DOCSLIB.ORG

Sequence and Series

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Sequence and Series Chapter 4 Sequences and Series–First View Recall that, for any set A, a sequence of elements of A is a function f : M A. Rather than using the notation f n for the elements that have been selected from A, since the domain is always the natural numbers, we use the notational convention an f n and denote sequences in any of the following forms: * an n1 an n+M or a1 a2 a3 a4 This is the only time that we use the set bracket notation in a different con- text. The distinction is made in the way that the indexing is communicated . For : * : ¿ an ,the an n1 is the constant sequence that “lists the term in nitely often,” : : : : while an : n + M is the set consisting of one element :. (When you read the last sentence, you should have come up with some version of “For ‘a sub n’ equal to :, the sequence of ‘a sub n’forn going from one to in¿nity is the constant sequence that “lists the term : in¿nitely often,” : : : while the set consisting of ‘a sub n’forn in the set of positive integers is the set consisting of : * one element ” i.e., the point is that you should not have skipped over the an n1 and an : n + M .) Most of your previous experience with sequences has been with sequences of real numbers, like | }* 3 1 1 2 3 5 8 13 21 34 55 ,... n 1 n1 | }* | }* | r s}* n2 3n 5 n3 1 log n nH 1n and sin . 3 n 47 n1 n 1 n1 n 8 n1 In this chapter, most of our sequences will be of elements in Euclidean n-space. In MAT127B, our second view will focus on sequence of functions. 123 124 CHAPTER 4. SEQUENCES AND SERIES–FIRST VIEW As children, our ¿rst exposure to sequences was made in an effort to teach us to look for patterns or to develop an appreciation for patterns that occur naturally. Excursion 4.0.1 For each of the following, ¿nd a description for the general term as a function of n + M that ¿ts the terms that are given. 2 4 8 16 32 64 1. 5 7 9 11 13 15 3 7 11 2. 1 9 81 729 5 9 13 ***An equation that works for (1) is 2n2n 31 while (2) needs a different for- mula for the odd terms and the even terms one pair that works is 2n 12n 11 for n even and 3n1 when n is odd.*** As part of the bigger picture, pattern recognition is important in areas of math- ematics or the mathematical sciences that generate and study models of various phenomena. Of course, the models are of value when they allow for analysis and/or making projections. In this chapter, we seek to build a deeper mathematical under- standing of sequences and series primary attention is on properties associated with convergence. After preliminary work with sequences in arbitrary metric spaces, we will restrict our attention to sequences of real and complex numbers. 4.1 Sequences and Subsequences in Metric Spaces If you recall the de¿nition of convergence from your frosh calculus course, you might notice that the de¿nition of a limit of a sequence of points in a metric space merely replaces the role formerly played by absolute value with its generalization, distance. 4.1. SEQUENCES AND SUBSEQUENCES IN METRIC SPACES 125 ¿ * De nition 4.1.1 Let pn n1 denote a sequence of elements of a metric space S d * and p0 be an element of S. The limit of pn n1 is p0 as n tends to (goes to or ap- proaches) in¿nity if and only if d e 1 + U F 0 " 2M MM + MF 1nn M " dpn p0 We write either pn p0 or lim pn p0. n* Remark 4.1.2 The description M M indicates that “limit of sequence proofs” require justi¿cation or speci¿cation of a means of prescribing how to ¿nd an M that “will work” corresponding to each 0. A function that gives us a nice way to specify M ’s is de¿ned by LxM inf j + ] : x n j z { 1 and is sometimes referred to as the ceiling function. Note, for example, that 2 1, L22M 2,andL5M 5. Compare this to the greatest integer