Theory of Infinite Sequences and Series

Preface
Contents
1 Sequences of Numbers
	1 Convergence and Introductory Examples
		1.1 Definition of a Sequence and Trivial (Pre-limit) Properties
		1.2 Convergence of a Sequence
	2 Common Properties of Convergent Sequences
		2.1 Uniqueness of the Limit
		2.2 Comparison Properties
		2.3 Arithmetic and Analytic Properties
	3 Special Properties of Convergent Sequences
		3.1 Convergence of Function and Corresponding Sequence
		3.2 Relationship Between Convergence and Boundedness
		3.3 Subsequences and Their Convergence. Bolzano-Weierstrass Theorem
		3.4 Cauchy Criterion for Convergence
		3.5 Sequences of the Arithmetic and Geometric Means
	4 Indeterminate Forms and Techniques of Their Solution
		4.1 Definition of Indeterminate Forms
		4.2 Techniques of Solution of Indeterminate Forms
		4.3 Various Indeterminate Forms and Examples
	Exercises
2 Series of Numbers
	1 Convergence and Introductory Examples
		1.1 Definition of a Series. Partial Sums and Convergence
		1.2 Elementary Examples of Series of Numbers
	2 Elementary Properties of Convergent Series
		2.1 Arithmetic Properties
			Linear Combination
			Product of Two Series
		2.2 Cauchy Criterion for Convergence
		2.3 Necessary Condition of Convergence (Divergence Test)
		2.4 Series and Its Remainder
			Convergence of the Original and Modified Series
			Criterion for Convergence Through Remainders
	3 Convergence of Positive Series
		3.1 General Criterion for Convergence
		3.2 Integral Test (Cauchy-Maclauren Test)
			Evaluation of the Remainder in the Integral Test
		3.3 The Comparison Tests
			Relationship Between the Comparison Tests with and Without Limit and Examples
			Complement: Nonexistence of an Universal Series for Comparison
		3.4 The Cauchy Condensation Test
			Complement: Schlömilch's Test
		3.5 D'Alembert's Tests (The Ratio Tests)
			Relationship Between the Tests with and Without Limit and Some Examples
		3.6 Cauchy's Tests (The Root Tests)
			Relationship Between the Tests with and Without Limit and Some Examples
		3.7 Comparison Between D'Alembert's and Cauchy's Tests
		3.8 Complement: Finer Forms of D'Alembert's and Cauchy's Tests
			Upper and Lower Limits and Their Properties
		3.9 Complement: The Kummer Chain of Tests
			The Kummer Tests with Upper and Lower Limits
			Restrictions in Application of the Kummer Hierarchy
		3.10 Complement: The Cauchy Chain of Tests
	4 Series of Different Types
		4.1 Alternating Series
		4.2 Dirichlet's and Abel's Tests
		4.3 Absolute and Conditional Convergence
			Tests of Absolute Convergence
		4.4 Product of Two Series
	5 Associative and Commutative Properties of Series
		5.1 Positive and Negative Parts of Series
		5.2 Associative Property of Convergent Series
		5.3 Commutative Property of Absolutely Convergent Series
		5.4 Commutative Property of Conditionally Convergent Series
	6 Complement: Double and Repeated Series
	Exercises
3 Sequences of Functions
	1 Pointwise Convergence and Introductory Examples
	2 Uniform and Non-uniform Convergence
		2.1 Concept of the Uniform and Non-uniform Convergence
		2.2 Arithmetic Properties of Uniform Convergence
			Complement: Product of Uniformly Convergent Sequences
		2.3 Cauchy Criterion for Uniform Convergence
	3 Dini's Theorem
	4 Properties of Limit Functions Under Uniform Convergence
		4.1 Boundedness of Limit Function
		4.2 Limit of the Limit Function
		4.3 Continuity of the Limit Function
		4.4 Integrability of the Limit Function (Integration by Parameter)
			Complement: Improper Integral
			Complement: Integrability with a Stronger Formulation and More Involved Proof
