Elementary Analysis: The Theory of Calculus

الغلاف الأمامي
Springer Science & Business Media, 17‏/04‏/2013 - 264 من الصفحات
Designed for students having no previous experience with rigorous proofs, this text on analysis can be used immediately following standard calculus courses. It is highly recommended for anyone planning to study advanced analysis, e.g., complex variables, differential equations, Fourier analysis, numerical analysis, several variable calculus, and statistics. It is also recommended for future secondary school teachers. A limited number of concepts involving the real line and functions on the real line are studied. Many abstract ideas, such as metric spaces and ordered systems, are avoided. The least upper bound property is taken as an axiom and the order properties of the real line are exploited throughout. A thorough treatment of sequences of numbers is used as a basis for studying standard calculus topics. Optional sections invite students to study such topics as metric spaces and Riemann-Stieltjes integrals.
 

المحتوى

Table of Contents Preface Chapter I Introduction 1 The Set ℕ of Natural Numbers
2 The Set ℚ of Rational Numbers
3 The Set ℝ of Real Numbers
4 The Completeness Axiom
5 The Symbols + and 6 A Development of
ChapterII Sequences 7 Limits of Sequences
8 A Discussion about Proofs
9 Limit Theorems for Sequences
19 Uniform Continuity
20 Limits of Functions
Continuity
Connectedness
Sequences and Series of Functions
23 Power Series
24 Uniform Convergence 25 More on Uniform Convergence
26 Differentiation and Integration of Power Series

10 Monotone Sequences and Cauchy Sequences
11 Subsequences
12 lim sups and lim infs 13 Some Topological Concepts in Metric Spaces
14 Series
15 Alternating Series and Integral Tests
16 Decimal Expansions of Real Numbers
Continuity 17 Continuous Functions
18 Properties of Continuous Functions
27 Weierstrasss Approximation Theorem
ChapterV Differentiation 28 Basic Properties of the Derivative
29 The Mean Value Theorem
30 LHospitals Rule
31 Taylors Theorem
Integration 32 The Riemann Integral 33 Properties of the Riemann Integral
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