A Radical Approach to Real Analysis (Ams/Maa Textbooks, 10)

In this second edition of the MAA classic, exploration continues to be an essential component. More than 60 new exercises have been added, and the chapters on Infinite Summations, Differentiability and Continuity, and Convergence of Infinite Series have been reorganized to make it easier to identify the key ideas. A Radical Approach to Real Analysis is an introduction to real analysis, rooted in and informed by the historical issues that shaped its development. It can be used as a textbook, as a resource for the instructor who prefers to teach a traditional course, or as a resource for the student who has been through a traditional course yet still does not understand what real analysis is about and why it was created. The book begins with Fourier's introduction of trigonometric series and the problems they created for the mathematicians of the early 19th century. It follows Cauchy's attempts to establish a firm foundation for calculus and considers his failures as well as his successes. It culminates with Dirichlet's proof of the validity of the Fourier series expansion and explores some of the counterintuitive results Riemann and Weierstrass were led to as a result of Dirichlet's proof. Preface Contents 1. Crisis in Mathematics: Fourier’s Series 1.1 Background to the Problem 1.2 Difficulties with the Solution 2. Infinite Summations 2.1 The Archimedean Understanding 2.2 Geometric Series. 2.3 Calculating π 2.4 Logarithms and the Harmonic Series 2.5 Taylor Series 2.6 Emerging Doubts 3. Differentiability and Continuity 3.1 Differentiability 3.2 Cauchy and the Mean Value Theorems 3.3 Continuity 3.4 Consequences of Continuity 3.5 Consequences of the Mean Value Theorem 4. The Convergence of Infinite Series 4.1 The Basic Tests of Convergence 4.2 Comparison Tests 4.3 The Convergence of Power Series 4.4 The Convergence of Fourier Series 5. Understanding Infinite Series 5.1 Groupings and Rearrangements 5.2 Cauchy and Continuity 5.3 Differentiation and Integration 5.4 Verifying Uniform Convergence 6. Return to Fourier Series 6.1 Dirichlet’s Theorem 6.2 The Cauchy Integral 6.3 The Riemann Integral 6.4 Continuity without Differentiability 7. Epilogue A. Explorations of the Infinite A.1 Wallis on π A.2 Bernoulli’s Numbers A.3 Sums of Negative Powers A.4 The Size of n! B. Bibliography C. Hints to Selected Exercises 2.1.6 2.4.10 3.1.4 3.3.18 3.4.13 4.1.3 4.3.2 4.4.12 5.4.1 6.1.6 6.3.16 A.1.12 Index d efghi jklm nopqrst uvwz Corrections Resources for A Radical Approach to Real Analysis (2nd edition) Chapter 1: Crises in Mathematics: Fourier's Series Derivation of Fourier’s Solution Laplace’s Equation How Fourier found the coeffcients for equation (1.7) Approximating Fourier's Solution (Maple code) The General Solution The Orthogonality Relation Fourier Series as Complex Power Series Maple code for exercises in section 1.2 Chapter 2: Infinite Summations The Quadrature of the Parabolic Segment The Archimedean Principle Explorations of the Alternating Harmonic Series (Maple code) Assigning Values to Divergent Series More Pi (Maple code) Newton’s Formula Explorations of the Harmonic Series Euler’s Solution to the Vibrating Drumhead Explorations of d'Alembert's Series (Maple code) Explorations of Lagrange's Remainder (Maple code) Maple code for exercises in section 2.1 Maple code for exercises in section 2.2 Maple code for exercises in section 2.3 Maple code for exercises in section 2.4 Maple code for exercises in section 2.5 Chapter 3: Differentiability and Continuity Newton-Raphson Method How to find and write a proof Continued Fractions The Marquis de l’Hospital Maple code for exercise in section 3.3 Maple code for exercises in section 3.4 Maple code for exercises in Newton-Raphson Method Chapter 4: The Convergence of Infinite Series Stirling's Formula (Maple code) Exponential Function Exponential Function (Maple code) Convergence in Norm Gauss’s Test Maple code for exercises in section 4.1 Maple code for exercises in section 4.2 Maple code for exercises in section 4.3 Maple code for exercises in section 4.4 Chapter 5: Understanding Infinite Series The Dilogarithm Maple code for exercises in section 5.1 Maple code for exercises in section 5.2 Maple code for exercises in section 5.3 Chapter 6: Return to Fourier Series Maple code for exercises in section 6.1 Maple code for exercises in section 6.2 Maple code for exercises in section 6.3 Maple code for exercises in section 6.4 Appendix A: Explorations of the Infinite Binomial Coefficients and Sums of nth Powers Maple code for exercises in section A.1 Maple code for exercises in section A.2 Maple code for exercises in section A.3 Maple code for exercises in section A.4 Acknowledgements Back Cover