Classical Analysis An Approach through Problems 1st Edition by Hongwei Chen ISBN 9781032302478 103230247X

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ISBN 10: 103230247X
ISBN 13: 9781032302478
Author: Hongwei Chen

Classical Analysis An Approach through Problems 1st Edition Table of contents:

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

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Tags: Hongwei Chen, Classical Analysis, Problems, Approach