دانلود کتاب Classical Analysis: An Approach through Problems

فهرست مطالب :

Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
Preface
1. Sequences
1.1. Completeness Theorems for the Real Number System
1.2. Stolz-Cesàro Theorem
1.3. Worked Examples
1.3.1. ε – N definition
1.3.2. Cauchy criterion
1.3.3. The squeeze theorem
1.3.4. Monotone convergence theorem
1.3.5. Upper and lower limits
1.3.6. Stolz-Cesàro theorem
1.3.7. Fixed-point theorems
1.3.8. Recursions with closed forms
1.3.9. Limits involving the harmonic numbers
1.4. Exercises
2. Infinite Numerical Series
2.1. Main Definitions and Basic Convergence Tests
2.2. Raabe and Logarithmic Tests
2.3. The Kummer, Bertrand, and Gauss Tests
2.4. More Sophisticated Tests Based on Monotonicity
2.5. On the Universal Test
2.6. Tests for General Series
2.7. Properties of Convergent Series
2.8. Infinite Products
2.9. Worked Examples
2.10. Exercises
3. Continuity
3.1. Definition of Continuity
3.2. Limits of Functions
3.3. Three Fundamental Theorems
3.4. From the Intermediate Value Theorem to Chaos
3.5. Monotone Functions
3.6. Worked Examples
3.7. Exercises
4. Differentiation
4.1. Derivatives
4.2. Fundamental Theorems of Differentiation
4.3. L\'Hôpital\'s Rules
4.4. Convex Functions
4.5. Taylor\'s Theorem
4.6. Worked Examples
4.7. Exercises
5. Integration
5.1. The Riemann Integral
5.2. Classes of Integrable Functions
5.3. The Mean Value Theorem
5.4. The Fundamental Theorems of Calculus
5.5. Worked Examples
5.6. Exercises
6. Sequences and Series of Functions
6.1. Pointwise and Uniform Convergence
6.2. Importance of Uniform Convergence
6.3. Two Other Convergence Theorems
6.4. Power Series
6.5. Weierstrass\'s Approximation Theorem
6.6. Worked Examples
6.7. Exercises
7. Improper and Parametric Integration
7.1. Improper Integrals
7.2. Integrals with Parameters
7.3. The Gamma Function
7.4. Worked Examples
7.5. Exercises
A. List of Problems from MAA
Bibliography
Index