		4.5 Differentiability of the Limit Function (Differentiation by Parameter)
			Complement: Differentiation with Stronger Formulation and More Involved Proof
	5 Complement: The Weierstrass Approximation Theorem
	Exercises
4 Series of Functions
	1 Pointwise Convergence and Introductory Examples
	2 Uniform and Non-uniform Convergence
		2.1 Concept of Uniform and Non-uniform Convergence
		2.2 Arithmetic Properties of Uniform Convergence
		2.3 The Cauchy Criterion for Uniform Convergence
		2.4 Uniform and Absolute Convergence
	3 Sufficient Conditions for Uniform Convergence of Series
		3.1 Comparison Tests
		3.2 Dirihlet's and Abel's Tests
		3.3 Dini's Theorem
	4 Properties of the Sum of Uniformly Convergent Series
		4.1 Boundedness of a Sum
		4.2 Limit of a Sum
		4.3 Continuity of a Sum
		4.4 Integrability of a Sum (Integration Term by Term)
		4.5 Differentiability of a Sum (Differentiation Term by Term)
	5 Complement: The Weierstrass Function—Everywhere Continuous and Nowhere Differentiable Function
	Exercises
5 Power Series
	1 Introduction
	2 Set of Convergence of a Power Series
		2.1 Convergence of a Power Series
		2.2 Determining the Radius of Convergence
			The D'Alembert and Cauchy Formulas
			Complement: The Cauchy-Hadamard Formula
		2.3 Convergence of the Series of Derivatives
		2.4 Behavior at the Endpoints of the Interval of Convergence
	3 Properties of Power Series and Their Sums
		3.1 Arithmetic Properties
			Product of Power Series
		3.2 Functional Properties
			Composition of Power Series
			Complement: Composite Series
			Change of the Central Point
		3.3 Analytic Properties
			Property 3: Continuity
			Property 4: Integrability
			Property 5: Differentiability
		3.4 Uniqueness of Power Series Expansion, Analytic Functions
			Even and Odd Functions
			Analytic Functions
	4 Taylor Series
		4.1 Taylor Coefficients and Taylor Series
		4.2 Relation Between the Taylor Series and Formula
		4.3 Conditions of Expansion in the Taylor Series
	5 Power Series Expansion of Elementary Functions
		5.1 Using Analytic Properties of Power Series
			1. Function 11-x and Its Derivatives and Integrals
			2. Function 11+x and Its Derivatives and Integrals
			3. Function 11+x2 and Its Derivatives and Integrals
		5.2 Finding the Sum of Power Series via Differential Relations
			1. Function 11-x
			2. Functions f(x)=(1+x)p, pN{0}
			3. Function f(x)=ex
		5.3 Method of the Taylor Coefficients
			1. Functions f(x)=(1+x)p, pR
			2. Function f(x)=11-x and Related Functions
			3. Function f(x)=ex
			4. Functions f(x)=sinx and f(x)=cosx
			5. Functions f(x)=tanx and f(x)=cotx
		5.4 Taylor Series for Various Functions
		5.5 The List of the Derived Formulas of Taylor Series
	6 Applications of Taylor Series
		6.1 Approximation of Functions
		6.2 Numerical Approximations
		6.3 Finding Sums of Series of Functions
		6.4 Sums of Series of Numbers
		6.5 Calculation of Limits
		6.6 Calculation of Integrals
		6.7 Solution of Ordinary Differential Equations
		6.8 Complement: The Number e Is Irrational
		6.9 Complement: The Number π Is Irrational
	7 Complement: Borel's Theorem
		7.1 Smooth Non-analytic Function
		7.2 Transition Function
		7.3 Borel's Theorem
	Exercises
Bibliography
	Textbooks on Calculus, Real Analysis and Infinite Series
	History of Analysis (Including Theory of Infinite Sequences and Series)
	Original Classical Sources on Real Analysis (Mentioned in the Text)
	Further Reading (Fourier Series, Divergent Series, Summability Methods, Multiple Series, Complex Series, Laurent Series, and More)
Index