A Radical Approach To Real Analysis (Classroom Resource Materials)

A Radical Approach to Real Analysis (Classroom Resource Materials)

This book is an undergraduate introduction to real analysis. Teachers can use it as a textbook for an innovative course, or as a resource for a traditional course. Students who have been through a traditional course, but do not understand what real analysis is about and why it was created, will find answers to many of their questions in this book. Although this is not a history of analysis, the author returns to the roots of the subject to make it more comprehensible. The book begins with Fouriers introduction of trigonometric series and the problems they created for the mathematicians of the early nineteenth century. Cauchys attempts to establish a firm foundation for calculus follow, and the author considers his failures and his successes. The book culminates with Dirichlets 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 Dirichlets proof. Mathematica ® commands and programs are included in the exercises. However, the reader may use any mathematical tool that has graphing capabilities, including the graphing calculator.