2.3. GEOMETRIC SERIES


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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
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