2.3. GEOMETRIC SERIES


 Buck McDonald
 1 years ago
 Views:
Transcription
1 6 CHAPTER INFINITE SERIES GEOMETRIC SERIES Oe of the most importat types of ifiite series are geometric series A geometric series is simply the sum of a geometric sequece, Fortuately, geometric series are also the easiest type of series to aalyze We dealt a little bit with geometric series i the last sectio; Example showed that ar, while Exercise 6 preseted Archimedes computatio that (Note that i this sectio we will sometimes begi our series at ad sometimes begi them at ) Geometric series are some of the oly series for which we ca ot oly determie covergece ad divergece easily, but also fid their sums, if they coverge: Geometric Series The geometric series a ar ar ar coverges to if r a r, ad diverges otherwise Proof If r, the the geometric series diverges by the Test for Divergece, so let us suppose that r Let s deote the sum of the first terms, s a ar ar ar, so rs ar ar ar ar
2 SECTION GEOMETRIC SERIES 7 Subtractig these two, we fid that s a ar ar ar rs ar ar ar ar r s a ar This allows us to solve for the partial sums s, s a ar r a r ar r Now we kow (see Example 5 of Sectio ) that for r, r as, so ar r 0 as as well, ad thus lim s lim a r ar r a r, provig the result A easy way to remember this theorem is geometric series We begi with two basic examples first term ratio betwee terms Example Compute 9 7 Solutio The first term is ad the ratio betwee terms is, so 9 7 first term ratio betwee terms 8, solvig the problem Example Compute 6 Solutio This series is geometric with commo ratio r a a,
3 8 CHAPTER INFINITE SERIES ad so it coverges because Its sum is 6 first term ratio betwee terms 9 6, which simplifies to 5 5 The use of the geometric series formula is of course ot limited to sigle geometric series, as our ext example demostrates Example Compute 9 5 Solutio We break this series ito two: (if both series coverge) The first of these series has commo ratio 5, so it coverges To aalyze the secod series, ote that 9 9 9, so this series has commo ratio 5 Sice both series coverge, we may proceed with the additio: This aswer simplifies to 7 6 If a geometric series ivolves a variable x, the it may oly coverge for certai values of x Where the series does coverge, it defies a fuctio of x, which we ca compute from the summatio formula Our ext example illustrates these poits Example For which values of x does the series of x, which fuctio does this series defie? x coverge? For those values Solutio The commo ratio of this series is r a a x x x Sice geometric series coverge if ad oly if r, we eed x for this series to coverge This expressio simplifies to x
4 SECTION GEOMETRIC SERIES 9 Where this series does coverge (ie, for x ), its sum ca be foud by the geometric series formula: x first term ratio betwee terms x x Although ote that this oly holds for x The geometric series formula may also be used to covert repeatig decimals ito fractios, as we show ext Example 5 Express the umber as a fractio i the form p q where p ad q have o commo factors Solutio Our first step is to express this umber as the sum of a geometric series Sice the decimal seems to repeat every digits, we ca write this as This series is geometric, so we ca use the formula to evaluate it: first term 000 ratio betwee terms Our iitial aswer is therefore Sice we were asked for a fractio with o commo factors betwee the umerator ad deomiator, we ow have to factor out the 9 which divides both 8 ad 999, leavig 8 To be sure that 8 ad have o commo factors, we eed to verify that 7, ad the prime 7 does ot divide 8 Our ext example i some sese goes i the other directio Here we are give a fractio ad asked to use geometric series to approximate its decimal expasio Example 6 Use geometric series to approximate the decimal expasio of 8 Solutio First we fid a umber ear 8 with a simple decimal expasio; 50 will work icely Now we express 8 as 50 times a fractio of the form r :
5 50 CHAPTER INFINITE SERIES Now we ca expad the fractio o the righthad side as a geometric series, Usig the first two terms of this series, we obtai the approximatio Example 7 Suppose that you draw a by square, the you joi the midpoits of its sides to draw aother square, the you joi the midpoits of that square s sides to draw aother square, ad so o, as show below Would you eed ifiitely may pecils to cotiue this process forever? Solutio If we look just at the upper lefthad corer of this figure, we see a triagle with two sides of legth ad a hypoteuse of legth : So the sequece of side legths of these rectagles is geometric with ratio :,,, Sice, the sum of the side legths of all these (ifiitely may) squares therefore coverges to
