# Foundations of Mathematical Analysis (Google eBook)

Springer, Dec 17, 2011 - Mathematics - 585 pages
Mathematical analysis is fundamental to the undergraduate curriculum not only because it is the stepping stone for the study of advanced analysis, but also because of its applications to other branches of mathematics, physics, and engineering at both the undergraduate and graduate levels. This self-contained textbook consists of eleven chapters, which are further divided into sections and subsections. Each section includes a careful selection of special topics covered that will serve to illustrate the scope and power of various methods in real analysis. The exposition is developed with thorough explanations, motivating examples, and illustrations conveying geometric intuition in a pleasant and informal style to help readers grasp difficult concepts. Key features include: * Questions and Exercises are provided at the end of each section, covering a broad spectrum of content with various levels of difficulty; * Some of the exercises are routine in nature while others are interesting, instructive, and challenging with hints provided for selected exercises; * Covers a broad spectrum of content with a range of difficulty that will enable students to learn techniques and standard analysis tools; * Introduces convergence, continuity, differentiability, the Riemann integral, power series, uniform convergence of sequences and series of functions, among other topics; * Examines various important applications throughout the book; * Figures throughout the book to demonstrate ideas and concepts are drawn using Mathematica. Foundations of Mathematical Analysis is intended for undergraduate students and beginning graduate students interested in a fundamental introduction to the subject. It may be used in the classroom or as a self-study guide without any required prerequisites.

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### Contents

 1 The Real Number System 1 2 Sequences Convergence and Divergence 22 3 Limits Continuity and Differentiability 71 4 Applications of Differentiability 115 5 Series Convergence and Divergence 147 6 Definite and Indefinite Integrals 208 7 Improper Integrals and Applications of Riemann Integrals 271 8 Power Series 331
 9 Uniform Convergence of Sequences of Functions 370 10 Fourier Series and Applications 429 11 Functions of Bounded Variation and RiemannStieltjes Integrals 468 References for Further Reading 506 Index of Notation 507 Appendix A Hints for Selected Questions and Exercises 513 Index 564 Copyright