2.7 Sequences, Sequences of Sets

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "2.7 Sequences, Sequences of Sets"

Transcription

1 2.7. SEQUENCES, SEQUENCES OF SETS Sequeces, Sequeces of Sets Sequeces Defiitio 190 (sequece Let S be some set. 1. A sequece i S is a fuctio f : K S where K = { N : 0 for some 0 N}. 2. For each K, we let x = f (. x is called the th term of the sequece f. For coveiece, we usually deote a sequece by {x } = 0 rather tha f. Some texts also use (x = 0. The startig poit, 0 is usually 1 i which case, we simply write {x } to deote a sequece. I theory, the startig poit does ot have to be 1. However, it is uderstood that whatever the startig poit is, the elemets x should be defied for ay 0. For example, if the geeral term of a sequece is x = 2 4, the, we must have 0 5. We ca thik of a sequece as a list of umbers. I this case, a sequece will look like: {x } = {x 1, x 2, x 3,...}. A sequece ca be give differet ways. { 1 1. List the elemets. For example, 2, 2 3, 3 } 4,.... From the elemets listed, the patter should be clear. 2. Give a formula to geerate the terms. For example, x = ( 1 2!. If the startig poit is ot specified, we use the smallest value of which will work. 3. A sequece ca be give recursively. The startig value of the sequece is give. The, a formula to geerate the th term from oe or more previous terms. For example, we could defie a sequece by givig: x 1 = 2 x +1 = 1 2 (x + 6 Aother example is the Fiboacci sequece defied by: x 1 = 1, x 2 = 1 x = x 1 + x 2 for 3 Like a fuctio, a sequece ca be plotted. However, sice the domai is a subset of Z, the plot will cosist of dots istead of a cotiuous curve. Sice a sequece is defied as a fuctio. everythig we defied for fuctios (bouds, supremum, ifimum,... also applies to sequeces. We restate those defiitios for coveiece.

2 68 CHAPTER 2. THE STRUCTURE OF R Defiitio 191 (Bouded Sequece As sequece (x is said to be bouded above if its rage is bouded above. It is bouded below if its rage is bouded below. It is bouded if its rage is bouded. If the domai of (x is { Z : k for some iteger k} the the above defiitio simply state that the set {x : k} must be bouded above, below or both. Defiitio 192 (Oe-to-oe Sequeces A sequece (x is said to be oeto-oe if wheever m the x x m. Defiitio 193 (Mootoe Sequeces Let (x be a sequece. 1. (x is said to be icreasig if x x +1 for every i the domai of the sequece. If we have x < x +1, we say the sequece is strictly icreasig. 2. (x is said to be decreasig if x x +1 for every i the domai of the sequece. 3. A sequece that is either icreasig or decreasig is said to be mootoe. If it is either strictly icreasig or strictly decreasig, we say it is strictly mootoe Sequeces of Sets ad Idexed Families of Sets Defiitio 194 Let A ad X be o-empty sets. dexed family of subsets of X with idex set s a fuctio f : A P (X (the power set of X. Like for sequeces, if f : A P (X, the for each α A, we let E α = f (α. We use a otatio similar to sequeces that is we deote the idexed family {E α } α A. If A = N as it is ofte the case (it will be for us, the {E α } α N is deoted {E } =1 or simply {E } ad it is called a sequece of subsets of X. Example 195 Cosider {N } =1 where N = {1, 2, 3,..., }. {N } =1 is a sequece of subsets of N. Example 196 For each N, defie I = {I } =1 is a sequece of subsets of R. { x R : 0 < x < 1 } ( = 0, 1. Example 197 For each x (0, 1, defie E x = {r Q : 0 r < x}. The, {E x } x (0,1 is a idexed family of subsets of Q. The idex set is (0, 1. The remaider of this sectio deals with sequeces of sets, though the results ad defiitios give ca be exteded to idexed families of subsets. Defiitio 198 (Uio ad Itersectio of a Sequece of Subsets Let {A } be a sequece of subsets of a set X.

3 2.7. SEQUENCES, SEQUENCES OF SETS We defie = A 1 A 2... A = {x X : x for some iteger 1 i } Similarly, we defie the uio of the etire sequece by = {x X : x for some iteger i} 2. Similarly, we defie = A 1 A 2... A = {x X : x for every iteger 1 i } ad = {x X : x for every iteger i} Example 199 Cosider {N } =1 where N = {1, 2, 3,..., }. The, ad N i = {1}. Example 200 For each N, defie I = prove that I i = If this were ot the case, that is if we had x N i = N { x R : 0 < x < 1 }. First, we I i this would mea that o matter what is, x < 1 or 1 > x which cotradicts the Archimedea priciple (theorem 173. We ext show that This because I I 1 for ay 1. I i = I 1

