Sequences, Series and Taylor Approximation (MA2712b, MA2730) Level 2 Teaching Team Current curator: Simon Shaw November 20, 2015 Contents 0 Introduction, Overview 6 1 Taylor Polynomials 10 1.1 Lecture 1: Taylor Polynomials, Definition . . . . . . . . . . . . . . . . . . . . . 10 1.1.1 Reminder from Level 1 about Differentiable Functions . . . . . . . . . . 11 1.1.2 Definition of Taylor Polynomials . . . . . . . . . . . . . . . . . . . . . . 11 1.2 Lectures 2 and 3: Taylor Polynomials, Examples . . . . . . . . . . . . . . . . . 13 x 1.2.1 Example: Compute and plot Tnf for f(x) = e . . . . . . . . . . . . 13 1.2.2 Example: Find the Maclaurin polynomials of f(x) = sinx . . . . . . 14 1.2.3 Find the Maclaurin polynomial T f for f(x) = sin(x2) . . . . . . . 15 11 1.2.4 Questions for Chapter 6: Error Estimates . . . . . . . . . . . . . . . . 15 1.3 Lecture 4 and 5: Calculus of Taylor Polynomials . . . . . . . . . . . . . . . . . 17 1.3.1 General Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.4 Lecture 6: Various Applications of Taylor Polynomials . . . . . . . . . . . . . 22 1.4.1 Relative Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.4.2 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.4.3 How to Calculate Complicated Taylor Polynomials? . . . . . . . . . . . 26 1.5 Exercise Sheet 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.5.1 Exercise Sheet 1a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.5.2 Feedback for Sheet 1a . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2 Real Sequences 40 2.1 Lecture 7: Definitions, Limit of a Sequence . . . . . . . . . . . . . . . . . . . . 40 2.1.1 Definition of a Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.1.2 Limit of a Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.1.3 Graphic Representations of Sequences . . . . . . . . . . . . . . . . . . . 43 2.2 Lecture 8: Algebra of Limits, Special Sequences . . . . . . . . . . . . . . . . . 44 2.2.1 Infinite Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 1 2.2.2 Algebra of Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.2.3 Some Standard Convergent Sequences . . . . . . . . . . . . . . . . . . . 46 2.3 Lecture 9: Bounded and Monotone Sequences . . . . . . . . . . . . . . . . . . 48 2.3.1 Bounded Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.3.2 Convergent Sequences and Closed Bounded Intervals . . . . . . . . . . 48 2.4 Lecture 10: Monotone Sequences . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.4.1 Convergence of Monotone, Bounded Sequences . . . . . . . . . . . . . . 50 2.5 Exercise Sheet 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.5.1 Exercise Sheet 2a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.5.2 Feedback for Sheet 2a . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3 A flipped classroom approach to Improper Integrals 58 3.1 Self-study for Lecture 11: Improper Integrals — Type 1 . . . . . . . . . . . . . 58 3.2 Self-study for Lecture 11: Improper Integrals — Type 2 . . . . . . . . . . . . . 61 3.3 Homework for improper integrals, Types 1 and 2 . . . . . . . . . . . . . . . . . 63 3.4 Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.4.1 Feedback on in-class exercises . . . . . . . . . . . . . . . . . . . . . . . 64 3.4.2 Homework feedback for improper integrals . . . . . . . . . . . . . . . . 66 4 Real Series 78 4.1 Lecture 12: Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.1.1 A Tale of a Rabbit and a Turtle following Zeno’s . . . . . . . . . . . . 78 4.1.2 Definition of a Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.1.3 Convergent Series, Geometric Series . . . . . . . . . . . . . . . . . . . . 80 4.2 Lecture 13: Important Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.2.1 A Criterion for Divergence . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.2.2 Telescopic Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.2.3 Harmonic Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.2.4 Algebra of Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.3 Lecture 14: Test for Convergence . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.3.1 Comparison Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.3.2 Integral Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.4 Lecture 15: Further Tests for Convergence . . . . . . . . . . . . . . . . . . . . 92 4.4.1 Comparing a Series with a Geometric Series: the Root and Ratio Tests 92 4.5 Exercise Sheet 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.5.1 Exercise Sheet 3a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.5.2 Additional Exercise Sheet 3b . . . . . . . . . . . . . . . . . . . . . . . . 97 4.5.3 Feedback for Sheet 3a . