60_090.qxd //0 : PM Page 606 606 CHAPTER 9 Ifiite Series Sectio 9. INFINITE SERIES The study of ifiite series was cosidered a ovelty i the fourteeth cetury. Logicia Richard Suiseth, whose ickame was Calculator, solved this problem. If throughout the first half of a give time iterval a variatio cotiues at a certai itesity, throughout the ext quarter of the iterval at double the itesity, throughout the followig eighth at triple the itesity ad so ad ifiitum; the the average itesity for the whole iterval will be the itesity of the variatio durig the secod subiterval (or double the itesity). This is the same as sayig that the sum of the ifiite series is. 8...... Series ad Covergece Uderstad the defiitio of a coverget ifiite series. Use properties of ifiite geometric series. Use the th-term Test for Divergece of a ifiite series. Ifiite Series Oe importat applicatio of ifiite sequeces is i represetig ifiite summatios. Iformally, if a is a ifiite sequece, the a a a a... a... is a ifiite series (or simply a series). The umbers a, a, a, are the terms of the series. For some series it is coveiet to begi the idex at 0 (or some other iteger). As a typesettig covetio, it is commo to represet a ifiite series as simply a. I such cases, the startig value for the idex must be take from the cotext of the statemet. To fid the sum of a ifiite series, cosider the followig sequece of partial sums. S a S a a S a a a S a a a... a Ifiite series If this sequece of partial sums coverges, the series is said to coverge ad has the sum idicated i the followig defiitio. Defiitios of Coverget ad Diverget Series For the ifiite series a, S a a... a. the th partial sum is give by If the sequece of partial sums S coverges to S, the the series coverges. The limit S is called the sum of the series. S a a... a... If S diverges, the the series diverges. a STUDY TIP As you study this chapter, you will see that there are two basic questios ivolvig ifiite series. Does a series coverge or does it diverge? If a series coverges, what is its sum? These questios are ot always easy to aswer, especially the secod oe. EXPLORATION Fidig the Sum of a Ifiite Series Fid the sum of each ifiite series. Explai your reasoig. a. 0. 0.0 0.00 0.000... b. 5 5 5 c. d. 00 0,000,000,000... 8 6... 0 00 000 0,000...
60_090.qxd //0 : PM Page 607 SECTION 9. Series ad Covergece 607 TECHNOLOGY Figure 9.5 shows the first 5 partial sums of the ifiite series i Example (a). Notice how the values appear to approach the lie y..5 EXAMPLE a. The series Coverget ad Diverget Series 8 6... has the followig partial sums. S S NOTE You ca geometrically determie the partial sums of the series i Example (a) usig Figure 9.6. 0 6 0 Figure 9.5 Figure 9.6 6 6 8 Because S 8 7 8 S 8... lim it follows that the series coverges ad its sum is. b. The th partial sum of the series... is give by S. Because the limit of S is, the series coverges ad its sum is. c. The series... diverges because S ad the sequece of partial sums diverges. The series i Example (b) is a telescopig series of the form b b b b b b b b 5.... Telescopig series FOR FURTHER INFORMATION To lear more about the partial sums of ifiite series, see the article Six Ways to Sum a Series by Da Kalma i The College Mathematics Joural. To view this article, go to the website www.matharticles.com. Note that b is caceled by the secod term, b is caceled by the third term, ad so o. Because the th partial sum of this series is S b b it follows that a telescopig series will coverge if ad oly if umber as. Moreover, if the series coverges, its sum is S b lim b. b approaches a fiite
60_090.qxd //0 : PM Page 608 608 CHAPTER 9 Ifiite Series EXAMPLE Writig a Series i Telescopig Form Fid the sum of the series. EXPLORATION I Proof Without Words, by Bejami G. Klei ad Irl C. Bives, the authors preset the followig diagram. Explai why the fial statemet below the diagram is valid. How is this result related to Theorem 9.6? T Solutio Usig partial fractios, you ca write a From this telescopig form, you ca see that the th partial sum is S 5.... So, the series coverges ad its sum is. That is,. lim S lim. Geometric Series The series give i Example (a) is a geometric series. I geeral, the series give by ar a ar ar... ar..., 0 a 0 Geometric series r r r is a geometric series with ratio r. r THEOREM 9.6 Covergece of a Geometric Series Q r r R r r. A geometric series with ratio r diverges if If 0 < r <, the the series coverges to the sum ar a 0 < r <. 0 r, P PQR TSP r r r... r Exercise take from Proof Without Words by Bejami G. Klei ad Irl C. Bives, Mathematics Magazie, October 988, by permissio of the authors. S Proof It is easy to see that the series diverges if If the a ar ar... r ±. r ±, S ar. Multiplicatio by r yields rs ar ar ar... ar. Subtractig the secod equatio from the first produces S rs a ar. Therefore, S r a r, ad the th partial sum is S a r r. If 0 < r <, it follows that r 0 as, ad you obtai lim S lim a r r a r lim r which meas that the series coverges ad its sum is a r. It is left to you to show that the series diverges if r >. a r
