a 3, a 4, ... are the terms of the sequence. The number a n is the nth term of the sequence, and the entire sequence is denoted by a n
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1 60_090.qxd //0 : PM Page CHAPTER 9 Ifiite Series Sectio 9. EXPLORATION Fidig Patters Describe a patter for each of the followig sequeces. The use your descriptio to write a formula for the th term of each sequece. As icreases, do the terms appear to be approachig a it? Explai your reasoig. a. b. c. d. e.,,, 8, 6,...,, 6,, 0,... 0, 0, 0 6, 0 0, 0 5,..., 9, 9 6, 6 5, 5 6,... 7, 5 0, 7, 9 6, 9,... NOTE Occasioally, it is coveiet to begi a sequece with a 0, so that the terms of the sequece become a 0, a, a, a,..., a,... STUDY TIP Some sequeces are defied recursively. To defie a sequece recursively, you eed to be give oe or more of the first few terms. All other terms of the sequece are the defied usig previous terms, as show i Example (d). Sequeces List the terms of a sequece. Determie whether a sequece coverges or diverges. Write a formula for the th term of a sequece. Use properties of mootoic sequeces ad bouded sequeces. Sequeces I mathematics, the word sequece is used i much the same way as i ordiary Eglish. To say that a collectio of objects or evets is i sequece usually meas that the collectio is ordered so that it has a idetified first member, secod member, third member, ad so o. Mathematically, a sequece is defied as a fuctio whose domai is the set of positive itegers. Although a sequece is a fuctio, it is commo to represet sequeces by subscript otatio rather tha by the stadard fuctio otatio. For istace, i the sequece,,,, a, Sequece is mapped oto a, is mapped oto a, ad so o. The umbers a, a, a,..., a,... are the terms of the sequece. The umber a is the th term of the sequece, ad the etire sequece is deoted by a. EXAMPLE Listig the Terms of a Sequece a. The terms of the sequece a are,,,,...,,,, b. The terms of the sequece b are, c. The terms of the sequece c are, d. The terms of the recursively defied sequece d, where d 5 ad d d 5 are 5, a,,, a,,,, a,, 5 5 0,, 9 7,...,...,, 5,, 6 5,, a, 0 5 5,, , ,....
2 60_090.qxd //0 : PM Page 595 SECTION 9. Sequeces 595 Limit of a Sequece The primary focus of this chapter cocers sequeces whose terms approach itig values. Such sequeces are said to coverge. For istace, the sequece,, 8, 6,,... coverges to 0, as idicated i the followig defiitio. y = a L + ε L L ε 5 6 M For > M, the terms of the sequece all lie withi uits of L. Figure 9. Defiitio of the Limit of a Sequece Let L be a real umber. The it of a sequece a is L, writte as a L if for each > 0, there exists M > 0 such that wheever > M. If the it L of a sequece exists, the the sequece coverges to L. If the it of a sequece does ot exist, the the sequece diverges. a L < Graphically, this defiitio says that evetually (for > M ad > 0) the terms of a sequece that coverges to L will lie withi the bad betwee the lies y L ad y L, as show i Figure 9.. If a sequece a agrees with a fuctio f at every positive iteger, ad if f x approaches a it L as x, the sequece must coverge to the same it L. THEOREM 9. Limit of a Sequece Let L be a real umber. Let f be a fuctio of a real variable such that f x L. x If a is a sequece such that f a for every positive iteger, the a L. EXAMPLE Fidig the Limit of a Sequece NOTE There are differet ways i which a sequece ca fail to have a it. Oe way is that the terms of the sequece icrease without boud or decrease without boud. These cases are writte symbolically as follows. Terms icrease without boud: a Terms decrease without boud: a Fid the it of the sequece whose th term is Solutio a. I Theorem 5.5, you leared that x x x e. So, you ca apply Theorem 9. to coclude that a e.
