Real Analysis

Acknowledgments

1. Archimedes and the Parabola

1.1 The Area of the Parabolic Segment

1.2 The Geometry of the Parabola

2. Fermat, Differentiation, and Integration

2.1 Fermatâs Calculus

3. Newtonâs Calculus (Part 1)

3.1 The Fractional Binomial Theorem

3.2 Areas and Infinite Series

3.3 Newtonâs Proofs

4. Newtonâs Calculus (Part 2)

4.1 The Solution of Differential Equations

4.2 The Solution of Algebraic Equations

Chapter Appendix. Mathematica implementations of Newtonâs algorithm

5. Euler

5.1 Trigonometric Series

6. The Real Numbers

6.1 An Informal Introduction

6.2 Ordered Fields

6.3 Completeness and Irrational Numbers

6.4 The Euclidean Process

6.5 Functions

7. Sequences and Their Limits

7.1 The Definitions

7.2 Limit Theorems

8. The Cauchy Property

8.1 Limits of Monotone Sequences

8.2 The Cauchy Property

9. The Convergence of Infinite Series

9.1 Stock Series

9.2 Series of Positive Terms

9.3 Series of Arbitrary Terms

9.4 The Most Celebrated Problem

10. Series of Functions

10.1 Power Series

10.2 Trigonometric Series

11. Continuity

11.1 An Informal Introduction

11.2 The Limit of a Function

11.3 Continuity

11.4 Properties of Continuous Functions

12. Differentiability

12.1 An Informal Introduction to Differentiation

12.2 The Derivative

12.3 The Consequences of Differentiability

12.4ãã Integrability

13. Uniform Convergence

13.1 Uniform and Non-Uniform Convergence

13.2 Consequences of Uniform Convergence

14. The Vindication

14.1 Trigonometric Series

14.2 Power Series

15. The Riemann Integral

15.1 Continuity Revisited

15.2 Lower and Upper Sums

15.3 Integrability

Appendix A. Excerpts from "Quadrature of the Parabola" by Archimedes

Appendix B. On a Method for Evaluation of Maxima and Minima by Pierre de Fermat

Appendix C. From a Letter to Henry Oldenburg on the Binomial Series (June 13, 1676) by Isaac Newton

Appendix D. From a Letter to Henry Oldenburg on the Binomial Series (October 24, 1676) by Isaac Newton

Appendix E. Excerpts from "Of Analysis by Equations of an Infinite Number of Terms" by Isaac Newton

Appendix F. Excerpts from "Subsiduum Calculi Sinuum" by Leonhard Euler)

Solutions to Selected Exercises

Bibliography

Index