6 SECTION GEOMETRIC SERIES 5 As the perimeter of a square is four times its side legth, the total perimeter of this ifiite costructio is 8, so we would oly eed fiitely may pecils to draw the figure forever Our last two examples are sigificatly more advaced that the previous examples O the other had, they are also more iterestig Example 8 (A Game of Chace) A gambler offers you a propositio He carries a fair coi, with two differet sides, heads ad tails ( fair here meas that it is equally likely to lad heads or tails), which he will toss If it comes up heads, he will pay you $ If it comes up tails, he will toss the coi agai If, o the secod toss, it comes up heads, he will pay you $, ad if it comes up tails agai he will toss it agai O the third toss, if it comes up heads, he will pay you $, ad if it comes up tails agai, he will toss it agai How much should you be willig to pay to play this game? Solutio Would you pay the gambler $ to play this game? Of course You ve got to wi at least a dollar Would you pay the gamble $ to play this game? Here thigs get more complicated, sice you have a 50% chace of losig moey if you pay $ to play, but you also have a 5% chace to get your $ back, ad a 5% chace to wi moey Would you pay the gambler oe millio dollars to play the game? The gambler asserts that there are ifiitely may positive itegers greater tha oe millio Thus (accordig to the gambler, at least) you have ifiitely may chaces of wiig more tha oe millio dollars! To figure out how much you should pay to play this game, we compute its average payoff (its expectatio) I the chart below, we write H for heads ad T for tails Outcome Probability Payoff Expected Wiigs H $ $ $050 TH $ $ $050 TTH 8 $ 8 $ $075 TTTH 6 $ 6 $ $05 TT T H $ $ $ There are several observatios we ca make about this table of payoffs First, the sum of the probabilities of the various outcomes is, which idicates that we have ideed listed all the (oegligible) outcomes But what if the coi ever lads heads? The probability of this happeig is lim 0, ad so it is safely igored Therefore, we igore this possibility
7 5 CHAPTER INFINITE SERIES To figure out how much you should pay to play this game, it seems reasoable to compute the sum of the expected wiigs, over all possible outcomes This sum is We demostrate two methods to compute this sum The first is elemetary but uses a clever trick, while the secod uses calculus s Applyig the geometric series formula, we ca write: If we ow add this array vertically, we obtai a equatio whose lefthad side is (it is the sum of a geometric series itself), ad whose righthad side,, is the sum of the expected wiigs, so a Because the expected wiigs are $, you should be willig to pay aythig less tha $ to play the game, because i the logru, you will make moey Now we preset a method usig calculus Sice we kow that x x x x x we ca differetiate this formula to get Now multiply both sides by x: x 0 x x x x x x x x x Fially, if we set x we obtai the sum we are lookig for:
8 SECTION GEOMETRIC SERIES 5 Oe might (quite rightly) complai that we do t kow that we ca take the derivative of a ifiite series i this way We cosider these issues i Sectio Example 9 (The St Petersburg Paradox ) Impressed with the calculatio of the expected wiigs, suppose that the gambler offers you a differet wager This time, he will pay $ for the outcome H, $ for the outcome TH,$ for the outcome TTH,$8 for the outcome TTTH, ad i geeral, $ if the coi lads tails times before ladig heads Computig the expected wiigs as before, we obtai the followig chart Outcome Probability Payoff Expected Wiigs H $ $050 TH $ $050 TTH 8 $ $050 TTTH 6 $8 $050 TT T H $ $050 The gambler ow suggests that you should be willig to pay ay amout of moey to play this game sice the expected wiigs, $050 $050 $050, are ifiite! Solutio That is why this game is referred to as a paradox Seemigly, there is o fair price to pay to play this game Beroulli, who itroduced this paradox, attempted to resolve it by assertig that moey has decliig margial utility This is ot i dispute; imagie what your life would be like with $ billio, ad the imagie what it would be like with $ billio Clearly the first $ billio makes a much bigger differece tha the secod $ billio, so it has higher utility However, the gambler ca simply adjust his payoffs, for example, he could offer you $ if the coi lads tails times before ladig heads, so eve takig ito accout the decliig margial utility of moey, there is always some payoff fuctio which results i a paradox A practical way out of this paradox is to ote that the gambler ca t keep his promises If we assume geerously that the gambler has $ billio, the the gambler caot pay you the full amout if the coi lads tails 0 times before ladig heads ( 9 56, 870, 9, but 0, 07, 7, 8) The payouts i this case will be as above up to 9, but the begiig at 0, all you ca wi is $ billio This gives a expected wiigs of 9 $050 0 $,000,000,000 $50 $,000,000,000 0 $66 This problem is kow as the St Petersburg Paradox because it was itroduced i a 78 paper of Daiel Beroulli (700 78) published i the St Petersburg Academy Proceedigs
