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Real Infinite Series

Daniel D. Bonar and Michael J. Khoury
Mathematical Association of America
Publication Date: 
Number of Pages: 
[Reviewed by
Underwood Dudley
, on

What a good book! It’s part of the MAA’s Classroom Resource Materials series and it fits it perfectly.

It starts with a leisurely 90-page treatment of series that includes all that is found in the usual calculus texts and more — Raabe’s test, for example, doesn’t appear in most calculus books. Anyone teaching or learning about series could go through it with profit. Students who are having difficulty with series could be directed to it. (Of course, students who are having difficulty often can’t understand mathematical prose no matter how clearly and well it is written, or refuse to read it, but there can be exceptions to this all too general rule.) Students who are doing well could have the pleasure of reinforcing their learning and seeing new things. Instructors can make their learning broader and deeper. The authors use the Schlömilch Condensation Test, of which I had never heard, to show that a series converges, but I was able to do that more quickly not using the test. So, I learned about a new test and had fun too.

Next come sixty pages of what the authors call Gems, results that are pretty, likely to be unfamiliar, and can be elegantly established. It may be familiar that 1 + (1/2) – (2/3) + (1/4) + (1/5) – (2/6) + (1/7) + (1/8) – (2/9) + … converges to ln 3, but how to show that, though it’s obvious after it’s explained, is probably not at everyone’s fingertips. They make for fascinating reading. It’s possible that a student exposed to the Gems could be infected with a life-long love for series, and for mathematics. This would not happen very often, but even if it occurred only a small integer number of times, the world would be a better place.

There follow series problems and solutions from the Putnam examinations. Ordinary mortals will not be able to solve many of the problems, but it is enjoyable to read their solutions. The authors then include materials mostly taken from MAA journals, including proofs without words and examples of fallacious reasoning, and then 101 true-false questions. They are valuable material for students and teachers, even for stealing items for examinations.

Reviewers are traditionally allowed one complaint, so I will make mine about the index. Besides spelling Gauss’s name wrong, it’s perfunctory. When I read that “we will see in Ch. 4 that this test [the Root Test] is actually stronger than the Ratio Test” I was intrigued: I always thought that they did exactly the same thing, namely disclose what geometric series a given series was behaving like. But Chapter 4 is sixty-two pages long, and the index is no help in telling me where to find this new result. A search that was more then perfunctory but less than exhaustive did not turn it up, so my curiosity may have to go unsatisfied. But after the first edition rapidly sells out, as it deserves to, the second edition can fix this.

A better Classroom Resource could hardly be imagined.

Woody Dudley has retired from DePauw University and is now living in Florida.

 Contents: 1 Introduction to Infinite Series 1 1.1 Definitions 1 1.2 Special Series 9 1.3 Intuition and Infinity 15 1.4 Basic Convergence Tests 22 1.5 General Series 35 2 More Sophisticated Techniques 41 2.1 The Work of Cauchy 41 2.2 Kummer's Results 46 2.3 The Tests of Raabe and Gauss 48 2.4 Logarithmic Scales 52 2.5 Tests of Abel 54 Appendix: Proofs of Bertrand's Tests 58 3 The Harmonic Series and Related Results 65 3.1 Divergence Proofs 66 3.2 Rate of Growth 72 3.3 The Alternating Harmonic Series 73 3.4 Selective Sums 79 3.5 Unexpected Appearances 85 4 Intriguing Results 89 4.1 Gems 90 5 Series and the Putnam Competition 151 5.1 The Problems 152 5.2 The Solutions 159 6 Final Diversions 193 6.1 Puzzles 193 6.2 Visuals 195 6.3 Fallacious Proofs 206 6.4 Fallacies, Flaws and Flimflam 209 6.5 Answers to Puzzles 215 Appendix A: 101 True or False Questions 219 Appendix B: Harmonic Series Article 243 Appendix C: References 247 Books on Infinite Series 247 Books with Excellent Material on Infinite Series 249 Sources for Excellent Problems Related to Infinite Series 250 Pleasurable Reading 251 Journal Articles 252 Index 259 About the Authors 263