Numbers and functions: steps into analysis
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge
Cambridge University Press
2015
|
| Edition: | Third edition |
| Subjects: | |
| Online Access: | Contributor biographical information Publisher description Table of contents only Inhaltsverzeichnis |
| Physical Description: | xxv, 347 Seiten Diagramme |
| ISBN: | 9781107444539 1107444535 |
Staff View
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Record in the Search Index
| _version_ | 1804175910104989696 |
|---|---|
| adam_text | Titel: Numbers and functions
Autor: Burn, Robert P
Jahr: 2015
Contents
Preface to the first edition
Preface to the second edition
Preface to the third edition
Glossary
PART I NUMBERS
1 Mathematical induction
Mathematical induction (qns 1-8)
Historical Note
Answers and comments
2 Inequalities
Positive numbers and their properties (qns 1-29)
Summary: Properties of order
Arithmetic mean and geometric mean (qns 30-39)
Completing the square (qns 40-42)
The sequence (1 + 1 /n)n (qns 43-49)
nth roots (qns 50, 51)
Summary: Results on inequalities
Absolute value (qns 52-66)
Summary: Results on absolute value
Historical Note
Answers and comments
3 Sequences: A first bite at infinity
Introduction (qns 1-3)
Monotonie sequences (qns 4)
Bounded sequences (qns 5-7)
Subsequences (qns 8-16)
page xiii
xxi
xxiii
xxiv
3
3
6
7
9
9
12
13
15
15
16
17
17
19
19
21
28
28
30
30
32
Sequences tending to infinity (qns 17, 18)
Archimedean order and the integer function (qns 19-23)
Summary: The language of sequences
Null sequences (qns 24-47)
Summary: Null sequences
Convergent sequences and their limits (qns 48-60)
Boundedness of convergent sequences (qns 61-63)
Quotients of convergent sequences (qns 64-69)
d Alembert s ratio test (qns 70-74)
Convergent sequences in closed intervals (qns 75-80)
Intuition and convergence (qns 81-83)
Summary: Convergent sequences
Historical Note
Answers and comments
Completeness: What the rational numbers lack
The Fundamental Theorem of Arithmetic (qns 1-3)
Dense sets of rational numbers on the number line (qns 4-10)
Infinite decimals (qns 11-17)
Irrational numbers (qns 18-21)
Infinity: countability (qns 22-30)
Summary
The completeness principle: infinite decimals are convergent
(qns 31, 32)
Bounded monotonic sequences (qns 33-36)
nth roots of positive real numbers, n a positive
integer (qns 37—41)
Nested closed intervals (qns 42)
Convergent subsequences of bounded sequences (qns 43-47)
Cluster points: the Bolzano-Weierstrass theorem (qns 48-54)
Cauchy sequences (qns 55-58)
Least upper bounds (sup) and greatest lowest bounds (inf)
(qns 59-82)
Upper bounds and greatest terms (qns 59-61)
Least upper bound (sup) (qns 62-66)
Lower bounds and least members (qns 67-70)
Greatest lower bound (inf) (qns 71-78)
sup, inf and completeness (qns 79-82)
lim sup and lim inf (qns 83-84)
Summary: Completeness
Contents
vii
Historical Note 91
Answers and comments 95
5 Series: Infinite sums 103
Sequences of partial sums (qns 1-10) 103
The null sequence test (qns 11-13) 106
Simple consequences of convergence (qns 14-22) 106
Summary: Convergence of series 107
Series of positive terms 108
First comparison test (qns 23-29) 108
The harmonic series (qn 30) 109
The convergence of E1 /n (qns 31, 32) 109
Cauchy s nth root test (qns 33-39) 110
d Alembert s ratio test (qns 40-50) 111
Second comparison test (qns 51-55) 112
Integral test (qns 56-61) 113
Summary: Series of positive terms 115
Series with positive and negative terms 116
Alternating series test (qns 62-65) 116
Absolute convergence (qns 66-70) 117
Conditional convergence (qn 71) 118
Rearrangements (qns 72-77) 118
Summary: Series of positive and negative terms 120
