(Almost) impossible integrals, sums, and series

Foreword
Preface
Contents
1 Integrals
	1.1 A Powerful Elementary Integral
	1.2 A Pair of Elementary Logarithmic Integrals We Might Find Very Useful for Solving the Problems in the Book
	1.3 Four Logarithmic Integrals Strongly Connected with the League of Harmonic Series
	1.4 Two Very Useful Classical Logarithmic Integrals That May Arise in the Calculation of Some Tough Integrals and Series
	1.5 A Couple of Practical Definite Integrals Expressed in Terms of the Digamma Function
	1.6 A Useful Special Generalized Integral Expressed in Terms of the Polylogarithm Function
	1.7 Two Little Tricky Classical Logarithmic Integrals
	1.8 A Special Trio of Integrals with log2(1-x) and log2(1+x)
	1.9 A Darn Integral in Disguise (Possibly Harder Than It Seems to Be?), an Integral with Two Squared Logarithms on the Half of the Unit Interval
	1.10 The Evaluation of a Class of Logarithmic Integrals Using a Slightly Modified Result from Table of Integrals, Series and Products by I.S. Gradshteyn and I.M. Ryzhik Together with a Series Result Elementarily Proved by Guy Bastien
	1.11 Logarithmic Integrals Containing an Infinite Series in the Integrand, Giving Values in Terms of Riemann Zeta Function
	1.12 Two Appealing Integral Representations of ζ(4) and ζ(2)G
	1.13 A Special Pair of Logarithmic Integrals with Connections in the Area of the Alternating Harmonic Series
	1.14 Another Special Pair of Logarithmic Integrals with Connections in the Area of the Alternating Harmonic Series
	1.15 A Class of Tricky and Useful Integrals with Consecutive Positive Integer Powers for the Logarithms
	1.16 A Double Integral and a Triple Integral, Beautifully Connected with the Advanced Harmonic Series
	1.17 Let\'s Take Two Double Logarithmic Integrals with Beautiful Values Expressed in Terms of the Riemann Zeta Function
	1.18 Interesting Integrals Containing the Inverse Tangent Function and the Logarithmic Function
	1.19 Interesting Integrals Involving the Inverse Tangent Function and Dilogarithm Function
	1.20 More Interesting Integrals Involving the Inverse Tangent Function and the Logarithmic Function: The First Part
	1.21 More Interesting Integrals Involving the Inverse Tangent Function and the Logarithmic Function: The Second Part
	1.22 Challenging Integrals Involving arctan(x), log(x), log(1-x), Li2(x), and Li2(x2)
	1.23 Two More Special Challenging Integrals Involving arctan(x), log(x), log(1+x), and Li2(-x)
	1.24 A Challenging Integral with the Inverse Tangent Function and an Excellent Generalization According to the Even Positive Powers of the Logarithm
	1.25 Let\'s Tango with an Exciting Integral Involving the Inverse Tangent Integral, the Lerch Transcendent Function, and the Logarithm with Odd Positive Powers
	1.26 A Superb Integral with Logarithms and the Inverse Tangent Function, and a Surprisingly Beautiful Generalization of It
	1.27 A Kind of Deviant Pair of Integrals with Logarithms and Polylogarithms, Using Symmetry
	1.28 Wonderful Integrals Containing the Logarithm and the Polylogarithm, Involving Beautiful Ideas About Symmetry: The First Part
	1.29 Wonderful Integrals Containing the Logarithm and the Polylogarithm, Involving Beautiful Ideas About Symmetry: The Second Part
	1.30 Wonderful Integrals Containing the Logarithm and the Polylogarithm, Involving Beautiful Ideas About Symmetry: The Third Part
	1.31 Two Families of Special Polylogarithmic Integrals Expressed in Terms of Infinite Series with the Generalized Harmonic Number and the Tails of Some Functions
	1.32 A Generalized Integral Beautifully Connected to a Spectacular (and Simultaneously Strange) Series
	1.33 A Special (and Possibly Slightly Daunting) Integral with*6pt Two Polylogarithms, Li2(to. eepicxx-1)to. and Li2(to. eepicxx+1)to.*6pt
	1.34 Exciting Challenging Triple Integrals with the Dilogarithm
	1.35 A Curious Integral with Polylogarithms Connected to a Double Integral with a Symmetrical Exponential Integrand
	1.36 Double Integrals Expressed in Terms of the Exponential Function and the Polylogarithm (of Orders 2, 3, 4, 5, and 6)
	1.37 Exponential Double Integrals with an Appealing Look
	1.38 A Generalized Double Integral Involving a Symmetrical Exponential Integrand and a Limit Related to It
	1.39 A Special Multiple Integral and a Limit of It Involving the Euler–Mascheroni Constant γ, the Euler\'s Number e, and the Famous π All at Once
	1.40 Some Curious Integrals Involving the Hyperbolic Tangent, Also Having Beautiful Connections with the Beta Function
	1.41 A Little Integral-Beast from Inside Interesting Integrals Together with a Similar Version of It Tamed by Real Methods
	1.42 Ramanujan\'s Integrals with Beautiful Connections with the Digamma Function and Frullani\'s Integral
	1.43 The Complete Elliptic Integral of the First Kind Ramanujan Is Asked to Calculate in the Movie The Man Who Knew Infinity Together with Another Question Originating from His Work
	1.44 The First Double Integral I Published in La Gaceta de la RSME, Together with Another Integral Similar to It
	1.45 An Out-of-Order Integral with an Integrand Expressed in Terms of an Infinite Series and a Generalization of It
	1.46 Pretty Charming Ramanujan-Like (Double) Integral Representations of the Riemann Zeta Function and Its Derivative
