This textbook covers the majority of traditional topics of infinite sequences and series, starting from the very beginning – the definition and elementary properties of sequences of numbers, and ending with advanced results of uniform convergence and power series.
The text is aimed at university students specializing in mathematics and natural sciences, and at all the readers interested in infinite sequences and series. It is designed for the reader who has a good working knowledge of calculus. No additional prior knowledge is required.
The text is divided into five chapters, which can be grouped into two parts: the first two chapters are concerned with the sequences and series of numbers, while the remaining three chapters are devoted to the sequences and series of functions, including the power series. Within each major topic, the exposition is inductive and starts with rather simple definitions and/or examples, becoming more compressed and sophisticated as the course progresses. Each key notion and result is illustrated with examples explained in detail. Some more complicated topics and results are marked as complements and can be omitted on a first reading.
The text includes a large number of problems and exercises, making it suitable for both classroom use and self-study. Many standard exercises are included in each section to develop basic techniques and test the understanding of key concepts. Other problems are more theoretically oriented and illustrate more intricate points of the theory, or provide counterexamples to false propositions which seem to be natural at first glance. Solutions to additional problems proposed at the end of each chapter are provided as an electronic supplement to this book.
Author(s): Ludmila Bourchtein
Publisher: Birkhäuser
Year: 2021
Language: English
Pages: 392
City: Cham
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
