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E-raamat: First Course In Analysis, A

(The Ohio State Univ At Newark, Usa)
  • Formaat: 208 pages
  • Ilmumisaeg: 29-Nov-2012
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • Keel: eng
  • ISBN-13: 9789814417877
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First Course In Analysis, ASuurem pilt
  • Formaat: 208 pages
  • Ilmumisaeg: 29-Nov-2012
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • Keel: eng
  • ISBN-13: 9789814417877
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This book is an introductory text on real analysis for undergraduate students. The prerequisite for this book is a solid background in freshman calculus in one variable. The intended audience of this book includes undergraduate mathematics majors and students from other disciplines who use real analysis. Since this book is aimed at students who do not have much prior experience with proofs, the pace is slower in earlier chapters than in later chapters. There are hundreds of exercises, and hints for some of them are included.
Preface vii
1 Sets, Functions, and Real Numbers
1(28)
1.1 Sets
1(3)
1.2 Functions
4(4)
1.3 Real Numbers
8(9)
1.4 Mathematical Induction
17(4)
1.5 Countability
21(4)
1.6 Additional Exercises
25(4)
2 Sequences
29(32)
2.1 Sequences of Real Numbers
29(8)
2.2 Properties of Limits
37(7)
2.3 The Bolzano-Weierstrass Theorem
44(7)
2.4 Limit Superior and Limit Inferior
51(6)
2.5 Additional Exercises
57(4)
3 Series
61(22)
3.1 Convergence of Series
62(5)
3.2 Comparison Tests
67(3)
3.3 Alternating Series Test
70(2)
3.4 Absolute Convergence
72(5)
3.5 Rearrangement of Series
77(3)
3.6 Additional Exercises
80(3)
4 Continuous Functions
83(32)
4.1 Limit Points
83(2)
4.2 Limits of Functions
85(5)
4.3 Continuity
90(3)
4.4 Extreme and Intermediate Value Theorems
93(3)
4.5 Uniform Continuity
96(3)
4.6 Monotone and Inverse Functions
99(5)
4.7 Functions of Bounded Variation
104(6)
4.8 Additional Exercises
110(5)
5 Differentiation
115(20)
5.1 The Derivative
115(7)
5.2 Mean Value Theorem
122(4)
5.3 Taylor's Theorem
126(5)
5.4 Additional Exercises
131(4)
6 Integration
135(22)
6.1 The Integral
135(6)
6.2 Integration via Tagged Partitions
141(5)
6.3 Basic Properties of Integrals
146(3)
6.4 Fundamental Theorem of Calculus
149(6)
6.5 Additional Exercises
155(2)
7 Sequences and Series of Functions
157(26)
7.1 Pointwise and Uniform Convergence
157(6)
7.2 Interchange of Limits
163(6)
7.3 Series of Functions
169(6)
7.4 Power Series
175(6)
7.5 Additional Exercises
181(2)
Hints for Selected Exercises 183(6)
List of Notations 189(2)
Bibliography 191(2)
Index 193
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