	1.47 The Elementary Calculation of a Fractional Part Integral Naturally Arising in an Exotic Triple Fractional Part Integral
	1.48 The Calculation of a Beautiful Triple Fractional Part Integral with a Cubic Power
	1.49 The Calculation of a Generalized Triple Fractional Part Integral with Positive Integer Powers
	1.50 A Pair of Cute Fractional Part Integrals Involving the Cotangent Function
	1.51 Playing with a Resistant Classical Integral Family to the Real Methods That Responds to the Tricks Involving the Use of the Cauchy–Schlömilch Transformation
	1.52 Calculating a Somewhat Strange-Looking Quartet of Integrals Involving the Trigonometric Functions
	1.53 Two Beautiful Representations of Catalan\'s Constant, G=1-132+152-172+192-@汥瑀瑯步渠
	1.54 Proving Two Equalities with Tough Integrals Involving Logarithms and Polylogarithms
	1.55 Tough Integrals with Logarithms, Polylogarithms, and Trigonometric and Hyperbolic Functions
	1.56 A Double Integral Hiding a Beautiful Idea About the Symmetry and (Possibly) an Unexpected Closed-Form
	1.57 An Exciting Representation of Catalan\'s Constant with Trigonometric Functions and Digamma Function
	1.58 Evaluating an Enjoyable Trigonometric Integral Involving the Complete Elliptic Integral of the First Kind at Its Roots
	1.59 Integrating Over an Infinite Product with Factors Containing the Secant and the Hyperbolic Secant with Powers of 2
	1.60 Linking Two Generalized Integrals Involving the Polylogarithm Function to Seductive Series
2 Hints
	2.1 A Powerful Elementary Integral
	2.2 A Pair of Elementary Logarithmic Integrals We Might Find Very Useful for Solving the Problems in the Book
	2.3 Four Logarithmic Integrals Strongly Connected with the League of Harmonic Series
	2.4 Two Very Useful Classical Logarithmic Integrals That May Arise in the Calculation of Some Tough Integrals and Series
	2.5 A Couple of Practical Definite Integrals Expressed in Terms of the Digamma Function
	2.6 A Useful Special Generalized Integral Expressed in Terms of the Polylogarithm Function
	2.7 Two Little Tricky Classical Logarithmic Integrals
	2.8 A Special Trio of Integrals with log2(1-x) and log2(1+x)
	2.9 A Darn Integral in Disguise (Possibly Harder Than It Seems to Be?), an Integral with Two Squared Logarithms on the Half of the Unit Interval
	2.10 The Evaluation of a Class of Logarithmic Integrals Using a Slightly Modified Result from Table of Integrals, Series, and Products by I.S. Gradshteyn and I.M. Ryzhik Together with a Series Result Elementarily Proved by Guy Bastien
	2.11 Logarithmic Integrals Containing an Infinite Series in the Integrand, Giving Values in Terms of Riemann Zeta Function
	2.12 Two Appealing Integral Representations of ζ(4) and ζ(2)G
	2.13 A Special Pair of Logarithmic Integrals with Connections in the Area of the Alternating Harmonic Series
	2.14 Another Special Pair of Logarithmic Integrals with Connections in the Area of the Alternating Harmonic Series
	2.15 A Class of Tricky and Useful Integrals with Consecutive Positive Integer Powers for the Logarithms
	2.16 A Double Integral and a Triple Integral, Beautifully Connected with the Advanced Harmonic Series
	2.17 Let\'s Take Two Double Logarithmic Integrals with Beautiful Values Expressed in Terms of the Riemann Zeta Function
	2.18 Interesting Integrals Containing the Inverse Tangent Function and the Logarithmic Function
	2.19 Interesting Integrals Involving the Inverse Tangent Function and Dilogarithm Function
	2.20 More Interesting Integrals Involving the Inverse Tangent Function and the Logarithmic Function: The First Part
	2.21 More Interesting Integrals Involving the Inverse Tangent Function and the Logarithmic Function: The Second Part
	2.22 Challenging Integrals Involving arctan(x), log(x), log(1-x), Li2(x), and Li2(x2)
	2.23 Two More Special Challenging Integrals Involving arctan(x), log(x), log(1+x), and Li2(-x)
	2.24 A Challenging Integral with the Inverse Tangent Function and an Excellent Generalization According to the Even Positive Powers of the Logarithm
	2.25 Let\'s Tango with an Exciting Integral Involving the Inverse Tangent Integral, the Lerch Transcendent Function, and the Logarithm with Odd Positive Powers
	2.26 A Superb Integral with Logarithms and the Inverse Tangent Function, and a Surprisingly Beautiful Generalization of It
	2.27 A Kind of Deviant Pair of Integrals with Logarithms and Polylogarithms, Using Symmetry
	2.28 Wonderful Integrals Containing the Logarithm and the Polylogarithm, Involving Beautiful Ideas About Symmetry: The First Part
	2.29 Wonderful Integrals Containing the Logarithm and the Polylogarithm, Involving Beautiful Ideas About Symmetry: The Second Part
	2.30 Wonderful Integrals Containing the Logarithm and the Polylogarithm, Involving Beautiful Ideas About Symmetry: The Third Part
	2.31 Two Families of Special Polylogarithmic Integrals Expressed in Terms of Infinite Series with the Generalized Harmonic Number and the Tails of Some Functions
	2.32 A Generalized Integral Beautifully Connected to a Spectacular (and Simultaneously Strange) Series
	2.33 A Special (and Possibly Slightly Daunting) Integral with*5pt Two Polylogarithms, Li2(to. eepicxx-1)to. and Li2(to. eepicxx+1)to.*5pt
