This comprehensive textbook is intended for a two-semester sequence in analysis. The first four chapters present a practical introduction to analysis by using the tools and concepts of calculus.The last five chapters present a first course in analysis. The presentation is clear and concise, allowing students to master the calculus tools that are crucial in understanding analysis.
From Calculus to Analysis prepares readers for their first analysis courseimportant because many undergraduate programs traditionally require such a course. Undergraduates and some advanced high-school seniors will find this text a useful and pleasant experience in the classroom or as a self-study guide. The only prerequisite is a standard calculus course.
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Table of ContentsPreface.- Ch. 1 Number Systems.- 1.1 The algebra of the reals.- 1.2 Natural numbers and integers.- .1.3 Rational numbers and real numbers.- 1.4 Power functions.- Ch. 2 Sequences and Series.- 2.1 Sequences.- 2.2 Montone sequences, Bolzano-Weirestrass theorem and operations on limits.- 2.3 Series.- 2.4 Absolute convergence.- Ch. 3 Power series and special functions.-3.1 Power series.-3.2 Tigonometric functions.- 3.3 Inverse trigonometric functions.- 3.4 Exponential and logarithmic functions.- Ch 4 Fifty Ways to Estimate the Number pi.-4.1 Power series expansions.- 4.2 Wallis' integrals, Euler's formula, and Stirling's formula.-4.3 Convergence of infinite products.- 4.4 The number pi is irrational.- Ch. 5 Continuity, Limits, and Differentiation.- 5.1 Continuity.- 5.2 Limits of functions and derivatives.- 5.3 Algebra of derivatives and mean value theorems.- 5.4 Intervals, continuity, and inverse functions.- Ch. 6 Riemann Integration.- 6.1 Construction of the integral.- 6.2 Properties of the integral.- 6.3 Uniform continuity.- Ch 7 Decimal Represenation of Numbers.- Ch 8 Countable and Uncountable Sets.- Further Readings.- Index.