9 5 CHAPTER INFINITE SERIES So by makig this rather iocuous assumptio, the game goes from beig priceless to beig worth the same as a ew Tshirt Aother valid poit is that you do t have ulimited moey, so there is a high probability that by repeatedly playig this game, you will go broke before you hit the rare but gigatic jackpot which makes you rich Oe amusig but evertheless accurate way to summarize the St Petersburg Paradox is therefore: If both you ad the gambler had a ifiite amout of time ad moey, you could ear aother ifiite amout of moey playig this game forever But would t you have better thigs to do with a ifiite supply of time ad moey? EXERCISES FOR SECTION Determie which of the series i Exercises 8 are geometric series Fid the sums of the geometric series Fid the sums of the series i Exercises e π 9 6 8! Determie which of the geometric series (or sums of geometric series) i Exercises 9 coverge Fid the sums of the coverget series 9 Determie which values of x the series i Exercises 7 0 coverge for Whe these series coverge, they defie fuctios of x What are these fuctios? x 7 8 x
10 SECTION GEOMETRIC SERIES x show below 0 x For Exercises, use geometric series to approximate these reciprocals, as i Example For Exercises 5 ad 6, suppose that a is a geometric series such that the sum of the first terms is 875 ad the sum of the first 6 terms is What is a 6? 6 What is a? 7 Write a series represetig the area of the Koch sowflake, ad fid its value 8 Write a series represetig the perimeter of the Koch sowflake Does this series coverge? 9 Two 50% marksme decide to fight i a duel i which they exchage shots util oe is hit What are the odds i favor of the ma who shoots first? 0 I the decimal system, some umbers have more tha oe expasio Verify this by showig that The egadecimal umber system is like the decimal system except that the base is 0 So, for example, i base 0 Prove that (like the decimal system) there are ouique expasios i the egadecimal system by showig that Exercises 7 ad 8 cosider the Koch sowflake Itroduced i90 by the Swedish mathematicia Helge vo Koch (870 9), the Koch sowflake is oe of the earliest fractals to have bee described We start with a equilateral triagle The we divide each of the three sides ito three equal lie segmets, ad replace the middle portio with a smaller equilateral triagle We the iterate this costructio, dividig each of the lie segmets of the ew figure ito thirds ad replacig the middle with a equilateral triagle, ad the iterate this agai ad agai, forever The first four iteratios are Exercises ad ask you to prove that there are ifiitely may primes, followig a proof of Euler Exercise provides the classic proof, due to Euclid Prove, usig the fact that every positive iteger has a uique prime factorizatio, that 5 where the righthad side is the product of all series of the form p for primes p This refers to the fact that every positive iteger ca be writte as a product p a pa p a k k for primes p,p,,p k ad oegative itegers a,a,, a k
11 56 CHAPTER INFINITE SERIES Use Exercise to prove that there are ifiitely may primes Hit: the left hadside diverges to, while every sum o the right hadside coverges (Euclid s proof) Suppose to the cotrary that there are oly fiitely may prime umbers, p,p,,p m Draw a cotradictio by cosiderig the umber p p p m 5 Compute Hit: cosider the derivative of x x For ow, assume that you ca differetiate this series as doe i Example 8 6 Suppose the gambler from Example 8 alters his game as follows If the coi lads tails a eve umber of times before ladig heads, you must pay him $ However, if the coi lads tails a odd umber of times before ladig heads, he pays you $ The gambler argues that there are just as may eve umbers as odd umbers, so the game is fair Would you be willig to play this game? Why or why ot? (Note that 0 is a eve umber) 7 Suppose the gambler alters his game oce more If the coi lads tails a eve umber of times before ladig heads, you must pay him $ However, if the coi lads tails a odd umber of times before ladig heads, he pays you $ Would you be willig to play this game? Why or why ot? 8 Suppose that after a few flips, you grow suspicious of the gambler s coi because it seems to lad heads more tha 50% of the time (but less tha00% of the time) Desig a procedure which will evertheless produce odds usig his ufair coi (The simplest solutio to the problem is attributed to Joh vo Neuma (90 957)) 9 Cosider the series where the sum is take over all positive itegers which do ot cotai a 9 i their decimal expasio Due to AJ Kemper i9, series like this are referred to as depleted harmoic series Show that this series coverges Hit: how may terms have deomiators betwee ad 9? Betwee0 ad 99? More geerally, betwee0 ad 0?
