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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.

Tabla de contenidos provista por Syndetics

• Preface
• 1 Crisis in mathematics: Fourier's series
• 2 Infinite summations
• 3 Differentiability and continuity
• 4 The convergence of infinite series
• 5 Understanding infinite series
• 6 Return to Fourier series
• 7 Epilogue
• A Explorations of the infinite
• B Bibliography
• C Hints to selected exercises

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CHOICE Review

The crisis that erupted in analysis in the early 19th century began with Fourier's use of trigonometric series in 1807. With this observation, Bressoud picks this as the pedagogically sound place to start a book on real analysis. Chapter 2 provides the context in which infinite series were understood and misunderstood prior to Fourier's work; then, beginning with chapter 3, the reader moves with the great pioneers, coming to appreciate the difficulties that troubled them, the errors they made in addressing those problems, and the accomplishment that our current understandings represent. The book is not a history, but encourages the student to experiment and try to get a least a little flavor of the historic struggle--and in contrast to the customary logical development of material in this field, the approach is indeed radical. It will appeal as a text; it should be in every library as a reference. Undergraduate. W. Roberts; Macalester College