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


 Sabrina Green
 3 years ago
 Views:
Transcription
1 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 (8.). Elemetary Cocepts Simply speakig, a sequece is simply a list of umbers, writte i a de ite order: fa ; a 2 ; a 3 ; :::a ; a + ; :::g where the elemets a i represet umbers. I this sectio, we oly cocetrate o i ite sequeces. Here are some geeral facts about sequeces studied i this class: Every umber a i the sequece has a successor a + ; the sequece ever stops sice we study i ite sequeces. Thik of the idex of a particular elemet as idicatig the positio of the elemet i the list. The idex ca also be associated with a formula whe the elemets i the list are geerated by oe (see below). As such, the idex of the umbers does ot have to start at, though it does most of the time. If we call 0 the value of the startig idex, the there is a umber a for every 0. Thus, we ca de e a fuctio f such that a = f () where is a atural umber. I this class, we will cocetrate o i ite sequeces of real umbers. A more formal de itio of a sequece is as follows: De itio (sequece) A sequece of real umbers is a realvalued fuctio f whose domai is a subset of the oegative itegers, that is a set of the form f 0 ; 0 + ; :::g where 0 is a iteger such that The umbers a = f () are called the terms of the sequece.
2 The typical otatio for a sequece is (a ), or fa g or fa g = where a deotes the geeral term of the sequece. Remark Whe a sequece is give by a fuctio ff ()g = 0, the fuctio f must be de ed for every 0. A sequece ca be give di eret ways The elemets of the sequece are give. A formula to geerate the terms of the sequece is give. A recursive formula to geerate the terms of the sequece is give. Example 2 The examples below illustrate sequeces give by listig all the elemets (it is really ot possible sice these are i ite sequeces),. f; 5; 0; 4; 98; 000; 0; 2; :::g. 2. ; 2 ; 3 ; :::. Though the elemets are listed, we ca also guess a formula to geerate them, what is it? 3. f ; ; ; ; ; ; :::g. Though the elemets are listed, we ca also guess a formula to geerate them, what is it? Example 3 The examples below illustrate sequeces give by a simple formula. Notice that does ot have to start at... The elemets of the sequece are: ; 2 ; 3 ; :::. The geeral = term of the sequece is a =. 2. p 3 =3. The elemets are 0; p ; p 2; p 3; :::. I this case, could ot start at. 3. f( ) g =2. The elemets of the sequece are f; ; ; ; :::g. Example 4 The examples below illustrate sequeces give by a recursive formula. a. =. We ca use this formula to geerate all the terms. a = 2a + 5 But the terms have to be geerated i order. For example, i order to get a 0, we eed to kow a 9, ad so o. Usig the formula, we get that a 2 = 2a + 5 = 7 2
3 Havig foud a 2, we ca ow geerate a 3 Ad so o. a 3 = 2a = 9 2. A special sequece geerated recursively is the Fiboacci 8 sequece, amed < f = after a Italia mathematicia. It is de ed as follows: f 2 =. : f = f + f 2 We ca use this formula to geerate the terms of the sequece. This sequece was devised to model rabbit populatio. f represets the umber of pairs of rabbits after moths, assumig that each moth, a pair of rabbit produces a ew pair which becomes productive at age 2 moths. Example 5 Other sequeces. Arithmetic sequece. A sequece is arithmetic if the di erece betwee two cosecutive terms is costat. Let r be this costat. The, we have a a 0 = r =) a = a 0 + r a 2 a = r =) a 2 = a + r = a 0 + 2r ::: a a = r =) a = a + r = a 0 + r Thus, there is a de ig formula for arithmetic sequeces. For example a arithmetic sequece startig at 2, such that the di erece betwee two cosecutive terms is 3 is de ed by: a = Geometric sequece. A sequece is geometric if the ratio of two cosecutive terms is costat. Let q deote this costat. The, we have a a 0 = q =) a = qa 0 a 2 a = q =) a 2 = qa = q 2 a 0 ::: a a = q =) a = qa = q a 0 Thus, there is a de ig formula for geometric sequeces. For example a geometric sequece startig at 2, such that the ratio betwee two cosecutive terms is 3 is de ed by: a = 2 3. Sequeces ca be plotted. However, the plot of a sequece will cosists of dots, sice they are oly de ed at the itegers. Figures, 2, 3, ad 4 show some sequeces beig plotted. 3