12 SECTION GEOMETRIC SERIES 57 ANSWERS TO SELECTED EXERCISES, SECTION Geometric series, coverges to Not a geometric series 5 Geometric series, coverges to 7 Geometric series, coverges to Diverges; the ratio is 9 Diverges The sum is The sum is Coverges for x, to the fuctio x x 9 Coverges for 5 x 5, to the fuctio 5 x 5x
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 pseries (the Compariso Test), but of
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 informationwhen 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 information1 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 informationModule 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 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 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 information3.2 Introduction to Infinite Series
3.2 Itroductio to Ifiite Series May of our ifiite sequeces, for the remaider of the course, will be defied by sums. For example, the sequece S m := 2. () is defied by a sum. Its terms (partial sums) are
More information8.5 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 informationLesson 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 informationRiemann 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, oegative fuctio o the closed iterval [a, b] Fid
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 informationARITHMETIC AND GEOMETRIC PROGRESSIONS
Arithmetic Ad Geometric Progressios Sequeces Ad ARITHMETIC AND GEOMETRIC PROGRESSIONS Successio of umbers of which oe umber is desigated as the first, other as the secod, aother as the third ad so o gives
More informationWinter Camp 2012 Sequences Alexander Remorov. Sequences. Alexander Remorov
Witer Camp 202 Sequeces Alexader Remorov Sequeces Alexader Remorov alexaderrem@gmail.com Warmup Problem : Give a positive iteger, cosider a sequece of real umbers a 0, a,..., a defied as a 0 = 2 ad =
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 information1 n. n > dt. t < n 1 + n=1
Math 05 otes C. Pomerace The harmoic sum The harmoic sum is the sum of recirocals of the ositive itegers. We kow from calculus that it diverges, this is usually doe by the itegral test. There s a more
More informationIn 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 informationThe 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 informationSection 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 informationORDERS 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 informationYour 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 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 informationThe geometric series and the ratio test
The geometric series ad the ratio test Today we are goig to develop aother test for covergece based o the iterplay betwee the it compariso test we developed last time ad the geometric series. A ote about
More information.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth
Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,
More informationChapter 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 informationContinued 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 information8.1 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 informationSection 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 informationSEQUENCES 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 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 informationMath 475, Problem Set #6: Solutions
Math 475, Problem Set #6: Solutios A (a) For each poit (a, b) with a, b oegative itegers satisfyig ab 8, cout the paths from (0,0) to (a, b) where the legal steps from (i, j) are to (i 2, j), (i, j 2),
More informationThe 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 informationHW 1 Solutions Math 115, Winter 2009, Prof. Yitzhak Katznelson
HW Solutios Math 5, Witer 2009, Prof. Yitzhak Katzelso.: Prove 2 + 2 2 +... + 2 = ( + )(2 + ) for all atural umbers. The proof is by iductio. Call the th propositio P. The basis for iductio P is the statemet
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 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 informationGeometric Sequences and Series. Geometric Sequences. Definition of Geometric Sequence. such that. a2 4
3330_0903qxd /5/05 :3 AM Page 663 Sectio 93 93 Geometric Sequeces ad Series 663 Geometric Sequeces ad Series What you should lear Recogize, write, ad fid the th terms of geometric sequeces Fid th partial
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 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 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, Series and Convergence with the TI 92. Roger G. Brown Monash University
Sequeces, Series ad Covergece with the TI 92. Roger G. Brow Moash Uiversity email: rgbrow@deaki.edu.au Itroductio. Studets erollig i calculus at Moash Uiversity, like may other calculus courses, are itroduced
More informationA Gentle Introduction to Algorithms: Part II
A Getle Itroductio to Algorithms: Part II Cotets of Part I:. Merge: (to merge two sorted lists ito a sigle sorted list.) 2. Bubble Sort 3. Merge Sort: 4. The BigO, BigΘ, BigΩ otatios: asymptotic bouds
More informationChapter Eleven. Taylor Series. (x a) k. c k. k= 0
Chapter Eleve Taylor Series 111 Power Series Now that we are kowledgeable about series, we ca retur to the problem of ivestigatig the approximatio of fuctios by Taylor polyomials of higher ad higher degree
More informationMocks.ie Maths LC HL Further Calculus mocks.ie Page 1
Maths Leavig Cert Higher Level Further Calculus Questio Paper By Cillia Fahy ad Darro Higgis Mocks.ie Maths LC HL Further Calculus mocks.ie Page Further Calculus ad Series, Paper II Q8 Table of Cotets:.
