Ball / Plumpton An Introduction to Real Analysis

The Commonwealth and International Library: Mathematical Topics
1. Auflage 2014
ISBN: 978-1-4831-5896-9
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 324 Seiten, Web PDF

An Introduction to Real Analysis presents the concepts of real analysis and highlights the problems which necessitate the introduction of these concepts. Topics range from sets, relations, and functions to numbers, sequences, series, derivatives, and the Riemann integral. This volume begins with an introduction to some of the problems which are met in the use of numbers for measuring, and which provide motivation for the creation of real analysis. Attention then turns to real numbers that are built up from natural numbers, with emphasis on integers, rationals, and irrationals. The chapters that follow explore the conditions under which sequences have limits and derive the limits of many important sequences, along with functions of a real variable, Rolle's theorem and the nature of the derivative, and the theory of infinite series and how the concepts may be applied to decimal representation. The book also discusses some important functions and expansions before concluding with a chapter on the Riemann integral and the problem of area and its measurement. Throughout the text the stress has been upon concepts and interesting results rather than upon techniques. Each chapter contains exercises meant to facilitate understanding of the subject matter. This book is intended for students in colleges of education and others with similar needs.

Autoren/Hrsg.

Ball, Derek G.

Plumpton, C.

Inhaltsverzeichnis

1;Front Cover;1
2;An Introduction to Real Analysis;4
3;Copyright Page;5
4;Table of Contents;6
5;PREFACE;8
6;INTRODUCTION. THE PURPOSE OF REAL ANALYSIS;10
7;CHAPTER 1. SETS, RELATIONS, AND FUNCTIONS;20
7.1;1.1. Sets;20
7.2;1.2. Relations and Functions;23
8;CHAPTER 2. NUMBERS;32
8.1;2.1. Natural numbers;32
8.2;2.2. Integers;35
8.3;2.3. Rationals;36
8.4;2.4. Real Numbers;39
8.5;2.5. Irrationals;49
8.6;2.6. Appendix;50
9;CHAPTER 3. SEQUENCES;55
9.1;3.1. Introduction;55
9.2;3.2. Limits of sequences;58
9.3;3.3. Elementary theorems about sequences;64
9.4;3.4. Behaviour of monotonie sequences;70
9.5;3.5. Sequences defined by recurrence relations;71
9.6;3.6. More sequences and their limits;74
9.7;3.7. Upper and lower limits;78
10;CHAPTER 4. SERIES;89
10.1;4.1. Introduction;89
10.2;4.2. Convergence of a series;92
10.3;4.3. More series, convergent and divergent;97
10.4;4.4 The comparison test;99
10.5;4.5. Decimal representation;102
10.6;4.6. Absolute convergence;108
10.7;4.7. Conditional convergence;113
10.8;4.8. Rearrangement of series;115
10.9;4.9. Multiplication of series;119
11;CHAPTER 5. FUNCTIONS OF A REAL VARIABLE;129
11.1;5.1. Introduction;129
11.2;5.2. Limits;136
11.3;5.3. Properties of limits;146
11.4;5.4. Continuity;148
11.5;5.5. The place of pathological functions in real analysis;154
11.6;5.6. The nature of discontinuities;155
11.7;5.7. Properties of continuous functions;160
12;CHAPTER 6. THE DERIVATIVE;168
12.1;6.1. Derivatives and their evaluation;168
12.2;6.2. Rolle's theorem and the nature of the derivative;181
12.3;6.3. Mean value theorems;186
12.4;6.4. Applications of derivatives;189
12.5;6.5. Taylor series;199
13;CHAPTER 7. SOME IMPORTANT FUNCTIONS AND EXPANSIONS;210
13.1;7.1. Power series;210
13.2;7.2. The exponential function;218
13.3;7.3. Trigonometric functions;225
13.4;7.4. Logarithmic functions;234
13.5;7.5. Infinite products;245
13.6;7.6. The binomial theorem;248
14;CHAPTER 8. THE RIEMANN INTEGRAL;253
14.1;8.1. Introduction;253
14.2;8.2. The Riemann integral;257
14.3;8.3. Integrability of monotonic functions;268
14.4;8.4. Continuous functions and the Riemann integral;271
14.5;8.5. Further applications of the fundamental theorem;280
14.6;8.6. Alternative approach to the logarithmic function;287
14.7;8.7. Infinite and improper integrals;294
14.8;8.8 Volumes of revolution;301
15;ANSWERS AND HINTS;309
16;INDEX;322