	2.34 Exciting Challenging Triple Integrals with the Dilogarithm
	2.35 A Curious Integral with Polylogarithms Connected to a Double Integral with a Symmetrical Exponential Integrand
	2.36 Double Integrals Expressed in Terms of the Exponential Function and the Polylogarithm (of Orders 2, 3, 4, 5, and 6)
	2.37 Exponential Double Integrals with an Appealing Look
	2.38 A Generalized Double Integral Involving a Symmetrical Exponential Integrand and a Limit Related to It
	2.39 A Special Multiple Integral and a Limit of It Involving the Euler–Mascheroni Constant γ, the Euler\'s Number e, and the Famous π All at Once
	2.40 Some Curious Integrals Involving the Hyperbolic Tangent, Also Having Beautiful Connections with the Beta Function
	2.41 A Little Integral-Beast from Inside Interesting Integrals Together with a Similar Version of It Tamed by Real Methods
	2.42 Ramanujan\'s Integrals with Beautiful Connections with the Digamma Function and Frullani\'s Integral
	2.43 The Complete Elliptic Integral of the First Kind Ramanujan Is Asked to Calculate in the Movie The Man Who Knew Infinity Together with Another Question Originating from His Work
	2.44 The First Double Integral I Published in La Gaceta de la RSME, Together with Another Integral Similar to It
	2.45 An Out-of-Order Integral with an Integrand Expressed in Terms of an Infinite Series and a Generalization of It
	2.46 Pretty Charming Ramanujan-Like (Double) Integral Representations of the Riemann Zeta Function and Its Derivative
	2.47 The Elementary Calculation of a Fractional Part Integral Naturally Arising in an Exotic Triple Fractional Part Integral
	2.48 The Calculation of a Beautiful Triple Fractional Part Integral with a Cubic Power
	2.49 The Calculation of a Generalized Triple Fractional Part Integral with Positive Integer Powers
	2.50 A Pair of Cute Fractional Part Integrals Involving the Cotangent Function
	2.51 Playing with a Resistant Classical Integral Family to the Real Methods That Responds to the Tricks Involving the Use of the Cauchy–Schlömilch Transformation
	2.52 Calculating a Somewhat Strange-Looking Quartet of Integrals Involving the Trigonometric Functions
	2.53 Two Beautiful Representations of Catalan\'s Constant, G=1-132+152-172+192-@汥瑀瑯步渠
	2.54 Proving Two Equalities with Tough Integrals Involving Logarithms and Polylogarithms
	2.55 Tough Integrals with Logarithms, Polylogarithms, Trigonometric, and Hyperbolic Functions
	2.56 A Double Integral Hiding a Beautiful Idea About the Symmetry and (Possibly) an Unexpected Closed-Form
	2.57 An Exciting Representation of Catalan\'s Constant with Trigonometric Functions and Digamma Function
	2.58 Evaluating an Enjoyable Trigonometric Integral Involving the Complete Elliptic Integral of the First Kind at Its Roots
	2.59 Integrating Over an Infinite Product with Factors Containing the Secant and the Hyperbolic Secant with Powers of 2
	2.60 Linking Two Generalized Integrals Involving the Polylogarithm Function to Seductive Series
	References
3 Solutions
	3.1 A Powerful Elementary Integral
	3.2 A Pair of Elementary Logarithmic Integrals We Might Find Very Useful for Solving the Problems in the Book
	3.3 Four Logarithmic Integrals Strongly Connected with the League of Harmonic Series
	3.4 Two Very Useful Classical Logarithmic Integrals That May Arise in the Calculation of Some Tough Integrals and Series
	3.5 A Couple of Practical Definite Integrals Expressed in Terms of the Digamma Function
	3.6 A Useful Special Generalized Integral Expressed in Terms of the Polylogarithm Function
	3.7 Two Little Tricky Classical Logarithmic Integrals
	3.8 A Special Trio of Integrals with log2(1-x) and log2(1+x)
	3.9 A Darn Integral in Disguise (Possibly Harder Than It Seems to Be?), an Integral with Two Squared Logarithms on the Half of the Unit Interval
	3.10 The Evaluation of a Class of Logarithmic Integrals Using a Slightly Modified Result from Table of Integrals, Series and Products by I.S. Gradshteyn and I.M. Ryzhik Together with a Series Result Elementarily Proved by Guy Bastien
	3.11 Logarithmic Integrals Containing an Infinite Series in the Integrand, Giving Values in Terms of Riemann Zeta Function
	3.12 Two Appealing Integral Representations of ζ(4) and ζ(2)G
	3.13 A Special Pair of Logarithmic Integrals with Connections in the Area of the Alternating Harmonic Series
	3.14 Another Special Pair of Logarithmic Integrals with Connections in the Area of the Alternating Harmonic Series
	3.15 A Class of Tricky and Useful Integrals with Consecutive Positive Integer Powers for the Logarithms
	3.16 A Double Integral and a Triple Integral, Beautifully Connected with the Advanced Harmonic Series
	3.17 Let\'s Take Two Double Logarithmic Integrals with Beautiful Values Expressed in Terms of the Riemann Zeta Function
	3.18 Interesting Integrals Containing the Inverse Tangent Function and the Logarithmic Function
	3.19 Interesting Integrals Involving the Inverse Tangent Function and Dilogarithm Function
	3.20 More Interesting Integrals Involving the Inverse Tangent Function and the Logarithmic Function: The First Part
	3.21 More Interesting Integrals Involving the Inverse Tangent Function and the Logarithmic Function: The Second Part