4 Figure : Plot of a =.2 Limit of a Sequece Give a sequece fa g, oe of the questios we try to aswer is: what is the behavior of a as!? Is a gettig closer ad closer to a umber? I other words, we wat to d lim! a. De itio 6 (limit of a sequece) A sequece fa g coverges to a umber L as goes to i ity if a ca be made as close as oe wats to L, simply by takig large eough. I this case, we write lim a = L. If lim a = L!! ad L is a ite umber, we say that fa g coverges. Otherwise, it diverges. Sometimes, we will make the distictio betwee diverges to i ity ad simply diverges. I the rst case, we still kow what the sequece is doig, it is gettig large without bouds. A sequece may diverge for several reasos. Its geeral term could get arbitrarily large (go to i ity), as show o gure 3. Its geeral term could also oscillate betwee di eret values without ever gettig close to aythig. This is the case of f( ) g, or f( ) l g as show o gure 4. 4
5 Figure 2: Plot of a = si Aother way to uderstad this is that if lim a = L the ja Lj goes to 0! whe!. It should also be oted that if lim a = L, the lim a + = L.!! I fact, lim a +p = L for ay positive iteger p. Graphically, the meaig of lim a = L is as follows. Cosider the sequece show o gure 5 whose plot is represeted by the dots. The sequece appears to have 3 as its limit, this is idicated by the horizotal solid lie through 3. If we draw a regio havig the lie y = 3 at its ceter, the sayig that lim a = 3 meas that there exists a certai value of (deote it 0 ) such that if > 0, the all the dots correspodig to the plot of the sequece will fall i the regio. O gure 5, we drew two regios, oe with dotted lies, the other oe with dashdot lies. We ca see that i both cases, after a while, the sequece always falls i the regio. Of course, the more arrow the regio is, the larger 0 will be. I other words, if we wat to guaratee that a is closer to 3, we have to look at a for larger values of. It appears that for the larger regio, the sequece falls i the regio if > 0. For the smaller regio, it happes whe > 22 (approximately). 5
6 Figure 3: Plot of l Let us rst state, but ot prove, a importat theorem. Theorem 7 If a sequece coverges, its limit is uique. We ow look at various techiques used whe computig the limit of a sequece. As oticed above, a sequece ca be give di eret ways. How its limit is computed depeds o the way a sequece is give. We begi with the easiest case, oe we are already familiar with from Calculus I. If the sequece is give by a fuctio, we ca, i may istaces, use our kowledge of dig the limit of a fuctio, to d the limit of a sequece. This is what the ext theorem tells us. Theorem 8 If lim x! f(x) = L ad a = f() the lim! a = L This theorem simply says that if we kow the fuctio which geerates the geeral term of the sequece, ad that fuctio coverges as x! the the sequece coverges to the same limit. We kow how to do the latter from Calculus I. 6
7 Figure 4: Plot of ( ) l Example 9 Fid lim +. The fuctio geeratig this sequece is f (x) = (l Hôpital s rule), lim + =. x. Sice lim x + x! Be careful, this theorem oly gives a de ite aswer if x x + = lim f(x) = L. If x! the fuctio diverges, we caot coclude. For example, cosider the sequece a = cos 2. The fuctio f such that a = f () is f (x) = cos 2x. This fuctio diverges as x!. Yet, a = cos 2 = for ay. So, lim a =. Theorem 0 If a b c for 0 ad lim b = L! lim a = lim c = L the!! This is the equivalet of the squeeze theorem. You will otice i the statemet of the theorem that the coditio a b c does ot have to be true for every. It simply has to be true from some poit o. Example Fid lim! 7