3 60_090.qxd //0 : PM Page CHAPTER 9 Ifiite Series The followig properties of its of sequeces parallel those give for its of fuctios of a real variable i Sectio.. THEOREM 9. Let a L ad Properties of Limits of Sequeces b K.. a. ca cl, c is ay real umber ± b L ± K. a. L b LK b 0 ad K 0 b K, a EXAMPLE Determiig Covergece or Divergece a. Because the sequece a has terms,,,,... that alterate betwee ad, the it a does ot exist. So, the sequece diverges. See Example (a), page 59. b. For b divide the umerator ad deomiator by to obtai, which implies that the sequece coverges to. See Example (b), page 59. EXAMPLE Usig L Hôpital s Rule to Determie Covergece Show that the sequece whose th term is a coverges. Solutio Cosider the fuctio of a real variable TECHNOLOGY Use a graphig utility to graph the fuctio i Example. Notice that as x approaches ifiity, the value of the fuctio gets closer ad closer to 0. If you have access to a graphig utility that ca geerate terms of a sequece, try usig it to calculate the first 0 terms of the sequece i Example. The view the terms to observe umerically that the sequece coverges to 0. f x Applyig L Hôpital s Rule twice produces x Because f a for every positive iteger, you ca apply Theorem 9. to coclude that x x. x x x 0. x x l x So, the sequece coverges to 0. l x 0. See Example (c), page 59. idicates that i the HM mathspace CD-ROM ad the olie Eduspace system for this text, you will fid a Ope Exploratio, which further explores this example usig the computer algebra systems Maple, Mathcad, Mathematica, ad Derive.
4 60_090.qxd //0 : PM Page 597 SECTION 9. Sequeces 597 The symbol! (read factorial ) is used to simplify some of the formulas developed i this chapter. Let be a positive iteger; the factorial is defied as As a special case, zero factorial is defied as 0!. From this defiitio, you ca see that!,!,! 6, ad so o. Factorials follow the same covetios for order of operatios as expoets. That is, just as x ad x imply differet orders of operatios,! ad! imply the followig orders. ad!....!!...! Aother useful it theorem that ca be rewritte for sequeces is the Squeeze Theorem from Sectio a ( )! For,! is squeezed betwee ad. Figure 9. NOTE Example 5 suggests somethig about the rate at which! icreases as. As Figure 9. suggests, both ad! approach 0 as. Yet! approaches 0 so much faster tha does that! 0.! I fact, it ca be show that for ay fixed umber k, k 0.! This meas that the factorial fuctio grows faster tha ay expoetial fuctio. EXAMPLE 5 Show that the sequece Usig the Squeeze Theorem coverges, ad fid its it. Solutio To apply the Squeeze Theorem, you must fid two coverget sequeces that ca be related to the give sequece. Two possibilities are a ad b both of which coverge to 0. By comparig the term! with,, you ca see that ad THEOREM 9. If a L b! factors factors This implies that for, <!, ad you have!, as show i Figure 9.. So, by the Squeeze Theorem it follows that 0.! Squeeze Theorem for Sequeces ad there exists a iteger N such that a c b for all > N, the c L. c!
5 60_090.qxd //0 : PM Page CHAPTER 9 Ifiite Series I Example 5, the sequece c has both positive ad egative terms. For this sequece, it happes that the sequece of absolute values, c, also coverges to 0. You ca show this by the Squeeze Theorem usig the iequality 0!,. I such cases, it is ofte coveiet to cosider the sequece of absolute values ad the apply Theorem 9., which states that if the absolute value sequece coverges to 0, the origial siged sequece also coverges to 0. THEOREM 9. For the sequece a, if a 0 Absolute Value Theorem the a 0. Proof Cosider the two sequeces ad Because both of these sequeces coverge to 0 ad a a a a a. you ca use the Squeeze Theorem to coclude that a coverges to 0. Patter Recogitio for Sequeces Sometimes the terms of a sequece are geerated by some rule that does ot explicitly idetify the th term of the sequece. I such cases, you may be required to discover a patter i the sequece ad to describe the th term. Oce the th term has bee specified, you ca ivestigate the covergece or divergece of the sequece. EXAMPLE 6 Fidig the th Term of a Sequece Fid a sequece a whose first five terms are,, 8 5, 6 7, 9,... ad the determie whether the particular sequece you have chose coverges or diverges. Solutio First, ote that the umerators are successive powers of, ad the deomiators form the sequece of positive odd itegers. By comparig a with, you have the followig patter.,, 5, 7, 5 9,..., Usig L Hôpital s Rule to evaluate the it of f x x x, you obtai x x x l x x So, the sequece diverges..
