Elementary Analysis: The Theory of CalculusSpringer Science & Business Media, 03/03/1980 - 351 من الصفحات Designed for students having no previous experience with rigorous proofs, this text on analysis can be used immediately following standard calculus courses. It is highly recommended for anyone planning to study advanced analysis, e.g., complex variables, differential equations, Fourier analysis, numerical analysis, several variable calculus, and statistics. It is also recommended for future secondary school teachers. A limited number of concepts involving the real line and functions on the real line are studied. Many abstract ideas, such as metric spaces and ordered systems, are avoided. The least upper bound property is taken as an axiom and the order properties of the real line are exploited throughout. A thorough treatment of sequences of numbers is used as a basis for studying standard calculus topics. Optional sections invite students to study such topics as metric spaces and Riemann-Stieltjes integrals. |
المحتوى
1 Introduction | 1 |
2 The Set Q of Rational Numbers | 6 |
3 The Set R of Real Numbers | 12 |
4 The Completeness Axiom | 19 |
5 The Symbols +0o and 0o | 27 |
6 A Development of R | 28 |
7 Limits of Sequences | 31 |
8 A Discussion about Proofs | 37 |
4 Sequences and Series of Functions | 171 |
24 Uniform Convergence | 177 |
25 More on Uniform Convergence | 184 |
26 Differentiation and Integration of Power Series | 192 |
27 Weierstrasss Approximation Theorem | 200 |
5 Differentiation | 205 |
29 The Mean Value Theorem | 213 |
30 LHospitals Rule | 222 |
9 Limit Theorems for Sequences | 43 |
10 Monotone Sequences and Cauchy Sequences | 54 |
11 Subsequences | 63 |
12 lim sups and lim infs | 75 |
13 Some Tbpological Concepts in Metric Spaces | 79 |
14 Series | 90 |
15 Alternating Series and Integral Tests | 100 |
16 Decimal Expansions of Real Numbers | 105 |
3 Continuity | 115 |
18 Properties of Continuous Functions | 126 |
19 Uniform Continuity | 132 |
20 Limits of Functions | 145 |
Continuity | 156 |
Connectedness | 164 |
31 Taylors Theorem | 230 |
6 Integration | 243 |
33 Properties of the Riemann Integral | 253 |
34 Fundamental Theorem of Calculus | 261 |
35 RiemannStieltjes Integrals | 268 |
36 Improper Integrals | 292 |
37 A Discussion of Exponents and Logarithms | 299 |
Appendix on Set Notation | 309 |
Answers | 311 |
References | 341 |
| 345 | |
| 347 | |
طبعات أخرى - عرض جميع المقتطفات
Elementary Analysis: The Theory of Calculus <span dir=ltr>Kenneth A. Ross</span> معاينة محدودة - 2013 |
Elementary Analysis: The Theory of Calculus <span dir=ltr>Kenneth A. Ross</span> لا تتوفر معاينة - 2010 |
عبارات ومصطلحات مألوفة
algebraic Analysis assume Bolzano-Weierstrass theorem bounded function Calculus Cauchy sequence Consider continuous at xo continuous function converges uniformly Corollary decimal expansion Dedekind cuts Definition diverges dom(f Example f is continuous f uniformly F-integrable f(xn finite fn(x follows function f ƒ and g ƒ is integrable Hence Hint implies f(x implies Sn induction inequality Lemma Let f Let f(x Let Sn LF(f lim inf sn lim sn lim sup sn limn limx limx→a f(x Mathematics Mean Value Theorem metric space monotonic N₁ nonnegative Note open interval P₁ partition polynomials power series properties r₁ radius of convergence rational numbers real number real-valued function Repeat Exercise Riemann integral Riemann-Stieltjes integral Root Test s₁ sequence Sn series converges Show that f shows that lim Sn+1 Sntn strictly increasing subsequence subsequential limits Suppose t₁ triangle inequality uniformly continuous upper bound