MATH10242 Sequences and Series Mike Prest1 School of Mathematics Alan Turing Building Room 1.120 [email protected] January 26, 2018 1These notes are a slightly modified version of notes developed by Prof. J. T. Stafford and, before him, Prof. A. J. Wilkie. Contents 0.1 Introduction . .3 1 Before We Begin 4 1.1 Some Reminders about Mathematical Notation . .4 1.1.1 Special sets . .4 1.1.2 Set theory notation . .4 1.1.3 Logical notation . .4 1.1.4 Greek letters . .5 1.1.5 Structure of the course . .5 1.1.6 Where we're headed and some things we'll see on the way . .6 1.1.7 Reading outside the Course Notes . .7 1.1.8 Basic properties of the real numbers . .7 1.1.9 The Integer Part (or `Floor') Function . .9 I Sequences 10 2 Convergence 11 2.1 What is a Sequence? . 11 2.2 The Triangle Inequality . 12 2.3 The Definition of Convergence . 13 2.4 The Completeness Property for R ........................... 15 2.5 Some General Theorems about Convergence . 18 2.6 Exponentiation - a digression ............................. 19 3 The Calculation of Limits 21 3.1 The Sandwich Rule . 21 3.2 The Algebra of Limits. 23 4 Some Special Sequences 27 4.1 Basic Sequences . 27 4.2 New Sequences from Old . 30 4.3 Newton's Method for Finding Roots of Equations - optional ............ 34 5 Divergence 36 5.1 Sequences that Tend to Infinity . 36 6 Subsequences 39 6.1 The Subsequence Test for Non-Convergence . 39 6.2 Cauchy Sequences and the Bolzano-Weierstrass Theorem . 41 6.2.1 Proofs for the section - optional ........................ 41 1 7 L'H^opital'sRule 45 7.1 L'H^opital'sRule . 45 II Series 48 8 Introduction to Series 49 8.1 The Basic Definitions . 50 9 Series with Non-Negative Terms 53 9.1 The Basic Theory of Series with Non-Negative Terms . 53 9.2 The Integral Test . 56 10 Series with Positive and Negative Terms 62 10.1 Alternating Series . 62 10.2 Absolute Convergence . 65 11 Power Series 67 11.1 The Radius of Convergence of a Power Series . 68 12 Further Results on Power Series - further reading 72 12.1 More General Taylor Series. 72 12.2 Rearranging Series . 73 12.3 Analytic Functions . 76 2 0.1 Introduction Maybe you can see that 1 1 1 1 1 + + + + ··· + + ··· = 2 2 4 8 2n and even that 1 1 1 1 3 1 + + + + ··· + + ··· = : 3 9 27 3n 2 But what exactly do these formulas mean? After all, we cannot really add infinitely many numbers together. You might recognize them as geometric progressions and know the general formula 1 1 + x + x2 + x3 + ··· + xn + ··· = : 1 − x But how do we prove this? And what if x ≥ 1? In this course we shall answer these, and related, questions. In particular, we shall give a rigorous definition of what it means to add up infinitely many numbers and then we shall find rules and procedures for finding the sum in a wide range of particular cases. Here are some more, rather remarkable, such formulas: 1 1 1 1 π2 1 + + + + ··· + + ··· = ; 4 9 16 n2 6 1 1 1 (−1)n+1 1 − + − + ··· + + ··· = log 2; 2 3 4 n 1 1 1 1 1 + + + + ··· + + ··· = 1: 2 3 4 n We shall prove the second and third of these formulas in this course unit, but the first one is too difficult and will be done in your lectures on real and complex analysis in the second year. Here, \real analysis" means the study of functions from real numbers to real numbers, from the point of view of developing a rigorous foundation for calculus (differentiation and integration) and for other infinite processes.1 The study of sequences and series is the first step in this programme. This also means there are two contrasting sides to this course. On the one hand we will develop the machinery to produce formulas like the ones above. On the other hand it is also crucial to understand the theory that lies behind that machinery. This rigorous approach forms the second aspect of the course|and is in turn the first step in providing the formal foundation of real analysis. 1and \complex analysis" refers to the complex numbers, not (necessarily) complexity in the sense of \compli- cated"! 