Limits, limits everywhere: the tools of mathematical analysis
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York
Oxford Univ. Press
2012
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| Edition: | 1. publ. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 200 S. Ill., graph. Darst. |
| ISBN: | 9780199640089 |
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Record in the Search Index
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| adam_text | Titel: Limits, limits everywhere
Autor: Applebaum, David
Jahr: 2012
Contents
PART I APPROACHING LIMITS
1. A Whole Lot of Numbers 2
1.1 Natural Numbers 2
1.2 Prime Numbers 3
1.3 The Integers 9
1.4 Exercises for Chapter 1 12
2. Let sGetReal 14
2.1 The Rational Numbers 14
2.2 Irrational Numbers 17
2.3 The Real Numbers 23
2.4 A First Look at Infinity 24
2.5 Exercises for Chapter 2 26
3. The Joy of Inequality 28
3.1 Greater or Less? 28
3.2 Intervals 33
3.3 The Modulusofa Number 34
3.4 Maxima and Minima 38
3.5 The Theorem ofthe Means 38
3.6 Getting Closer 41
3.7 Exercises for Chapter 3 42
k. Where Do You Go To, My Lovely? 45
4.1 Limits 45
4.2 Bounded Sequences 54
4.3 The Algebra of Limits 56
4.4 Fibonacci Numbers and the Golden Section 59
4.5 Exercises for Chapter 4 62
5. Bounds for Glory 65
5.1 Bounded Sequences Revisited 65
5.2 Monotone Sequences 69
5.3 An Old Friend Returns 71
CONTENTS
5.4 Finding Square Roots 73
5.5 Exercises for Chapter 5 75
6. You Cannot be Series 78
6.1 What are Series? 78
6.2 The Sigma Notation 79
6.3 Convergence of Series 82
6.4 Nonnegative Series 84
6.5 The Comparison Test 88
6.6 Geometrie Series 92
6.7 The Ratio Test 95
6.8 General Infinite Series 98
6.9 Conditional Convergence 99
6.10 Regrouping and Rearrangements 102
6.11 Real Numbers and Decimal Expansions 104
6.12 Exercises for Chapter 6 105
PART II EXPLORING LIMITS
7. Wonderful Numbers - e, n and y 110
7.1 The Number e 110
7.2 The Number Jt 118
7.3 The Number y 123
8. Infinite Products 126
8.1 Convergence of Infinite Products 126
8.2 Infinite Products and Prime Numbers 130
8.3 Diversion - Complex Numbers and the Riemann Hypothesis 134
9. Continued Fractions 138
9.1 Euclid s Algorithm 138
9.2 Rational and Irrational Numbers as Continued Fractions 139
10. How Infinite Can You Get? 145
11. Constructing the Real Numbers 151
11.1 DedekindCuts 151
11.2 Cauchy Sequences 152
11.3 Completeness 153
12. Where to Next in Analysis? The Calculus 157
12.1 Functions 157
12.2 Limits and Continuity 162
xw
CONTENTS
12.3 Differentiation 163
12.4 Integration 166
13. Some Brief Remarks About the History of Analysis 171
Further Reading 175
Appendices 181
Appendix 1: The Binomial Theorem 181
Appendix 2: The Language of Set Theory 183
Appendix 3: Proof by Mathematical Induction 186
Appendix 4: The Algebra of Numbers 188
Hints and Solutions to Selected Exercise 190
Index 195
XV
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| spelling | Applebaum, David 1956- Verfasser (DE-588)136277659 aut Limits, limits everywhere the tools of mathematical analysis David Applebaum 1. publ. New York Oxford Univ. Press 2012 XV, 200 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Grenzwert Mathematik (DE-588)4316404-3 gnd rswk-swf Analysis (DE-588)4001865-9 gnd rswk-swf Unterhaltungsmathematik (DE-588)4124357-2 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Grenzwert Mathematik (DE-588)4316404-3 s Analysis (DE-588)4001865-9 s Unterhaltungsmathematik (DE-588)4124357-2 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024852685&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Applebaum, David 1956- Limits, limits everywhere the tools of mathematical analysis Grenzwert Mathematik (DE-588)4316404-3 gnd Analysis (DE-588)4001865-9 gnd Unterhaltungsmathematik (DE-588)4124357-2 gnd |
| subject_GND | (DE-588)4316404-3 (DE-588)4001865-9 (DE-588)4124357-2 (DE-588)4123623-3 |
| title | Limits, limits everywhere the tools of mathematical analysis |
| title_auth | Limits, limits everywhere the tools of mathematical analysis |
| title_exact_search | Limits, limits everywhere the tools of mathematical analysis |
| title_full | Limits, limits everywhere the tools of mathematical analysis David Applebaum |
| title_fullStr | Limits, limits everywhere the tools of mathematical analysis David Applebaum |
| title_full_unstemmed | Limits, limits everywhere the tools of mathematical analysis David Applebaum |
| title_short | Limits, limits everywhere |
| title_sort | limits limits everywhere the tools of mathematical analysis |
| title_sub | the tools of mathematical analysis |
| topic | Grenzwert Mathematik (DE-588)4316404-3 gnd Analysis (DE-588)4001865-9 gnd Unterhaltungsmathematik (DE-588)4124357-2 gnd |
| topic_facet | Grenzwert Mathematik Analysis Unterhaltungsmathematik Lehrbuch |
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