A concrete introduction to real analysis / Robert Carlson.

Format:

Book

Author/Creator:

Carlson, Robert, 1951-

Contributor:

Alumni and Friends Memorial Book Fund.

Series:

Monographs and textbooks in pure and applied mathematics

Language:

English

Subjects (All):

Mathematical analysis.

Physical Description:

296 pages : illustrations ; 24 cm.

Place of Publication:

Boca Raton, FL : Chapman & Hall/CRC, 2006.

Summary:

Avoiding unnecessary abstractions to provide an accessible presentation of the material, A Concrete Introduction to Real Analysis supplies the crucial transition from a calculations-focused treatment of mathematics to a proof-centered approach. Providing a substantial rethinking of the presentation of real analysis as well as drawing from the history of mathematics and practical applications, this volume uses problems emerging from calculus to introduce themes of estimation, approximation, and convergence.

Beginning with standard calculus techniques and ending with the formal treatment of logic, real numbers, and functions, the book covers discrete calculus, selected area computations, Taylor's theorem, infinite sequences and series, limits, continuity and differentiability of functions, the Riemann integral, and much more. It contains a large collection of examples and exercises, ranging from simple problems that allow students to check their understanding of the concepts to challenging problems that develop new material.

Providing a solid foundation in analysis, A Concrete Introduction to Real Analysis demonstrates that the mathematical treatments described in the text will be valuable both for students planning to study more analysis and for those who are less inclined to take another analysis class.

Contents:

1 Discrete Calculus 1

1.2 Proof by induction 2

1.3 A calculus of sums and differences 6

1.4 Sums of powers 14

2 Selected Area Computations 25

2.2 Areas under power function graphs 26

2.3 The computation of [pi] 31

2.4 Natural logarithms 35

2.5 Stirling's formula 41

3 Limits and Taylor's Theorem 55

3.2 Limits of infinite sequences 56

3.2.2 Properties of limits 60

3.3 Series representations 65

3.4 Taylor series 68

3.4.1 Taylor polynomials 69

3.4.2 Taylor's Theorem 73

3.4.3 The remainder 76

3.4.4 Additional results 82

4 Infinite Series 93

4.1.1 Bounded monotone sequences 95

4.2 Positive series 97

4.3 General series 101

4.3.1 Absolute convergence 102

4.3.2 Alternating series 104

4.3.3 Power series 106

4.4 Grouping and rearrangement 108

5 A Bit of Logic 119

5.1 Some mathematical philosophy 119

5.2 Propositional logic 122

5.3 Predicates and quantifiers 127

5.4 Proofs 131

5.4.1 Axioms for propositional logic 132

5.4.2 Additional rules of inference 135

5.4.3 Adding hypotheses 136

5.4.4 Proof by contradiction 138

6 Real Numbers 145

6.1 Field axioms 146

6.2 Order axioms 149

6.3 Completeness axioms 154

6.4 Subsequences and compact intervals 161

6.5 Products and fractions 164

6.5.1 Infinite products 164

6.5.2 Continued fractions 169

7 Functions 181

7.3 Limits and continuity 184

7.3.1 Limits 184

7.3.2 Continuity 190

7.3.3 Uniform continuity 195

7.4 Derivatives 198

7.4.1 Computation of derivatives 199

7.4.2 The Mean Value Theorem 205

7.4.3 Contractions 209

7.4.4 Convexity 212

8 Integrals 223

8.2 Integrable functions 226

8.3 Properties of integrals 235

8.4 Numerical computation of integrals 241

8.4.1 Endpoint Riemann sums 242

8.4.2 More sophisticated integration procedures 244

9 More Integrals 255

9.2 Improper integrals 256

9.2.1 Integration of positive functions 258

9.2.2 Absolutely convergent integrals 262

9.2.3 Conditionally convergent integrals 264

9.3 Integrals with parameters 268

9.3.1 Sample computations 268

9.3.2 Some analysis in two variables 270

9.3.3 Functions defined by Riemann integration 273

9.3.4 Functions defined by improper integrals 278.

Notes:

Includes bibliographical references (pages 291-292) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Alumni and Friends Memorial Book Fund.

ISBN:

1584886544

OCLC:

64594399

Publisher Number:

9781584886549

Online:

Publisher description