Sharpening Mathematical Analysis Skills

This book is a book of problems and worked out solutions covering all areas of real analysis. It should be of use to a variety of readers including, those teaching the traditional Calculus sequence, researchers who routinely deal with series, and mathematical laypeople who wish to enjoy some punchy, beautiful identities and revel at some mathematical jokes. 

 

To those teaching the traditional Calculus sequence, the book has about 400 problems (with solutions) covering (in single and multiple variables) Taylor series, Maclaurin series, derivative formulas including the chain rule and the generalized Leibnitz formula, traditional Calculus functions such as the Hessian, Jacobian, Divergence, Gradient, and  Laplacian,  extrema problems, polar and spherical coordinates, classical graphs, higher-order derivatives, some differential and integral equations, and some topics not always covered in depth, including homogeneous functions (Euler’s identity) and implicit function problems. These problems resemble in flavor and difficulty those found in traditional Calculus texts. There are also several unusual problems involving the floor and fractional part function as they appear in integrals.

 

There are also chapters on sequences, series, and power series with several hundred problems covering traditional Calculus topics including geometric series, alternating series, telescoping, Cauchy products, generating functions including those for the classical recursions (Stirling, Fibonacci numbers), the squeeze theorem, treatment of oo/oo, 0/0, the criteria of Cauchy D'Alembert, Stolz-Cesaro, and the Cauchy root test. Some rarer convergence tests not always mentioned in Calculus texts such as those of Chen, Raabe-Duhamel, and Ivan are also mentioned.  

 

However, the chapters on sequences, series, and power series are sometimes biased toward the number theorist. Those interested in number-theoretic topics, including recursions, generating functions for the classical recursive sequences (Fibonacci, Lucas, Stirling), and the  Riemann Zeta function will find numerous problems for practice here and many delightful identities for pi, the Riemann zeta function, and the harmonic numbers. 

 

Finally, the book should be of interest to mathematical laypeople who simply want to enjoy some good punchy problems and unexpected identities. I was quite surprised in a rather technical book to find amusing identities such as the fact that 16/64 = ¼ can be obtained by canceling the 6s. And no, it is not a coincidence, since there are others like 95/19=5, 98/49=2 The book is filled with several such mathematical jokes. Additionally, emotional words like beautiful, gem, exotic, fabulous, pearls, artistry, and remarkable fill the book and correctly so. 

 

The book consists of two parts, one for problems and one for solutions, each having about 250 pages. Thus the solutions are frequently high-quality step-by-step derivations. 

 

The book is more than a collection of problems. It has a bibliography with 150 items not all of which are references to problems published in journals. For example, Chen’s result, A New Ratio Test for Positive Monotone Series, College Math Journal, 44(2), 2013, pp. 139-141, presents a new test for convergence not found in current Calculus books. Several new proofs – for example of Nesbitt’s Inequality or the de Moivre Trigonometric Identity – are presented. Chapter 8 presents two new proofs of the Sandham-Yeung series. Several of the references are simply good reference texts on analysis. 

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Russel Jay Hendel, RHendel@Towson.Edu,  holds a Ph.D. in theoretical mathematics and an Associateship from the Society of Actuaries. He teaches at Towson University. His interests include discrete number theory, graph theory, applications of technology to education, problem writing, theory of pedagogy, actuarial science, and the interaction between mathematics, art, and poetry.)