function, which is sometimes referred to as the Àoor function. | }* 2 Example 4.1.3 The sequence has the limit 0 in U. We can take M 1 2, t u t nu n1 z { 1 3 700 M 200, and M 234. Of course, three examples 100 350 3 z { 2 does not a proof make. In general, for 0,letM .Thenn M implies that z { 2 o 2 n 0 n n n n 1 2 n2n which, by Proposition 1.2.9 (#7) and (#5), implies that and n n . n 2 n n Using the de¿nition to prove that the limit of a sequence is some point in the metric space is an example of where our scratch work towards ¿nding a proof might 126 CHAPTER 4. SEQUENCES AND SERIES–FIRST VIEW be quite different from the proof that is presented. This is because we are allowed to “work backwards” with our scratch work, but we can not present a proof that starts with the bound that we want to prove. We illustrate this with the following excursion. Excursion 4.1.4 After reading the presented| } scratch work, ¿ll in what is missing to * 1 in F complete the proof of the claim that converges to i in . n 1 n1 (a) Scratch work towards a proof.Becausei+ F,itsuf¿ces to show that 1 in lim i. Suppose 0 is given. Then n* n 1 n n n n n n T T n n n n n n n1 in n n1 in i n 1n n 1 i n 2 2 n in n n n n n 1 n 1 n 1 n 1 n T T 2 2 whenever n. So taking M will work. (b) A proof. For 0,letM .Thenn+ M and n M T 1 2 implies that n which is equivalent to . Because 2 T 4 T 2 n 1 n and 2 0, we also know that . Consequently, if n 3 n M , then n n n n n n T n n n n n n n1 in n n1 in i n 1 n n 1 i n 2 n in n n n n n 1 n 1 n 1 n 1 5 Since 0 was arbitrary, we conclude that v t t n n uuw n n n1 in n 1 0 " 2M M + MF 1n n M " n in n 1 1 in i.e., . Finally, i 0 1 + F and lim n* n 1 | 6 } * 1 in F i yields that converges to i in . n 1 n1 4.1. SEQUENCES AND SUBSEQUENCES IN METRIC SPACES 127 T T T 2 2 T 2 ***Acceptable responses are (1) ,(2) ,(3)n 1, (4) 2, (5) ,(6) n n 1 in lim i.*** n* n 1 ¿ * De nition 4.1.5 The sequence pn n1 of elements in a metric space S is said to converge (or be convergent) in S if there is a point p0 + S such that lim pn p0 n* it is said to diverge in S if it does not converge in S . Remark 4.1.6 Notice that a sequence in a metric space S will be divergent in S if | }* 2 its limit is a point that is not in S. In our previous example, we proved that n | }* n 1 2 converges to 0 in U consequently, is convergent in Euclidean 1-space. n | }* n 1 2 b c On the other hand, is divergent in U x + U : x 0 d where d n n1 denotes the Euclidean metric on U,dx y x y . Our ¿rst result concerning convergent sequences is metric spaces assures us of the uniqueness of the limits when they exist. * Lemma 4.1.7 Suppose pn n1 is a sequence of elements in a metric space S d . Then rK L s 1p1q p q + S F lim pn p F lim pn q " q p . n* n* Space for scratch work. Excursion 4.1.8 Fill in what is missing in order to complete the proof of the lemma. * Proof. Let pn n1 be a sequence of elements in a metric space S d for which there exists p and q in S such that lim pn p and lim pn q. Suppose n* n* 128 CHAPTER 4. SEQUENCES AND SERIES–FIRST VIEW 1 the p / q. Then d p q 0 and we let d p q. Because lim pn p and 2 n* 0, there exists a positive integer M1 such that n M1 " d pn p similarly, lim pn q yields the existence of a positive integer M2 such that n* . 1 Now, let M max M1 M2 . It follows from the symmetry property and the triangular inequality for metrics that n M implies that d p q n d p pn 2 d p q 2 3 which contradicts the trichotomy law. Since we have reached a contradiction, we conclude that as needed. Therefore, the limit of any convergent sequence 4 in a metric space is unique. ***Acceptable ¿ll-ins are: (1) n M2 " d pn q ,(2)d pn q (3) 1 d p q,(4)p q.*** 2 ¿ * De nition 4.1.9 The sequence pn n1 of elements in a metric space S d is bounded if and only if d e 2M2x M 0 F x + S F 1nn + M " d x pnM .