8 Figure 5: Limit of a sequece Sice! is oly de ed for itegers, we caot d a fuctio f such that a = f (). Thus, we caot use the previous theorem. We use the squeeze theorem istead. We otice that: 0! = 2 3 ::: ::: By the squeeze theorem, it follows that lim! = 0. Theorem 2 If lim! ja j = 0 the lim! a = 0 This is a applicatio of the squeeze theorem usig the fact that ja j a ja j. This theorem is ofte useful whe the geeral term of a sequece cotais ( ) as suggests the ext example. 8
9 Example 3 Fid lim a for a = ( ). First, we otice that ja j = ( ) = which coverges to 0 by the previous example. Hece, by theorem 2, a! 0 also. We ow look at a theorem which is very importat. Ulike the other theorems, we ca prove this oe as it is ot very di cult. Theorem 4 The sequece fx g coverges to 0 if jxj <. It coverges to if x =. It diverges otherwise. Proof. We cosider several cases. case : x =. The, x = =. Thus, the sequece coverges to. case 2: x =. The, x = ( ) which diverges. case 3: x = 0. The, x = 0 = 0. Thus, the sequece coverges to 0. case 4: jxj < ad x 6= 0. The, l jxj < 0. Thus, l jxj = l jxj!. Thus, jxj! 0 as!. Hece, by the previous theorem, x! 0 as!. case 5: x >. The, x!. case 6: x <. x oscillate betwee positive ad egative values, which are gettig larger i absolute value. Thus it also diverges. Fially, we state a theorem which is the equivalet for sequeces of the limit rules for fuctios. Theorem 5 Suppose that fa g ad fb g coverge, ad that c is a costat. The:. lim! (a + b ) = lim! a + lim! b 2. lim (a b ) = lim a lim!! 3. lim (a b ) = lim a!! 4. lim a! b = lim! a lim! 5. lim! ca = c lim! a 6. lim! c = c 7. lim! ap = h! b lim! b providig lim b b 6= 0! i p lim a if p > 0 ad a > 0.! 9
10 Sice we oly cosider limits as!, we will omit it ad simply write lim a. The way we d lim a depeds greatly o how the sequece is give. Example 6 Fid lim a for a =. I this case, the fuctio which geerates the terms of the sequece is f (x) = x. Sice lim x! x = 0, it follows that lim a = 0 by theorem 8. Example 7 Fid lim l. From a previous example, lim + + =, therefore lim l = l = 0. + Example 8 Fid lim ( ). Sice the terms of this sequece are f ; ; ; ; :::g, they oscillate but ever get close to aythig. The sequece diverges. I cotrast, the sequece of the rst example also oscillated. But it also got closer ad closer to 0. Example 9 Fid lim si. Though we ca d a fuctio to express the geeral term of this sequece, sice si x diverges, we caot use theorem 8 to try to compute the limit of this sequece. We ote that si Therefore Sice lim 0. si = 0, by the squeeze theorem for sequeces, it follows that lim si = Example 20 Fid lim l The fuctio de ig the geeral term is f (x) = l x x. Sice l x lim x! x It follows that lim l = 0. = lim x! = lim x! = 0 x x by l Hôpital s rule 0
11 Example 2 Assumig that the sequece give recursively by ( a = 2 a + = 2 (a + 6) coverges, d its limit. Let L = lim a. If we take the limit o both sides of the relatio de ig the sequece, we have lim a + = lim 2 (a + 6) So, lim a = 6. L = (L + 6) 2 2L = L + 6 L = 6.3 Icreasig, Decreasig ad Bouded Sequeces De itio 22 (icreasig, decreasig sequeces) A sequece fa g is said to be. icreasig if ad oly if a < a + for each oegative iteger. 2. odecreasig if ad oly if a a + for each oegative iteger. 3. decreasig if ad oly if a > a + for each oegative iteger. 4. oicreasig if ad oly if a a + for each oegative iteger. 5. mootoic if ay of these four properties holds. To show that a sequece is icreasig, we ca try oe of the followig:. Show that a < a + for all. 2. Show that a a + < 0 for all. 3. If a > 0 for all, the show that 4. If f () = a, show that f 0 (x) > 0 5. By iductio. Example 23 Let a = a a + < for all. +. Show fa g is icreasig.
