Ulteriori informazioni
This text for courses in real analysis or advanced calculus is designed specifically to present advanced calculus topics within a framework that will help students more effectively write and analyze proofs. The authors' comprehensive yet accessible presentation for one- or two-term courses offers a balanced depth of topic coverage and mathematical rigor.
Sommario
1. Proofs, Sets, and Functions
Proofs
Sets
Functions
Mathematical Induction
2. The Structure of R
Algebraic and Other Properties of R
The Completeness Axiom
The Rational Numbers Are Dense in R
Cardinality
3. Sequences
Convergence
Limit Theorems
Subsequences
Monotone Sequences
Bolzano-Weierstrass Theorems
Cauchy Sequences
Limits at Infinity
Limit Superior and Limit Inferior
4. Continuity
Continuous Functions
Continuity and Sequences
Limits of Functions
Consequences of Continuity
Uniform Continuity
Discontinuities and Monotone Functions
5. Differentiation
The Derivative
Mean Value Theorems
Taylor's Theorem
L'Hpital's Rule
6. Riemann Integration
Existence of the Riemann Integral
Riemann Sums
Properties of the Riemann Integral
Families of Riemann Integrable Functions
Fundamental Theorem of Calculus
Improper Integrals
7. Infinite Series
Convergence and Divergence
Absolute and Conditional Convergence
Regrouping and Rearranging Series
Multiplication of Series
8. Sequences and Series of Functions
Function Sequences
Preservation Theorems
Series of Functions
Weierstrass Approximation Theorem
9. Power Series
Convergence
Taylor Series
10. The Riemann-Stieltjes Theorem
Monotone Increasing Integrators
Families of Intergrable Functions
Riemann-Stieltjes Sums
Functions of Bounded Variation
Integrators of Bounded Variations
11. The Topology of R
Open and Closed Sets
Neighborhoods and Accumulation Points
Compact Sets
Connected Sets
Continuous Functions
Bibliography. Hints and Answers. Index
Riassunto
This text for courses in real analysis or advanced calculus is designed specifically to present advanced calculus topics within a framework that will help students more effectively write and analyze proofs. The authors' comprehensive yet accessible presentation for one- or two-term courses offers a balanced depth of topic coverage and mathematical rigor.
