A Radical Approach to Real Analysis

by Bressoud, David M.

• ISBN13: 9780883857472
• ISBN10: 0883857472
• Edition: 2nd
• Format: Hardcover
• Copyright: 2007-02-28
• Publisher: Mathematical Assn of Amer

Summary

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.

Author Biography

David Bressoud is DeWitt Wallace Professor of Mathematics at Macalester College. he served in the Peace Corps, teaching math and science at the Clare Hall School in Antigua, West Indies before styding with Emil Grosswald at Tample and then teaching at Penn State for 17 years, eight of them as a full professor. He chaired the Mathematics Dpeartment at Macalester from 1995 until 2001. He has held visiting positions at the Institute for Advanced Study, the University of Wisconsin-Madison, the University of Minnesota, UniversitT Louis Pasteur (Strasbourg, France), and the State College Area High School. He has received the MAA Distinguished Teaching Award (Allegheny Mountain Section), the MAA Beckenbach Book Award from Proofs and Confirmations, and has been a P=lya Lecturer for the MAA. He is a recipient of Macalester's Jefferson Award. He has published over fifty reseach articles in nuymber theory, combinatorics, and special functions. Other books include Factorization and Primality Testing, Second Year Calculus from Celestial Mechanics to Special Relativity, the first and second editions of A Radical Approach to Real Analysis, and, with Stan Wagon, A Course in Computational Number Theory. David Bressoud chairs the MAA Committee on the Undergraduate Program in Mathematics. He has chaired the AP Calculus Development Committee and has served as Director of the FIPSE-supported program Quantitative Methods for Public Policy. He has been active in the activities and programs of both the Mathematical Association of America and the American Mathematical Society.

Table of Contents

Preface	ix
Crisis in Mathematics: Fourier's Series	1	(8)
Background to the Problem	1	(3)
Difficulties with the Solution	4	(5)
Infinite Summations	9	(48)
The Archimedean Understanding	9	(8)
Geometric Series	17	(5)
Calculating π	22	(6)
Logarithms and the Harmonic Series	28	(10)
Taylor Series	38	(12)
Emerging Doubts	50	(7)
Differentiability and Continuity	57	(60)
Differentiability	58	(13)
Cauchy and the Mean Value Theorems	71	(7)
Continuity	78	(17)
Consequences of Continuity	95	(10)
Consequences of the Mean Value Theorem	105	(12)
The Convergence of Infinite Series	117	(54)
The Basic Tests of Convergence	118	(11)
Comparison Tests	129	(16)
The Convergence of Power Series	145	(13)
The Convergence of Fourier Series	158	(13)
Understanding Infinite Series	171	(46)
Groupings and Rearrangements	172	(9)
Cauchy and Continuity	181	(10)
Differentiation and Integration	191	(12)
Verifying Uniform Convergence	203	(14)
Return to Fourier Series	217	(50)
Dirichlet's Theorem	218	(18)
The Cauchy Integral	236	(12)
The Riemann Integral	248	(10)
Continuity without Differentiability	258	(9)
Epilogue	267	(4)
A. Explorations of the Infinite	271	(32)
Wallis on π	271	(6)
Bernoulli's Numbers	277	(7)
Sums of Negative Powers	284	(9)
The Size of n!	293	(10)
B. Bibliography	303	(2)
C. Hints to Selected Exercises	305	(12)
Index	317