12 Method : look at a a + Method 2: Let f (x) = a a + = = ( + 2) ( + ) ( + ) = < x x +. The f 0 (x) = (x + ) 2 > 0 De itio 24 (bouded sequeces) A sequece fa g is said to be bouded from above if there exists a umber M such that a M for all. M is called a upper boud of the sequece. A sequece fa g is said to be bouded from below if there exists a umber m such that a m for all. m is called a lower boud of the sequece. A sequece is bouded if it is bouded from above ad below. Example 25 Cosider the sequece a = cos. Sice cos, a is bouded from above by ad bouded from below by. So, a is bouded. Example 26 Cosider the sequece. We have 0 <. Thus the = sequece is bouded below by 0 ad above by. Remark 27 Clearly, if M is a upper boud of a sequece, the ay umber larger tha M is also a upper boud. So, if a sequece is bouded from above, it has i itely may upper bouds. Similarly, if a sequece is bouded below by m, the ay umber less tha m is also a lower boud. Theorem 28 coverge. Theorem 29 coverge. A sequece which is icreasig ad bouded from above must A sequece which is decreasig ad bouded from below must We use this theorem to prove that certai sequeces have a limit. Before we do this, let us review iductio. This cocept is ofte eeded to show a sequece is icreasig or bouded. Theorem 30 (Iductio) Let P () deote a statemet about atural umbers with the followig properties: 2
13 . The statemet is true whe = i.e. P () is true. This is call the base case. 2. P (k + ) is true wheever P (k) is true for ay iteger k. The, P () is true for all 2 N. Example 3 Prove that the sequece de ed by a = 2 ad a + = 2 (a + 6) is icreasig ad bouded. Fid its limit. Icreasig: We show by iductio that a + > a. Base case. a 2 = 2 (a + 6) = 2 (2 + 6) = 4 > a. Assume the result is true for ay iteger k, that is assume a k+ > a k. Show the result is true for k + that is a k+ > a k+. a k+2 = 2 (a k+ + 6) > 2 (a k + 6) sice a k+ > a k = a k+ The result follows by iductio. Bouded: We show by iductio that a 6. Base case: a = 2 6. Assume the result is true for ay iteger k, that is a k 6, show the result is also true for k + that is a k+ 6. The result follows by iductio. a k 6 =) a k =) 2 (a k + 6) 6 =) a k+ 6 Limit: We have already computed the limit of such a sequece. Now that we kow the limit exists sice fa g i icreasig ad bouded above, the idea is to give it a ame, say lim a = L ad d what L is usig the limit 3
14 rules as follows: a + = 2 (a + 6) () lim (a + ) = lim 2 (a + 6) () L = 2 lim (a + 6) () L = 2 (lim a + lim 6) () L = (L + 6) 2 () 2L = L + 6 () L = 6 So lim a = 6 We ish with a importat remark. Remark 32 Cosider a sequece fa g. If fa g has a limit, it must be bouded. Try to explai why. Not every bouded sequece has a limit. Give a example of a bouded sequece with o limit..4 Thigs to Kow ad Problems Assiged Be able to write the terms of a sequece o matter which way the sequece is preseted. Be able to tell if a sequece coverges or diverges. If it coverges, be ale to d its limit. Be able to tell if a sequece is icreasig or decreasig. Related problems assiged:. o pages 565, 566 #, 3, 5, 7, 9,, 3, 5, 7, 9, 2, 23, 25, 35, 39, 4, Let fa g be a sequece such that a = f () for some fuctio f. Suppose that lim f (x) does ot exist. Does it mea that fa g x! diverge? Explai or give a couter example. 3. Explai why if a sequece has a limit the it must be bouded. 4. Give examples of bouded sequeces which do ot have limits. 4
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 informationLecture 4: Cauchy sequences, BolzanoWeierstrass, and the Squeeze theorem
Lecture 4: Cauchy sequeces, BolzaoWeierstrass, 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 informationSAMPLE 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 informationSequences 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 informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More informationSection 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 informationExample 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 informationConvexity, 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 informationProperties 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 informationFIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. 1. Powers of a matrix
FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. Powers of a matrix We begi with a propositio which illustrates the usefuless of the diagoalizatio. Recall that a square matrix A is diogaalizable if
More informationBasic Elements of Arithmetic Sequences and Series
MA40S PRECALCULUS 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 informationAsymptotic 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 informationSoving Recurrence Relations
Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree
More informationCS103X: 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 sixfigure salaries i whole dollar amouts are there that cotai at least
More informationTheorems About Power Series
Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real oegative umber R, called the radius