	3.22 Challenging Integrals Involving arctan(x), log(x), log(1-x), Li2(x), and Li2(x2)
	3.23 Two More Special Challenging Integrals Involving arctan(x), log(x), log(1+x), and Li2(-x)
	3.24 A Challenging Integral with the Inverse Tangent Function and an Excellent Generalization According to the Even Positive Powers of the Logarithm
	3.25 Let\'s Tango with an Exciting Integral Involving the Inverse Tangent Integral, the Lerch Transcendent Function, and the Logarithm with Odd Positive Powers
	3.26 A Superb Integral with Logarithms and the Inverse Tangent Function, and a Surprisingly Beautiful Generalization of It
	3.27 A Kind of Deviant Pair of Integrals with Logarithms and Polylogarithms, Using Symmetry
	3.28 Wonderful Integrals Containing the Logarithm and the Polylogarithm, Involving Beautiful Ideas About Symmetry: The First Part
	3.29 Wonderful Integrals Containing the Logarithm and the Polylogarithm, Involving Beautiful Ideas About Symmetry: The Second Part
	3.30 Wonderful Integrals Containing the Logarithm and the Polylogarithm, Involving Beautiful Ideas About Symmetry: The Third Part
	3.31 Two Families of Special Polylogarithmic Integrals Expressed in Terms of Infinite Series with the Generalized Harmonic Number and the Tails of Some Functions
	3.32 A Generalized Integral Beautifully Connected to a Spectacular (and Simultaneously Strange) Series
	3.33 A Special (and Possibly Slightly Daunting)*3pt Integral with Two Polylogarithms, Li2(xx-1) and Li2(xx+1)*3pt
	3.34 Exciting Challenging Triple Integrals with the Dilogarithm
	3.35 A Curious Integral with Polylogarithms Connected to a Double Integral with a Symmetrical Exponential Integrand
	3.36 Double Integrals Expressed in Terms of the Exponential Function and the Polylogarithm (of Orders 2, 3, 4, 5, and 6)
	3.37 Exponential Double Integrals with an Appealing Look
	3.38 A Generalized Double Integral Involving a Symmetrical Exponential Integrand and a Limit Related to It
	3.39 A Special Multiple Integral and a Limit of It Involving the Euler–Mascheroni Constant γ, the Euler\'s Number e, and the Famous π All at Once
	3.40 Some Curious Integrals Involving the Hyperbolic Tangent, Also Having Beautiful Connections with the Beta Function
	3.41 A Little Integral-Beast from Inside Interesting Integrals Together with a Similar Version of It Tamed by Real Methods
	3.42 Ramanujan\'s Integrals with Beautiful Connections with the Digamma Function and Frullani\'s Integral
	3.43 The Complete Elliptic Integral of the First Kind Ramanujan Is Asked to Calculate in the Movie The Man Who Knew Infinity Together with Another Question Originating from His Work
	3.44 The First Double Integral I Published in La Gaceta de la RSME, Together with Another Integral Similar to It
	3.45 An Out-of-Order Integral with an Integrand Expressed in Terms of an Infinite Series and a Generalization of It
	3.46 Pretty Charming Ramanujan-Like (Double) Integral Representations of the Riemann Zeta Function and Its Derivative
	3.47 The Elementary Calculation of a Fractional Part Integral Naturally Arising in an Exotic Triple Fractional Part Integral
	3.48 The Calculation of a Beautiful Triple Fractional Part Integral with a Cubic Power
	3.49 The Calculation of a Generalized Triple Fractional Part Integral with Positive Integer Powers
	3.50 A Pair of Cute Fractional Part Integrals Involving the Cotangent Function
	3.51 Playing with a Resistant Classical Integral Family to the Real Methods That Responds to the Tricks Involving the Use of the Cauchy–Schlömilch Transformation
	3.52 Calculating a Somewhat Strange-Looking Quartet of Integrals Involving the Trigonometric Functions
	3.53 Two Beautiful Representations of Catalan\'s Constant, G=1-132+152-172+192-@汥瑀瑯步渠
	3.54 Proving Two Equalities with Tough Integrals Involving Logarithms and Polylogarithms
	3.55 Tough Integrals with Logarithms, Polylogarithms, and Trigonometric and Hyperbolic Functions
	3.56 A Double Integral Hiding a Beautiful Idea About the Symmetry and (Possibly) an Unexpected Closed-Form
	3.57 An Exciting Representation of Catalan\'s Constant with Trigonometric Functions and Digamma Function
	3.58 Evaluating an Enjoyable Trigonometric Integral Involving the Complete Elliptic Integral of the First Kind at Its Roots
	3.59 Integrating Over an Infinite Product with Factors Containing the Secant and the Hyperbolic Secant with Powers of 2
	3.60 Linking Two Generalized Integrals Involving the Polylogarithm Function to Seductive Series
	References
4 Sums and Series
	4.1 The First Series Submitted by Ramanujan to the Journal of the Indian Mathematical Society
	4.2 Starting from an Elementary Integral Result and Deriving Two Classical Series in a New Way
	4.3 An Extraordinary Series with the Tail of the Riemann Zeta Function Connected to the Inverse Sine Series
	4.4 The Evaluation of a Series Involving the Tails of the Series*3pt Representations of the Functions log(11-x) and x arcsin(x)1-x2*5pt
	4.5 A Breathtaking Infinite Series Involving the Binomial Coefficient and Expressing a Beautiful Closed-Form
	4.6 An Eccentric Multiple Series Having the Roots in the Realm of the Botez–Catalan Identity
	4.7 Two Classical Series with Fibonacci Numbers, One Related to the Arctan Function