4 70 CHAPTER 2. THE STRUCTURE OF R Results about fiite itersectio ad uio of sets remai true i this settig. I other words, we have the equivalet of theorems 16 (distributive properties, 20 (De Morga s laws, 60 (direct image of a set ad 62 (iverse image of a set. We list the theorem here but leave their proof as exercises. Theorem 201 (Distributive Laws Let E ad E be subsets of a set X. The, 1. E E i = (E E i 2. E E i = (E E i Proof. See problems. Theorem 202 (De Morga s Laws Let {E } be a sequece of subsets of X. The, ( c E i = ( c E i = E c i E c i Proof. See problems. Theorem 203 Let f : X Y 1. If {E } is a sequece of subsets of X, the f E i = f (E i f E i f (E i 2. If {G } is a sequece of subsets of Y, the f 1 G i = f 1 (G i f 1 G i = f 1 (G i

5 2.7. SEQUENCES, SEQUENCES OF SETS 71 Proof. See problems. Defiitio 204 (Cotractig ad Expadig Sequeces of Sets Suppose that (A is a sequece of sets. 1. We say that (A is expadig if we have A A +1 for every i the domai of the sequece. 2. We say that (A is cotractig if we have A +1 A for every i the domai of the sequece Exercises [ ] 1 1. Let I =, 1. Evaluate 2. Let I = 3. Let I = 4. Prove theorem Prove theorem 202. =1 [ 1 + 1, 5 2 ]. Evaluate =1 ( 1 1, Evaluate =1 6. Prove theorem 203. I particular, for part 2 of the theorem, explai why we do ot have equality. Give a ecessary coditio to have equality ad justify your aswer. 7. Explai why if {A } is a expadig sequece of subsets of R, the {R \ A } is a cotractig sequece. 8. Let {A } be a sequece of sets. (a Prove that if we defie I I I B = for each, the sequece {B } is a cotractig sequece of sets. i=

6 72 CHAPTER 2. THE STRUCTURE OF R (b Prove that if we defie B = for each, the sequece {B } is a expadig sequece of sets. i= (c Prove that if we defie B = for each, the sequece {B } is a cotractig sequece of sets. (d Prove that if we defie B = for each, the sequece {B } is a expadig sequece of sets.

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008 I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces

More information

Infinite Sequences and Series

Infinite Sequences and Series CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...

More information

Measurable Functions

Measurable Functions Measurable Fuctios Dug Le 1 1 Defiitio It is ecessary to determie the class of fuctios that will be cosidered for the Lebesgue itegratio. We wat to guaratee that the sets which arise whe workig with these

More information

SAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx

SAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval

More information

Advanced Probability Theory

Advanced Probability Theory Advaced Probability Theory Math5411 HKUST Kai Che (Istructor) Chapter 1. Law of Large Numbers 1.1. σ-algebra, measure, probability space ad radom variables. This sectio lays the ecessary rigorous foudatio

More information

1 The Binomial Theorem: Another Approach

1 The Binomial Theorem: Another Approach The Biomial Theorem: Aother Approach Pascal s Triagle I class (ad i our text we saw that, for iteger, the biomial theorem ca be stated (a + b = c a + c a b + c a b + + c ab + c b, where the coefficiets

More information

Sequences II. Chapter 3. 3.1 Convergent Sequences

Sequences II. Chapter 3. 3.1 Convergent Sequences Chapter 3 Sequeces II 3. Coverget Sequeces Plot a graph of the sequece a ) = 2, 3 2, 4 3, 5 + 4,...,,... To what limit do you thik this sequece teds? What ca you say about the sequece a )? For ǫ = 0.,

More information

8 The Poisson Distribution

8 The Poisson Distribution 8 The Poisso Distributio Let X biomial, p ). Recall that this meas that X has pmf ) p,p k) p k k p ) k for k 0,,...,. ) Agai, thik of X as the umber of successes i a series of idepedet experimets, each

More information

Module 4: Mathematical Induction

Module 4: Mathematical Induction Module 4: Mathematical Iductio Theme 1: Priciple of Mathematical Iductio Mathematical iductio is used to prove statemets about atural umbers. As studets may remember, we ca write such a statemet as a predicate

More information

Homework 1 Solutions

Homework 1 Solutions Homewor 1 Solutios Math 171, Sprig 2010 Please sed correctios to herya@math.staford.edu 2.2. Let h : X Y, g : Y Z, ad f : Z W. Prove that (f g h = f (g h. Solutio. Let x X. Note that ((f g h(x = (f g(h(x

More information

0,1 is an accumulation

0,1 is an accumulation Sectio 5.4 1 Accumulatio Poits Sectio 5.4 Bolzao-Weierstrass ad Heie-Borel Theorems Purpose of Sectio: To itroduce the cocept of a accumulatio poit of a set, ad state ad prove two major theorems of real