More information1. C. The formula for the confidence interval for a population mean is: x t, which was
s 1. C. The formula for the cofidece iterval for a populatio mea is: x t, which was based o the sample Mea. So, x is guarateed to be i the iterval you form.. D. Use the rule : pvalue
More informationHere are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio
More informationDiscrete 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 informationChapter 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 information3. 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 informationOur aim is to show that under reasonable assumptions a given 2πperiodic 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 informationHypothesis testing. Null and alternative hypotheses
Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate
More information4.3. The Integral and Comparison Tests
4.3. THE INTEGRAL AND COMPARISON TESTS 9 4.3. The Itegral ad Compariso Tests 4.3.. The Itegral Test. Suppose f is a cotiuous, positive, decreasig fuctio o [, ), ad let a = f(). The the covergece or divergece
More informationSEQUENCES AND SERIES
Chapter 9 SEQUENCES AND SERIES Natural umbers are the product of huma spirit. DEDEKIND 9.1 Itroductio I mathematics, the word, sequece is used i much the same way as it is i ordiary Eglish. Whe we say
More informationAP Calculus BC 2003 Scoring Guidelines Form B
AP Calculus BC Scorig Guidelies Form B The materials icluded i these files are iteded for use by AP teachers for course ad exam preparatio; permissio for ay other use must be sought from the Advaced Placemet
More information1 Computing the Standard Deviation of Sample Means
Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.
More informationLecture 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 BruMikowski iequality for boxes. Today we ll go over the
More informationChapter 6: Variance, the law of large numbers and the MonteCarlo method
Chapter 6: Variace, the law of large umbers ad the MoteCarlo 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 informationMath C067 Sampling Distributions
Math C067 Samplig Distributios Sample Mea ad Sample Proportio Richard Beigel Some time betwee April 16, 2007 ad April 16, 2007 Examples of Samplig A pollster may try to estimate the proportio of voters
More informationI. Chisquared Distributions
1 M 358K Supplemet to Chapter 23: CHISQUARED DISTRIBUTIONS, TDISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad tdistributios, we first eed to look at aother family of distributios, the chisquared distributios.
More informationDepartment of Computer Science, University of Otago
Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS200609 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly
More informationTrigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is
0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values
More informationElementary Theory of Russian Roulette
Elemetary Theory of Russia Roulette iterestig patters of fractios Satoshi Hashiba Daisuke Miematsu Ryohei Miyadera Itroductio. Today we are goig to study mathematical theory of Russia roulette. If some
More informationWeek 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable
Week 3 Coditioal probabilities, Bayes formula, WEEK 3 page 1 Expected value of a radom variable We recall our discussio of 5 card poker hads. Example 13 : a) What is the probability of evet A that a 5
More informationDetermining the sample size
Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factors
More informationThe following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles
The followig eample will help us uderstad The Samplig Distributio of the Mea Review: The populatio is the etire collectio of all idividuals or objects of iterest The sample is the portio of the populatio
More informationMARTINGALES AND A BASIC APPLICATION
MARTINGALES AND A BASIC APPLICATION TURNER SMITH Abstract. This paper will develop the measuretheoretic approach to probability i order to preset the defiitio of martigales. From there we will apply this
More informationRepeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern.
5.5 Fractios ad Decimals Steps for Chagig a Fractio to a Decimal. Simplify the fractio, if possible. 2. Divide the umerator by the deomiator. d d Repeatig Decimals Repeatig Decimals are decimal umbers
More information5 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 informationINFINITE SERIES KEITH CONRAD
INFINITE SERIES KEITH CONRAD. Itroductio The two basic cocepts of calculus, differetiatio ad itegratio, are defied i terms of limits (Newto quotiets ad Riema sums). I additio to these is a third fudametal
More informationUniversity of California, Los Angeles Department of Statistics. Distributions related to the normal distribution
Uiversity of Califoria, Los Ageles Departmet of Statistics Statistics 100B Istructor: Nicolas Christou Three importat distributios: Distributios related to the ormal distributio Chisquare (χ ) distributio.
