# Resources for the Study of Real Analysis

###### Robert L. Brabenec
Publisher:
MAA
Publication Date:
2004
Number of Pages:
231
Format:
Hardcover
Series:
Classroom Resource Materials
Price:
48.95
ISBN:
0-88385-737-5
Category:
Textbook
[Reviewed by
Ioana Mihaila
, on
03/1/2005
]

Resources for the Study of Real Analysis is an eclectic collection of problems, calculus results, and history tidbits, written for both students and instructors of analysis. The book is structured into four parts.

The first section of the book, entitled “Review of Calculus”, is mostly directed at students. It contains an outline of topics and a summary of important ideas and techniques. I like the straightforward presentation of this chapter, noticeable in statements like “know that $\frac10=\pm\infty$”.

Part II, “Analysis Problems” makes up the bulk of the book. Some of the problems are standard analysis fare, for example the Cantor middle third set, the Gamma function, and various proofs that $\displaystyle\sum_{n=1}^\infty=\frac{\pi^2}{6}$, others are more unusual, such as Problem 32 on “summability of divergent series”. The order of the problems is somewhat arbitrary, and the level of difficulty varies as well. Some problems, especially in the first two sections of Part II are meant to check the students’ understanding of basic concepts. For example, problems 8 and 9 (on topological notions and uniform continuity, respectively) ask the students to “fill in the table” with the relevant information. Other problems are better suited as projects (perhaps worked in groups) — especially the enrichment problems in the third section. I was pleased to find in this part of the book a comparison of the root and ratio tests for series, along with an introduction of the Raabe test.

“Essays”, the third part of the book has again three different sections. The first and largest is dedicated to a streamlined history of analysis, and includes biographical sketches of the founders of analysis. These essays guide the reader through the key points in the development of analysis, and make a nice starting point for student writing projects. Annotated papers supplied in the fourth part of the book, “Selected Readings”, supplement the historical information on L’Hopital, Bernoulli, Lebesgue, and others.

Overall, the book is what the title promises: a great resource for teaching and studying analysis. I would use this book in conjunction with others that I consider very valuable, such as Calculus Gems, by George Simmons (which Brabenec mentions in his annotated bibliography), Problems in Mathematical Analysis, by Kaczor and Nowak, Counterexamples in Analysis, by Gelbaum and Olmsted, and some history of mathematics resources. The presentation of the material is clear and the exercises are arranged with sufficient care to make the reading easy enough for students taking their first real analysis course. However, there is a lack of cohesion within the parts of the book, and as such this collection of problems and other analysis facts is probably more useful for instructors looking to spice up their courses, than for students.

Ioana Mihaila (imihaila@csupomona.edu) is Assistant Professor of Mathematics at Cal Poly Pomona. Her research area is analysis, and she is interested in mathematics competitions.