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) = sin x ...... 14 2 1.2.3 Find the Maclaurin polynomial T11f for f(x) = sin(x ) ....... 15 1.2.4 QuestionsforChapter6: ErrorEstimates . 15 1.3 Lecture 4 and 5: Calculus of Taylor Polynomials . .. 17 1.3.1 GeneralResults............................... 17 1.4 Lecture 6: Various Applications of Taylor Polynomials . ... 22 1.4.1 RelativeExtrema .............................. 22 1.4.2 Limits .................................... 24 1.4.3 How to Calculate Complicated Taylor Polynomials? . 26 1.5 ExerciseSheet1................................... 29 1.5.1 ExerciseSheet1a .............................. 29 1.5.2 FeedbackforSheet1a ........................... 33 2 Real Sequences 40 2.1 Lecture 7: Definitions, Limit of a Sequence . ... 40 2.1.1 DefinitionofaSequence .......................... 40 2.1.2 LimitofaSequence............................. 41 2.1.3 Graphic Representations of Sequences . .. 43 2.2 Lecture 8: Algebra of Limits, Special Sequences . ..... 44 2.2.1 InfiniteLimits................................ 44 1 2.2.2 AlgebraofLimits.............................. 44 2.2.3 Some Standard Convergent Sequences . .. 46 2.3 Lecture 9: Bounded and Monotone Sequences . ..... 48 2.3.1 BoundedSequences............................. 48 2.3.2 Convergent Sequences and Closed Bounded Intervals . .... 48 2.4 Lecture10:MonotoneSequences . ... 49 2.4.1 Convergence of Monotone, Bounded Sequences . .... 50 2.5 ExerciseSheet2................................... 54 2.5.1 ExerciseSheet2a .............................. 54 2.5.2 FeedbackforSheet2a ........................... 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 Homeworkforimproperintegrals,Types1and2. .... 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 Lecture12:Series.................................. 78 4.1.1 A Tale of a Rabbit and a Turtle following Zeno’s . 78 4.1.2 DefinitionofaSeries ............................ 79 4.1.3 Convergent Series, Geometric Series . 80 4.2 Lecture13:ImportantSeries. ... 83 4.2.1 ACriterionforDivergence .. .. .. 83 4.2.2 TelescopicSeries .............................. 84 4.2.3 HarmonicSeries............................... 85 4.2.4 AlgebraofSeries .............................. 86 4.3 Lecture14:TestforConvergence . .... 88 4.3.1 ComparisonTest .............................. 89 4.3.2 IntegralTest................................. 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 ExerciseSheet3................................... 96 4.5.1 ExerciseSheet3a .............................. 96 4.5.2 Additional Exercise Sheet 3b . 97 4.5.3 FeedbackforSheet3a ........................... 99 2 5 Deeper Results on Sequences and Series 103 5.1 Lecture 16: (ǫ, N)-DefinitionofLimits. .. .. 103 5.1.1 Practical Aspects of Estimates of Convergent Sequences . ....... 104 5.1.2 DivergentSequences ............................ 106 5.2 Extra curricular material: Error Estimates from the Tests . ........ 108 5.2.1 ErrorEstimatesfromtheRatioandRootTests . 108 5.2.2 ErrorEstimatesfortheIntegralTest . 108 5.3 Lecture 17: Absolute Convergence of Series and the Leibnitz Criterion of Con- vergence ....................................... 111 5.3.1 AlternatingSequences . .. .. .. 111 5.4 ExerciseSheet4................................... 116 5.4.1 ExerciseSheet4a .............................. 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 FeedbackforExerciseSheet4a . 119 6 Approximation with Taylor Polynomials 123 6.1 Lecture 18: Taylor’s theorem and errorestimates . ...... 123 6.2 TaylorTheorem................................... 124 6.3 Estimates Using Taylor Polynomial . 128 6.3.1 How to compute e inafewdecimalplaces?. 128 6.3.2 How good is the approximation of T4f for f(x) = cos(x)? ...... 129 6.3.3 Error in the Approximation sin x ≈ x .................. 130 6.4 EstimatingIntegrals ................................ 131 6.5 Extra curricular material: Estimating n for Taylor Polynomials . 135 6.6 ExerciseSheet5................................... 138 6.6.1 ExerciseSheet5a .............................. 138 6.6.2 ExerciseSheet5b.............................. 139 6.6.3 Miscellaneous Exercises . 140 6.6.4 FeedbackforExerciseSheet5a . 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 Lecture21: MoreonPowerandTaylorSeries . 155 7.3.1 TheBinomialTheorem........................... 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 ln 2 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 MoreabouttheFibonacciNumbers . 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- entiableFunctions ............................. 170 7.7 ExerciseSheet6................................... 172 7.7.1 ExerciseSheet6a .............................. 172 7.7.2 FeedbackforExerciseSheet6a . 175 7.7.3 Additional Exercise Sheet 6b . 180 7.7.4 ExerciseSheet0c .............................. 181 7.7.5 FeedbackforExerciseSheet0c. 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 andperturbations. 14 1.3 Maclaurin polynomials of sin x ........................... 15 x3 1.4 The graph near x =0 of g(x) = cos(x2) esin x ln(1 x) . ....... 24 − − − − 3 ( 1)n − 2.1 The first few values of a sequence alternating about the limit, like 1 + n . 43 2.2 The first few values of a sequence converging in a random fashion to the limiting cos n value, like 1 + n . ................................ 43 2.3 The first few values of a sequence increasing to 1, like (1 1 ). ........ 51 − n 1 2.4 The first few values of a sequence decreasing to 1, like (1 + n ). ........ 51 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