More information

Lecture 7: Borel Sets and Lebesgue Measure

Lecture 7: Borel Sets and Lebesgue Measure EE50: Probability Foudatios for Electrical Egieers July-November 205 Lecture 7: Borel Sets ad Lebesgue Measure Lecturer: Dr. Krisha Jagaatha Scribes: Ravi Kolla, Aseem Sharma, Vishakh Hegde I this lecture,

More information

Riemann Sums y = f (x)

Riemann Sums y = f (x) Riema Sums Recall that we have previously discussed the area problem I its simplest form we ca state it this way: The Area Problem Let f be a cotiuous, o-egative fuctio o the closed iterval [a, b] Fid

More information

Chapter Suppose you wish to use the Principle of Mathematical Induction to prove that 1 1! + 2 2! + 3 3! n n! = (n + 1)! 1 for all n 1.

Chapter Suppose you wish to use the Principle of Mathematical Induction to prove that 1 1! + 2 2! + 3 3! n n! = (n + 1)! 1 for all n 1. Chapter 4. Suppose you wish to prove that the followig is true for all positive itegers by usig the Priciple of Mathematical Iductio: + 3 + 5 +... + ( ) =. (a) Write P() (b) Write P(7) (c) Write P(73)

More information

Section 9.2 Series and Convergence

Section 9.2 Series and Convergence Sectio 9. Series ad Covergece Goals of Chapter 9 Approximate Pi Prove ifiite series are aother importat applicatio of limits, derivatives, approximatio, slope, ad cocavity of fuctios. Fid challegig atiderivatives

More information

8.1 Arithmetic Sequences

8.1 Arithmetic Sequences MCR3U Uit 8: Sequeces & Series Page 1 of 1 8.1 Arithmetic Sequeces Defiitio: A sequece is a comma separated list of ordered terms that follow a patter. Examples: 1, 2, 3, 4, 5 : a sequece of the first

More information

Lesson 12. Sequences and Series

Lesson 12. Sequences and Series Retur to List of Lessos Lesso. Sequeces ad Series A ifiite sequece { a, a, a,... a,...} ca be thought of as a list of umbers writte i defiite order ad certai patter. It is usually deoted by { a } =, or

More information

Review for College Algebra Final Exam

Review for College Algebra Final Exam Review for College Algebra Fial Exam (Please remember that half of the fial exam will cover chapters 1-4. This review sheet covers oly the ew material, from chapters 5 ad 7.) 5.1 Systems of equatios i

More information

Section 11.3: The Integral Test

Section 11.3: The Integral Test Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult

More information

1 Set Theory and Functions

1 Set Theory and Functions Set Theory ad Fuctios. Basic De itios ad Notatio A set A is a collectio of objects of ay kid. We write a A to idicate that a is a elemet of A: We express this as a is cotaied i A. We write A B if every

More information

Discrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13

Discrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13 EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Note 13 Itroductio At this poit, we have see eough examples that it is worth just takig stock of our model of probability ad may

More information

{{1}, {2, 4}, {3}} {{1, 3, 4}, {2}} {{1}, {2}, {3, 4}} 5.4 Stirling Numbers

{{1}, {2, 4}, {3}} {{1, 3, 4}, {2}} {{1}, {2}, {3, 4}} 5.4 Stirling Numbers . Stirlig Numbers Whe coutig various types of fuctios from., we quicly discovered that eumeratig the umber of oto fuctios was a difficult problem. For a domai of five elemets ad a rage of four elemets,

More information

Winter Camp 2012 Sequences Alexander Remorov. Sequences. Alexander Remorov

Winter Camp 2012 Sequences Alexander Remorov. Sequences. Alexander Remorov Witer Camp 202 Sequeces Alexader Remorov Sequeces Alexader Remorov alexaderrem@gmail.com Warm-up Problem : Give a positive iteger, cosider a sequece of real umbers a 0, a,..., a defied as a 0 = 2 ad =

More information

The second difference is the sequence of differences of the first difference sequence, 2

The second difference is the sequence of differences of the first difference sequence, 2 Differece Equatios I differetial equatios, you look for a fuctio that satisfies ad equatio ivolvig derivatives. I differece equatios, istead of a fuctio of a cotiuous variable (such as time), we look for

More information

Asymptotic Growth of Functions

Asymptotic Growth of Functions CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll

More information

Chapter 7. In the questions below, describe each sequence recursively. Include initial conditions and assume that the sequences begin with a 1.

