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An Infinite Series Approach to Calculus

Susan Bassein
Publish or Perish, Inc.
Publication Date: 
Number of Pages: 
[Reviewed by
Stan VerNooy
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This book covers most of the traditional topics in a single-variable calculus course, as well as a small amount of material on partial derivatives of functions of two variables. However, it covers those topics in a decidedly non-traditional manner, and in a non-traditional sequence.

The author believes that, first, students absorb new information best when it is presented as an extension of material that they already know — a contention which in my opinion can hardly be disputed. Second, she believes that the topic of infinite series is easier to connect with the beginning calculus student's existing knowledge than the more traditional initial topic of limits of continuous functions — a contention which I find highly plausible, but which others might want to argue with. Finally, she believes that the approach in this book is more consistent with the view that math and science are a search for intellectual harmony with the world rather than an attempt to subdue it. I personally see little if any basis for that third belief, but a reader's appreciation of this book is unlikely to be influenced by her or his position on that issue.

The book is not a heavy tome along the lines of of Stewart, Anton, Larson, Thomas, etc. It is 361 pages long, including the index and appendices. (By contrast, the single-variable portion of the Larson/Hostetler text has 760 pages.) In terms of topics covered, it is much closer to the recent trend toward a "streamlined" version of calculus. There are no deep discussions of (for example) trigonometric integrals or trigonometric substitutions, partial fractions, or implicit differentiation.

However, what separates this book from other calculus books is not the list of topics omitted, but the unique approach to the entire subject. By defining limits (and almost everything else) in terms of the discrete steps in an infinite series, the book provides new insights on almost every page. For teachers who have spent decades teaching calculus in the traditional way, reading this book is likely to freshen our ideas even if we don't use it as a text. The author's approach is to define the real number system in terms of infinite decimal expansions. Limits are then defined by looking at the successive truncations of the decimal expansions of the numbers in the sequence. There are no deltas or epsilons. The derivative is defined as the limit of the difference quotient, as usual, but of course the conception of a limit is not the same as it is in the traditional textbooks.

The traditional formula for the derivative of a polynomial is presented (and proven), and that rule plays a more fundamental role in this book than it does in traditional books. That's because the exponential and trigonometric functions are defined by their MacLaurin series (which of course are not referred to as MacLaurin series). Those definitions enable us to differentiate them as infinite polynomials. The logarithmic function is then defined as the inverse of the exponential, rather than as the antiderivative of 1/x.

This book's departure from the traditional is not limited to the sequence of topics. Almost every topic is approached from a more conceptual and less mechanical viewpoint than traditional texts. There are fewer routine, mechanical exercises than in those traditional texts. Most of the exercises are thought-provoking (and thought-requiring), and many of the major ideas and theorems are found in the exercises rather than in the body of the text. For example, the statement and proof of the Mean Value Theorem are an exercise. This book was published in 1993, but as far as I know has received much less attention than some other reform texts, such as Hughes-Hallett/Gleason, et. al. The author claims considerable success in using her approach in her classes at Mills College, but I would be interested in learning of other people's experiences with it. The ideas in this book deserve wider dissemination and more experimentation.

Stan VerNooy is a full-time member of the mathematics faculty at the Community College of Southern Nevada in Henderson, Nevada. He received his MA in mathematics from the University of Maryland, College Park, in 1973; but he developed his true love for the teaching of mathematics while working as a graduate teaching assistant at the University of Oregon more than a decade later, while earning no degree whatsoever.

PART 1    Making sense of infinity
                     1 Discrete models
                     2  Functions and continuity of the line
                     3  Infinite series
                     4  Functions and power series

PART 2    Differentiation: rate of change
                     5  Linear and quadratic functions
                     6  Derivatives of Polynomials
                     7  Power series and differentiability
                     8  Differential equations
                     9  Compositions and quotients of functions
                    10  Inverse functions
                    11  Extreme values
                    12  Graphing
                    13 Functions of two variables

PART 3    Integration
                    14  Anti-derivatives
                    15  Areas
                    16  Techniques of integration
                    17  Lengths
                    18  Volumes and surface areas
                    19 Approximation of integrals
                    20 Improper integrals
                    21 Analytic functions
                    22 Techniques in solving differential equations