One Variable Advanced Calculus

One Variable Advanced Calculus Kenneth Kuttler March 9, 2014 2 Contents 1 Introduction 7 2 The Real And Complex Numbers 9 2.1 The Number Line And Algebra Of The Real Numbers .......... 9 2.2 Exercises ................................... 13 2.3 Set Notation ................................. 13 2.4 Order ..................................... 14 2.5 Exercises ................................... 18 2.6 The Binomial Theorem ........................... 19 2.7 Well Ordering Principle And Archimedean Property . 20 2.8 Exercises ................................... 24 2.9 Completeness of R .............................. 27 2.10 Existence Of Roots .............................. 28 2.11 Exercises ................................... 29 2.12 The Complex Numbers ............................ 31 2.13 Exercises ................................... 33 3 Set Theory 35 3.1 Basic Definitions ............................... 35 3.2 The Schroder Bernstein Theorem ...................... 37 3.3 Equivalence Relations ............................ 40 3.4 Exercises ................................... 41 4 Functions And Sequences 43 4.1 General Considerations ............................ 43 4.2 Sequences ................................... 46 4.3 Exercises ................................... 47 4.4 The Limit Of A Sequence .......................... 49 4.5 The Nested Interval Lemma ......................... 52 4.6 Exercises ................................... 53 4.7 Compactness ................................. 55 4.7.1 Sequential Compactness ....................... 55 4.7.2 Closed And Open Sets ........................ 55 4.7.3 Compactness And Open Coverings . 59 4.8 Exercises ................................... 60 4.9 Cauchy Sequences And Completeness ................... 62 4.9.1 Decimals ............................... 64 4.9.2 lim sup and lim inf .......................... 66 4.9.3 Shrinking Diameters ......................... 69 4.10 Exercises ................................... 70 3 4 CONTENTS 5 Infinite Series Of Numbers 75 5.1 Basic Considerations ............................. 75 5.1.1 Absolute Convergence ........................ 78 5.2 Exercises ................................... 82 5.3 More Tests For Convergence ......................... 83 5.3.1 Convergence Because Of Cancellation . 83 5.3.2 Ratio And Root Tests ........................ 85 5.4 Double Series ................................. 87 5.5 Exercises ................................... 92 6 Continuous Functions 95 6.1 Equivalent Formulations Of Continuity . 101 6.2 Exercises ................................... 102 6.3 The Extreme Values Theorem . 104 6.4 The Intermediate Value Theorem . 105 6.5 Connected Sets ................................ 108 6.6 Exercises ................................... 110 6.7 Uniform Continuity .............................. 111 6.8 Exercises ................................... 112 6.9 Sequences And Series Of Functions . 113 6.10 Sequences Of Polynomials, Weierstrass Approximation . 117 6.11 Ascoli Arzela Theorem∗ . 120 6.12 Exercises ................................... 122 7 The Derivative 125 7.1 Limit Of A Function ............................. 125 7.2 Exercises ................................... 130 7.3 The Definition Of The Derivative . 131 7.4 Continuous And Nowhere Differentiable . 135 7.5 Finding The Derivative . 138 7.6 Mean Value Theorem And Local Extreme Points . 140 7.7 Exercises ................................... 141 7.8 Mean Value Theorem ............................. 143 7.9 Exercises ................................... 146 7.10 Derivatives Of Inverse Functions . 147 7.11 Derivatives And Limits Of Sequences . 149 7.12 Exercises ................................... 150 8 Power Series 153 8.1 Functions Defined In Terms Of Series . 153 8.2 Operations On Power Series . 155 8.3 The Special Functions Of Elementary Calculus . 157 8.3.1 The Functions, sin; cos; exp . 157 8.3.2 ln And logb .............................. 162 8.4 The Binomial Theorem . 164 8.5 Exercises ................................... 166 8.6 L'Hôpital'sRule ............................... 168 8.6.1 Interest Compounded Continuously . 173 8.7 Exercises ................................... 173 8.8 Multiplication Of Power Series . 175 8.9 Exercises ................................... 177 8.10 The Fundamental Theorem Of Algebra . 179 8.11 Some Other Theorems ............................ 181 CONTENTS 5 9 The Riemann And Riemann Stieltjes Integrals 187 9.1 The Darboux Stieltjes Integral . 187 9.1.1 Upper And Lower Darboux Stieltjes Sums . 187 9.2 Exercises ................................... 191 9.2.1 Functions Of Darboux Integrable Functions . 193 9.2.2 Properties Of The Integral . 196 9.2.3 Fundamental Theorem Of Calculus . 200 9.3 Exercises ................................... 