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 2 5 Deeper Results on Sequences and Series 103 5.1 Lecture 16: (ǫ,N)-Definition of Limits . . . . . . . . . . . . . . . . . . . . . . 103 5.1.1 Practical Aspects of Estimates of Convergent Sequences . . . . . . . . . 104 5.1.2 Divergent Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.2 Extra curricular material: Error Estimates from the Tests . . . . . . . . . . . . 108 5.2.1 Error Estimates from the Ratio and Root Tests . . . . . . . . . . . . . 108 5.2.2 Error Estimates for the Integral Test . . . . . . . . . . . . . . . . . . . 108 5.3 Lecture 17: Absolute Convergence of Series and the Leibnitz Criterion of Con- vergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.3.1 Alternating Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.4 Exercise Sheet 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.4.1 Exercise Sheet 4a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.4.2 Additional Exercise Sheet 4b . . . . . . . . . . . . . . . . . . . . . . . . 117 5.4.3 Short Feedback for Exercise Sheet 4a . . . . . . . . . . . . . . . . . . . 117 5.4.4 Short Feedback for the Additional Exercise Sheet 4b . . . . . . . . . . 118 5.4.5 Feedback for Exercise Sheet 4a . . . . . . . . . . . . . . . . . . . . . . 119 6 Approximation with Taylor Polynomials 123 6.1 Lecture 18: Taylor’s theorem and error estimates . . . . . . . . . . . . . . . . 123 6.2 Taylor Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 6.3 Estimates Using Taylor Polynomial . . . . . . . . . . . . . . . . . . . . . . . . 128 6.3.1 How to compute e in a few decimal places? . . . . . . . . . . . . . . . . 128 6.3.2 How good is the approximation of T f for f(x) = cos(x)? . . . . . . 129 4 6.3.3 Error in the Approximation sinx ≈ x . . . . . . . . . . . . . . . . . . 130 6.4 Estimating Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 6.5 Extra curricular material: Estimating n for Taylor Polynomials . . . . . . . . 135 6.6 Exercise Sheet 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 6.6.1 Exercise Sheet 5a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 6.6.2 Exercise Sheet 5b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 6.6.3 Miscellaneous Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 6.6.4 Feedback for Exercise Sheet 5a . . . . . . . . . . . . . . . . . . . . . . 144 3 7 Power and Taylor Series 147 7.1 Lecture 19: About Taylor and Maclaurin Series . . . . . . . . . . . . . . . . . 147 7.1.1 Some Special Examples of Maclaurin Series . . . . . . . . . . . . . . . 147 7.1.2 Convergence Issues about Taylor Series . . . . . . . . . . . . . . . . . . 148 7.1.3 Valid Taylor Series Expansions . . . . . . . . . . . . . . . . . . . . . . 149 7.2 Lecture 20: Power Series, Radius of Convergence . . . . . . . . . . . . . . . . . 151 7.2.1 Behaviour at the Boundary of the Interval of Convergence . . . . . . . 153 7.2.2 Elementary Calculus of Taylor Series . . . . . . . . . . . . . . . . . . . 154 7.3 Lecture 21: More on Power and Taylor Series . . . . . . . . . . . . . . . . . . 155 7.3.1 The Binomial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 7.4 Extra curricular material: General Theorem About Taylor Series . . . . . . . . 159 7.4.1 Examples of Power Series Revisited . . . . . . . . . . . . . . . . . . . . 160 7.4.2 Leibniz’ Formulas For ln2 and π/4 . . . . . . . . . . . . . . . . . . . . 161 7.5 Extra curricular material: Taylor Series and Fibonacci Numbers . . . . . . . . 163 7.5.1 Taylor Series in Number Theory . . . . . . . . . . . . . . . . . . . . . . 163 7.5.2 Taylor’s Formula and Fibonacci Numbers . . . . . . . . . . . . . . . . . 164 7.5.3 More about the Fibonacci Numbers . . . . . . . . . . . . . . . . . . . . 165 7.6 Extra curricular Christmas treat: Series of Functions Can Be Difficult Objects 167 7.6.1 What Can Go ‘Wrong’ with Taylor Approximation? . . . . . . . . . . . 167 7.6.2 The Day That All Chemistry Stood Still . . . . . . . . . . . . . . . . . 168 7.6.3 Series Can Define Bizarre Functions: Continuous but Nowhere Differ- entiable Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 7.7 Exercise Sheet 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 7.7.1 Exercise Sheet 6a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 7.7.2 Feedback for Exercise Sheet 6a . . . . . . . . . . . . . . . . . . . . . . 175 7.7.3 Additional Exercise Sheet 6b . . . . . . . . . . . . . . . . . . . . . . . . 180 7.7.4 Exercise Sheet 0c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 7.7.5 Feedback for Exercise Sheet 0c . . . . . . . . . . . . . . . . . . . . . . . 183 4 List of Figures 1.1 The Maclaurin polynomials of degree 0, 1 and 2 of ex . . . . . . . . . . . . . . 