This is a great resource of concepts and challenging problems that is useful for teaching and exploring infinite series at any level, in calculus, in analysis, or as preparation for the Putnam Examination.
— David Bressoud, DeWitt Wallace Professor, Macalester College
This is an introductory treatment of infinite series of real numbers, bringing the reader from basic definitions and tests to advanced results. An up-to-date presentation is given, making infinite series accessible, interesting, and useful to a wide audience, including students, teachers, and researchers.
In its most basic setting, infinite series is the vehicle mathematicians use to extend finite addition to “infinite addition.” Real Infinite Series presents elementary and advanced tests for convergence or divergence, information about the harmonic series, the alternating harmonic series, and closely related series. One chapter offers 107 concise, crisp, surprising results about infinite series. Recognizing the interest in problem solving that abounds with students of mathematics, the authors devote a chapter to problems on infinite series, and solutions, which appeared on the annual William Lowell Putnam Mathematical Competition.
The lighter side of infinite series is treated in the concluding chapter where three puzzles, eighteen visuals (what Martin Gardner calls “look-see” diagrams), and several fallacious proofs are made available.
Three appendices provide a listing of true or false statements, answers to why the harmonic series is so named, and an extensive list of published works devoted entirely or partially to infinite series.
Table of Contents
1. Introduction to Infinite Series
2. More Sophisticated Techniques
3. The Harmonic Series and Related Results
4. Intriguing Results
5. Series and the Putnam Competition
6. Final Diversions
Appendix A: 101 True or False Questions
Appendix B: Harmonic Series Article
Appendix C: References
About the Authors
About the Authors
Daniel Donald Bonar was born in Murraysville, West Virginia on July 7, 1983, the son of Nelson Edward Bonar II and Ada Polk Bonar. He graduated from Ravenswood High School and was awarded a four-year full-tuition scholarship to West Virginia University where he received the BS in Chemical Engineering in 1960. While at WVU, he was a member of the physics, chemistry, and chemical engineering honoraries and served as President of Tau Beta Pi, the engineering honorary. Two National Science Foundation Fellowships supported his graduate work in mathematics. He received the MS from WVU in 1961 with a major in mathematics and a minor in physics. His PhD work was in complex analysis at Ohio State University in Granville, OH where he has been teaching mathematics, statistics, and computer science.
Awards received by Don Bonar include the Richard King Mellon Foundation Award for excellence in teaching and scholarship in 1973 and the Sears-Roebuck Teaching Excellence and Community Leadership Award in 1991. In 1995 he was appointed to the newly created George R. Stibitz Distinguished Professorship in Mathematics and Computer Science. In 1999 Don was inducted into the Academy of Chemical Engineers at West Virginia University. He is the author of a book entitled On Annular Functions, and is co-author on several research papers. He has published joint work with the internationally acclaimed Hungarian mathematician Paul Erdős. Community service includes membership on the Granville Foundation, the Granville Development Commission, the Licking County (OH) Joint Vacation School Board, and serving as President of the Granville Exempted Village School Board.
Don and his wife Martha Baker Bonar are the parents of Mary Martha, a student in medical school at Ohio University. The Bonars enjoy time at their farm, family-owned since 1869, in West Virginia.
Michael John Khoury, Jr. was born to Michael Sr. and Amy Khoury in Cuyahoga Falls, OH on April 28, 1981. He graduated as co-valedictorian from Brother Rice High School in Bloomfield Hills, MI. He studied for four years at Denison University in Granville, OH under the Wells Scholarship. He was a member of the mathematics and the education honoraries and president of the former; he also served as a departmental fellow in the Mathematics and Computer Science Department. Michael Khoury graduated in 2003 with a BS in Mathematics and Education, as well as the President Medal (awarded to only six members of his class). He is currently pursuing a PhD in number theory at the Ohio State University with the support of the National Science Foundation.
Michael Khoury was a prize winner in the Putnam Mathematical Competition, and participated twice in the Math Olympiad Summer Program in Lincoln, NE, a four-week program open to the top performers on the USA Math Olympiad.