	4.8 Two New Infinite Series with Fibonacci Numbers, Related to the Arctan Function
	4.9 Useful Series Representations of log(1+x)log(1-x) and arctan(x)log(1+x2) from the Notorious Table of Integrals, Series, and Products by I.S. Gradshteyn and I.M. Ryzhik
	4.10 A Group of Five Useful Generating Functions Related to the Generalized Harmonic Numbers
	4.11 Four Members from a Neat Group of Generating Functions Expressed in Terms of Polylogarithm Function
	4.12 Two Elementary Harmonic Sums Arising in the Calculation of Harmonic Series
	4.13 A Strong Generalized Sum, Making a Very Good Cocktail Together with the Identities Generated by The Master Theorem of Series
	4.14 Four Elementary Sums with Harmonic Numbers, Very Useful in the Calculation of the Harmonic Series of Weight 7
	4.15 The Master Theorem of Series, a New Very Useful Theorem in the Calculation of Many Difficult (Harmonic) Series
	4.16 The First Application of The Master Theorem of Series on the (Generalized) Harmonic Numbers
	4.17 The Second Application of The Master Theorem of Series on the Harmonic Numbers
	4.18 The Third Application of The Master Theorem of Series on the Harmonic Numbers
	4.19 The Fourth Application of The Master Theorem of Series on the (Generalized) Harmonic Numbers
	4.20 Cool Identities with Ingredients Like the Generalized Harmonic Numbers and the Binomial Coefficient
	4.21 Special (and Very Useful) Pairs of Classical Euler Sums Arising in Many Difficult Harmonic Series
	4.22 Another Perspective on the Famous Quadratic Series of Au-Yeung Which Leads to an Elementary Solution
	4.23 Treating a Big Brother Series of the Quadratic Series of Au-Yeung by Elementary Means
	4.24 Calculating Two More Elder Brother Series of the Quadratic Series of Au-Yeung, This Time the Versions with the Powers 4 and 5 in Denominator*3pt
	4.25 An Advanced Harmonic Series of Weight 5, n=1∞ Hn Hn(2)n2, Attacked with a Special Class of Sums
	4.26 An Advanced Harmonic Series of Weight 5, n=1∞ Hn3n2, Attacked with a Special Identity
	4.27 The Evaluation of an Advanced Cubic Harmonic Series*3pt of Weight 6, n=1∞ (Hnn)3, Treated with Both The Master*3pt Theorem of Series and Special Logarithmic Integrals of Powers Two and Three
	4.28 Another Evaluation of an Advanced Harmonic Series*3pt of Weight 6, n=1∞Hn Hn(2)n3, Treated with The Master Theorem of Series*2pt
	4.29 And Now a Series of Weight 6, n=1∞Hn Hn(3)n2, Treated with*3pt Both The Master Theorem of Series and Special Logarithmic Integrals
	4.30 An Appealing Exotic Harmonic Series of Weight 6,*5pt n=1∞ Hn2 Hn(2)n2, Derived by Elementary Series Manipulations*2pt
	4.31 Another Appealing Exotic Harmonic Series of Weight 6, n=1∞Hn4n2, Derived by Elementary Series Manipulations
	4.32 Four Sums with Harmonic Series Involving the Generalized Harmonic Numbers of Order 1, 2, 3, 4, 5, and 6, Originating from The Master Theorem of Series
	4.33 Awesomely Wicked Sums of Series of Weight 7,*3pt n=1∞ Hn Hn(2)n4 and n=1∞ Hn Hn(3)n3, Originating from a Strong*3pt Generalized Sum: The First Part
	4.34 Awesomely Wicked Sums of Series of Weight 7,*3pt n=1∞ Hn Hn(2)n4 and n=1∞ Hn(2) Hn(3)n2, Originating from a Strong*3pt Generalized Sum: The Second Part
	4.35 Awesomely Wicked Sums of Series of Weight 7,*5pt n=1∞ Hn Hn(2)n4, n=1∞ Hn2 Hn(3)n2, and n=1∞ Hn2 Hn(2)n3, Derivation*3pt Based upon a New Identity: The Third Part
	4.36 Deriving More Useful Sums of Harmonic Series of Weight 7
	4.37 Preparing the Weapons of The Master Theorem of Series to Breach the Fortress of the Challenging Harmonic Series of Weight 7: The 1st Episode
	4.38 Preparing the Weapons of The Master Theorem of Series to Breach the Fortress of the Challenging Harmonic Series of Weight 7: The 2nd Episode
	4.39 Preparing the Weapons of The Master Theorem of Series to Breach the Fortress of the Challenging Harmonic Series of Weight 7: The 3rd Episode
	4.40 Calculating the Harmonic Series of Weight 7,*5pt n=1∞Hn2 Hn(2)n3, with the Weapons of The Master Theorem of Series*3pt
	4.41 The Calculation of Two Good-Looking Pairs of Harmonic Series: The Series n=1∞Hnn2k=1n Hkk3, n=1∞Hnn3k=1n Hkk2*6pt and n=1∞Hn2n2k=1n Hkk2, n=1∞Hnn2k=1n Hk2k2*2pt
	4.42 The Calculation of an Essential Harmonic Series of Weight*3pt 7: The Series n=1∞Hn Hn(2)n4*3pt
	4.43 Plenty of Challenging Harmonic Series of Weight 7 Obtained by Combining the Previous Harmonic Series of Weight 7 with Various Harmonic Series Identities (Derivations by Series Manipulations Only)
	4.44 A Member of a Glamorous Series Family Containing the Harmonic Number and the Tail of the Riemann Zeta Function
	4.45 More Members of a Glamorous Series Family Containing the Harmonic Number and the Tail of the Riemann Zeta Function
	4.46 Two Series Generalizations with the Generalized Harmonic Numbers and the Tail of the Riemann Zeta Function
	4.47 The Art of Mathematics with a Series Involving the Product of the Tails of ζ(2) and ζ(3)
	4.48 The Art of Mathematics with Another Splendid Series Involving the Product of the Tails of ζ(2) and ζ(3)
	4.49 Expressing Polylogarithmic Values by Combining the Alternating Harmonic Series and the Non-alternating Harmonic Series with Integer Powers of 2 in Denominator
	4.50 Cool Results with Cool Series Involving Summands with the Harmonic Number and the Integer Powers of 2