More informationAP Calculus AB 2006 Scoring Guidelines Form B
AP Calculus AB 6 Scorig Guidelies Form B The College Board: Coectig Studets to College Success The College Board is a otforprofit membership associatio whose missio is to coect studets to college success
More informationLesson 17 Pearson s Correlation Coefficient
Outlie Measures of Relatioships Pearso s Correlatio Coefficiet (r) types of data scatter plots measure of directio measure of stregth Computatio covariatio of X ad Y uique variatio i X ad Y measurig
More information5.3. 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 informationCS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations
CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad
More informationA probabilistic proof of a binomial identity
A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two
More informationApproximating 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 informationDescriptive Statistics
Descriptive Statistics We leared to describe data sets graphically. We ca also describe a data set umerically. Measures of Locatio Defiitio The sample mea is the arithmetic average of values. We deote
More information1. 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 informationSampling Distribution And Central Limit Theorem
() Samplig Distributio & Cetral Limit Samplig Distributio Ad Cetral Limit Samplig distributio of the sample mea If we sample a umber of samples (say k samples where k is very large umber) each of size,
More informationModified Line Search Method for Global Optimization
Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o
More information0.7 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 KolmogorovSmirov 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 informationMeasures of Spread and Boxplots Discrete Math, Section 9.4
Measures of Spread ad Boxplots Discrete Math, Sectio 9.4 We start with a example: Example 1: Comparig Mea ad Media Compute the mea ad media of each data set: S 1 = {4, 6, 8, 10, 1, 14, 16} S = {4, 7, 9,
More informationListing 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 informationMaximum 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 informationCHAPTER 3 DIGITAL CODING OF SIGNALS
CHAPTER 3 DIGITAL CODING OF SIGNALS Computers are ofte used to automate the recordig of measuremets. The trasducers ad sigal coditioig circuits produce a voltage sigal that is proportioal to a quatity
More informationExploratory Data Analysis
1 Exploratory Data Aalysis Exploratory data aalysis is ofte the rst step i a statistical aalysis, for it helps uderstadig the mai features of the particular sample that a aalyst is usig. Itelliget descriptios
More informationWHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER?
WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER? JÖRG JAHNEL 1. My Motivatio Some Sort of a Itroductio Last term I tought Topological Groups at the Göttige Georg August Uiversity. This
More informationBINOMIAL EXPANSIONS 12.5. In this section. Some Examples. Obtaining the Coefficients
652 (1226) 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 informationCHAPTER 7: Central Limit Theorem: CLT for Averages (Means)
CHAPTER 7: Cetral Limit Theorem: CLT for Averages (Meas) X = the umber obtaied whe rollig oe six sided die oce. If we roll a six sided die oce, the mea of the probability distributio is X P(X = x) Simulatio:
More informationNotes on exponential generating functions and structures.
Notes o expoetial geeratig fuctios ad structures. 1. The cocept of a structure. Cosider the followig coutig problems: (1) to fid for each the umber of partitios of a elemet set, (2) to fid for each the
More informationGCSE STATISTICS. 4) How to calculate the range: The difference between the biggest number and the smallest number.
GCSE STATISTICS You should kow: 1) How to draw a frequecy diagram: e.g. NUMBER TALLY FREQUENCY 1 3 5 ) How to draw a bar chart, a pictogram, ad a pie chart. 3) How to use averages: a) Mea  add up all
More informationSEQUENCES 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 informationa 4 = 4 2 4 = 12. 2. Which of the following sequences converge to zero? n 2 (a) n 2 (b) 2 n x 2 x 2 + 1 = lim x n 2 + 1 = lim x
0 INFINITE SERIES 0. Sequeces Preiary Questios. What is a 4 for the sequece a? solutio Substitutig 4 i the expressio for a gives a 4 4 4.. Which of the followig sequeces coverge to zero? a b + solutio
More informationConfidence Intervals. CI for a population mean (σ is known and n > 30 or the variable is normally distributed in the.