Chapter 7. In the questions below, describe each sequence recursively. Include initial conditions and assume that the sequences begin with a 1. Use the followig to aswer questios -6: Chapter 7 I the questios below, describe each sequece recursively Iclude iitial coditios ad assume that the sequeces begi with a a = 5 As: a = 5a,a = 5 The Fiboacci

More information

Math Discrete Math Combinatorics MULTIPLICATION PRINCIPLE:

Math Discrete Math Combinatorics MULTIPLICATION PRINCIPLE: Math 355 - Discrete Math 4.1-4.4 Combiatorics Notes MULTIPLICATION PRINCIPLE: If there m ways to do somethig ad ways to do aother thig the there are m ways to do both. I the laguage of set theory: Let

More information

Department of Computer Science, University of Otago

Department of Computer Science, University of Otago Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS-2006-09 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly

More information

if A S, then X \ A S, and if (A n ) n is a sequence of sets in S, then n A n S,

if A S, then X \ A S, and if (A n ) n is a sequence of sets in S, then n A n S, Lecture 5: Borel Sets Topologically, the Borel sets i a topological space are the σ-algebra geerated by the ope sets. Oe ca build up the Borel sets from the ope sets by iteratig the operatios of complemetatio

More information

NATIONAL SENIOR CERTIFICATE GRADE 12

NATIONAL SENIOR CERTIFICATE GRADE 12 NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages ad iformatio sheet. Please tur over Mathematics/P DBE/04 NSC Grade Eemplar INSTRUCTIONS

More information

Intro to Sequences / Arithmetic Sequences and Series Levels

Intro to Sequences / Arithmetic Sequences and Series Levels Itro to Sequeces / Arithmetic Sequeces ad Series Levels Level : pg. 569: #7, 0, 33 Pg. 575: #, 7, 8 Pg. 584: #8, 9, 34, 36 Levels, 3, ad 4(Fiboacci Sequece Extesio) See Hadout Check for Uderstadig Level

More information

x(x 1)(x 2)... (x k + 1) = [x] k n+m 1

x(x 1)(x 2)... (x k + 1) = [x] k n+m 1 1 Coutig mappigs For every real x ad positive iteger k, let [x] k deote the fallig factorial ad x(x 1)(x 2)... (x k + 1) ( ) x = [x] k k k!, ( ) k = 1. 0 I the sequel, X = {x 1,..., x m }, Y = {y 1,...,

More information

when n = 1, 2, 3, 4, 5, 6, This list represents the amount of dollars you have after n days. Note: The use of is read as and so on.

when n = 1, 2, 3, 4, 5, 6, This list represents the amount of dollars you have after n days. Note: The use of is read as and so on. Geometric eries Before we defie what is meat by a series, we eed to itroduce a related topic, that of sequeces. Formally, a sequece is a fuctio that computes a ordered list. uppose that o day 1, you have

More information

Sequences and Series

Sequences and Series CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their

More information

Properties of MLE: consistency, asymptotic normality. Fisher information.

Properties of MLE: consistency, asymptotic normality. Fisher information. Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout

More information

Lecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)

Lecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009) 18.409 A Algorithmist s Toolkit October 27, 2009 Lecture 13 Lecturer: Joatha Keler Scribe: Joatha Pies (2009) 1 Outlie Last time, we proved the Bru-Mikowski iequality for boxes. Today we ll go over the

More information

CS103X: Discrete Structures Homework 4 Solutions

CS103X: Discrete Structures Homework 4 Solutions CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible six-figure salaries i whole dollar amouts are there that cotai at least

More information

MATH 361 Homework 9. Royden Royden Royden

MATH 361 Homework 9. Royden Royden Royden MATH 61 Homework 9 Royde..9 First, we show that for ay subset E of the real umbers, E c + y = E + y) c traslatig the complemet is equivalet to the complemet of the traslated set). Without loss of geerality,

More information

A Simplified Binet Formula for k-generalized Fibonacci Numbers

A Simplified Binet Formula for k-generalized Fibonacci Numbers A Simplified Biet Formula for k-geeralized Fiboacci Numbers Gregory P. B. Dresde Departmet of Mathematics Washigto ad Lee Uiversity Lexigto, VA 440 dresdeg@wlu.edu Abstract I this paper, we preset a particularly

More information

Convexity, Inequalities, and Norms

Convexity, Inequalities, and Norms Covexity, Iequalities, ad Norms Covex Fuctios You are probably familiar with the otio of cocavity of fuctios. Give a twicedifferetiable fuctio ϕ: R R, We say that ϕ is covex (or cocave up) if ϕ (x) 0 for

More information

1.3 Binomial Coefficients

1.3 Binomial Coefficients 18 CHAPTER 1. COUNTING 1. Biomial Coefficiets I this sectio, we will explore various properties of biomial coefficiets. Pascal s Triagle Table 1 cotais the values of the biomial coefficiets ( ) for 0to

More information

4.1 Sigma Notation and Riemann Sums

4.1 Sigma Notation and Riemann Sums 0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas

More information

3. Greatest Common Divisor - Least Common Multiple

3. Greatest Common Divisor - Least Common Multiple 3 Greatest Commo Divisor - Least Commo Multiple Defiitio 31: The greatest commo divisor of two atural umbers a ad b is the largest atural umber c which divides both a ad b We deote the greatest commo gcd