	4.51 Eight Harmonic Series Involving the Integer Powers of 2 in Denominator
	4.52 Let\'s Calculate Three Classical Alternating Harmonic Series,*3pt n=1∞(-1)n-1 Hnn3, n=1∞ (-1)n-1Hn(2)n2, and n=1∞(-1)n-1 Hn2n2*3pt
	4.53 Then, Let\'s Calculate Another Pair of Classical Alternating Harmonic Series, n=1∞(-1)n-1 Hnn2 and n=1∞(-1)n-1 Hnn4
	4.54 A Nice Challenging Trio of Alternating Harmonic Series,*3pt n=1∞ (-1)n-1Hn(2)n3, n=1∞ (-1)n-1Hn(3)n2, and n=1∞ (-1)n-1Hn(4)n*3pt
	4.55 Encountering an Alternating Harmonic Series of Weight 5*3pt with an Eye-Catching Closed-Form, n=1∞ (-1)n-1Hn2n3*3pt
	4.56 Encountering Another Alternating Harmonic Series of*3pt Weight 5 with a Dazzling Closed-Form, n=1∞ (-1)n-1Hn3n2*3pt
	4.57 Yet Another Encounter with a Superb Alternating*3pt Harmonic Series of Weight 5, n=1∞ (-1)n-1Hn Hn(2)n2*3pt
	4.58 Fascinating Sums of Two Alternating Harmonic Series Involving the Generalized Harmonic Number
	4.59 An Outstanding Sum of Series Representation of the Particular Value of the Riemann Zeta Function, ζ(4)
	4.60 An Excellent Representation of the Particular Value of the Riemann Zeta Function, ζ(4), with a Triple Series Involving the Factorials and the Generalized Harmonic Numbers
5 Hints
	5.1 The First Series Submitted by Ramanujan to the Journal of the Indian Mathematical Society
	5.2 Starting from an Elementary Integral Result and Deriving Two Classical Series in a New Way
	5.3 An Extraordinary Series with the Tail of the Riemann Zeta Function Connected to the Inverse Sine Series
	5.4 The Evaluation of a Series Involving the Tails of the Series*3pt Representations of the Functions log(11-x) and x arcsin(x)1-x2*2pt
	5.5 A Breathtaking Infinite Series Involving the Binomial Coefficient and Expressing a Beautiful Closed-Form
	5.6 An Eccentric Multiple Series Having the Roots in the Realm of the Botez–Catalan Identity
	5.7 Two Classical Series with Fibonacci Numbers, One Related to the Arctan Function
	5.8 Two New Infinite Series with Fibonacci Numbers, Related to the Arctan Function
	5.9 Useful Series Representations of log(1+x)log(1-x) and arctan(x)log(1+x2) from the Notorious Table of Integrals, Series, and Products by I.S. Gradshteyn and I.M. Ryzhik
	5.10 A Group of Five Useful Generating Functions Related to the Generalized Harmonic Numbers
	5.11 Four Members from a Neat Group of Generating Functions Expressed in Terms of Polylogarithm Function
	5.12 Two Elementary Harmonic Sums Arising in the Calculation of Harmonic Series
	5.13 A Strong Generalized Sum, Making a Very Good Cocktail Together with the Identities Generated by The Master Theorem of Series
	5.14 Four Elementary Sums with Harmonic Numbers, Very Useful in the Calculation of the Harmonic Series of Weight 7
	5.15 The Master Theorem of Series, a New Very Useful Theorem in the Calculation of Many Difficult (Harmonic) Series
	5.16 The First Application of The Master Theorem of Series on the (Generalized) Harmonic Numbers
	5.17 The Second Application of The Master Theorem of Series on the Harmonic Numbers
	5.18 The Third Application of The Master Theorem of Series on the Harmonic Numbers
	5.19 The Fourth Application of The Master Theorem of Series on the (Generalized) Harmonic Numbers
	5.20 Cool Identities with Ingredients Like the Generalized Harmonic Numbers and the Binomial Coefficient
	5.21 Special (and Very Useful) Pairs of Classical Euler Sums Arising in Many Difficult Harmonic Series
	5.22 Another Perspective on the Famous Quadratic Series of Au-Yeung Which Leads to an Elementary Solution
	5.23 Treating a Big Brother Series of the Quadratic Series of Au-Yeung by Elementary Means
	5.24 Calculating Two More Elder Brother Series of the Quadratic Series of Au-Yeung, This Time the Versions with the Powers 4 and 5 in Denominator
	5.25 An Advanced Harmonic Series of Weight 5, n=1∞ Hn Hn(2)n2, Attacked with a Special Class of Sums
	5.26 An Advanced Harmonic Series of Weight 5, n=1∞ Hn3n2, Attacked with a Special Identity
	5.27 The Evaluation of an Advanced Cubic Harmonic Series*3pt of Weight 6, n=1∞ (Hnn)3, Treated with Both The Master*3pt Theorem of Series and Special Logarithmic Integrals of Powers Two and Three
	5.28 Another Evaluation of an Advanced Harmonic Series of*3pt Weight 6, n=1∞Hn Hn(2)n3, Treated with The Master Theorem of Series*3pt
	5.29 And Now a Series of Weight 6, n=1∞Hn Hn(3)n2, Treated with*3pt Both The Master Theorem of Series and Special Logarithmic Integrals
	5.30 An Appealing Exotic Harmonic Series of Weight 6,*3pt n=1∞ Hn2 Hn(2)n2, Derived by Elementary Series Manipulations*3pt
	5.31 Another Appealing Exotic Harmonic Series of Weight 6,*3pt n=1∞Hn4n2, Derived by Elementary Series Manipulations*3pt
	5.32 Four Sums with Harmonic Series Involving the Generalized Harmonic Numbers of Order 1, 2, 3, 4, 5, and 6, Originating from The Master Theorem of Series
	5.33 Awesomely Wicked Sums of Series of Weight 7,*5pt n=1∞ Hn Hn(2)n4 and n=1∞ Hn Hn(3)n3, Originating from a Strong*2pt Generalized Sum: The First Part
	5.34 Awesomely Wicked Sums of Series of Weight 7,*5pt n=1∞ Hn Hn(2)n4 and n=1∞ Hn(2) Hn(3)n2, Originating from a Strong*2pt Generalized Sum: The Second Part
