You are here

Limits, Limits Everywhere: The Tools of Mathematical Analysis

David Applebaum
Oxford University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Mehdi Hassani
, on

There is no doubt that the concept of “limit” is fundamental in calculus and analysis. Many other essential topics, such as derivative and integral, are defined as limits of some kind, as are topics in analysis like sequences and series. The notion of limit is also hidden in heart of some primordial matters such as the notion of a “number”. These are some facts that the author of the book under review is focused on.

The book is divided into two parts. The first part consist of the initial six chapters, in which the author talks about preliminary concepts: numbers (natural, prime, rational, irrational, and real) and inequalities. He also writes about the limit of sequences and series, and proves some standard facts about them. All of chapters 1–6 contain exercises at the end. The second part of the book consists of the seven remaining chapters, in which the author explores some facts related to limits, such as special numbers, infinite products, and continued fractions. Some basic concepts, including constructing the real numbers and the limit of functions, are studied in these chapters. Chapters 7–13 don’t contain exercises. The book ends with some short appendices followed by hints and solutions to selected exercises.

The book is infected by a number of errors. Some of these errors are serious, and one cannot condone them. Here, I list some of them for careful readers.

  • On the top of the page 82, it is claimed that Figure 6.1 represents some initial triangular numbers, but the figure just represents simply the first few odd numbers.
  • On the page 116, where the author talks about the Bernoulli numbers \(B_n\), he claims that \(B_{2k+2}\) is always negative. In fact the Bernoulli numbers of even order change sign alternately.
  • On the page 129, where the author talks about Wallis’s infinite product for \(\pi\), he writes \[ \frac{\pi}{2}=\frac{2^24^26^2\cdots}{3^25^27^2\cdots}, \quad\text{and}\quad \frac{\pi}{2}=\prod_{n=1}^\infty\frac{(2n)^2}{(2n+1)^2}.\] This is a strangely wrong statement of Wallis’s product! I do not have enough space to describe the details and state the true formula, so let’s just point out that \(\frac{\pi}{2}\) is strictly larger than 1, but \(\frac{(2n)^2}{(2n+1)^2}\) is always strictly less than 1 for any positive integer \(n\).
  • On the page 130, the author writes in the initial line of Proof 1 of Theorem 8.2.1, “by a binomial series expansion”. Obviously, here he uses a geometric series, not the binomial series.
  • On the line 4 of the page 131, the author writes “the sum is over all natural numbers whose prime factors are no larger than \(N\).” In fact the sum that author talking about is over all natural numbers whose prime factors are strictly less than \(N\).
  • On the page 136, line 28, when the author writes about the analytic continuation of the Riemann zeta function for \(\Re(s)<1\), he writes “there is no clean formula”. But in fact there are several ways to continue this important function analytically for \(\Re(s)<1\) with clear and clean formulas.
  • On the page 137 the author writes some notes about the prime number theorem (PNT), and the Riemann hypothesis (RH). After introducing the logarithmic integral function for \(x>2\) by \(\mathrm{li}(x)=\int_2^x\frac{1}{\log_e(y)}dy\), he writes “and we have the stronger form of the prime number theorem: \(\lim_{n\to\infty}\frac{\pi(n)}{ \mathrm{li}(n)}=1\)”. This is an equivalent form of PNT, and not actually “stronger.” In what follows, the author mentions an explicit approximation for the prime counting function \(\pi(x)\) in terms of \(\mathrm{li}(x)\) due to Schoenfeld (1976). Here, the note is that Schoenfeld defines \(\mathrm{li}(x)\) as the Cauchy Principal Value (CPV) of the above mentioned integral starting from 0 instead of 2. This means that there is a small difference between Schoenfield’s \(\mathrm{li}(x)\) and the author’s, which might well have an effect on the cited explicit formula.

This book is not a text book, but after correcting the above mentioned errors and other possible ones, it may be helpful for college students to understand and find more about the concept of limit.

Mehdi Hassani is a faculty member at the Department of Mathematics, Zanjan University, Iran. His fields of interest are Elementary, Analytic and Probabilistic Number Theory.

I Approaching Limits
1. A Whole Lot of Numbers
2. Let's Get Real
3. The Joy of Inequality
4. Where Do You Go To, My Lovely
5. Bounds for Glory
6. You Cannot be Series
II Exploring Limits
7. Wonderful Numbers
8. Infinite Products
9. Continued Fractions
10. How Infinite Can You Get?
11. Constructing the Real Numbers
12. Where to Next in Analysis? The Calculus
13. Some Brief Remarks About the History of Analysis
Further Reading
1. The Binomial Theorem
2. The Language of Set Theory
3. Proof by Mathematical Induction
4. The Algebra of Numbers
Hints and Selected Solutions