More information

The Field Q of Rational Numbers

The Field Q of Rational Numbers Chapter 3 The Field Q of Ratioal Numbers I this chapter we are goig to costruct the ratioal umber from the itegers. Historically, the positive ratioal umbers came first: the Babyloias, Egyptias ad Grees

More information

Continued Fractions continued. 3. Best rational approximations

Continued Fractions continued. 3. Best rational approximations Cotiued Fractios cotiued 3. Best ratioal approximatios We hear so much about π beig approximated by 22/7 because o other ratioal umber with deomiator < 7 is closer to π. Evetually 22/7 is defeated by 333/06

More information

REAL ANALYSIS 2016/2017 (MATH20111) MARCUS TRESSL. Lecture Notes. Homepage of the course:

REAL ANALYSIS 2016/2017 (MATH20111) MARCUS TRESSL. Lecture Notes. Homepage of the course: REAL ANALYSIS 206/207 (MATH20) MARCUS TRESSL Lecture Notes Homepage of the course: http://persoalpages.machester.ac.uk/staff/marcus.tressl/teachig/realaalysis/idex.php Cotets. Sequeces.. Defiitio ad first

More information

Lecture 17 Two-Way Finite Automata

Lecture 17 Two-Way Finite Automata This is page 9 Priter: Opaque this Lecture 7 Two-Way Fiite Automata Two-way fiite automata are similar to the machies we have bee studyig, except that they ca read the iput strig i either directio. We

More information

To get the next Fibonacci number, you add the previous two. numbers are defined by the recursive formula. F 1 F n+1

To get the next Fibonacci number, you add the previous two. numbers are defined by the recursive formula. F 1 F n+1 Liear Algebra Notes Chapter 6 FIBONACCI NUMBERS The Fiboacci umbers are F, F, F 2, F 3 2, F 4 3, F, F 6 8, To get the ext Fiboacci umber, you add the previous two umbers are defied by the recursive formula

More information

MA2108S Tutorial 5 Solution

MA2108S Tutorial 5 Solution MA08S Tutorial 5 Solutio Prepared by: LuJigyi LuoYusheg March 0 Sectio 3. Questio 7. Let x := / l( + ) for N. (a). Use the difiitio of limit to show that lim(x ) = 0. Proof. Give ay ɛ > 0, sice ɛ > 0,

More information

MARTINGALES AND A BASIC APPLICATION

MARTINGALES AND A BASIC APPLICATION MARTINGALES AND A BASIC APPLICATION TURNER SMITH Abstract. This paper will develop the measure-theoretic approach to probability i order to preset the defiitio of martigales. From there we will apply this

More information

ON MINIMAL COLLECTIONS OF INDEXES. Egor A. Timoshenko

ON MINIMAL COLLECTIONS OF INDEXES. Egor A. Timoshenko ON MINIMAL COLLECTIONS OF INDEXES Egor A. Timosheko We deote s [ +1 ], l [ ], M C s C; l idexes built for the case of colums (i.e., ordered subsets of the set {1,,..., }) will be called -idexes. The legth

More information

Our aim is to show that under reasonable assumptions a given 2π-periodic function f can be represented as convergent series

Our aim is to show that under reasonable assumptions a given 2π-periodic function f can be represented as convergent series 8 Fourier Series Our aim is to show that uder reasoable assumptios a give -periodic fuctio f ca be represeted as coverget series f(x) = a + (a cos x + b si x). (8.) By defiitio, the covergece of the series

More information

Exponential function: For a > 0, the exponential function with base a is defined by. f(x) = a x

Exponential function: For a > 0, the exponential function with base a is defined by. f(x) = a x MATH 11011 EXPONENTIAL FUNCTIONS KSU AND THEIR APPLICATIONS Defiitios: Expoetial fuctio: For a > 0, the expoetial fuctio with base a is defied by fx) = a x Horizotal asymptote: The lie y = c is a horizotal

More information

5 Boolean Decision Trees (February 11)

5 Boolean Decision Trees (February 11) 5 Boolea Decisio Trees (February 11) 5.1 Graph Coectivity Suppose we are give a udirected graph G, represeted as a boolea adjacecy matrix = (a ij ), where a ij = 1 if ad oly if vertices i ad j are coected

More information

BINOMIAL EXPANSIONS 12.5. In this section. Some Examples. Obtaining the Coefficients

BINOMIAL EXPANSIONS 12.5. In this section. Some Examples. Obtaining the Coefficients 652 (12-26) Chapter 12 Sequeces ad Series 12.5 BINOMIAL EXPANSIONS I this sectio Some Examples Otaiig the Coefficiets The Biomial Theorem I Chapter 5 you leared how to square a iomial. I this sectio you

More information

Analysis Notes (only a draft, and the first one!)