More informationFourier Series and the Wave Equation Part 2
Fourier Series ad the Wave Equatio Part There are two big ideas i our work this week. The first is the use of liearity to break complicated problems ito simple pieces. The secod is the use of the symmetries
More information4.1 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 information1. a n = 2. a n = 3. a n = 4. a n = 5. a n = 6. a n =
Versio PREVIEW Homework Berg (5860 This pritout should have 9 questios. Multiplechoice questios may cotiue o the ext colum or page fid all choices before aswerig. CalCb0b 00 0.0 poits Rewrite the fiite
More informationI. Why is there a time value to money (TVM)?
Itroductio to the Time Value of Moey Lecture Outlie I. Why is there the cocept of time value? II. Sigle cash flows over multiple periods III. Groups of cash flows IV. Warigs o doig time value calculatios
More informationInfinite 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 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 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 information1 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 informationSequences 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 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 information0,1 is an accumulation
Sectio 5.4 1 Accumulatio Poits Sectio 5.4 BolzaoWeierstrass ad HeieBorel 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 informationLecture Notes CMSC 251
We have this messy summatio to solve though First observe that the value remais costat throughout the sum, ad so we ca pull it out frot Also ote that we ca write 3 i / i ad (3/) i T () = log 3 (log ) 1
More informationDivergence of p 1/p. Adrian Dudek. adrian.dudek[at]anu.edu.au
Divergece of / Adria Dudek adria.dudek[at]au.edu.au Whe I was i high school, my maths teacher cheekily told me that it s ossible to add u ifiitely may umbers ad get a fiite umber. She the illustrated this
More information5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized?
5.4 Amortizatio Questio 1: How do you fid the preset value of a auity? Questio 2: How is a loa amortized? Questio 3: How do you make a amortizatio table? Oe of the most commo fiacial istrumets a perso
More informationMath 115 HW #4 Solutions
Math 5 HW #4 Solutios From 2.5 8. Does the series coverge or diverge? ( ) 3 + 2 = Aswer: This is a alteratig series, so we eed to check that the terms satisfy the hypotheses of the Alteratig Series Test.
More informationRecursion and Recurrences
Chapter 5 Recursio ad Recurreces 5.1 Growth Rates of Solutios to Recurreces Divide ad Coquer Algorithms Oe of the most basic ad powerful algorithmic techiques is divide ad coquer. Cosider, for example,
More information7 b) 0. Guided Notes for lesson P.2 Properties of Exponents. If a, b, x, y and a, b, 0, and m, n Z then the following properties hold: 1 n b
Guided Notes for lesso P. Properties of Expoets If a, b, x, y ad a, b, 0, ad m, Z the the followig properties hold:. Negative Expoet Rule: b ad b b b Aswers must ever cotai egative expoets. Examples: 5
More informationChapter Gaussian Elimination
Chapter 04.06 Gaussia Elimiatio After readig this chapter, you should be able to:. solve a set of simultaeous liear equatios usig Naïve Gauss elimiatio,. lear the pitfalls of the Naïve Gauss elimiatio
More informationx(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 informationHomework 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 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 information8.3 POLAR FORM AND DEMOIVRE S THEOREM
SECTION 8. POLAR FORM AND DEMOIVRE S THEOREM 48 8. POLAR FORM AND DEMOIVRE S THEOREM Figure 8.6 (a, b) b r a 0 θ Complex Number: a + bi Rectagular Form: (a, b) Polar Form: (r, θ) At this poit you ca add,
More informationMA2108S 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 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 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 informationGrade 7. Strand: Number Specific Learning Outcomes It is expected that students will:
Strad: Number Specific Learig Outcomes It is expected that studets will: 7.N.1. Determie ad explai why a umber is divisible by 2, 3, 4, 5, 6, 8, 9, or 10, ad why a umber caot be divided by 0. [C, R] [C]
More informationReview for College Algebra Final Exam
Review for College Algebra Fial Exam (Please remember that half of the fial exam will cover chapters 14. This review sheet covers oly the ew material, from chapters 5 ad 7.) 5.1 Systems of equatios i
