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Excursions in Classical Analysis

Hongwei Chen
Mathematical Association of America
Publication Date: 
Number of Pages: 
Classroom Resource Materials
[Reviewed by
Henry Ricardo
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The author’s intention in this enlightening and inspiring book is to introduce the reader to advanced problem solving techniques via case studies. It has twenty-one relatively short independent chapters that contain “kernels of sophisticated ideas connected to important current research” and expose the principles underlying these ideas. Chen is a devotee of Pólya’s rubric for problem solving and practices what he preaches:

…I have begun each topic by categorizing and identifying the problem at hand, then indicated which technique I will use and why, and ended by making a worthwhile discovery or proving a memorable result. I often take the reader through a method which presents rough estimates before I derive finer ones, and I demonstrate how the more easily solved special cases often lead to insights that drive improvements of existing results. Readers will clearly see how mathematical proofs evolve ― from the specific to the general and from the simplified scenario to the theoretical framework.

Each chapter begins with an appropriate quotation and ends with exercises and a brief list of references. (Chapter 18 references a paper written by the author’s son when he was an undergraduate.) Many of the examples and problems are taken from the Monthly or the Putnam exam; and there are 14 open problems. Selected problem solutions are provided at the back of the book. There aren’t many figures, but each page is dense with (beautiful) mathematical formulas.

The first three chapters discuss inequalities and are relatively elementary. Chapter 2 presents the author’s new approach to proving inequalities based on the fact that

where E is a subset of R. Together with convexity, Chen’s method yields proofs of the AGM and Cauchy-Schwarz inequalities, as well as a new elementary proof Ky Fan’s inequality connecting weighted arithmetic, geometric, and harmonic means. As is the case for several of the chapters, this work is based on the author’s published work. In Chapter 3, the author shows that six standard means can be represented by the function

for different values of t. By showing that f(t) is strictly increasing, Chen derives inequalities connecting the six means. I remember being impressed by the author’s paper (Mathematics Magazine, December 2005) on which this exposition is based.

Chapter 6 treats generating functions, Fibonacci numbers, harmonic numbers, Bernoulli numbers, and the Euler-Maclaurin summation formula. Even though there’s nothing particularly new here, the material is developed very nicely.

In Chapter 18, following a paper of D. H. Lehmer, the author defines an “interesting” series as one for which there is a simple explicit formula for its nth term and which has a sum with a closed form expressed in terms of well-known constants ― for example, the Gregory-Leibniz series for π/4 or the series defining Apery’s constant, z(3). Here, as in Lehmer’s paper, Chen focuses on series involving reciprocal binomial coefficients. The author goes beyond Lehmer’s focus on series involving the central binomial coefficient and ends this chapter with a search for new formulas for π employing the methods of experimental mathematics.

Some readers may recognize Chen’s name as an avid proposer and solver in several journal problem sections. The last chapter presents Chen’s solutions to eight integral problems that have appeared in the Monthly, half of which have not had their solutions published at the time of this book’s debut.

This book is similar to Excursions in Calculus: An Interplay of the Continuous and the Discrete by Robert M. Young. However, whereas Young’s book (also invoking the spirit of Pólya) is intended as a supplement to calculus texts, the work under review is for the most part clearly meant to supplement advanced calculus or real analysis courses.

Chen’s book is a wonderful updated tour of classical analysis and would serve as an excellent source of undergraduate enrichment/research problems. It recalls the type of gems in classical analysis, number theory, and combinatorics I first encountered in the books of Pólya and Szegö as an undergraduate many years ago. Peruse the Table of Contents and see if some of the topics and subtopics don’t grab you.



Henry Ricardo ( has retired from Medgar Evers College (CUNY), but continues to serve as Governor of the Metropolitan NY Section of the MAA. He is the author of A Modern Introduction to Differential Equations (Second Edition). His linear algebra text was published in October 2009 by CRC Press.


1 Two Classical Inequalities
1.1 AM-GM Inequality
1.2 Cauchy-Schwarz Inequality

2 A New Approach for Proving Inequalities

3 Means Generated by an Integral

4 The L’Hôpital Monotone Rule

5 Trigonometric Identities via Complex Numbers
5.1 A Primer of complex numbers
5.2 Finite Product Identities
5.3 Finite Summation Identities
5.4 Euler’s Infinite Product
5.5 Sums of inverse tangents
5.6 Two Applications

6 Special Numbers
6.1 Generating Functions
6.2 Fibonacci Numbers
6.3 Harmonic numbers
6.4 Bernoulli Numbers

7 On a Sum of Cosecants
7.1 A well-known sum and its generalization
7.2 Rough estimates
7.3 Tying up the loose bounds
7.4 Final Remarks

8 The Gamma Products in Simple Closed Forms

9 On the Telescoping Sums
9.1 The sum of products of arithmetic sequences
9.2 The sum of products of reciprocals of arithmetic sequences
9.3 Trigonometric sums
9.4 Some more telescoping sums

10 Summation of Subseries in Closed Form

11 Generating Functions for Powers of Fibonacci Numbers

12 Identities for the Fibonacci Powers

13 Bernoulli Numbers via Determinants

14 On Some Finite Trigonometric Power Sums
14.1 Sums involving sec2p(kπ/n)
14.2 Sums involving csc2p(kπ/n)
14.3 Sums involving tan2p(kπ/n)
14.4 Sums involving cot2p(kπ/n)

15 Power Series of (arcsin x)2
15.1 First Proof of the Series (15.1)
15.2 Second Proof of the Series (15.1)

16 Six Ways to Sum ζ(2)
16.1 Euler’s Proof
16.2 Proof by Double Integrals
16.3 Proof by Trigonometric Identities
16.4 Proof by Power Series
16.5 Proof by Fourier Series
16.6 Proof by Complex Variables

17 Evaluations of Some Variant Euler Sums

18 Interesting Series Involving Binomial Coefficients
18.1 An integral representation and its applications
18.2 Some Extensions
18.3 Searching for new formulas for π

19 Parametric Differentiation and Integration
Example 1
Example 2
Example 3
Example 4
Example 5
Example 6
Example 7
Example 8
Example 9
Example 10

20 Four Ways to Evaluate the Poisson Integral
20.1 Using Riemann Sums
20.2 Using A Functional Equation
20.3 Using Parametric Differentiation
20.4 Using Infinite Series

21 Some Irresistible Integrals
21.1 Monthly Problem 10611
21.2 Monthly Problem 11206
21.3 Monthly Problem 11275
21.4 Monthly Problem 11277
21.5 Monthly Problem 11322
21.6 Monthly Problem 11329
21.7 Monthly Problem 11331
21.8 Monthly Problem 11418



Solutions to Selected Problems