Analysis Notes (only a draft, and the first one!) Aalysis Notes (oly a draft, ad the first oe!) Ali Nesi Mathematics Departmet Istabul Bilgi Uiversity Kuştepe Şişli Istabul Turkey aesi@bilgi.edu.tr Jue 22, 2004 2 Cotets 1 Prelimiaries 9 1.1 Biary Operatio...........................

More information

1. MATHEMATICAL INDUCTION

1. MATHEMATICAL INDUCTION 1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1

More information

Mathematicians have been fascinated by the majestic simplicity of the Fibonacci

Mathematicians have been fascinated by the majestic simplicity of the Fibonacci Joh Holde Tutoa3000@aol.com Ivertig the iboacci Sequece Mathematicias have bee fasciated by the majestic simplicity of the iboacci Sequece for ceturies. It starts as a simple,,, 3, 5, 8,3,... computed

More information

Arithmetic Sequences

Arithmetic Sequences . Arithmetic Sequeces Essetial Questio How ca you use a arithmetic sequece to describe a patter? A arithmetic sequece is a ordered list of umbers i which the differece betwee each pair of cosecutive terms,

More information

A CHARACTERIZATION OF MINIMAL ZERO-SEQUENCES OF INDEX ONE IN FINITE CYCLIC GROUPS

A CHARACTERIZATION OF MINIMAL ZERO-SEQUENCES OF INDEX ONE IN FINITE CYCLIC GROUPS INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5(1) (2005), #A27 A CHARACTERIZATION OF MINIMAL ZERO-SEQUENCES OF INDEX ONE IN FINITE CYCLIC GROUPS Scott T. Chapma 1 Triity Uiversity, Departmet

More information

Chapter 5: Inner Product Spaces

Chapter 5: Inner Product Spaces Chapter 5: Ier Product Spaces Chapter 5: Ier Product Spaces SECION A Itroductio to Ier Product Spaces By the ed of this sectio you will be able to uderstad what is meat by a ier product space give examples

More information

5.3. Generalized Permutations and Combinations

5.3. Generalized Permutations and Combinations 53 GENERALIZED PERMUTATIONS AND COMBINATIONS 73 53 Geeralized Permutatios ad Combiatios 53 Permutatios with Repeated Elemets Assume that we have a alphabet with letters ad we wat to write all possible

More information

THE COMPLETENESS OF CONVERGENT SEQUENCES SPACE OF FUZZY NUMBERS. Hee Chan Choi

THE COMPLETENESS OF CONVERGENT SEQUENCES SPACE OF FUZZY NUMBERS. Hee Chan Choi Kagweo-Kyugki Math. Jour. 4 (1996), No. 2, pp. 117 124 THE COMPLETENESS OF CONVERGENT SEQUENCES SPACE OF FUZZY NUMBERS Hee Cha Choi Abstract. I this paper we defie a ew fuzzy metric θ of fuzzy umber sequeces,

More information

Approximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find

Approximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find 1.8 Approximatig Area uder a curve with rectagles 1.6 To fid the area uder a curve we approximate the area usig rectagles ad the use limits to fid 1.4 the area. Example 1 Suppose we wat to estimate 1.

More information

Maximum Likelihood Estimators.

Maximum Likelihood Estimators. Lecture 2 Maximum Likelihood Estimators. Matlab example. As a motivatio, let us look at oe Matlab example. Let us geerate a radom sample of size 00 from beta distributio Beta(5, 2). We will lear the defiitio

More information

MATH 140A - HW 5 SOLUTIONS

MATH 140A - HW 5 SOLUTIONS MATH 40A - HW 5 SOLUTIONS Problem WR Ch 3 #8. If a coverges, ad if {b } is mootoic ad bouded, rove that a b coverges. Solutio. Theorem 3.4 states that if a the artial sums of a form a bouded sequece; b

More information

4 n. n 1. You shold think of the Ratio Test as a generalization of the Geometric Series Test. For example, if a n ar n is a geometric sequence then

4 n. n 1. You shold think of the Ratio Test as a generalization of the Geometric Series Test. For example, if a n ar n is a geometric sequence then SECTION 2.6 THE RATIO TEST 79 2.6. THE RATIO TEST We ow kow how to hadle series which we ca itegrate (the Itegral Test), ad series which are similar to geometric or p-series (the Compariso Test), but of

More information

6 Borel sets in the light of analytic sets

6 Borel sets in the light of analytic sets Tel Aviv Uiversity, 2012 Measurability ad cotiuity 86 6 Borel sets i the light of aalytic sets 6a Separatio theorem................. 86 6b Borel bijectios.................... 87 6c A o-borel aalytic set

More information

ORDERS OF GROWTH KEITH CONRAD

ORDERS OF GROWTH KEITH CONRAD ORDERS OF GROWTH KEITH CONRAD Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really wat to uderstad their behavior It also helps you better grasp topics i calculus

More information

8.5 Alternating infinite series

8.5 Alternating infinite series 65 8.5 Alteratig ifiite series I the previous two sectios we cosidered oly series with positive terms. I this sectio we cosider series with both positive ad egative terms which alterate: positive, egative,

More information

Overview of some probability distributions.