13 1.2 Maclaurin polynomials of ex and perturbations . . . . . . . . . . . . . . . . . . 14 1.3 Maclaurin polynomials of sinx . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 x3 1.4 The graph near x = 0 of g(x) = cos(x2) esinx ln(1 x) . . . . . . . . 24 − − − − 3 2.1 The first few values of a sequence alternating about the limit, like 1+ (−1)n. . 43 n 2.2 The first fewvaluesofasequence converging inarandomfashionto thelimiting value, like 1+ cosn. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 n 2.3 The first few values of a sequence increasing to 1, like (1 1). . . . . . . . . 51 − n 2.4 The first few values of a sequence decreasing to 1, like (1+ 1). . . . . . . . . 51 n 4.1 The area under the curve is less than the area under the line y = f(k 1) and − it is greater than the area under the line y = f(k). . . . . . . . . . . . . . . . 90 7.1 An innocent looking function with an unexpected Taylor series. . . . . . . . . 169 7.2 Example (7.10) of a continuous nowhere differentiable real function . . . . . . 170 5 Chapter 0 Introduction, Overview The first 12 lectures (Chapters 1-3) contribute to the study blocks MA2730 (for M, FM, MSM and MCS) and MA2712b (for MMS and MCC). Those blocks feed into the assessment blocks MA2812 and MA2815 (for M, FM, MSM and MCS) or MA2810 (for MMS and MCC). The purpose of those lectures is to make you familiar with important concepts in Calculus and Analysis, namely those of sequences and series as well as Taylor1 polynomials and series. In the first three chapters, you shall be introduced to elementary ideas about these concepts, so you could apprehend them as well as follow and perform relevant calculations. Students studying the full Analysis study block (MA2730) will continue, revisiting, broadening and deepening these concepts. In particular, you will be given the means to use more formal definitions and prove the results stated in the following first set of lectures. Most of Calculus (MA1711) and Fundamentals of Mathematics (MA1712) may be used in this set of lectures. For quick reference, we have put some essential background material of Level 1 in a revision section on Blackboard. You will have two lectures a week with one seminar (the class is split in four seminar groups). The 24 lectures are split into 6 chapters; each section of those chapters corresponding to a lecture. There will be one set of exercise sheets per chapter, obviously most of these sheets lasting for a few weeks. In Exercise Sheet number Na, N = 1,...,6, we expect to give you some short feedback first (to check your answers), finally rather detailed feedback. Additional exercises are given in Exercise Sheets Nb, N = 1,...,6. Hence there will only be short feedback for them. Chapter 1 Taylor Polynomials (5 Lectures) Taylor polynomials are approximating a given differentiable function f, say, in a neighbour- hood of a given point x = a, say, by a polynomial of degree n, say. This means that the notation Taf for such polynomial looks cumbersome, but it encodes the full information we n need to know about them. Because polynomials are often easier to manipulate than more complex functions, we can replace those by one of their appropriate Taylor polynomial 1B. Taylor (1685-1731) was an English mathematician who worked on analysis, geometry and mechanics (vibrating string). He introduced the ‘calculus of finite differences’. In that context, he invented integration bypartsandgavehisversionofwhatisknownasTaylor’sTheorem,althoughthetheoremwasalreadyknown. He got embroiled in the disputes between English and continental mathematicians about following Newton’s or Leibnitz’s versions of calculus. He worked also on the foundation of descriptive and projective geometry but did not elaborate although he gave some deep results. 6 1. to evaluate limits, 2. to determine the type of a degenerate critical point or, even, 3. to approximate integrals of complicated functions. In the first two operations, the result of the replacement will be an exact value, in the last it will only be an approximation. In the five lectures of this chapter, we shall first define Taylor polynomials, then give examples of calculation of those Taylor polynomial. This involves the calculation of many derivatives. You should pay attention to the Product, Quotient and Chain Rules for differentiation. In the second lecture we shall also give a first idea about the error Raf = f Taf made by n − n replacing a function f by one of its Taylor polynomial Taf. In practical term, the calculation n ofaTaylor polynomial ofacomplexfunctioncanbesimplified byusing calculus rulestoobtain the calculus