harmonic Numbers and Harmonic Numbers H2n
4.58 Nice Series with the Reciprocal of the Central Binomial Coefficient and the Generalized Harmonic Number
4.59 Marvellous Binoharmonic Series Forged with Nice Ideas
4.60 Presenting an Appealing Triple Infinite Series Together with an Esoteric-Looking Functional Equation
5 Hints
5.1 A Remarkable IMC Limit Problem Involving a Curious Sum with the Reciprocal of a Product with Two Logarithms
5.2 Two Series with Tail Involving the Double Factorial, Their Generalizations, and a ζ(2) Representation
5.3 Six Enjoyable Sums Involving the Reciprocal of the Central Binomial Coefficient and Two Series Derived from Them
5.4 A Great Time with a Special Binoharmonic Series
5.5 A Panel of (Very) Useful Cauchy Products of Two Series: From Known Cauchy Products to Less Known Ones
5.6 Good-to-Know Generating Functions: The First Part
5.7 Good-to-Know Generating Functions: The Second Part
5.8 Good-to-Know Generating Functions: The Third Part
5.9 Good-to-Know Generating Functions: The Fourth Part
5.10 Good-to-Know Generating Functions: The Fifth Part
5.11 Good-to-Know Generating Functions: The Sixth Part
5.12 Good-to-Know Generating Functions: The Seventh Part
5.13 Two Nice Sums Related to the Generalized Harmonic Numbers, an Asymptotic Expansion Extraction, a Neat Representation of log2(2), and a Curious Power Series
5.14 Opening the World of Harmonic Series with Beautiful Series That Require Athletic Movements During Their Calculations: The First (Enjoyable) Part
5.15 Opening the World of Harmonic Series with Beautiful Series That Require Athletic Movements During Their Calculations: The Second (Enjoyable) Part
5.16 A Special Harmonic Series in Disguise Involving Nice Tricks
5.17 A Few Nice Generalized Series: Most of Them May Be Seen as Applications of The Master Theorem of Series
5.18 Useful Relations Involving Polygamma with the Argument n/2 and the Generalized Skew-Harmonic Numbers
5.19 A Key Classical Generalized Harmonic Series
5.20 Revisiting Two Classical Challenging Alternating Harmonic Series, Calculated by Exploiting a Beta Function Form
5.21 A Famous Classical Generalization with Alternating Harmonic Series, Derived by a New Special Way
5.22 Seven Useful Generalized Harmonic Series
5.23 A Special Challenging Harmonic Series of Weight 4, Involving Harmonic Numbers of the Type H2n
5.24 Two Useful Atypical Harmonic Series of Weight 4 with Denominators of the Type (2n+1)2
5.25 Another Special Challenging Harmonic Series of Weight 4, Involving Harmonic Numbers of the Type H2n
5.26 A First Uncommon Series with the Tail of the Riemann Zeta Function ζ(2)-H2n(2), Related to Weight 4 Harmonic Series
5.27 A Second Uncommon Series with the Tail of the Riemann Zeta Function ζ(2)-Hn(2), Related to Weight 4 Harmonic Series
5.28 A Third Uncommon Series with the Tail of the Riemann Zeta Function ζ(2)-H2n(2), Related to Weight 4 Harmonic Series
5.29 A Fourth Uncommon Series with the Tail of the Riemann Zeta Function ζ(2)-Hn(2), Related to Weight 4 Harmonic Series
5.30 A Fifth Uncommon Series with the Tail of the Riemann Zeta Function ζ(2)-H2n(2), Related to Weight 4 Harmonic Series
5.31 A Sixth Uncommon Series with the Tail of the Riemann Zeta Function ζ(2)-Hn(2), Related to Weight 4 Harmonic Series
5.32 A Seventh Uncommon Series with the Tail of the Riemann Zeta Function ζ(2)-H2n(2), Related to Weight 4 Harmonic Series
5.33 On the Calculation of an Essential Harmonic Series of Weight 5, Involving Harmonic Numbers of the Type H2n
5.34 More Helpful Atypical Harmonic Series of Weight 5 with Denominators of the Type (2n+1)2 and (2n+1)3
5.35 On the Calculation of Another Essential Harmonic Series of Weight 5, Involving Harmonic Numbers of the Type H2n
5.36 A First Unusual Series with the Tail of the Riemann Zeta Function ζ(3)-H2n(3), Related to Weight 5 Harmonic Series
5.37 A Second Unusual Series with the Tail of the Riemann Zeta Function ζ(3)-Hn(3), Related to Weight 5 Harmonic Series
5.38 A Third Unusual Series with the Tail of the Riemann Zeta Function ζ(3)-H2n(3), Related to Weight 5 Harmonic Series
5.39 A Fourth Unusual Series with the Tail of the Riemann Zeta Function ζ(3)-Hn(3), Related to Weight 5 Harmonic Series