203 9.4 The Riemann Stieltjes Integral . 205 9.4.1 Change Of Variables . 210 9.4.2 Uniform Convergence And The Integral . 212 9.4.3 A Simple Procedure For Finding Integrals . 212 9.4.4 General Riemann Stieltjes Integrals . 213 9.4.5 Stirling's Formula . 218 9.5 Exercises ................................... 221 10 Fourier Series 233 10.1 The Complex Exponential . 233 10.2 Definition And Basic Properties . 233 10.3 The Riemann Lebesgue Lemma . 237 10.4 Dini's Criterion For Convergence . 241 10.5 Integrating And Differentiating Fourier Series . 244 10.6 Ways Of Approximating Functions . 248 10.6.1 Uniform Approximation With Trig. Polynomials . 249 10.6.2 Mean Square Approximation . 251 10.7 Exercises ................................... 254 11 The Generalized Riemann Integral 261 11.1 Definitions And Basic Properties . 261 11.2 Integrals Of Derivatives . 271 11.3 Exercises ................................... 273 A Construction Of Real Numbers 275 Copyright ⃝c 2007, 6 CONTENTS Chapter 1 Introduction The difference between advanced calculus and calculus is that all the theorems are proved completely and the role of plane geometry is minimized. Instead, the notion of completeness is of preeminent importance. Silly gimmicks are of no significance at all. Routine skills involving elementary functions and integration techniques are supposed to be mastered and have no place in advanced calculus which deals with the fundamental issues related to existence and meaning. This is a subject which places calculus as part of mathematics and involves proofs and definitions, not algorithms and busy work. An orderly development of the elementary functions is included but it is assumed the reader is familiar enough with these functions to use them in problems which illustrate some of the ideas presented. 7 8 CHAPTER 1. INTRODUCTION Chapter 2 The Real And Complex Numbers 2.1 The Number Line And Algebra Of The Real Num- bers To begin with, consider the real numbers, denoted by R, as a line extending infinitely far in both directions. In this book, the notation, ≡ indicates something is being defined. Thus the integers are defined as Z ≡ {· · · − 1; 0; 1; · · · g ; the natural numbers, N ≡ f1; 2; · · · g and the rational numbers, defined as the numbers which are the quotient of two integers. n o m Q ≡ such that m; n 2 Z; n =6 0 n are each subsets of R as indicated in the following picture. −4 −3 −2 −1 0 1 2 3 4 - 1=2 1 As shown in the picture, 2 is half way between the number 0 and the number, 1: By analogy, you can see where to place all the other rational numbers. It is assumed that R has the following algebra properties, listed here as a collection of assertions called axioms. These properties will not be proved which is why they are called axioms rather than theorems. In general, axioms are statements which are regarded as true. Often these are things which are \self evident" either from experience or from some sort of intuition but this does not have to be the case. Axiom 2.1.1 x + y = y + x; (commutative law for addition) Axiom 2.1.2 x + 0 = x; (additive identity). 9 10 CHAPTER 2. THE REAL AND COMPLEX NUMBERS Axiom 2.1.3 For each x 2 R; there exists −x 2 R such that x + (−x) = 0; (existence of additive inverse). Axiom 2.1.4 (x + y) + z = x + (y + z) ; (associative law for addition). Axiom 2.1.5 xy = yx; (commutative law for multiplication). Axiom 2.1.6 (xy) z = x (yz) ; (associative law for multiplication). Axiom 2.1.7 1x = x; (multiplicative identity). Axiom 2.1.8 For each x =6 0; there exists x−1 such that xx−1 = 1:(existence of multiplicative inverse). Axiom 2.1.9 x (y + z) = xy + xz:(distributive law). These axioms are known as the field axioms and any set (there are many others besides R) which has two such operations satisfying the above axioms is called a field. Division( ) and subtraction are defined in the usual way by x − y ≡ x + (−y) and x=y ≡ x y−1 : It is assumed that the reader is completely familiar with these axioms in the sense that he or she can do the usual algebraic manipulations taught in high school and junior high algebra courses. The axioms listed above are just a careful statement of exactly what is necessary to make the usual algebraic manipulations valid. A word of advice regarding division and subtraction is in order here. Whenever you feel a little confused about an algebraic expression which involves division or subtraction, think of division as multiplication by the multiplicative inverse as just indicated and think( of) subtraction as addition of the additive inverse. Thus, when you see x=y; think x y−1 and when you see x − y; think x + (−y) : In many cases the source of confusion will disappear almost magically. The reason for this is that subtraction and division do not satisfy the associative law. This means there

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