# An Introduction to Real Analysis

## 1st Edition

### The Commonwealth and International Library: Mathematical Topics

Authors:
Editors:
eBook ISBN: 9781483158969
Imprint: Pergamon
Published Date: 1st January 1973
Page Count: 324
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## Description

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.

Preface

Introduction. The Purpose of Real Analysis

1. Sets, Relations, and Functions

1.1 Sets

1.2 Relations and Functions

Exercises

2. Numbers

2.1 Natural Numbers

2.2 Integers

2.3 Rationale

2.4 Real Numbers

2.5 Irrationals

2.6 Appendix

3. Sequences

3.1 Introduction

3.2 Limits of Sequences

3.4 Behavior of Monotonic Sequences

3.5 Sequences Defined by Recurrence Relations

3.6 More Sequences and Their Limits

3.7 Upper and Lower Limits

Exercises

4. Series

4.1 Introduction

4.2 Convergence of a Series

4.3 More Series, Convergent and Divergent

4.4 The Comparison Test

4.5 Decimal Representation

4.6 Absolute Convergence

4.7 Conditional Convergence

4.8 Rearrangement of Series

4.9 Multiplication of Series

Exercises

5. Functions of a Real Variable

5.1 Introduction

5.2 Limits

5.3 Properties of Limits

5.4 Continuity

5.5 The Place of Pathological Functions Real Analysis

5.6 The Nature of Discontinuities

5.7 Properties of Continuous Functions

Exercises

6. The Derivative

6.1 Derivatives and their Evaluation

6.2 Rolle's Theorem and the Nature of the Derivative

6.3 Mean Value Theorems

6.4 Applications of Derivatives

6.5 Taylor Series

Exercises

7. Some Important Functions and Expansions

7.1 Power Series

7.2 The Exponential Function

7.3 Trigonometric Functions

7.4 Logarithmic Functions

7.5 Infinite Products

7.6 The Binomial Theorem

Exercises

8. The Riemann Integral

8.1 Introduction

8.2 The Riemann Integral

8.3 Integrability of Monotonic Functions

8.4 Continuous Functions and the Riemann Integral

8.5 Further Applications of the Fundamental Theorem

8.6 Alternative Approach to the Logarithmic Function

8.7 Infinite and Improper Integrals

8.8 Volumes of Revolution

Exercises

Index

No. of pages:
324
Language:
English
Published:
1st January 1973
Imprint:
Pergamon
eBook ISBN:
9781483158969