5.40 A Fifth Unusual Series with the Tail of the Riemann Zeta Function ζ(3)-H2n(3), Related to Weight 5 Harmonic Series
5.41 A Sixth Unusual Series with the Tail of the Riemann Zeta Function ζ(3)-Hn(3), Related to Weight 5 Harmonic Series
5.42 A Seventh Unusual Series with the Tail of the Riemann Zeta Function ζ(3)-H2n(3), Related to Weight 5 Harmonic Series
5.43 Three More Spectacular Harmonic Series of Weight 5, Involving Harmonic Numbers of the Type H2n and H2n(2)
5.44 Two Atypical Sums of Series, One of Them Involving the Product of the Generalized Harmonic Numbers Hn(3) Hn(6)
5.45 Amazing, Unexpected Relations with Alternating and Non-alternating Harmonic Series of Weights 5 and 7
5.46 A Quintet of Advanced Harmonic Series of Weight 5 Involving Skew-Harmonic Numbers
5.47 Fourier Series Expansions of the Bernoulli Polynomials
5.48 Stunning Fourier Series with log(sin(x)) and log(cos(x)) Raised to Positive Integer Powers, Related to Harmonic Numbers
5.49 More Stunning Fourier Series, Related to Atypical Harmonic Numbers (Skew-Harmonic Numbers)
5.50 And More Stunning Fourier Series, Related to Atypical Harmonic Numbers (Skew-Harmonic Numbers)
5.51 Yet Other Stunning Fourier Series, This Time with the Coefficients Mainly Kept in an Integral Form
5.52 A Pair of (Very) Challenging Alternating Harmonic Series with a Weight 4 Structure, Involving Harmonic Numbers of the Type H2n
5.53 Important Tetralogarithmic Values and More (Curious) Challenging Alternating Harmonic Series with a Weight 4 Structure, Involving Harmonic Numbers H2n
5.54 Two Alternating Euler Sums Involving Special Tails, a Joint Work with Moti Levy, plus Two Newer Ones
5.55 A (Very) Hard Nut to Crack (An Alternating Harmonic Series with a Weight 4 Structure, Involving Harmonic Numbers of the Type H2n)
5.56 Another (Very) Hard Nut to Crack (An Alternating Harmonic Series with a Weight 4 Structure, Involving Harmonic Numbers of the Type H2n)
5.57 Two Harmonic Series with a Wicked Look, Involving Skew-Harmonic Numbers and Harmonic Numbers H2n
5.58 Nice Series with the Reciprocal of the Central Binomial Coefficient and the Generalized Harmonic Number
5.59 Marvellous Binoharmonic Series Forged with Nice Ideas
5.60 Presenting an Appealing Triple Infinite Series Together with an Esoteric-Looking Functional Equation
6 Solutions
6.1 A Remarkable IMC Limit Problem Involving a Curious Sum with the Reciprocal of a Product with Two Logarithms
6.2 Two Series with Tail Involving the Double Factorial, Their Generalizations, and a ζ(2) Representation
6.3 Six Enjoyable Sums Involving the Reciprocal of the Central Binomial Coefficient and Two Series Derived from Them
6.4 A Great Time with a Special Binoharmonic Series
6.5 A Panel of (Very) Useful Cauchy Products of Two Series: From Known Cauchy Products to Less Known Ones
6.6 Good-to-Know Generating Functions: The First Part
6.7 Good-to-Know Generating Functions: The Second Part
6.8 Good-to-Know Generating Functions: The Third Part
6.9 Good-to-Know Generating Functions: The Fourth Part
6.10 Good-to-Know Generating Functions: The Fifth Part
6.11 Good-to-Know Generating Functions: The Sixth Part
6.12 Good-to-Know Generating Functions: The Seventh Part
6.13 Two Nice Sums Related to the Generalized Harmonic Numbers, an Asymptotic Expansion Extraction, a Neat Representation of log2(2), and a Curious Power Series
6.14 Opening the World of Harmonic Series with Beautiful Series That Require Athletic Movements During Their Calculations: The First (Enjoyable) Part
6.15 Opening the World of Harmonic Series with Beautiful Series That Require Athletic Movements During Their Calculations: The Second (Enjoyable) Part
6.16 A Special Harmonic Series in Disguise Involving Nice Tricks
6.17 A Few Nice Generalized Series: Most of Them May Be Seen as Applications of The Master Theorem of Series
6.18 Useful Relations Involving Polygamma with the Argument n/2 and the Generalized Skew-Harmonic Numbers
6.19 A Key Classical Generalized Harmonic Series
6.20 Revisiting Two Classical Challenging Alternating Harmonic Series, Calculated by Exploiting a Beta Function Form
6.21 A Famous Classical Generalization with Alternating Harmonic Series, Derived by a New Special Way
6.22 Seven Useful Generalized Harmonic Series
