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Solving Problems in Mathematical Analysis, Part I

Tomasz Radożycki
Publication Date: 
Number of Pages: 
Problem Books in Mathematics
Problem Book
[Reviewed by
Allen Stenger
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This is a compendium of worked problems, nearly all of which would be considered calculus rather than mathematical analysis. The problems are more challenging than usually found in modern calculus courses, and the solutions are very clear and extremely detailed. There are also numerous “exercises for independent work,” many of which gives the final answer without showing the work. The book is a translation from a work published in Polish in 2010. This is the first of three volumes.
The present volumes deal with (as indicated in its subtitle) “sets, functions, limits, derivatives, integrals, sequences and series.” Sets are studied in two senses: as a basis for defining functions, and as metric spaces. In both cases, the discussion focuses on definitions and verifying whether a set or function has specified properties, rather than on theorems or applications of these concepts. The “function” part of this is already studied these days in high-school algebra, and recapped in precalculus, and the treatment here doesn’t go beyond what is usually covered there. The “metric space” part of this is what is usually covered in the beginning of a real analysis course.  There is a very detailed chapter on mathematical induction, with difficult propositions that can be proved by this technique. This is an interesting chapter and has much more impressive examples of this technique than are typically found in books. On the other hand, it’s not well integrated with the rest of the book.
The chapters on limits, derivatives, sequences, and integrals involve mostly standard techniques, although they are applied to especially complicated examples. The chapter on convergence of series and the chapter on convergence of series of functions is relatively skimpy and covers examples of only a few techniques, mostly comparison of series and the ratio and root tests. Like the rest of the book it does cover how to handle very complicated expressions. 
One notable omission of the book is that there is no numerical work and no technology.

The most obvious competitor to this book is Schaum’s Outlines: Calculus, although the two books are not very similar. The most important differences are that the present book handles much more difficult problems, that it explains the solutions in great detail, and that it assumes you have already studied the techniques being used so it doesn’t have to explain them.  Schaum’s takes a speedy, no-frills approach to calculus; it does explain briefly the techniques that are being used, and it usually only works a couple of problems per section in detail, with a much longer list of problems with answers given and problems for independent work.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web site is His mathematical interests are number theory and classical analysis.