6 60_090.qxd //0 : PM Page 599 SECTION 9. Sequeces 599 Without a specific rule for geeratig the terms of a sequece or some kowledge of the cotext i which the terms of the sequece are obtaied, it is ot possible to determie the covergece or divergece of the sequece merely from its first several terms. For istace, although the first three terms of the followig four sequeces are idetical, the first two sequeces coverge to 0, the third sequece coverges to 9, ad the fourth sequece diverges. a :,, 8, 6,...,,... b :,, 8, 5,..., 6 6,... c :,, 8, 7 6,..., 9 5 8,... d :,, 8, 0,...,,... 6 The process of determiig a th term from the patter observed i the first several terms of a sequece is a example of iductive reasoig. EXAMPLE 7 Fidig the th Term of a Sequece Determie a th term for a sequece whose first five terms are, 8, 6 6, 80, 0,... ad the decide whether the sequece coverges or diverges. Solutio Note that the umerators are less tha. So, you ca reaso that the umerators are give by the rule. Factorig the deomiators produces This suggests that the deomiators are represeted by!. Fially, because the sigs alterate, you ca write the th term as a!. From the discussio about the growth of!, it follows that a 0.! Applyig Theorem 9., you ca coclude that a 0. So, the sequece a coverges to 0....
7 60_090.qxd //0 : PM Page CHAPTER 9 Ifiite Series Mootoic Sequeces ad Bouded Sequeces a So far you have determied the covergece of a sequece by fidig its it. Eve if you caot determie the it of a particular sequece, it still may be useful to kow whether the sequece coverges. Theorem 9.5 provides a test for covergece of sequeces without determiig the it. First, some preiary defiitios are give. a a Defiitio of a Mootoic Sequece a a {a } = { + ( ) } A sequece a is mootoic if its terms are odecreasig a a a... a... or if its terms are oicreasig a a a... a.... (a) Not mootoic EXAMPLE 8 Determiig Whether a Sequece Is Mootoic b c (b) Mootoic {c } = c c c c {b } = { + } { } (c) Not mootoic Figure 9. b b b b Determie whether each sequece havig the give th term is mootoic. a. a b. b c. c Solutio a. This sequece alterates betwee ad. So, it is ot mootoic. b. This sequece is mootoic because each successive term is larger tha its predecessor. To see this, compare the terms b ad b. [Note that, because is positive, you ca multiply each side of the iequality by ad without reversig the iequality sig.] b <? <? b <? 0 < Startig with the fial iequality, which is valid, you ca reverse the steps to coclude that the origial iequality is also valid. c. This sequece is ot mootoic, because the secod term is larger tha the first term, ad larger tha the third. (Note that if you drop the first term, the remaiig sequece c, c, c,... is mootoic.) Figure 9. graphically illustrates these three sequeces. NOTE I Example 8(b), aother way to see that the sequece is mootoic is to argue that the derivative of the correspodig differetiable fuctio f x x x is positive for all x. This implies that f is icreasig, which i tur implies that a is icreasig.