Cofidece Itervals A cofidece iterval is a iterval whose purpose is to estimate a parameter (a umber that could, i theory, be calculated from the populatio, if measuremets were available for the whole populatio).
More informationS. Tanny MAT 344 Spring 1999. be the minimum number of moves required.
S. Tay MAT 344 Sprig 999 Recurrece Relatios Tower of Haoi Let T be the miimum umber of moves required. T 0 = 0, T = 7 Iitial Coditios * T = T + $ T is a sequece (f. o itegers). Solve for T? * is a recurrece,
More informationThe 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 informationLesson 15 ANOVA (analysis of variance)
Outlie Variability betwee group variability withi group variability total variability Fratio Computatio sums of squares (betwee/withi/total degrees of freedom (betwee/withi/total mea square (betwee/withi
More informationChapter 7 Methods of Finding Estimators
Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of
More informationLearning objectives. Duc K. Nguyen  Corporate Finance 21/10/2014
1 Lecture 3 Time Value of Moey ad Project Valuatio The timelie Three rules of time travels NPV of a stream of cash flows Perpetuities, auities ad other special cases Learig objectives 2 Uderstad the timevalue
More information1 Correlation and Regression Analysis
1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio
More informationIrreducible polynomials with consecutive zero coefficients
Irreducible polyomials with cosecutive zero coefficiets Theodoulos Garefalakis Departmet of Mathematics, Uiversity of Crete, 71409 Heraklio, Greece Abstract Let q be a prime power. We cosider the problem
More informationMathematical goals. Starting points. Materials required. Time needed
Level A1 of challege: C A1 Mathematical goals Startig poits Materials required Time eeded Iterpretig algebraic expressios To help learers to: traslate betwee words, symbols, tables, ad area represetatios
More informationTHE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n
We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample
More informationNATIONAL 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 informationYour organization has a Class B IP address of 166.144.0.0 Before you implement subnetting, the Network ID and Host ID are divided as follows:
Subettig Subettig is used to subdivide a sigle class of etwork i to multiple smaller etworks. Example: Your orgaizatio has a Class B IP address of 166.144.0.0 Before you implemet subettig, the Network
More informationPart  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 informationChapter 5 O A Cojecture Of Erdíos Proceedigs NCUR VIII è1994è, Vol II, pp 794í798 Jeærey F Gold Departmet of Mathematics, Departmet of Physics Uiversity of Utah Do H Tucker Departmet of Mathematics Uiversity
More informationIncremental calculation of weighted mean and variance
Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically
More informationEkkehart Schlicht: Economic Surplus and Derived Demand
Ekkehart Schlicht: Ecoomic Surplus ad Derived Demad Muich Discussio Paper No. 200617 Departmet of Ecoomics Uiversity of Muich Volkswirtschaftliche Fakultät LudwigMaximiliasUiversität Müche Olie at http://epub.ub.uimueche.de/940/
More informationAnalysis 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 informationInteger Factorization Algorithms
Iteger Factorizatio Algorithms Coelly Bares Departmet of Physics, Orego State Uiversity December 7, 004 This documet has bee placed i the public domai. Cotets I. Itroductio 3 1. Termiology 3. Fudametal
More informationBetting on Football Pools
Bettig o Football Pools by Edward A. Beder I a pool, oe tries to guess the wiers i a set of games. For example, oe may have te matches this weeked ad oe bets o who the wiers will be. We ve put wiers i
More informationPresent Value Factor To bring one dollar in the future back to present, one uses the Present Value Factor (PVF): Concept 9: Present Value
Cocept 9: Preset Value Is the value of a dollar received today the same as received a year from today? A dollar today is worth more tha a dollar tomorrow because of iflatio, opportuity cost, ad risk Brigig
More informationOutput Analysis (2, Chapters 10 &11 Law)
B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should
More informationCHAPTER 3 THE TIME VALUE OF MONEY
CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all
More informationTHE ARITHMETIC OF INTEGERS.  multiplication, exponentiation, division, addition, and subtraction
THE ARITHMETIC OF INTEGERS  multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do. THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive,
More informationChapter 7: Confidence Interval and Sample Size
Chapter 7: Cofidece Iterval ad Sample Size Learig Objectives Upo successful completio of Chapter 7, you will be able to: Fid the cofidece iterval for the mea, proportio, ad variace. Determie the miimum
More informationConfidence Intervals for One Mean
Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a
More informationEscola Federal de Engenharia de Itajubá
Escola Federal de Egeharia de Itajubá Departameto de Egeharia Mecâica PósGraduação em Egeharia Mecâica MPF04 ANÁLISE DE SINAIS E AQUISÇÃO DE DADOS SINAIS E SISTEMAS Trabalho 02 (MATLAB) Prof. Dr. José
More informationHow Euler Did It. In a more modern treatment, Hardy and Wright [H+W] state this same theorem as. n n+ is perfect.
