Contents. Preface xi. vii

Transcription

1 Preface xi 1. Real Numbers and Monotone Sequences Introduction; Real numbers Increasing sequences Limit of an increasing sequence Example: the number e Example: the harmonic sum and Euler s number γ Decreasing sequences; Completeness property Estimations and Approximations Introduction; Inequalities Estimations Proving boundedness Absolute values; estimating size Approximations The terminology for n large The Limit of a Sequence Definition of limit Uniqueness of limits; the K-ǫ principle Infinite limits Limit of a n Writing limit proofs Some limits involving integrals Another limit involving an integral The Error Term The error term The error in the geometric series; Applications A sequence converging to 2: Newton s method The sequence of Fibonacci fractions Limit Theorems for Sequences Limits of sums, products, and quotients Comparison theorems Location theorems Subsequences; Non-existence of limits Two common mistakes 71 vii

2 viii Introduction to Analysis 6. The Completeness Property Introduction; Nested intervals Cluster points of sequences The Bolzano-Weierstrass theorem Cauchy sequences Completeness property for sets Infinite Series Series and sequences Elementary convergence tests The convergence of series with negative terms Convergence tests: ratio and n-th root tests The integral and asymptotic comparison tests Series with alternating signs: Cauchy s test Rearranging the terms of a series Power Series Introduction; Radius of convergence Convergence at the endpoints; Abel summation Operations on power series: addition Multiplication of power series Functions of One Variable Functions Algebraic operations on functions Some properties of functions Inverse functions The elementary functions Local and Global Behavior Intervals; estimating functions Approximating functions Local behavior Local and global properties of functions Continuity and Limits of Functions Continuous functions Limits of functions Limit theorems for functions Limits and continuous functions Continuity and sequences The Intermediate Value Theorem The existence of zeros Applications of Bolzano s theorem Graphical continuity Inverse functions 179

3 ix 13. Continuous Functions on Compact Intervals Compact intervals Bounded continuous functions Extremal points of continuous functions The mapping viewpoint Uniform continuity Differentiation: Local Properties The derivative Differentiation formulas Derivatives and local properties Differentiation: Global Properties The mean-value theorem Applications of the mean-value theorem Extension of the mean-value theorem L Hospital s rule for indeterminate forms Linearization and Convexity Linearization Applications to convexity Taylor Approximation Taylor polynomials Taylor s theorem with Lagrange remainder Estimating error in Taylor approximation Taylor series Integrability Introduction; Partitions Integrability Integrability of monotone and continuous functions Basic properties of integrable functions The Riemann Integral Refinement of partitions Definition of the Riemann integral Riemann sums Basic properties of integrals The interval addition property Piecewise continuous and monotone functions Derivatives and Integrals The first fundamental theorem of calculus Existence and uniqueness of antiderivatives Other relations between derivatives and integrals The logarithm and exponential functions Stirling s formula Growth rate of functions 280

4 x Introduction to Analysis 21. Improper Integrals Basic definitions Comparison theorems The gamma function Absolute and conditional convergence Sequences and Series of Functions Pointwise and uniform convergence Criteria for uniform convergence Continuity and uniform convergence Integration term-by-term Differentiation term-by-term Power series and analytic functions Infinite Sets and the Lebesgue Integral Introduction; infinite sets Sets of measure zero Measure zero and Riemann-integrability Lebesgue integration Continuous Functions on the Plane Introduction; Norms and distances in R Convergence of sequences Functions on R Continuous functions Limits and continuity Compact sets in R Continuous functions on compact sets in R Point-sets in the Plane Closed sets in R Compactness theorem in R Open sets Integrals with a Parameter Integrals depending on a parameter Differentiating under the integral sign Changing the order of integration Differentiating Improper Integrals Introduction Pointwise vs. uniform convergence of integrals Continuity theorem for improper integrals Integrating and differentiating improper integrals Differentiating the Laplace transform 393

5 xi Appendix A. Sets, Numbers, and Logic 399 A.0 Sets and numbers 399 A.1 If-then statements 403 A.2 Contraposition and indirect proof 406 A.3 Counterexamples 408 A.4 Mathematical induction 411 B. Quantifiers and Negation 418 B.1 Introduction; Quantifiers 418 B.2 Negation 421 B.3 Examples involving functions 423 C. Picard s Method 427 C.1 Introduction 427 C.2 The Picard iteration theorems 428 C.3 Fixed points 430 D. Applications to Differential Equations 434 D.1 Introduction 434 D.2 Discreteness of the zeros 435 D.3 Alternation of zeros 437 D.4 Reduction to normal form 439 D.5 Comparison theorems for zeros 440 E. Existence and Uniqueness of ODE Solutions 445 E.1 Picard s method of successive approximations 445 E.2 Local existence of solutions to y = f(x,y) 447 E.3 The uniqueness of solutions 450 E.4 Extending the existence and uniqueness theorems 452 F. Compactness via Open Sets 455 Index 458