8 60_090.qxd //0 : PM Page 60 SECTION 9. Sequeces 60 NOTE All three sequeces show i Figure 9. are bouded. To see this, cosider the followig. a b 0 c Defiitio of a Bouded Sequece. A sequece a is bouded above if there is a real umber M such that a M for all. The umber M is called a upper boud of the sequece.. A sequece a is bouded below if there is a real umber N such that N a for all. The umber N is called a lower boud of the sequece.. A sequece a is bouded if it is bouded above ad bouded below. Oe importat property of the real umbers is that they are complete. Iformally, this meas that there are o holes or gaps o the real umber lie. (The set of ratioal umbers does ot have the completeess property.) The completeess axiom for real umbers ca be used to coclude that if a sequece has a upper boud, it must have a least upper boud (a upper boud that is smaller tha all other upper bouds for the sequece). For example, the least upper boud of the sequece a,,,, 5,...,,... is. The completeess axiom is used i the proof of Theorem 9.5. THEOREM 9.5 Bouded Mootoic Sequeces If a sequece a is bouded ad mootoic, the it coverges. L a a a a a a L a a a 5 5 Every bouded odecreasig sequece coverges. Figure 9. Proof Assume that the sequece is odecreasig, as show i Figure 9.. For the sake of simplicity, also assume that each term i the sequece is positive. Because the sequece is bouded, there must exist a upper boud M such that a a a... a... M. From the completeess axiom, it follows that there is a least upper boud L such that a a a... a... L. For > 0, it follows that L < L, ad therefore L caot be a upper boud for the sequece. Cosequetly, at least oe term of a is greater tha L. That is, L < a N for some positive iteger N. Because the terms of a are odecreasig, it follows that a N a for > N. You ow kow that L < a N a L < L, for every > N. It follows that a L < for > N, which by defiitio meas that a coverges to L. The proof for a oicreasig sequece is similar. EXAMPLE 9 Bouded ad Mootoic Sequeces a. The sequece a is both bouded ad mootoic ad so, by Theorem 9.5, must coverge. b. The diverget sequece b is mootoic, but ot bouded. (It is bouded below.) c. The diverget sequece c is bouded, but ot mootoic.
9 60_090.qxd //0 : PM Page CHAPTER 9 Ifiite Series Exercises for Sectio 9. I Exercises 0, write the first five terms of the sequece.. a.. a. a 7. a a 0. a I Exercises, write the first five terms of the recursively defied sequece.. a, a k a k. a! 5. a 6. a si a a, a k k a k. a. a 6, a k a, a k a k k I Exercises 5 0, match the sequece with its graph. [The graphs are labeled (a), (b), (c), (d), (e), ad (f).] See for worked-out solutios to odd-umbered exercises. I Exercises, use a graphig utility to graph the first 0 terms of the sequece.. a. a. a. a 60.5 I Exercises 5 0, write the ext two apparet terms of the sequece. Describe the patter you used to fid these terms. 5., 5, 8,,... 6.,, 9, 5, , 0, 0, 0,... 8.,,, 8,... 9., 0.,, 9,, 8,..., 7 8,... I Exercises 6, simplify the ratio of factorials. 0!.. 8!!..!! 5. 6.! 7 5!!!!!! (a) a (b) a I Exercises 7, fid the it (if possible) of the sequece. 7. a 8. a a 0. a. a. a cos si (c) (e) a a (d) (f) a a I Exercises 6, use a graphig utility to graph the first 0 terms of the sequece. Use the graph to make a iferece about the covergece or divergece of the sequece. Verify your iferece aalytically ad, if the sequece coverges, fid its it.. a. a 5. a 6. a cos a a a 0. a a! 6 8 a 8 0 I Exercises 7 68, determie the covergece or divergece of the sequece with the give th term. If the sequece coverges, fid its it. 7. a 8. a 9. a 50. a 5. a a 5...!