Load more

Recommended publications
  • Lecture Notes M 517 Introduction to Analysis Donald J. Estep
    Lecture Notes M 517 Introduction to Analysis Donald J. Estep Department of Mathematics Colorado State University Fort Collins, CO 80523 [email protected] http://www.math.colostate.edu/∼estep Contents Chapter 1. Introduction 1 1.1. Review of Real Numbers 1 1.2. Functions and Sets 3 1.3. Unions and Intersection 5 1.4. Cardinality 5 Chapter 2. Metric Spaces 10 2.1. Quick Reivew of Rn 10 2.2. Definition and Initial Examples of a Metric Space 11 2.3. Components of a Metric Space 15 Chapter 3. Compactness 20 3.1. Compactness Arguments and Uniform Boundedness 20 3.2. Sequential Compactness 24 3.3. Separability 25 3.4. Notions of Compactness 28 3.5. Some Properties of Compactness 32 3.6. Compact Sets in Rn 33 Chapter 4. Cauchy Sequences in Metric Spaces 35 4.1. A Few Facts and Boundedness 35 4.2. Cauchy Sequences 36 4.3. Completeness 37 4.4. Cauchy Sequences and Compactness 40 Chapter 5. Sequences in Rn 42 5.1. Arithmetic Properties and Convergence 42 5.2. Sequences in R and Order. 45 5.3. Series 46 Chapter 6. Continuous Funtions on Metric Spaces 48 6.1. Limit of a Function 49 6.2. Continuous Functions 50 6.3. Uniform Continuity 52 6.4. Continuity and Compactness 53 6.5. Rn-valued Continuous Functions 54 Chapter 7. Sequences of Functions and C([a, b]) 57 7.1. Convergent Sequences of Functions 57 7.2. Uniform Convergence: C([a, b]) is Closed, and Complete 61 7.3. C([a, b]) is Separable 63 ii CONTENTS iii 7.4.
    [Show full text]
  • Generalized Tikhonov Regularization: Topological Aspects and Necessary Conditions
    Generalized Tikhonov Regularization: Topological Aspects and Necessary Conditions Von der Carl-Friedrich-Gauß-Fakult¨at der Technischen Universit¨atCarolo-Wilhelmina zu Braunschweig zur Erlangung des Grades einer Doktorin der Naturwissenschaften (Dr. rer. nat.) genehmigte Dissertation von Nadja Sofia Worliczek geboren am 17.08.1978 in Wien/Osterreich¨ Eingereicht am: 17.12.2013 Disputation am: 31.03.2014 1. Referent: Prof. Dr. Dirk Lorenz 2. Referent: Prof. Dr. Thorsten Hohage 2014 ii Contents Contents iii Introductionv 1 Similarity and convergence issues1 1.1 Preliminaries on topological spaces...............1 1.2 Preliminaries on sequential convergence spaces.........5 1.3 Prametrics............................. 11 1.3.1 The prametric topology................. 12 1.3.2 Sequential convergence structures induced by prametrics 17 2 Ill-posed inverse problems and regularization 23 2.1 Inverse problems and terms of solution............. 23 2.2 Well-posedness and stability................... 29 2.3 Regularization........................... 33 3 Generalized Tikhonov regularization 39 3.1 Basic notation........................... 39 3.2 Formalization of Tikhonov setups................ 40 3.3 Some necessary conditions on regularizing Tikhonov setups.. 43 4 Necessary conditions on a common set of sufficient condi- tions 47 4.1 A common set of sufficient conditions.............. 47 4.2 CONV and CONT........................ 50 4.2.1 Bottom slice topologies.................. 51 4.2.2 Necessary conditions................... 55 5 The special case of Bregman discrepancies 59 5.1 Bregman distances - definition and basic properties...... 60 5.2 Bregman prametrics, topological issues and Tikhonov regular- ization............................... 66 iii CONTENTS Conclusion 73 Notation 75 Index 78 Bibliography 81 Zusammenfassung (Summary in German) 87 iv Introduction Context, motivation and subject An inverse problem is the problem of retrieving some quantity of interest from some measured data, that is in some way indirectly related to our quantity of interest, e.g.