3 Chapter 1 Before We Begin 1.1 Some Reminders about Mathematical Notation 1.1.1 Special sets We use the following notation throughout the course. R - the set of real numbers; + + R - the set of strictly positive real numbers, i.e. R = fx 2 R : x > 0g; Q - the set of rational numbers; Z - the set of integers (positive, negative and 0); + N (or Z ) - the set of natural numbers, or positive integers fx 2 Z : x > 0g. (In this ≥0 course, we do not count 0 as a natural number. We can use some other notation like Z for the set of integers greater than or equal to 0.) ; - the empty set. 1.1.2 Set theory notation The expression \x 2 X" means x is an element (or member) of the set X. For sets A, B, we write A ⊆ B to mean that A is a subset of B (i.e. every element of A is also an element of B). Thus ; ⊆ N ⊆ Z ⊆ Q ⊆ R: Standard intervals in R: if a; b 2 R with a ≤ b, then • (a; b) =fx 2 R : a < x < bg; • [a; b) =fx 2 R : a ≤ x < bg; • (a; b] =fx 2 R : a < x ≤ bg; • [a; b] =fx 2 R : a ≤ x ≤ bg; • (a; 1) =fx 2 R : a < xg; • [a; 1) =fx 2 R : a ≤ xg; • (−∞; b) =fx 2 R : x < bg; • (−∞; b] =fx 2 R : x ≤ bg; • (−∞; 1) = R. 1.1.3 Logical notation The expression \8x :::" means \for all x :::" and \9x :::" means \there exists at least one x such that :::". These are usually used in the context \8x 2 A :::" meaning \for all elements x of the set A ...", and \9x 2 A :::" meaning \there exists at least one element x in the set A such that :::". Thus, for example, \8x 2 R x > 1" means \for all real numbers x, x is greater than 1" (which happens to be false) and \9x x > 1" means \there exists a real number x such that x is greater than 1" (which happens to be true). 4 1.1.4 Greek letters The two most commonly used Greek letters in this course are δ (delta) and (epsilon). They are reserved exclusively for (usually small) positive real numbers. Others are α (alpha), β (beta), γ (gamma), λ (lambda), θ (theta-usually an angle), η (eta) and Σ (capital sigma - the summation sign which will be used when we come to study series in Part II). 1.1.5 Structure of the course This course unit has two lectures each week; each student has one tutorial per week - you will be assigned to one of the four tutorial groups (this should be shown in your space on the student system). Lectures: Tuesday 10:00{11:00, except for the first week of term, when it will be 11:00-12:00, in Crawford House, Theatre 1; Thursday 12:00{1:00 in Simon Building, Theatre E. Tutorials: These start on 5th February. You will be assigned to one of these: Tutorial 1: Monday 11:00{12:00 in Simon Building, Room 3.40; Tutorial 2: Monday 15:00{16:00 in Samuel Alexander Building, SG.16; Tutorial 3: Tuesday 11:00{12:00 in University Place, 1.219; Tutorial 4: Monday 16:00{17:00 in Samuel Alexander Building, SG.16. The tutorials start in Week 2 and, typically, Week n tutorials concentrate on the subject matter introduced in Week n − 1. I will put the weekly exercise sheets on my teaching webpage website. (Complete course notes and some links are already there.) It is very important that you work at these sheets before the weekly tutorials; try to have a serious attempt at all the problems. But don't waste time going round in circles: try to recognise when you're doing that, have a break/do something else and, maybe, when you come back to the question you'll see your way around what seemed to be a problem. In general, for each lecture or tutorial hour, you should expect to spend two to three hours working at understanding the material and testing/strengthening your understanding by doing problems. It is not enough to read and summarise the notes; it is only by testing your understanding by doing examples that you will really understand the material. Moreover, the exam questions will be similar to these problems! When you get really stuck on something, discuss it with someone! As well as me and the people who help in the tutorials, there are your fellow-students - they are an excellent resource and bear in mind that one of the most useful exercises you can do is to try to explain something to someone else. Office Hour: see my teaching website http://www.maths.manchester.ac.uk/∼mprest/teaching.html Assessment: 1. Final exam carrying 80% of the course weight. 2. Coursework carries 20% of the course weight and will consist of: an in-class test on Thursday 15th March. 5 1.1.6 Where we're headed and some things we'll see on the way In Part I we aim to understand the behaviour of infinite sequences of real numbers, meaning what happens to the terms as we go further and further on in the sequence.