10 60_090.qxd //0 : PM Page 60 SECTION 9. Sequeces a a l a 58.! 59. a 60.! a a 6. a p 6. e, p > 0 a k, a si 68. a a a a l a 0.5!! a si a cos I Exercises 95 98, (a) use Theorem 9.5 to show that the sequece with the give th term coverges ad (b) use a graphig utility to graph the first 0 terms of the sequece ad fid its it. 95. a a a a 99. Let a be a icreasig sequece such that a. Explai why a has a it. What ca you coclude about the it? 00. Let a be a mootoic sequece such that a. Discuss the covergece of a. If a coverges, what ca you coclude about its it? 0. Compoud Iterest Cosider the sequece A whose th term is give by A P r I Exercises 69 8, write a expressio for the th term of the sequece. (There is more tha oe correct aswer.) 69.,, 7, 0, , 7,, 5,... 7.,, 7,,,... 7.,, 9, 6, ,,,,,, 5, 5 6,... 8, ,,,, 5, ,, 7 5 8, 6,, ,, 5, 5 6,...,, 6,, 0,...,, 5, 5 7,..., x, x, x 6, x, x 5 0,... 8.,, 70, 0,0,,68,800,... 8., 6, 0, 500, 6,880,... I Exercises 8 9, determie whether the sequece with the give th term is mootoic. Discuss the boudedess of the sequece. Use a graphig utility to cofirm your results. 8. a 8. a 85. a 86. a e a a a si a cos 9. a a a cos a si where P is the pricipal, A is the accout balace after moths, ad r is the iterest rate compouded aually. (a) Is A a coverget sequece? Explai. (b) Fid the first 0 terms of the sequece if P $9000 ad r Compoud Iterest A deposit of $00 is made at the begiig of each moth i a accout at a aual iterest rate of % compouded mothly. The balace i the accout after moths is A (a) Compute the first six terms of the sequece A. (b) Fid the balace i the accout after 5 years by computig the 60th term of the sequece. (c) Fid the balace i the accout after 0 years by computig the 0th term of the sequece. Writig About Cocepts 0. I your ow words, defie each of the followig. (a) Sequece (b) Covergece of a sequece (c) Mootoic sequece (d) Bouded sequece 0. The graphs of two sequeces are show i the figures. Which graph represets the sequece with alteratig sigs? Explai. a 6 a 6
11 60_090.qxd //0 : PM Page CHAPTER 9 Ifiite Series Writig About Cocepts (cotiued) I Exercises 05 08, give a example of a sequece satisfyig the coditio or explai why o such sequece exists. (Examples are ot uique.) 05. A mootoically icreasig sequece that coverges to A mootoically icreasig bouded sequece that does ot coverge 07. A sequece that coverges to 08. A ubouded sequece that coverges to Govermet Expeditures A govermet program that curretly costs taxpayers $.5 billio per year is cut back by 0 percet per year. (a) Write a expressio for the amout budgeted for this program after years. (b) Compute the budgets for the first years. (c) Determie the covergece or divergece of the sequece of reduced budgets. If the sequece coverges, fid its it. 0. Iflatio If the rate of iflatio is % per year ad the average price of a car is curretly $6,000, the average price after years is P $6, Compute the average prices for the ext 5 years.. Modelig Data The umber of edagered ad threateed species i the Uited States from 996 through 00 is show i the table, where represets the year, with 6 correspodig to 996. (Source: U.S. Fish ad Wildlife Service) a (a) Use the regressio capabilities of a graphig utility to fid a model of the form a b c d, for the data. Use the graphig utility to plot the poits ad graph the model. (b) Use the model to predict the umber of edagered ad threateed species i the year Modelig Data The aual sales a (i millios of dollars) for Avo Products, Ic. from 99 through 00 are give below as ordered pairs of the form, a, where represets the year, with correspodig to 99. (Source: 00 Avo Products, Ic. Aual Report), 8,, 67, 5, 9, 6, 8, 7, 5079, 8, 5, 9, 589, 0, 568,, 5958,, 67 a , 7,..., (a) Use the regressio capabilities of a graphig utility to fid a model of the form a b c, for the data. Graphically compare the poits ad the model. (b) Use the model to predict sales i the year Comparig Expoetial ad Factorial Growth Cosider the