Overview of some probability distributions. Lecture Overview of some probability distributios. I this lecture we will review several commo distributios that will be used ofte throughtout the class. Each distributio is usually described by its probability

More information

B1. Fourier Analysis of Discrete Time Signals

B1. Fourier Analysis of Discrete Time Signals B. Fourier Aalysis of Discrete Time Sigals Objectives Itroduce discrete time periodic sigals Defie the Discrete Fourier Series (DFS) expasio of periodic sigals Defie the Discrete Fourier Trasform (DFT)

More information

SEQUENCES AND SERIES. Chapter Nine

SEQUENCES AND SERIES. Chapter Nine Chapter Nie SEQUENCES AND SERIES I this chapter, we look at ifiite lists of umbers, called sequeces, ad ifiite sums, called series. I Sectio 9., we study sequeces. I Sectio 9.2, we begi with a particular

More information

Chapter One BASIC MATHEMATICAL TOOLS

Chapter One BASIC MATHEMATICAL TOOLS Chapter Oe BAIC MATHEMATICAL TOOL As the reader will see, the study of the time value of moey ivolves substatial use of variables ad umbers that are raised to a power. The power to which a variable is

More information

Entropy Rates of a Stochastic Process

Entropy Rates of a Stochastic Process Etropy Rates of a Stochastic Process Best Achievable Data Compressio Radu Trîmbiţaş October 2012 1 Etropy Rates of a Stochastic Process Etropy rates The AEP states that H(X) bits suffice o the average

More information

SEQUENCES AND SERIES CHAPTER

SEQUENCES AND SERIES CHAPTER CHAPTER SEQUENCES AND SERIES Whe the Grat family purchased a computer for $,200 o a istallmet pla, they agreed to pay $00 each moth util the cost of the computer plus iterest had bee paid The iterest each

More information

Confidence Intervals for the Mean of Non-normal Data Class 23, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom

Confidence Intervals for the Mean of Non-normal Data Class 23, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom Cofidece Itervals for the Mea of No-ormal Data Class 23, 8.05, Sprig 204 Jeremy Orloff ad Joatha Bloom Learig Goals. Be able to derive the formula for coservative ormal cofidece itervals for the proportio

More information

Chapter 6: Variance, the law of large numbers and the Monte-Carlo method

Chapter 6: Variance, the law of large numbers and the Monte-Carlo method Chapter 6: Variace, the law of large umbers ad the Mote-Carlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value

More information

Lecture 4: Cauchy sequences, Bolzano-Weierstrass, and the Squeeze theorem

Lecture 4: Cauchy sequences, Bolzano-Weierstrass, and the Squeeze theorem Lecture 4: Cauchy sequeces, Bolzao-Weierstrass, ad the Squeeze theorem The purpose of this lecture is more modest tha the previous oes. It is to state certai coditios uder which we are guarateed that limits

More information

Example 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).

Example 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here). BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook - Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly

More information

Measure Theory, MA 359 Handout 1

Measure Theory, MA 359 Handout 1 Measure Theory, M 359 Hadout 1 Valeriy Slastikov utum, 2005 1 Measure theory 1.1 Geeral costructio of Lebesgue measure I this sectio we will do the geeral costructio of σ-additive complete measure by extedig

More information

Section IV.5: Recurrence Relations from Algorithms

Section IV.5: Recurrence Relations from Algorithms Sectio IV.5: Recurrece Relatios from Algorithms Give a recursive algorithm with iput size, we wish to fid a Θ (best big O) estimate for its ru time T() either by obtaiig a explicit formula for T() or by

More information

5. SEQUENCES AND SERIES

5. SEQUENCES AND SERIES 5. SEQUENCES AND SERIES 5.. Limits of Sequeces Let N = {0,,,... } be the set of atural umbers ad let R be the set of real umbers. A ifiite real sequece u 0, u, u, is a fuctio from N to R, where we write

More information

Listing terms of a finite sequence List all of the terms of each finite sequence. a) a n n 2 for 1 n 5 1 b) a n for 1 n 4 n 2

Listing terms of a finite sequence List all of the terms of each finite sequence. a) a n n 2 for 1 n 5 1 b) a n for 1 n 4 n 2 74 (4 ) Chapter 4 Sequeces ad Series 4. SEQUENCES I this sectio Defiitio Fidig a Formula for the th Term The word sequece is a familiar word. We may speak of a sequece of evets or say that somethig is

More information

Binet Formulas for Recursive Integer Sequences

Binet Formulas for Recursive Integer Sequences Biet Formulas for Recursive Iteger Sequeces Homer W. Austi Jatha W. Austi Abstract May iteger sequeces are recursive sequeces ad ca be defied either recursively or explicitly by use of Biet-type formulas.

