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Limits, Limits Everywhere : the Tools of Mathematical Analysis /
A quantity can be made smaller and smaller without it ever vanishing. This fact has profound consequences for science, technology, and even the way we think about numbers. In this book, we will explore this idea by moving at an easy pace through an account of elementary real analysis and, in particu...
| Clasificación: | Libro Electrónico |
|---|---|
| Autor principal: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Oxford :
OUP Oxford,
2012.
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Cover; Contents; PART I: APPROACHING LIMITS; 1. A Whole Lot of Numbers; 1.1 Natural Numbers; 1.2 Prime Numbers; 1.3 The Integers; 1.4 Exercises for Chapter 1; 2. Let's Get Real; 2.1 The Rational Numbers; 2.2 Irrational Numbers; 2.3 The Real Numbers; 2.4 A First Look at Infinity; 2.5 Exercises for Chapter 2; 3. The Joy of Inequality; 3.1 Greater or Less?; 3.2 Intervals; 3.3 The Modulus of a Number; 3.4 Maxima and Minima; 3.5 The Theorem of the Means; 3.6 Getting Closer; 3.7 Exercises for Chapter 3; 4. Where Do You Go To, My Lovely?; 4.1 Limits; 4.2 Bounded Sequences; 4.3 The Algebra of Limits.
- 4.4 Fibonacci Numbers and the Golden Section4.5 Exercises for Chapter 4; 5. Bounds for Glory; 5.1 Bounded Sequences Revisited; 5.2 Monotone Sequences; 5.3 An Old Friend Returns; 5.4 Finding Square Roots; 5.5 Exercises for Chapter 5; 6. You Cannot be Series; 6.1 What are Series?; 6.2 The Sigma Notation; 6.3 Convergence of Series; 6.4 Nonnegative Series; 6.5 The Comparison Test; 6.6 Geometric Series; 6.7 The Ratio Test; 6.8 General Infinite Series; 6.9 Conditional Convergence; 6.10 Regrouping and Rearrangements; 6.11 Real Numbers and Decimal Expansions; 6.12 Exercises for Chapter 6.
- PART II: EXPLORING LIMITS7. Wonderful Numbers
- e, p and?; 7.1 The Number e; 7.2 The Number p; 7.3 The Number?; 8. Infinite Products; 8.1 Convergence of Infinite Products; 8.2 Infinite Products and Prime Numbers; 8.3 Diversion
- Complex Numbers and the Riemann Hypothesis; 9. Continued Fractions; 9.1 Euclid's Algorithm; 9.2 Rational and Irrational Numbers as Continued Fractions; 10. How Infinite Can You Get?; 11. Constructing the Real Numbers; 11.1 Dedekind Cuts; 11.2 Cauchy Sequences; 11.3 Completeness; 12. Where to Next in Analysis? The Calculus; 12.1 Functions; 12.2 Limits and Continuity.
- 12.3 Differentiation12.4 Integration; 13. Some Brief Remarks About the History of Analysis; Further Reading; Appendices; Appendix 1: The Binomial Theorem; Appendix 2: The Language of Set Theory; Appendix 3: Proof by Mathematical Induction; Appendix 4: The Algebra of Numbers; Hints and Solutions to Selected Exercise; Index; A; B; C; D; E; F; G; H; I; J; K; L; M; N; O; P; Q; R; S; T; U; V; W; Z.