	5.35 Awesomely Wicked Sums of Series of Weight 7,*5pt n=1∞ Hn Hn(2)n4, n=1∞ Hn2 Hn(3)n2, and n=1∞ Hn2 Hn(2)n3, Derivation*3pt Based upon a New Identity: The Third Part
	5.36 Deriving More Useful Sums of Harmonic Series of Weight 7
	5.37 Preparing the Weapons of The Master Theorem of Series to Breach the Fortress of the Challenging Harmonic Series of Weight 7: The 1st Episode
	5.38 Preparing the Weapons of The Master Theorem of Series to Breach the Fortress of the Challenging Harmonic Series of Weight 7: The 2nd Episode
	5.39 Preparing the Weapons of The Master Theorem of Series to Breach the Fortress of the Challenging Harmonic Series of Weight 7: The 3rd Episode*3pt
	5.40 Calculating the Harmonic Series of Weight 7, n=1∞Hn2 Hn(2)n3, with the Weapons of The Master Theorem of Series
	5.41 The Calculation of Two Good-Looking Pairs of Harmonic Series: The Series n=1∞Hnn2k=1n Hkk3, n=1∞Hnn3k=1n Hkk2*6pt and n=1∞Hn2n2k=1n Hkk2, n=1∞Hnn2k=1n Hk2k2
	5.42 The Calculation of an Essential Harmonic Series of Weight*3pt 7: The Series n=1∞Hn Hn(2)n4*3pt
	5.43 Plenty of Challenging Harmonic Series of Weight 7 Obtained by Combining the Previous Harmonic Series of Weight 7 with Various Harmonic Series Identities (Derivations by Series Manipulations Only)
	5.44 A Member of a Glamorous Series Family Containing the Harmonic Number and the Tail of the Riemann Zeta Function
	5.45 More Members of a Glamorous Series Family Containing the Harmonic Number and the Tail of the Riemann Zeta Function
	5.46 Two Series Generalizations with the Generalized Harmonic Numbers and the Tail of the Riemann Zeta Function
	5.47 The Art of Mathematics with a Series Involving the Product of the Tails of ζ(2) and ζ(3)
	5.48 The Art of Mathematics with Another Splendid Series Involving the Product of the Tails of ζ(2) and ζ(3)
	5.49 Expressing Polylogarithmic Values by Combining the Alternating Harmonic Series and the Non-alternating Harmonic Series with Integer Powers of 2 in Denominator
	5.50 Cool Results with Cool Series Involving Summands with the Harmonic Number and the Integer Powers of 2
	5.51 Eight Harmonic Series Involving the Integer Powers of 2 in Denominator
	5.52 Let\'s Calculate Three Classical Alternating Harmonic Series,*3pt n=1∞(-1)n-1 Hnn3, n=1∞ (-1)n-1Hn(2)n2, and n=1∞(-1)n-1 Hn2n2*3pt
	5.53 Then, Let\'s Calculate Another Pair of Classical Alternating Harmonic Series, n=1∞(-1)n-1 Hnn2 and n=1∞(-1)n-1 Hnn4*3pt
	5.54 A Nice Challenging Trio of Alternating Harmonic Series,*4pt n=1∞ (-1)n-1Hn(2)n3, n=1∞ (-)n-1Hn(3)n2, and n=1∞ (-1)n-1Hn(4)n
	5.55 Encountering an Alternating Harmonic Series of Weight 5*4pt with an Eye-Catching Closed-Form, n=1∞ (-1)n-1Hn2n3*3pt
	5.56 Encountering Another Alternating Harmonic Series of Weight 5 with a Dazzling Closed-Form, n=1∞ (-1)n-1Hn3n2*3pt
	5.57 Yet Another Encounter with a Superb Alternating*4pt Harmonic Series of Weight 5, n=1∞ (-1)n-1Hn Hn(2)n2*3pt
	5.58 Fascinating Sums of Two Alternating Harmonic Series Involving the Generalized Harmonic Number
	5.59 An Outstanding Sum of Series Representation of the Particular Value of the Riemann Zeta Function, ζ(4)
	5.60 An Excellent Representation of the Particular Value of the Riemann Zeta Function, ζ(4), with a Triple Series Involving the Factorials and the Generalized Harmonic Numbers
6 Solutions
	6.1 The First Series Submitted by Ramanujan to the Journal of the Indian Mathematical Society
	6.2 Starting from an Elementary Integral Result and Deriving Two Classical Series in a New Way
	6.3 An Extraordinary Series with the Tail of the Riemann Zeta Function Connected to the Inverse Sine Series
	6.4 The Evaluation of a Series Involving the Tails of the Series*3pt Representations of the Functions log(11-x) and x arcsin(x)1-x2*3pt
	6.5 A Breathtaking Infinite Series Involving the Binomial Coefficient and Expressing a Beautiful Closed-Form
	6.6 An Eccentric Multiple Series Having the Roots in the Realm of the Botez–Catalan Identity
	6.7 Two Classical Series with Fibonacci Numbers, One Related to the Arctan Function
	6.8 Two New Infinite Series with Fibonacci Numbers, Related to the Arctan Function
	6.9 Useful Series Representations of log(1+x)log(1-x) and arctan(x)log(1+x2) from the Notorious Table of Integrals, Series, and Products by I.S. Gradshteyn and I.M. Ryzhik
	6.10 A Group of Five Useful Generating Functions Related to the Generalized Harmonic Numbers
	6.11 Four Members from a Neat Group of Generating Functions Expressed in Terms of Polylogarithm Function
	6.12 Two Elementary Harmonic Sums Arising in the Calculation of Harmonic Series
	6.13 A Strong Generalized Sum, Making a Very Good Cocktail Together with the Identities Generated by The Master Theorem of Series
	6.14 Four Elementary Sums with Harmonic Numbers, Very Useful in the Calculation of the Harmonic Series of Weight 7
	6.15 The Master Theorem of Series, a New Very Useful Theorem in the Calculation of Many Difficult (Harmonic) Series
	6.16 The First Application of The Master Theorem of Series on the (Generalized) Harmonic Numbers
	6.17 The Second Application of The Master Theorem of Series on the Harmonic Numbers
	6.18 The Third Application of The Master Theorem of Series on the Harmonic Numbers
	6.19 The Fourth Application of The Master Theorem of Series on the (Generalized) Harmonic Numbers
