AMS / MAA CLASSROOM RESOURCE MATERIALS VOL 56 Real Infinite Series Daniel D. Bonar Michael J. Khoury Real Infinite Series Originally published by The Mathematical Association of America, 2006. ISBN: 978-1-4704-4782-3 LCCN: 2005937268 Copyright © 2006, held by the Amercan Mathematical Society Printed in the United States of America. Reprinted by the American Mathematical Society, 2018 The American Mathematical Society retains all rights except those granted to the United States Government. ⃝1 The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10 9 8 7 6 5 4 3 2 23 22 21 20 19 18 10.1090/clrm/056 AMS/MAA CLASSROOM RESOURCE MATERIALS VOL 56 Real Infinite Series Daniel D. Bonar Michael J. Khoury, Jr. Council on Publications Roger Nelsen, Chair Classroom Resource Materials Editorial Board Zaven A. Karian, Editor William C. Bauldry Stephen B Maurer Gerald Bryce Douglas Meade George Exner Judith A. Palagallo William J. Higgins Wayne Roberts Paul Knopp Kay B. Somers 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. 101 Careers in Mathematics, 2nd edition 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 R C. Kenschaft Essentials of Mathematics, Margie Hale Exploratory Examples for Real Analysis, Joanne E. Snow and Kirk E. Weller Fourier Series, Rajendra Bhatia Geometry From Africa: Mathematical and Educational Explorations, Paulus Gerdes Historical Modules for the Teaching and Learning of Mathematics (CD), edited by Victor Katz and Karen Dee Michalowicz Identification Numbers and Check Digit Schemes, Joseph Kirtland Interdisciplinary Lively Application Projects, edited by Chris Amey Inverse Problems: Activities for Undergraduates, Charles 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 Mathematical Connections: A Companion for Teachers and Others, A1 Cuoco 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 text­ book), Richard B. Thompson and Christopher G. Lamoureux Mathematics for Business Decisions Part 2: Calculus and Optimization (electronic text­ book), Richard B. Thompson and Christopher G. Lamoureux Ordinary Differential Equations: A Brief Eclectic Tour, David A. Sanchez Oval Track and Other Permutation Puzzles, John O. Kiltinen A Primer of Abstract 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 Real Infinite Series, Daniel D. Bonar and Michael Khoury, Jr. 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 Simula­ tion, David Williamson, Marilou Mendel, Julie Tarr, and Deborah Yoklic Student Manual for Mathematics for Business Decisions Part 2: Calculus and Optimiza­ tion, David Williamson, Marilou Mendel, Julie Tarr, and Deborah Yoklic Teaching Statistics Using Baseball, Jim Albert Topology Now!, Robert Messer and Philip Straffin Understanding our Quantitative World, Janet Andersen and Todd Swanson Writing Projects for Mathematics Courses: Crushed Clowns, Cars, and Coffee to Go, Annalisa Crannell, Gavin LaRose, Thomas Ratliff, Elyn Rykken Preface The theory of infinite series is an especially interesting mathematical construct due to its wealth of surprising results. In its most basic setting, infinite series is the vehicle we use to extend finite addition to “infinite addition.” The material we present deals almost exclu­ sively with real infinite series whose terms are constants. It is possible to spend an entire professional career studying infinite series, and there is no shortage of books written for such specialists. However, the authors of this book expect no such prerequisites from our readers. We believe that the general mathematician can and should understand infinite se­ ries beyond the Calculus II level; tragically, most existing presentations are either very basic or exclusively for the specialist. Between these covers, the reader will find discus­ sions that are deep enough and rich enough to be useful to anyone involved in mathematics or the sciences, but that are accessible to all. Indeed, even those who are not mathemati­ cians will be able to appreciate infinite series on an aesthetic level. This book is designed to be a resource on infinite series that may prove helpful to 1 ) teachers of high school cal­ culus, 2) teachers of college level calculus, 3) undergraduate math majors who have had a taste of infinite series by way of one or two chapters in a beginning calculus text and need to review that material while learning much more, or 4) students who have studied infinite series in advanced calculus or in elementary real analysis and have a thirst for more information on the topic. This book may also serve as a resource or supplemental text for a course on real infinite series for either advanced undergraduates or beginning graduate students. It may be of value in connection with a special topics course, or as one of several texts for a seminar on infinite series. Chapter 1 is devoted to a review of the basic definitions and theory of infinite series as found about midway through a thousand page beginning calculus text. This presentation is self-contained, so the reader who has not taken calculus or whose recollection of the idea of “limits” is shaky should in no way feel discouraged. The various tests for conver- gence/divergence such as the Comparison Test, the Limit Comparison Test, the Ratio Test, the Root Test, the Integral Test, etc. are presented in this chapter. While the mathematical content does not differ from that in many standard texts, the presentation is designed to give some understanding of the intuitive notion of an infinite series and to reconcile the definitions with an intuitive perspective. Most students and mathematicians know basic tests such as the Ratio and Root Tests— they are taught in most advanced calculus classes and are easy to apply and to remember. However, several more powerful tests that are only slightly more technical are largely un­ known. This is particularly unfortunate because many quite natural series elude analysis by these means; we believe that this can give the false impression that the study of in­ finite series has very limited applications for the nonspecialist. Chapter 2 is given over to these more delicate and sophisticated tests for convergence/divergence which one can use if the basic tests of Chapter 1 are insufficient; the reader will see that the greatly in­ creased power comes with only small costs in complexity. Semi-well-known tests such as Raabe’s Test, Rummer’s Test, Cauchy’s Condensation Test, Abel’s Test, and Dirichlet’s Test are discussed, and examples are offered to illustrate their worth, as well as tests such as Bertrand’s Test, whose obscurity belies their elegance and power. The harmonic series and the alternating harmonic series have intrigued mathematicians for centuries. Chapter 3 is largely devoted to facts concerning these two series and similar results for closely related series. Even exploring the origin of the name “harmonic” leads to interesting discoveries. While it is generally known that the harmonic series diverges while the alternating harmonic series converges, we believe that this exposition makes these statements more tangible, and we include several proofs of different types. This chapter deals with examples of rearranging series and some clever techniques for summing special series exactly. The exploration of the harmonic and alternating harmonic series gives some insight into more general series. In some sense we see Chapter 4 being the heart of the book. We work through 107 gems of mathematics relating in one way or another to infinite series. One purpose of this chapter is to demonstrate that the theory of infinite series is full of surprising and intriguing results. These results have many forms: they may show the exact sum of a series, they may exhibit a neat proof of a common sense result, or they may present a particularly slick problem­ solving technique. The gems presented here could be used as nontraditional examples in a classroom setting, or simply for the reader’s own enjoyment, which is justification enough. Some of these gems even allow us to push the material of the preceding chapters even further, giving the reader a taste of extensions such as multiple series and power series. The next chapter is dedicated to the annual Putnam Mathematics Competition, which has been in existence since 1938. In Chapter 5 we have reproduced nearly all the infinite series problems offered on those examinations. The reader who is so inclined may choose to take this as a challenge; however, solutions to the problems, as found in the literature, are also provided, in many cases accompanied by alternate solutions or commentary provided by the authors. Students of mathematics can learn a nice assortment of problem solving techniques by studying these problems and their solutions. The final official chapter is offered to the reader as a parting gift, and is fittingly called “Final Diversions.” Here we explore the lighter side of infinite series. We provide three puzzles, including the classic pebbling problem.