Power series 121
Applications of d Alembert s ratio test and Cauchy s nth root
test for absolute convergence (qns 78-90) 121
Radius of convergence (qns 91-101) 122
Cauchy-Hadamard formula (qns 102-107) 124
The Cauchy product (qns 108-113) 125
Summary: Power series and the Cauchy product 127
Historical Note 127
Answers and comments 130
PART II FUNCTIONS
6 Functions and continuity: Neighbourhoods, limits of functions 141
Functions (qn 1) 141
The domain of a function 141
The ränge and co-domain of a function (qns 2—4) 142
Bijections and inverse functions (qns 5-6) 143
Summary: Functions 143
viii
Contents
Continuity (qns 7-11) 144
Definition of continuity by sequences (qns 12-18) 145
Examples of discontinuity (qns 19, 20) 146
Sums and products of continuous functions (qns 21-31) 147
Continuity in less familiar settings (qns 32-35) 148
A squeeze rule (qns 36, 37) 149
Continuity of composite functions and quotients of continuous
functions (qns 38-55) 149
Summary: Continuity by sequences 152
Neighbourhoods (qns 56-63) 152
One-sided limits 157
Definition of one-sided limits by sequences (qns 73-82) 157
Definition of one-sided limits by neighbourhoods
(qns 83-85) 159
Two-sided limits 160
Definition of continuity by limits (qns 86-92) 160
Theorems on limits (qns 93-99) 161
Limits as x - oo and when f(x) - oo (qns 100, 101) 163
Summary: Continuity by neighbourhoods and limits 163
Historical Note 164
Answers 167
7 Continuity and completeness: Functions on intervals 176
Monotonie functions: one-sided limits (qns 1-7) 176
Intervals (qns 8-11) 177
Intermediate Value Theorem (qns 12-21) 179
Inverses of continuous functions (qns 22-28) 180
Continuous functions on a closed interval (qns 29-36) 182
Uniform continuity (qns 37—45) 184
Extension of functions on Q to functions on R (qns 46-48) 186
Summary 188
Historical Note 189
Answers 191
8 Derivatives: Tangents 197
Definition of derivative (qns 1-9) 197
Sums of functions (qns 10, 11) 199
The produet rule (qns 13-16) 199
The quotient rule (qn 17) 200
The chain rule (qn 18) 200
Differentiability and continuity (qns 12, 9-25) 201
Contents
ix
Derived functions (qns 26-34) 205
Second derivatives (qns 35-38) 207
Inverse functions (qns 39—45) 208
Derivatives at end points (qn 46) 209
Summary 210
Historical Note 210
Answers 213
9 Differentiation and completeness: Mean Value Theorems,
Taylor s Theorem 218
Rolle s Theorem (qns 1-11) 218
An intermediate value theorem for derivatives (qn 12) 220
The Mean Value Theorem (qns 13-24) 220
Cauchy s Mean Value Theorem (qn 25) 224
de l Höpital s rule (qns 26-30) 224
Summary: Rolle s Theorem and Mean Value Theorem 226
The Second and Third Mean Value Theorems (qns 31-34) 227
Taylor s Theorem or nth Mean Value Theorem (qns 35, 36) 228
Maclaurin s Theorem (qns 37^16) 229
Summary: Taylor s Theorem 232
Historical Note 233
Answers 236
10 Integration: The Fundamental Theorem of Calculus 244
Areas with curved boundaries (qns 1-5) 244
Monotonie functions (qns 6-9) 247
The definite integral (qns 10, 11) 249
Step functions (qns 12-15) 250
Lower integral and upper integral (qns 16-22) 252
The Riemann integral (qns 23-25) 254
Summary: Definition of the Riemann integral 255
Theorems on integrability (qns 26-36) 256
Integration and continuity (qns 37-44) 259
Mean Value Theorem for integrals (qn 45) 261
Integration on subintervals (qns 46-48) 261
Summary: Properties of the Riemann integral 261
Indefinite integrals (qns 49-53) 262
The Fundamental Theorem of Calculus (qns 54-56) 263
Integration by parts (qns 57-59) 264
Integration by Substitution (qn 60) 265
X
Improper integrals (qns 61-68)
Summary: The Fundamental Theorem of Calculus
Historical Note
