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...

Ausführliche Beschreibung

Bibliographische Detailangaben

1. Verfasser:	Bressoud, David M. 1950- (VerfasserIn)
Format:	Elektronisch E-Book
Sprache:	English
Veröffentlicht:	Washington, DC MAA Press, an imprint of the American Mathematical Society [2007]
Ausgabe:	Second edition
Schriftenreihe:	AMS/MAA textbooks vol. 10
Schlagworte:	Reelle Funktion Analysis Mathematical analysis Electronic books
Online-Zugang:	DE-91 Volltext
Zusammenfassung:	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. Cover -- copyright page -- 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 -- Appendix 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! -- Appendix B Bibliography -- Appendix C Hints to Selected Exercises -- Index -- Back cover.
Beschreibung:	1 Online-Ressource (xv, 322 Seiten) Illustrationen, Diagramme
ISBN:	9781614446231