	6.20 Cool Identities with Ingredients Like the Generalized Harmonic Numbers and the Binomial Coefficient
	6.21 Special (and Very Useful) Pairs of Classical Euler Sums Arising in Many Difficult Harmonic Series
	6.22 Another Perspective on the Famous Quadratic Series of Au-Yeung Which Leads to an Elementary Solution
	6.23 Treating a Big Brother Series of the Quadratic Series of Au-Yeung by Elementary Means
	6.24 Calculating Two More Elder Brother Series of the Quadratic Series of Au-Yeung, This Time the Versions with the Powers 4 and 5 in Denominator
	6.25 An Advanced Harmonic Series of Weight 5, n=1∞ Hn Hn(2)n2, Attacked with a Special Class of Sums
	6.26 An Advanced Harmonic Series of Weight 5, n=1∞ Hn3n2, Attacked with a Special Identity
	6.27 The Evaluation of an Advanced Cubic Harmonic Series*4pt of Weight 6, n=1∞ (Hnn)3, Treated with Both The Master*3pt Theorem of Series and Special Logarithmic Integrals of Powers Two and Three
	6.28 Another Evaluation of an Advanced Harmonic Series of*3pt Weight 6, n=1∞Hn Hn(2)n3, Treated with The Master Theorem*3pt of Series*3pt
	6.29 And Now a Series of Weight 6, n=1∞Hn Hn(3)n2, Treated with*3pt Both The Master Theorem of Series and Special Logarithmic Integrals
	6.30 An Appealing Exotic Harmonic Series of Weight 6,*5pt n=1∞ Hn2 Hn(2)n2, Derived by Elementary Series Manipulations
	6.31 Another Appealing Exotic Harmonic Series of Weight 6,*3pt n=1∞Hn4n2, Derived by Elementary Series Manipulations*3pt
	6.32 Four Sums with Harmonic Series Involving the Generalized Harmonic Numbers of Order 1, 2, 3, 4, 5, and 6, Originating from The Master Theorem of Series
	6.33 Awesomely Wicked Sums of Series of Weight 7,*3pt n=1∞ Hn Hn(2)n4 and n=1∞ Hn Hn(3)n3, Originating from a Strong*3pt Generalized Sum: The First Part
	6.34 Awesomely Wicked Sums of Series of Weight 7,*3pt n=1∞ Hn Hn(2)n4 and n=1∞ Hn(2) Hn(3)n2, Originating from a Strong*3pt Generalized Sum: The Second Part
	6.35 Awesomely Wicked Sums of Series of Weight 7,*7pt n=1∞ Hn Hn(2)n4, n=1∞ Hn2 Hn(3)n2 and n=1∞ Hn2 Hn(2)n3, Derivation*3pt Based upon a New Identity: The Third Part
	6.36 Deriving More Useful Sums of Harmonic Series of Weight 7
	6.37 Preparing the Weapons of The Master Theorem of Series to Breach the Fortress of the Challenging Harmonic Series of Weight 7: The 1st Episode
	6.38 Preparing the Weapons of The Master Theorem of Series to Breach the Fortress of the Challenging Harmonic Series of Weight 7: The 2nd Episode
	6.39 Preparing the Weapons of The Master Theorem of Series to Breach the Fortress of the Challenging Harmonic Series of Weight 7: The 3rd Episode
	6.40 Calculating the Harmonic Series of Weight 7, n=1∞Hn2 Hn(2)n3, with the Weapons of The Master Theorem of Series
	6.41 The Calculation of Two Good-Looking Pairs of Harmonic Series: The Series n=1∞Hnn2k=1n Hkk3, n=1∞Hnn3k=1n Hkk2*6pt and n=1∞Hn2n2k=1n Hkk2, n=1∞Hnn2k=1n Hk2k2*3pt
	6.42 The Calculation of an Essential Harmonic Series of Weight*3pt 7: The Series n=1∞Hn Hn(2)n4*3pt
	6.43 Plenty of Challenging Harmonic Series of Weight 7 Obtained by Combining the Previous Harmonic Series of Weight 7 with Various Harmonic Series Identities (Derivations by Series Manipulations Only)
	6.44 A Member of a Glamorous Series Family Containing the Harmonic Number and the Tail of the Riemann Zeta Function
	6.45 More Members of a Glamorous Series Family Containing the Harmonic Number and the Tail of the Riemann Zeta Function
	6.46 Two Series Generalizations with the Generalized Harmonic Numbers and the Tail of the Riemann Zeta Function
	6.47 The Art of Mathematics with a Series Involving the Product of the Tails of ζ(2) and ζ(3)
	6.48 The Art of Mathematics with Another Splendid Series Involving the Product of the Tails of ζ(2) and ζ(3)
	6.49 Expressing Polylogarithmic Values by Combining the Alternating Harmonic Series and the Non-alternating Harmonic Series with Integer Powers of 2 in Denominator
	6.50 Cool Results with Cool Series Involving Summands with the Harmonic Number and the Integer Powers of 2
	6.51 Eight Harmonic Series Involving the Integer Powers of 2 in Denominator
	6.52 Let\'s Calculate Three Classical Alternating Harmonic Series,*3pt n=1∞(-1)n-1 Hnn3, n=1∞ (-1)n-1Hn(2)n2, and n=1∞(-1)n-1 Hn2n2*3pt
	6.53 Then, Let\'s Calculate Another Pair of Classical Alternating Harmonic Series, n=1∞(-1)n-1 Hnn2 and n=1∞(-1)n-1 Hnn4*3pt
	6.54 A Nice Challenging Trio of Alternating Harmonic Series,*6pt n=1∞ (-1)n-1Hn(2)n3, n=1∞ (-1)n-1Hn(3)n2, and n=1∞ (-1)n-1Hn(4)n*3pt
	6.55 Encountering an Alternating Harmonic*5pt Series of Weight 5 with an Eye-Catching Closed-Form, n=1∞ (-1)n-1Hn2n3*3pt
	6.56 Encountering Another Alternating Harmonic Series of Weight 5 with a Dazzling Closed-Form, n=1∞ (-1)n-1Hn3n2*3pt
	6.57 Yet Another Encounter with a Superb Alternating*4pt Harmonic Series of Weight 5, n=1∞ (-1)n-1Hn Hn(2)n2*3pt
	6.58 Fascinating Sums of Two Alternating Harmonic Series Involving the Generalized Harmonic Number
	6.59 An Outstanding Sum of Series Representation of the Particular Value of the Riemann Zeta Function, ζ(4)
	6.60 An Excellent Representation of the Particular Value of the Riemann Zeta Function, ζ(4), with a Triple Series Involving the Factorials and the Generalized Harmonic Numbers
	References
Index