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Resources for the Study of Real Analysis 4 AMS / MAA PRESS AMS / MAA TEXTBOOKS VOL 4 Resources for the Study VOL of Real Analysis AMS / MAA TEXTBOOKS 4 Robert L Brabenec The book offers even more than its title suggests: a true trove of resources for students (and their teachers) who face the exciting—but often rough— passage from the routine calculations of elementary calculus to the Analysis of Real the Study for Resources deeper arguments and insights of real analysis. There are gems here for average students who need review; for the quickest students who need mathematical challenges; for teachers who need ideas; and for everyone with a taste for mathematical highlights and culture. —Paul Zorn, St. Olaf College Brabenec has provided a rich smorgasbord of mathematical analysis— L Brabenec Robert including a host of problems, historical essays, and selected readings— from which no one should go away hungry. —William Dunham, Muhlenberg College This book is a collection of materials gathered by the author while teaching real analysis over a period of years. It is intended for use as a supplement to a traditional analysis textbook, or to provide material for seminars or independent study in analysis and its historical development. The book includes historical and biographical information, a wide range of problem types, selected readings on a variety of topics, and many references for additional study. Since all these materials are collected into a single book, teachers and students can easily choose items most suitable for their purpose. Teachers may use the book as a supplement to their courses, while students may read much of the book on their own. No other book has been written specifically as a supplement for a real analysis course. AMSMAA / PRESS Trim Size 7 x 10 243 pages • Spine: 1/2 in • 50lb paper • Softcover • 4-color Process (CMYK) Resources for the Study of Real Analysis 10.1090/text/004 Resources for the Study of Real Analysis by Robert L. Brabenec Wheaton College Published and Distributed by THE MATHEMATICAL ASSOCIATION OF AMERICA CLASSROOM RESOURCE MATERIALS Classroom Resource Materials is intended to provide supplementary classroom material for students-laboratory exercises, projects, historical information, textbooks with unusual approaches for presenting mathematical ideas, career information, etc. Council on Publications Roger Nelsen, Chair Zaven A. Karlan, Editor William Bauldry Daniel E. Kullman Gerald Bryce Stephen B Maurer Sheldon P. Gordon Douglas Meade William J. Higgins Judith A. Palagallo Mic Jackson Wayne Roberts Paul Knopp 101 Careers in Mathematics, edited by Andrew Sterrett Archimedes: What Did He Do Besides Cry Eureka?, Sherman Stein Calculus Mysteries and Thrillers, R. Grant Woods Combinatorics: A Problem Oriented Approach, Daniel A. Marcus Conjecture and Proof, Miklos Laczkovich A Course in Mathematical Modeling, Douglas Mooney and Randall Swift Cryptological Mathematics, Robert Edward Lewand Elementary Mathematical Models, Dan Kalman Environmental Mathematics in the Classroom, edited by B. A. Fusaro and P. C. Kenschaft Essentials of Mathematics: Introduction to Theory, Proof, and the Professional Culture, Margie Hale Exploratory Examples for Real Analysis, Joanne E. Snow and Kirk E. Weller Geometry from Africa: Mathematical and Educational Explorations, Paulus Gerdes Identification Numbers and Check Digit Schemes, Joseph Kirtland Interdisciplinary Lively Application Projects, edited by Chris Amey Inverse Problems: Activities for Undergraduates, C. W. Groetsch Laboratory Experiences in Group Theory, Ellen Maycock Parker Learn from the Masters, Frank Swetz, John Fauvel, Otto Bekken, Bengt Johansson, and Victor Katz Math through the Ages: A Gentle History for Teachers and Others (Expanded Edition), William P. Berlinghoff and Fernando Q. Gouvea Mathematical Evolutions, edited by Abe Shenitzer and John Stillwell Mathematical Modeling in the Environment, Charles Hadlock Mathematics for Business Decisions Part 1: Probability and Simulation (electronic textbook), Richard B. Thompson and Christopher G. Lamoureux Mathematics for Business Decisions Part 2: Calculus and Optimization (electronic textbook), Richard B. Thompson and Christopher G. Lamoureux Ordinary Differential Equations: A Brief Eclectic Tour, David A. Sanchez Oval Track and Other Permutation Puzzles, Robert B. Ash A Primer ofAbstract Mathematics, Robert B. Ash Proofs Without Words, Roger B. Nelsen Proofs Without Words II, Roger B. Nelsen A Radical Approach to Real Analysis, David M. Bressoud Resources for the Study of Real Analysis, Robert L. Brabenec She Does Math!, edited by Marla Parker Solve This: Math Activities for Students and Clubs, James S. Tanton Student Manual for Mathematics for Business Decisions Part 1: Probability and Simulation, David Williamson, Marilou Mendel, Julie Tarr, and Deborah Yoklic Student Manual for Mathematics for Business Decisions Part 2: Calculus and Optimization, David Williamson, Marilou Mendel, Julie Tarr, and Deborah Yoklic Tepching Statistics Using Baseball, Jim Albert Writing Projects for