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Limits, series, and fractional part integrals. Problems in mathematical analysis. (English) Zbl 1279.26017

Problem Books in Mathematics. New York, NY: Springer (ISBN 978-1-4614-6761-8/hbk; 978-1-4614-6762-5/ebook). xviii, 274 p. (2013).
This book contains a collection of unusual problems and solutions in mathematical analysis in the area of limits, series and fractional part integrals. The book is made up of three chapters each specializing on specific topics, viz: limits, series and fractional part integrals.
The first chapter deals with a collection of non-standard problems on limits of sequences and integrals. The problems in this chapter specialize in different aspects of calculus such as asymptotic behaviour of sequences to the evaluation of limits involving special functions, integrals and finite sums. The problems vary in difficulty and solutions are provided while in some cases, generous hints are provided for the interested readers to solve them.
The second chapter focuses on the collection and evaluation of problems of exotic integrals involving a fractional part usually called the fractional part integral. The novelty of these problems is that it requires a host of mathematical techniques that involve elegant collections between integrals, infinite series, exotic constants and special functions.
The last chapter contains a bouquet of problems with a flavour towards computational aspects of infinite series and special products. The problems here helps the readers to discover new connections between a series and an integral and also enable evaluation of sums using the modern tools of analysis. This chapter also offers to the readers a collection of interesting and unconventional problems for solutions.
This book is written in a way that each chapter contains a section of difficult problems which are designated as “open problems” and generous hints and links to relevant literature are provided in some cases to assist the interested reader to solve them.
This book is indispensable for graduate students of mathematics, physics, engineering and other researchers interested in exploring the powerful techniques of mathematical analysis in their research work.

MSC:

26-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to real functions
40-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to sequences, series, summability
00A07 Problem books
26A33 Fractional derivatives and integrals
26A03 Foundations: limits and generalizations, elementary topology of the line
26A06 One-variable calculus
40A25 Approximation to limiting values (summation of series, etc.)
11Y60 Evaluation of number-theoretic constants
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