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Monthly Problem Gems

Hongwei Chen
Chapman and Hall/CRC
Publication Date: 
Number of Pages: 
Problem Book
[Reviewed by
Henry Ricardo
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In the author’s words, “Each selected problem has a central theme, contains gems of sophisticated ideas connected to important current research, and opens new vistas in the understanding of mathematics.”
A problem and its solution are springboards that Chen uses to launch into related areas of mathematics and encourage the reader to transition from problem solving to creative research.  The spirit of George Pólya hovers over this book.  As in his earlier book Excursions in Classical Analysis, heuristic reasoning is emphasized as a way of developing insight and intuition.  Most of the solutions are the author’s, even when they were not featured in the Monthly. My only quibble is that there are only seven figures in the book.
The first four chapters focus on key themes of analysis—limits, infinite series, integrations, and inequalities. In Chapter 4, for example, there is an illuminating discussion of “a bound of divisor sums related to the Riemann hypothesis,” a weaker version of an inequality equivalent to the famous unsolved problem, which uses Mathematica to verify certain number-theoretic estimates. The fifth chapter is a miscellany consisting of mean value theorems, (0, 1)–matrices, weighted trigonometric sums, Dirichlet series, and other topics, with the intention “to facilitate a natural transition from problem-solving to independent exploration of new results.”  
Appendix A is a reference list of all problems treated, including the “problems for additional practice.”  Putnam problems and problems in Mathematics Magazine and The College Mathematics Journal are noted.  Appendix B is a glossary of definitions and theorems, ranging from Abel’s limit theorem to the Wallis product formula.  Lists of useful inequalities, power series, and Fourier series follow the alphabetical glossary. The book ends with a ninety-four-item bibliography and a helpful Index.
Although most of the featured problems in Monthly Problem Gems have appeared since the publication of Excursion in Classical Analysis, there is some overlap.  For example, problem 10611, an error function inequality, appears in both books, with the same solution.  However, in MPG, Chen adds related material such as an asymptotic series for the error function, as well as a discussion of inequalities involving the complementary error function  Similarly, problem 11329 (about log gamma integrals) is treated in both books, but with added heuristic comments in the new book.  Furthermore, the author gives some background to the problem and provides information about the Laplace transform of the digamma function and examples of other definite integrals in which the integrand is a combination of powers, logarithms, and trigonometric functions. 
To quote Abraham Lincoln (who had no opportunity to read this book): “People who like this sort of thing will find this the sort of thing they like.”  In this case, the word ‘people’ refers to good undergraduate math students, participants in mathematical competitions, teachers, regular readers of journal problem columns, and all who want to improve their understanding of classical analysis. 


Henry Ricardo ([email protected]) has retired from Medgar Evers College (CUNY) as Professor of Mathematics.  He is currently affiliated with the Westchester Area Math Circle. He is the author of A Modern Introduction to Differential Equations and A Modern Introduction to Linear Algebra.  His first featured solutions to Monthly problems appeared in 1967.

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A less expensive paperback edition is available. Libraries, however, should purchase the hard cover version.