An Introduction to Real Analysis
The Commonwealth and International Library: Mathematical Topics
Published on 17. May 2014
324 pages
978-1-4831-5896-9 (ISBN)
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.
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Content
PrefaceIntroduction. The Purpose of Real Analysis1. Sets, Relations, and Functions 1.1 Sets 1.2 Relations and Functions Exercises2. Numbers 2.1 Natural Numbers 2.2 Integers 2.3 Rationale 2.4 Real Numbers 2.5 Irrationals 2.6 Appendix3. Sequences 3.1 Introduction 3.2 Limits of Sequences 3.3 Elementary Theorems about 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 Exercises4. 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 Exercises5. 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 Exercises6. 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 Exercises7. 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 Exercises8. 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 ExercisesAnswersIndex