sequece a 0!. (a) Fid two cosecutive terms that are equal i magitude. (b) Are the terms followig those foud i part (a) icreasig or decreasig? (c) I Sectio 8.7, Exercises 65 70, it was show that for large values of the idepedet variable a expoetial fuctio icreases more rapidly tha a polyomial fuctio. From the result i part (b), what iferece ca you make about the rate of growth of a expoetial fuctio versus a factorial fuctio for large iteger values of?. Compute the first six terms of the sequece a. If the sequece coverges, fid its it. 5. Compute the first six terms of the sequece a. If the sequece coverges, fid its it. 6. Prove that if s coverges to L ad L > 0, the there exists a umber N such that s > 0 for > N. True or False? I Exercises 7 0, determie whether the statemet is true or false. If it is false, explai why or give a example that shows it is false. 7. If a coverges to ad b coverges to, the a b coverges to If a coverges, the a a If >, the!!. 0. If a coverges, the a coverges to 0.. Fiboacci Sequece I a study of the progey of rabbits, Fiboacci (ca. 70 ca. 0) ecoutered the sequece ow bearig his ame. It is defied recursively by a a a, where a ad a. (a) Write the first terms of the sequece. (b) Write the first 0 terms of the sequece defied by b a a, (c) Usig the defiitio i part (b), show that b b.. (d) The golde ratio ca be defied by. Show that ad solve this equatio for.,,...,
12 60_090.qxd //0 : PM Page 605 SECTION 9. Sequeces 605. Cojecture Let x 0 ad cosider the sequece x give by the formula x x x, Use a graphig utility to compute the first 0 terms of the sequece ad make a cojecture about the it of the sequece.. Cosider the sequece (a) Compute the first five terms of this sequece. (b) Write a recursio formula for a, for. (c) Fid a.. Cosider the sequece (a) Compute the first five terms of this sequece. (b) Write a recursio formula for a, for. (c) Fid a. 5. Cosider the sequece a where a k, a k a, ad k > 0. (a) Show that a is icreasig ad bouded. (b) Prove that exists. (c) Fid a. 6. Arithmetic-Geometric Mea Let a 0 > b 0 > 0. Let a be the arithmetic mea of a 0 ad b 0 ad let b be the geometric mea of a 0 ad b 0. a a 0 b 0 b a 0 b 0 a Arithmetic mea Geometric mea Now defie the sequeces a ad b as follows. a a b,,....,,,... 6, 6 6, 6 6 6,... b a b (a) Let a 0 0 ad b 0. Write out the first five terms of a ad b. Compare the terms of b. Compare a ad b. What do you otice? (b) Use iductio to show that a > a > b > b, for a 0 > b 0 > 0. (c) Explai why a ad b are both coverget. (d) Show that a b. 7. (a) Let fx si x ad a si. Show that a f0. (b) Let fx be differetiable o the iterval 0, ad f0 0. Cosider the sequece a, where a f. Show that a f0. 8. Cosider the sequece a r. Decide whether a coverges for each value of r. (a) r (b) r (c) r (d) For what values or r does the sequece r 9. (a) Show that l x dx < l! for y y = lx (b) Draw a graph similar to the oe above that shows l! < l x dx. (c) Use the results of parts (a) ad (b) to show that for >. e <! <, e coverge? (d) Use the Squeeze Theorem for Sequeces ad the result of part (c) to show that! e. (e) Test the result of part (d) for 0, 50, ad Cosider the sequece (a) Write the first five terms of a. (b) Show that l by iterpretig as a Riema a a sum of a defiite itegral. k. Prove, usig the defiitio of the it of a sequece, that 0.. Prove, usig the defiitio of the it of a sequece, that for < r <. r 0. Complete the proof of Theorem 9.5. x k. a Putam Exam Challege. Let x, 0, be a sequece of ozero real umbers such that x x x for,,,.... Prove that there exists a real umber a such that x ax x, for all. 5. Let T 0, T, T 6, ad, for, T T T 8T. The first 0 terms of the sequece are,, 6,, 0, 5, 78, 568, 0,576, 6,9. Fid, with proof, a formula for T of the form T A B, where A ad B are well-kow sequeces. These problems were composed by the Committee o the Putam Prize Competitio. The Mathematical Associatio of America. All rights reserved.
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