Amicable umbers November 005 How Euler Did It by Ed Sadifer Six is a special umber. It is divisible by, ad 3, ad, i what at first looks like a strage coicidece, 6 = + + 3. The umber 8 shares this remarkable
More informationHypergeometric Distributions
7.4 Hypergeometric Distributios Whe choosig the startig lieup for a game, a coach obviously has to choose a differet player for each positio. Similarly, whe a uio elects delegates for a covetio or you
More informationNormal Distribution.
Normal Distributio www.icrf.l Normal distributio I probability theory, the ormal or Gaussia distributio, is a cotiuous probability distributio that is ofte used as a first approimatio to describe realvalued
More informationLecture 4: Cheeger s Inequality
Spectral Graph Theory ad Applicatios WS 0/0 Lecture 4: Cheeger s Iequality Lecturer: Thomas Sauerwald & He Su Statemet of Cheeger s Iequality I this lecture we assume for simplicity that G is a dregular
More informationClass Meeting # 16: The Fourier Transform on R n
MATH 18.152 COUSE NOTES  CLASS MEETING # 16 18.152 Itroductio to PDEs, Fall 2011 Professor: Jared Speck Class Meetig # 16: The Fourier Trasform o 1. Itroductio to the Fourier Trasform Earlier i the course,
More informationA RANDOM PERMUTATION MODEL ARISING IN CHEMISTRY
J. Appl. Prob. 45, 060 070 2008 Prited i Eglad Applied Probability Trust 2008 A RANDOM PERMUTATION MODEL ARISING IN CHEMISTRY MARK BROWN, The City College of New York EROL A. PEKÖZ, Bosto Uiversity SHELDON
More informationChapter 14 Nonparametric Statistics
Chapter 14 Noparametric Statistics A.K.A. distributiofree statistics! Does ot deped o the populatio fittig ay particular type of distributio (e.g, ormal). Sice these methods make fewer assumptios, they
More informationCenter, Spread, and Shape in Inference: Claims, Caveats, and Insights
Ceter, Spread, ad Shape i Iferece: Claims, Caveats, ad Isights Dr. Nacy Pfeig (Uiversity of Pittsburgh) AMATYC November 2008 Prelimiary Activities 1. I would like to produce a iterval estimate for the
More informationSection 8.3 : De Moivre s Theorem and Applications
The Sectio 8 : De Moivre s Theorem ad Applicatios Let z 1 ad z be complex umbers, where z 1 = r 1, z = r, arg(z 1 ) = θ 1, arg(z ) = θ z 1 = r 1 (cos θ 1 + i si θ 1 ) z = r (cos θ + i si θ ) ad z 1 z =
More informationPerfect Packing Theorems and the AverageCase Behavior of Optimal and Online Bin Packing
SIAM REVIEW Vol. 44, No. 1, pp. 95 108 c 2002 Society for Idustrial ad Applied Mathematics Perfect Packig Theorems ad the AverageCase Behavior of Optimal ad Olie Bi Packig E. G. Coffma, Jr. C. Courcoubetis
More information