of Taylor polynomials. To justify those calculations we need the information we got in the second lecture about the error term. Those are the topics of the third lecture. We end the chapter with two lectures on the application of Taylor polynomials to calculate limits and to study the type of degenerate critical points of real functions. Chapter 2: Real Sequences (3 Lectures) In the second chapter, we consider sequences (of real numbers) and their limits. Sequences are ordered countable sets a ,a ,... of real numbers. In this chapter we shall concentrate 1 2 on special sequences defined by real functions such that a = f(n), for all n N, where n ∈ f : [1, ) R. This will allow us to use what you did with limits of functions at Level 1. ∞ → This is not a significant restriction because, in practice, we shall use sequences of this type. In lecture six, the limit of such sequence will then be defined as equal to lim f(x). In lecture x →∞ seven, we use the algebra of limits of functions to determine an essentially equivalent algebra of limits of sequences as well as establishing some important limits. Those results are important because, when they hold, they free us to have to go back to the initial definition. The chapter ends with lecture eight where we discuss some important qualitative properties ofsequences, inparticular the fundamental theorem stating that bounded AND monotone sequences converge. Chapter 3: Improper Integrals (2 Lectures) This material has been pulled through from Level 1 in order to leave more room behind. It deals with the evaluation of definite integrals in two cases. Type 1: where the interval of integration if infinite; and, Type 2: where the integrand becomes infinite at a vertical asymptote. 7 Chapter 4: Real Series (4 Lectures) The first half of the course finishes in Chapter 3 with the study of series of real numbers. ∞ Series are infinite sums of sequences, say a . You have already seen that a geometric n n=1 X series converges if and only if its common ratio is strictly smaller than 1 in modulus. In lecture nine, we give the definition of the convergence of a series, using the convergence of the sequence of partial sums. In the next lecture ten, we look at other important series and show that there exists an algebra of convergent series. This leads to the discussion of many tests to assess the convergence or divergence of series without calculating their sum. In the last two lectures of the chapter, eleven and twelve, we discuss the Comparison, Integral, Ratio and Root Tests, usually by comparing with known convergent or divergent series, like the geometric or harmonic series. In this theory, it means that we establish first the convergence or divergence of a series, then establish its exact sum (when possible), or give estimates for it. This finishes the first half of the block. We now move to deepen our understanding of the first 12 lectures. Chapter 5: Additional, Deeper Results on Sequences and Series (2 Lectures) In this chapter we use a more general definition of limit of sequences that works for any real sequence: the (ǫ,N)-definition. In lecture thirteen you shall learn how to use it, and show that it is equal to our previous definition in the context of Chapter 2. In lectures fourteen and fifteen, we use the more subtle definition to study the convergence of alternating series, series of real numbers that converge but, with a divergent series of the modulus of the terms. We shall also see (mainly in exercises) that such series behave in an unexpected manner. Chapter 6: Approximation with Taylor Polynomials (3 Lectures) Chapter 5 is entirely devoted to the estimate of errors made by replacing functions by their Taylorpolynomials. InlecturesixteenwestateandproveTaylor’sTheorem, givinganestimate of the error term Raf with greater precision than in Chapter 1, so that we can estimate other n error terms, in particular in lecture eighteen, where we discuss approximation of integrals. All those estimates lead to the next, and final, chapter. Chapter 7: Power and Taylor Series (4 Lectures) This final chapter is about functions defined as series (with an infinite sum depending on ∞ xn powers of the variable x, like ). This series happens to converge to ex and its finite n! n=0 X 8 initial segments correspond to the Taylor polynomials T ex of ex. From the other side, as we n let n tend to , what happens to Taf(x)? How far is it from f(x)? Those questions are ∞ n studied in the last 5 lectures. In lectures twenty one and twenty two, we derive fundamental properties shared by all power series. We finish in the last two lectures by applying those results to Taylor series. 9