More informationBuilding Blocks Problem Related to Harmonic Series
TMME, vol3, o, p.76 Buildig Blocks Problem Related to Harmoic Series Yutaka Nishiyama Osaka Uiversity of Ecoomics, Japa Abstract: I this discussio I give a eplaatio of the divergece ad covergece of ifiite
More informationNumerical Solution of Equations
School of Mechaical Aerospace ad Civil Egieerig Numerical Solutio of Equatios T J Craft George Begg Buildig, C4 TPFE MSc CFD Readig: J Ferziger, M Peric, Computatioal Methods for Fluid Dyamics HK Versteeg,
More informationLinear 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 informationUnit 8 Rational Functions
Uit 8 Ratioal Fuctios Algebraic Fractios: Simplifyig Algebraic Fractios: To simplify a algebraic fractio meas to reduce it to lowest terms. This is doe by dividig out the commo factors i the umerator ad
More informationThe 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 informationTILE PATTERNS & GRAPHING
TILE PATTERNS & GRAPHING LESSON 1 THE BIG IDEA Tile patters provide a meaigful cotext i which to geerate equivalet algebraic expressios ad develop uderstadig of the cocept of a variable. Such patters are
More informationTAYLOR 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 information8 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 informationApproximating the Sum of a Convergent Series
Approximatig the Sum of a Coverget Series Larry Riddle Ages Scott College Decatur, GA 30030 lriddle@agesscott.edu The BC Calculus Course Descriptio metios how techology ca be used to explore covergece
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 informationDivide 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 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 informationBinet 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 Biettype formulas.
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 informationSample. Activity Library: Volume II. Activity Collections. Featuring realworld context collections:
Activity Library: Volume II Sample Activity Collectios Featurig realworld cotext collectios: Arithmetic II Fractios, Percets, Decimals III Fractios, Percets, Decimals IV Geometry I Geometry II Graphig
More information5. 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 informationChapter 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 information6.5 Probability Calculations in Hashing
6.5. PROBABILITY CALCULATIONS IN HASHING 245 6.5 Probability Calculatios i Hashig We ca use our kowledge of probability ad expected values to aalyze a umber of iterestig aspects of hashig icludig:. expected
More informationSearching Algorithm Efficiencies
Efficiecy of Liear Search Searchig Algorithm Efficiecies Havig implemeted the liear search algorithm, how would you measure its efficiecy? A useful measure (or metric) should be geeral, applicable to ay
More informationEquation of a line. Line in coordinate geometry. Slopeintercept form ( 斜 截 式 ) Intercept form ( 截 距 式 ) Pointslope form ( 點 斜 式 )
Chapter : Liear Equatios Chapter Liear Equatios Lie i coordiate geometr I Cartesia coordiate sstems ( 卡 笛 兒 坐 標 系 統 ), a lie ca be represeted b a liear equatio, i.e., a polomial with degree. But before
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 informationCALCULUS. Taimur Khalid
CALCULUS Taimur Khalid 5/20/2015 Chapter 1 What is Calculus? What is it? I m sure you ve all already heard about what calculus is from the other studets i my calculus class, but just i case you eed a refresher,
More informationTHE LEAST SQUARES REGRESSION LINE and R 2
THE LEAST SQUARES REGRESSION LINE ad R M358K I. Recall from p. 36 that the least squares regressio lie of y o x is the lie that makes the sum of the squares of the vertical distaces of the data poits from
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 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 informationOnestep equations. Vocabulary
Review solvig oestep equatios with itegers, fractios, ad decimals. Oestep equatios Vocabulary equatio solve solutio iverse operatio isolate the variable Additio Property of Equality Subtractio Property
More informationMath : Sequences and Series
EPProgram  Strisuksa School  Roiet Math : Sequeces ad Series Dr.Wattaa Toutip  Departmet of Mathematics Kho Kae Uiversity 00 :Wattaa Toutip wattou@kku.ac.th http://home.kku.ac.th/wattou. Sequeces
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 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 information