More information

Linear Algebra II. Notes 6 25th November 2010

Linear Algebra II. Notes 6 25th November 2010 MTH6140 Liear Algebra II Notes 6 25th November 2010 6 Quadratic forms A lot of applicatios of mathematics ivolve dealig with quadratic forms: you meet them i statistics (aalysis of variace) ad mechaics

More information

Metric, Normed, and Topological Spaces

Metric, Normed, and Topological Spaces Chapter 13 Metric, Normed, ad Topological Spaces A metric space is a set X that has a otio of the distace d(x, y) betwee every pair of poits x, y X. A fudametal example is R with the absolute-value metric

More information

0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5

0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5 Sectio 13 Kolmogorov-Smirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.

More information

Divide and Conquer, Solving Recurrences, Integer Multiplication Scribe: Juliana Cook (2015), V. Williams Date: April 6, 2016

Divide and Conquer, Solving Recurrences, Integer Multiplication Scribe: Juliana Cook (2015), V. Williams Date: April 6, 2016 CS 6, Lecture 3 Divide ad Coquer, Solvig Recurreces, Iteger Multiplicatio Scribe: Juliaa Cook (05, V Williams Date: April 6, 06 Itroductio Today we will cotiue to talk about divide ad coquer, ad go ito

More information

Your grandmother and her financial counselor

Your grandmother and her financial counselor Sectio 10. Arithmetic Sequeces 963 Objectives Sectio 10. Fid the commo differece for a arithmetic sequece. Write s of a arithmetic sequece. Use the formula for the geeral of a arithmetic sequece. Use the

More information

π d i (b i z) (n 1)π )... sin(θ + )

π d i (b i z) (n 1)π )... sin(θ + ) SOME TRIGONOMETRIC IDENTITIES RELATED TO EXACT COVERS Joh Beebee Uiversity of Alaska, Achorage Jauary 18, 1990 Sherma K Stei proves that if si π = k si π b where i the b i are itegers, the are positive

More information

1 Notes on Little s Law (l = λw)

1 Notes on Little s Law (l = λw) Copyright c 29 by Karl Sigma Notes o Little s Law (l λw) We cosider here a famous ad very useful law i queueig theory called Little s Law, also kow as l λw, which asserts that the time average umber of

More information

Basic Elements of Arithmetic Sequences and Series

Basic Elements of Arithmetic Sequences and Series MA40S PRE-CALCULUS UNIT G GEOMETRIC SEQUENCES CLASS NOTES (COMPLETED NO NEED TO COPY NOTES FROM OVERHEAD) Basic Elemets of Arithmetic Sequeces ad Series Objective: To establish basic elemets of arithmetic

More information

The Euler Totient, the Möbius and the Divisor Functions

The Euler Totient, the Möbius and the Divisor Functions The Euler Totiet, the Möbius ad the Divisor Fuctios Rosica Dieva July 29, 2005 Mout Holyoke College South Hadley, MA 01075 1 Ackowledgemets This work was supported by the Mout Holyoke College fellowship

More information

The Stable Marriage Problem

The Stable Marriage Problem The Stable Marriage Problem William Hut Lae Departmet of Computer Sciece ad Electrical Egieerig, West Virgiia Uiversity, Morgatow, WV William.Hut@mail.wvu.edu 1 Itroductio Imagie you are a matchmaker,

More information

Permutations, the Parity Theorem, and Determinants

Permutations, the Parity Theorem, and Determinants 1 Permutatios, the Parity Theorem, ad Determiats Joh A. Guber Departmet of Electrical ad Computer Egieerig Uiversity of Wiscosi Madiso Cotets 1 What is a Permutatio 1 2 Cycles 2 2.1 Traspositios 4 3 Orbits

More information

TAYLOR SERIES, POWER SERIES

TAYLOR SERIES, POWER SERIES TAYLOR SERIES, POWER SERIES The followig represets a (icomplete) collectio of thigs that we covered o the subject of Taylor series ad power series. Warig. Be prepared to prove ay of these thigs durig the

More information

Algebra Vocabulary List (Definitions for Middle School Teachers)

Algebra Vocabulary List (Definitions for Middle School Teachers) Algebra Vocabulary List (Defiitios for Middle School Teachers) A Absolute Value Fuctio The absolute value of a real umber x, x is xifx 0 x = xifx < 0 http://www.math.tamu.edu/~stecher/171/f02/absolutevaluefuctio.pdf

More information

Part - I. Mathematics

Part - I. Mathematics Part - I Mathematics CHAPTER Set Theory. Objectives. Itroductio. Set Cocept.. Sets ad Elemets. Subset.. Proper ad Improper Subsets.. Equality of Sets.. Trasitivity of Set Iclusio.4 Uiversal Set.5 Complemet

More information