    [Show full text]
  • Real Analysis
    Real Analysis Andrew Kobin Fall 2012 Contents Contents Contents 1 Introduction 1 1.1 The Natural Numbers . .1 1.2 The Rational Numbers . .1 1.3 The Real Numbers . .6 1.4 A Note About Infinity . .8 2 Sequences and Series 9 2.1 Sequences . .9 2.2 Basic Limit Theorems . 12 2.3 Monotone Sequences . 16 2.4 Subsequences . 19 2.5 lim inf and lim sup . 22 2.6 Series . 23 2.7 The Integral Test . 28 2.8 Alternating Series . 30 3 Functions 32 3.1 Continuous Functions . 32 3.2 Properties of Continuous Functions . 35 3.3 Uniform Continuity . 37 3.4 Limits of Functions . 41 3.5 Power Series . 44 3.6 Uniform Convergence . 46 3.7 Applications of Uniform Convergence . 49 4 Calculus 52 4.1 Differentiation and Integration of Power Series . 52 4.2 The Derivative . 54 4.3 The Mean Value Theorem . 57 4.4 Taylor's Theorem . 61 4.5 The Integral . 63 4.6 Properties of Integrals . 67 i 1 Introduction 1 Introduction These notes were compiled from a semester of lectures at Wake Forest University by Dr. Sarah Raynor. The primary text for the course is Ross's Elementary Analysis: The Theory of Calculus. In the introduction, we develop an axiomatic presentation for the real numbers. Subsequent chapters explore sequences, continuity, functions and finally a rigorous study of single-variable calculus. 1.1 The Natural Numbers The following notation is standard. N = f1; 2; 3; 4;:::g the natural numbers Z = f:::; −3; −2; −1; 0; 1; 2; 3;:::g the integers n p o Q = q j p; q 2 Z; q 6= 0 the rational numbers R ? the real numbers Note that N ⊂ Z ⊂ Q ⊂ R.
    [Show full text]
  • Mathematical Analysis Master of Science in Electrical Engineering
    Mathematical Analysis Master of Science in Electrical Engineering Erivelton Geraldo Nepomuceno Department of Electrical Engineering Federal University of São João del-Rei August 2015 Prof. Erivelton (UFSJ) Mathematical Analysis August 2015 1 / 89 Teaching Plan Content 1 The Real and Complex Numbers Systems 2 Basic Topology 3 Numerical Sequences and Series 4 Continuity 5 Dierentiation 6 Sequences and Series of Functions 7 IEEE Standard for Floating-Point Arithmetic 8 Interval Analysis Prof. Erivelton (UFSJ) Mathematical Analysis August 2015 2 / 89 References Rudin, W. (1976), Principles of mathematical analysis, McGraw-Hill New York. Lima, E. L. (2014). Análise Real - Volume 1 - Funções de Uma Variável. 12 ed. Rio de Janeiro: IMPA. Overton, M. L. (2001), Numerical Computing with IEEE oating point arithmetic, SIAM. Institute of Electrical and Electronic Engineering (2008), 754-2008 IEEE standard for oating-point arithmetic. Goldberg, D. (1991), What Every Computer Scientist Should Know About Floating-point Arithmetic, Computing Surveys 23(1), 548. Moore, R. E. (1979), Methods and Applications of Interval Analysis, Philadelphia: SIAM. Nepomuceno, E. G (2014). Convergence of recursive functions on computers. The Journal of Engineering q, Institution of Engineering and Technology, 1-3. Prof. Erivelton (UFSJ) Mathematical Analysis August 2015 3 / 89 Assessment Item Value Date Observation N1 - Exam 1 100 02/09/2015 Chapters 1 and 2 N2 - Exam 2 100 14/10/2015 Chapters 3 and 4. N3 - Exam 3 100 04/11/2015 Chapters 5 and 6. N4 - Exam 4 100 02/12/2015 Chapters 7 and 8. Ns - Seminar 100 09/12/2015 Paper + Presentation. Ne - Especial 100 16/12/2015 Especial Exam Table 1: Assesment Schedule 2(N + N + N + N + N ) Score: S = 1 2 3 4 s 500 S + N With N the nal score is: S = e , otherwise S = S: e f 2 f If Sf ≥ 6:0 then Succeed.
    [Show full text]