6.23 A Special Challenging Harmonic Series of Weight 4, Involving Harmonic Numbers of the Type H2n
6.24 Two Useful Atypical Harmonic Series of Weight 4 with Denominators of the Type (2n+1)2
6.25 Another Special Challenging Harmonic Series of Weight 4, Involving Harmonic Numbers of the Type H2n
6.26 A First Uncommon Series with the Tail of the Riemann Zeta Function ζ(2)-H2n(2), Related to Weight 4 Harmonic Series
6.27 A Second Uncommon Series with the Tail of the Riemann Zeta Function ζ(2)-Hn(2), Related to Weight 4 Harmonic Series
6.28 A Third Uncommon Series with the Tail of the Riemann Zeta Function ζ(2)-H2n(2), Related to Weight 4 Harmonic Series
6.29 A Fourth Uncommon Series with the Tail of the Riemann Zeta Function ζ(2)-Hn(2), Related to Weight 4 Harmonic Series
6.30 A Fifth Uncommon Series with the Tail of the Riemann Zeta Function ζ(2)-H2n(2), Related to Weight 4 Harmonic Series
6.31 A Sixth Uncommon Series with the Tail of the Riemann Zeta Function ζ(2)-Hn(2), Related to Weight 4 Harmonic Series
6.32 A Seventh Uncommon Series with the Tail of the Riemann Zeta Function ζ(2)-H2n(2), Related to Weight 4 Harmonic Series
6.33 On the Calculation of an Essential Harmonic Series of Weight 5, Involving Harmonic Numbers of the Type H2n
6.34 More Helpful Atypical Harmonic Series of Weight 5 with Denominators of the Type (2n+1)2 and (2n+1)3
6.35 On the Calculation of Another Essential Harmonic Series of Weight 5, Involving Harmonic Numbers of the Type H2n
6.36 A First Unusual Series with the Tail of the Riemann Zeta Function ζ(3)-H2n(3), Related to Weight 5 Harmonic Series
6.37 A Second Unusual Series with the Tail of the Riemann Zeta Function ζ(3)-Hn(3), Related to Weight 5 Harmonic Series
6.38 A Third Unusual Series with the Tail of the Riemann Zeta Function ζ(3)-H2n(3), Related to Weight 5 Harmonic Series
6.39 A Fourth Unusual Series with the Tail of the Riemann Zeta Function ζ(3)-Hn(3), Related to Weight 5 Harmonic Series
6.40 A Fifth Unusual Series with the Tail of the Riemann Zeta Function ζ(3)-H2n(3), Related to Weight 5 Harmonic Series
6.41 A Sixth Unusual Series with the Tail of the Riemann Zeta Function ζ(3)-Hn(3), Related to Weight 5 Harmonic Series
6.42 A Seventh Unusual Series with the Tail of the Riemann Zeta Function ζ(3)-H2n(3), Related to Weight 5 Harmonic Series
6.43 Three More Spectacular Harmonic Series of Weight 5, Involving Harmonic Numbers of the Type H2n and H2n(2)
6.44 Two Atypical Sums of Series, One of Them Involving the Product of the Generalized Harmonic Numbers Hn(3) Hn(6)
6.45 Amazing, Unexpected Relations with Alternating and Non-alternating Harmonic Series of Weight 5 and 7
6.46 A Quintet of Advanced Harmonic Series of Weight 5 Involving Skew-Harmonic Numbers
6.47 Fourier Series Expansions of the Bernoulli Polynomials
6.48 Stunning Fourier Series with log(sin(x)) and log(cos(x)) Raised to Positive Integer Powers, Related to Harmonic Numbers
6.49 More Stunning Fourier Series, Related to Atypical Harmonic Numbers (Skew-Harmonic Numbers)
6.50 And More Stunning Fourier Series, Related to Atypical Harmonic Numbers (Skew-Harmonic Numbers)
6.51 Yet Other Stunning Fourier Series, This Time with the Coefficients Mainly Kept in an Integral Form
6.52 A Pair of (Very) Challenging Alternating Harmonic Series with a Weight 4 Structure, Involving Harmonic Numbers of the Type H2n
6.53 Important Tetralogarithmic Values and More (Curious) Challenging Alternating Harmonic Series with a Weight 4 Structure, Involving Harmonic Numbers H2n
6.54 Two Alternating Euler Sums Involving Special Tails, a Joint Work with Moti Levy, Plus Two Newer Ones
6.55 A (Very) Hard Nut to Crack (An Alternating Harmonic Series with a Weight 4 Structure, Involving Harmonic Numbers of the Type H2n)
6.56 Another (Very) Hard Nut to Crack (An Alternating Harmonic Series with a Weight 4 Structure, Involving Harmonic Numbers of the Type H2n)
6.57 Two Harmonic Series with a Wicked Look, Involving Skew-Harmonic Numbers and Harmonic Numbers H2n
6.58 Nice Series with the Reciprocal of the Central Binomial Coefficient and the Generalized Harmonic Number
6.59 Marvellous Binoharmonic Series Forged with Nice Ideas
6.60 Presenting an Appealing Triple Infinite Series Together with an Esoteric-Looking Functional Equation
References
