A Radical Approach to Real Analysis: Second Edition (Classroom Resource Materials)

This document was uploaded by one of our users. The uploader already confirmed that they had the permission to publish it. If you are author/publisher or own the copyright of this documents, please report to us by using this DMCA report form.

Simply click on the Download Book button.

Yes, Book downloads on Ebookily are 100% Free.

Sometimes the book is free on Amazon As well, so go ahead and hit "Search on Amazon"

In the second edition of this 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, or 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.

Author(s): David M. Bressoud
Edition: 2
Publisher: The Mathematical Association of Americaa
Year: 2006

Language: English
Pages: 341
Tags: Математика;Математический анализ;

Preface......Page 10
Contents......Page 16
1.1 Background to the Problem......Page 18
1.2 Difficulties with the Solution......Page 21
2.1 The Archimedean Understanding......Page 26
2.2 Geometric Series......Page 34
2.3 Calculating Pi......Page 39
2.4 Logarithms and the Harmonic Series......Page 45
2.5 Taylor Series......Page 55
2.6 Emerging Doubts......Page 67
3 Differentiability and Continuity......Page 74
3.1 Differentiability......Page 75
3.2 Cauchy and the Mean Value Theorems......Page 88
3.3 Continuity......Page 95
3.4 Consequences of Continuity......Page 112
3.5 Consequences of the Mean Value Theorem......Page 122
4 The Convergence of Infinite Series......Page 134
4.1 The Basic Tests of Convergence......Page 135
4.2 Comparison Tests......Page 146
4.3 The Convergence of Power Series......Page 162
4.4 The Convergence of Fourier Series......Page 175
5 Understanding Infinite Series......Page 188
5.1 Groupings and Rearrangements......Page 189
5.2 Cauchy and Continuity......Page 198
5.3 Differentiation and Integration......Page 208
5.4 Verifying Uniform Convergence......Page 220
6 Return to Fourier Series......Page 234
6.1 Dirichlet's Theorem......Page 235
6.2 The Cauchy Integral......Page 253
6.3 The Riemann Integral......Page 265
6.4 Continuity without Differentiability......Page 275
7 Epilogue......Page 284
A.1 Wallis on Pi......Page 288
A.2 Bernoulli's Numbers......Page 294
A.3 Sums of Negative Powers......Page 301
A.4 The Size of n!......Page 310
B Bibliography......Page 320
C Hints to Selected Exercises......Page 322
Index......Page 334