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This text gives a rigorous treatment of the foundations of
calculus. In contrast to more traditional approaches, infinite
sequences and series are placed at the forefront. The approach
taken has not only the merit of simplicity, but students are well
placed to understand and appreciate more sophisticated concepts in
advanced mathematics. The authors mitigate potential difficulties
in mastering the material by motivating definitions, results and
proofs. Simple examples are provided to illustrate new material and
exercises are included at the end of most sections. Noteworthy
topics include: an extensive discussion of convergence tests for
infinite series, Wallis's formula and Stirling's formula, proofs of
the irrationality of and e and a treatment of Newton's method as a
special instance of finding fixed points of iterated functions.
This text gives a rigorous treatment of the foundations of
calculus. In contrast to more traditional approaches, infinite
sequences and series are placed at the forefront. The approach
taken has not only the merit of simplicity, but students are well
placed to understand and appreciate more sophisticated concepts in
advanced mathematics. The authors mitigate potential difficulties
in mastering the material by motivating definitions, results and
proofs. Simple examples are provided to illustrate new material and
exercises are included at the end of most sections. Noteworthy
topics include: an extensive discussion of convergence tests for
infinite series, Wallis's formula and Stirling's formula, proofs of
the irrationality of and e and a treatment of Newton's method as a
special instance of finding fixed points of iterated functions.
This text provides a detailed presentation of the main results for
infinite products, as well as several applications. The target
readership is a student familiar with the basics of real analysis
of a single variable and a first course in complex analysis up to
and including the calculus of residues. The book provides a
detailed treatment of the main theoretical results and applications
with a goal of providing the reader with a short introduction and
motivation for present and future study. While the coverage does
not include an exhaustive compilation of results, the reader will
be armed with an understanding of infinite products within the
course of more advanced studies, and, inspired by the sheer beauty
of the mathematics. The book will serve as a reference for students
of mathematics, physics and engineering, at the level of senior
undergraduate or beginning graduate level, who want to know more
about infinite products. It will also be of interest to instructors
who teach courses that involve infinite products as well as
mathematicians who wish to dive deeper into the subject. One could
certainly design a special-topics class based on this book for
undergraduates. The exercises give the reader a good opportunity to
test their understanding of each section.
Although students of analysis are familiar with real and complex
numbers, few treatments of analysis deal with the development of
such numbers in any depth. An understanding of number systems at a
fundamental level is necessary for a deeper grasp of analysis.
Beginning with elementary concepts from logic and set theory, this
book develops in turn the natural numbers, the integers and the
rational, real and complex numbers. The development is motivated by
the need to solve polynomial equations, and the book concludes by
proving that such equations have solutions in the complex number
system.
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