Mathematics Courses: Crushed Clowns, Cars, and Coffee to Go, Annalisa Crannell, Gavin LaRose, Thomas Ratliff, and Elyn Rykken MAA Service Center P. 0. Box 91112 Washington, DC 20090-1112 1-800-331-lMAA FAX: 1-301-206-9789 www.maa.org This book is dedicated to the hundreds of Wheaton College students who have taken my course in real analysis during the past forty years. Their enthusiasm and contributions enriched these classes and helped to shape the material in this book. Contents Preface xi Review of Calculus 1 An Outline of a Traditional Course in Calculus 1 Calculus Review Problems 6 II Analysis Problems 19 Introduction 19 Section 1. Basic Problems Problem l. Properties of Even and Odd Functions 19 Problem 2. Use of the Cancellation Principle 22 Problem 3. Finding Maclaurin Series Representations 26 Problem 4. Evaluating Jd xk dx by Cavalieri Sums 29 Section 2. Supplementary Problems Problem 5. The Pervasive Nature of Sequences 31 Problem 6. Counterexamples 36 Problem 7. Some Examples of Unusual Functions 37 Problem 8. Identifying Interior, Exterior, Boundary, and Limit Points 39 Problem 9. Exploring Uniform Continuity 40 Problem 10. Comparing the Ratio Test, the Root Test, and Raabe's Test 42 Problem 11. Various Proofs that L:'= 1 ;!,: = 1~t 47 viii Part . Contents Problem 12. Finding a Value for I::= 1 n\ 51 Problem 13. Rearrangements of Conditionally Convergent Series 52 Problem 14. Finding Approximations for rr 61 Problem 15. A Function with Derivatives of All Orders, but No Maclaurin Series 68 Problem 16. Evaluating Jd xk dx by Fermat's Method 69 Problem 17. Two Proofs that Jd (x lnx)ll dx = (-l)nn!/(n + l)<n+l) 70 Problem 18. Evaluating Jd xx dx by Bernoulli's Method 72 Problem 19. Some Interesting Sequences of Functions 73 Problem 20. Comparing Uniform Continuity and Uniform Convergence 74 Problem 21. Questions for Sequences of Functions and for Series of Functions 75 Section 3 .. Enrichment Problems Problem 22. Generalizing the Fibonacci Sequence 79 Problem 23. Euler's Proof that e Is Irrational 81 Problem 24. Subseries of the Harmonic Series 83 Problem 25. Finding Functions whose Maclaurin Series is I::= 1 n k xn 87 Problem 26. A Pascal-like Triangle 91 Problem 27. Finding Areas without Antiderivatives 95 Problem 28. Questions from the Euler-Stirling Correspondence 100 Problem 29. The Gamma Function 102 Problem 30. Bernoulli Numbers 109 Problem 31. A Look at Fourier Series 112 Problem 32. Summability of Divergent Series 115 Problem 33. Lebesgue Measure 121 Problem 34. Cantor's Middle Third Set 123 Ill Essays 125 Section 1. History and Biography Essay 1. The Time of Archimedes and Other Giants (pre-1660) 126 Essay 2. The Time of Newton and Leibniz (1660-1690) 129 Essay 3. The Time of Euler and the Bernoullis (1690-1790) 132 Essay 4. The Time of Cauchy, Fourier, and Lagrange (1790-1850) 135 Essay 5. The Time of Riemann and Weierstrass (1850-1900) 139 Essay 6. The Time of Lebesgue and Hardy (1900-1940) 142 Section 2. New Looks at Calculus Content Essay 7. Obtaining the Derivative Formulas 147 Essay 8. Tests for Convergence of Series 155 Section 3. General Topics for Analysis Essay 9. Proof Techniques in Analysis 163 Essay 10. Sets and Topology 170 Part . Contents Ix IV Selected Readings 175 Introduction 175 Selection 1. John Bernoulli and the Marquis de l'Hopital 177 Selection 2. The Euler-Stirling Correspondence 179 Selection 3. Developing Rigor in Calculus 187 Selection 4. The Contribution of Lebesgue 198 Selection 5. The Bernoulli Numbers and Some Wonderful Discoveries of Euler 202 Annotated Bibliography 207 Addltlonal References 217 Index 221 Preface This book is a collection of materials I have gathered while teaching a real analysis course every year for more than thirty-five years. I prepared it with the hope it will benefit and enrich the experience both of students who take a real analysis course, as well as those who teach it. This collection is intended to supplement a traditional real analysis textbook, where teachers and students may choose items of interest to them. It is my conviction that such supplementary materials have a much greater chance of being used in an analysis course if they are readily available in one place. Part I contains materials that provide the greatest benefit if read before the real analysis course begins. Because such a course assumes knowledge of topics from calculus of one real variable, review of this material in advance frees the student to concentrate on new content and theoretical emphases in analysis without having to revisit the calculus at the same time. The outline of a traditional calculus course allows a student to check necessary topics for review, whereas the calculus review problems offer the opportunity to refresh necessary skills which may have lain dormant for some time. Many hints are provided with the problems to encourage students to carry out this review. The problems in Part II are intended to supplement the ones usually found in an analysis text. There is a wide variety of problem types, and each one has exercises for the student to attempt.
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