Answers
11 Indices and circle functions
Exponential and logarithmic functions
Positive integers as indices (qns 1-3)
Positive rationals as indices (qns 4-7)
Rational numbers as indices (qns 8-17)
Real numbers as indices (qns 18-24)
Natural logarithms (qns 25-31)
Exponential and logarithmic limits (qns 32-38)
Summary: Exponential and logarithmic functions
Circular or trigonometric functions
Length of a line segment (qns 39—42)
Are length (qns 43-48)
Are cosine (qns 49, 50)
Cosine and sine (qns 51-58)
Tangent (qns 59-62)
Summary: Circular or trigonometric functions
Historical Note
Answers
12 Sequences of functions
Pointwise limit functions (qns 1-14)
Uniform convergence (qns 15-19)
Uniform convergence and continuity (qns 20-23)
Uniform convergence and Integration (qns 24-31)
Summary: Uniform convergence, continuity and
integration
Uniform convergence and differentiation (qns 32-34)
Uniform convergence of power series (qns 35—4-4)
The Binomial Theorem for any real index
(qn 45)
The blancmange funetion (qn 46)
Summary: Differentiation and the AZ-test
Historical Note
Answers
Contents
265
267
267
270
279
279
279
280
280
282
283
284
285
286
287
287
289
289
290
291
292
294
300
301
303
304
305
307
308
309
312
312
316
316
318
Contents xi
APPENDICES
Appendix 1 Properties of the real numbers 327
Appendix 2 Geometry and intuition 330
Appendix 3 Questions for Student investigation and discussion 332
Bibliography 337
Index 342
|
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| spelling | Burn, Robert P. 1934- Verfasser (DE-588)171996968 aut Numbers and functions steps into analysis R.P. Burn Third edition Cambridge Cambridge University Press 2015 xxv, 347 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Mathematical analysis Analysis (DE-588)4001865-9 gnd rswk-swf Funktionalanalysis (DE-588)4018916-8 gnd rswk-swf 1\p (DE-588)4151278-9 Einführung gnd-content Funktionalanalysis (DE-588)4018916-8 s DE-604 Analysis (DE-588)4001865-9 s 2\p DE-604 3\p DE-604 http://www.loc.gov/catdir/enhancements/fy1503/2014046495-b.html Contributor biographical information http://www.loc.gov/catdir/enhancements/fy1503/2014046495-d.html Publisher description http://www.loc.gov/catdir/enhancements/fy1503/2014046495-t.html Table of contents only HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028773592&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Burn, Robert P. 1934- Numbers and functions steps into analysis Mathematical analysis Analysis (DE-588)4001865-9 gnd Funktionalanalysis (DE-588)4018916-8 gnd |
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| title | Numbers and functions steps into analysis |
| title_auth | Numbers and functions steps into analysis |
| title_exact_search | Numbers and functions steps into analysis |
| title_full | Numbers and functions steps into analysis R.P. Burn |
| title_fullStr | Numbers and functions steps into analysis R.P. Burn |
| title_full_unstemmed | Numbers and functions steps into analysis R.P. Burn |
| title_short | Numbers and functions |
| title_sort | numbers and functions steps into analysis |
| title_sub | steps into analysis |
| topic | Mathematical analysis Analysis (DE-588)4001865-9 gnd Funktionalanalysis (DE-588)4018916-8 gnd |
| topic_facet | Mathematical analysis Analysis Funktionalanalysis Einführung |
| url | http://www.loc.gov/catdir/enhancements/fy1503/2014046495-b.html http://www.loc.gov/catdir/enhancements/fy1503/2014046495-d.html http://www.loc.gov/catdir/enhancements/fy1503/2014046495-t.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028773592&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT burnrobertp numbersandfunctionsstepsintoanalysis |