60_090.qxd //0 : PM Page 609 SECTION 9. Series ad Covergece 609 TECHNOLOGY Try usig a graphig utility or writig a computer program to compute the sum of the first 0 terms of the sequece i Example (a). You should obtai a sum of about 5.99999. EXAMPLE a. The geometric series 0 Coverget ad Diverget Geometric Series has a ratio of r with a. Because 0 < r <, the series coverges ad its sum is S 0 b. The geometric series... a r 6. 0 9 7 8... r, has a ratio of r. Because the series diverges. The formula for the sum of a geometric series ca be used to write a repeatig decimal as the ratio of two itegers, as demostrated i the ext example. EXAMPLE A Geometric Series for a Repeatig Decimal Use a geometric series to write 0.08 as the ratio of two itegers. Solutio For the repeatig decimal 0.08, you ca write 0.080808... 8 0 8 0 8 0 6 8 0 8... 8. 0 0 0 For this series, you have a 80 ad r 0. So, 0.080808... a 80 r 0 8 99. Try dividig 8 by 99 o a calculator to see that it produces 0.08. The covergece of a series is ot affected by removal of a fiite umber of terms from the begiig of the series. For istace, the geometric series ad both coverge. Furthermore, because the sum of the secod series is you ca coclude that the sum of the first series is S 0 5 8 8. 0 a r,
60_090.qxd //0 : PM Page 60 60 CHAPTER 9 Ifiite Series STUDY TIP As you study this chapter, it is importat to distiguish betwee a ifiite series ad a sequece. A sequece is a ordered collectio of umbers a, a, a,..., a,... whereas a series is a ifiite sum of terms from a sequece a a... a.... The followig properties are direct cosequeces of the correspodig properties of limits of sequeces. THEOREM 9.7 Properties of Ifiite Series If a A, b B, ad c is a real umber, the the followig series coverge to the idicated sums.. ca ca. a b A B. a b A B th-term Test for Divergece The followig theorem states that if a series coverges, the limit of its th term must be 0. NOTE Be sure you see that the coverse of Theorem 9.8 is geerally ot true. That is, if the sequece a coverges to 0, the the series a may either coverge or diverge. THEOREM 9.8 If a Limit of th Term of a Coverget Series coverges, the lim a 0. Proof Assume that a lim S L. The, because S S a ad lim S lim S L it follows that L lim S lim S a lim S lim a L lim a which implies that a coverges to 0. The cotrapositive of Theorem 9.8 provides a useful test for divergece. This th-term Test for Divergece states that if the limit of the th term of a series does ot coverge to 0, the series must diverge. THEOREM 9.9 th-term Test for Divergece a If lim 0, the diverges. a
60_090.qxd //0 : PM Page 6 SECTION 9. Series ad Covergece 6 EXAMPLE 5 Usig the th-term Test for Divergece STUDY TIP The series i Example 5(c) will play a importat role i this chapter. 7 6 5 D... You will see that this series diverges eve though the th term approaches 0 as approaches. 5 6 7 The height of each bouce is three-fourths the height of the precedig bouce. Figure 9.7 i a. For the series, you have lim. So, the limit of the th term is ot 0, ad the series diverges. b. For the series! you have!, lim So, the limit of the th term is ot 0, ad the series diverges. c. For the series you have, Because the limit of the th term is 0, the th-term Test for Divergece does ot apply ad you ca draw o coclusios about covergece or divergece. (I the ext sectio, you will see that this particular series diverges.) EXAMPLE 6 lim 0. Boucig Ball Problem A ball is dropped from a height of 6 feet ad begis boucig, as show i Figure 9.7. The height of each bouce is three-fourths the height of the previous bouce. Fid the total vertical distace traveled by the ball. Solutio Whe the ball hits the groud for the first time, it has traveled a distace of D 6 feet. For subsequet bouces, let D i be the distace traveled up ad dow. For example, ad are as follows. D 6 6 Up D Up Dow D 6 6 Dow By cotiuig this process, it ca be determied that the total vertical distace is D 6... 6 6 6 9 6 9 feet. 0!!. 0 D 0
60_090.qxd //0 : PM Page 6 6 CHAPTER 9 Ifiite Series I Exercises 6, fid the first five terms of the sequece of partial sums... I Exercises 7 6, verify that the ifiite series diverges. 7. 8. 0 0 9. 000.055 0..0 0 0.... 5. 6.! I Exercises 7, match the series with the graph of its sequece of partial sums. [The graphs are labeled (a), (b), (c), (d), (e), ad (f).] Use the graph to estimate the sum of the series. Cofirm your aswer aalytically. (a) Exercises for Sectio 9. 9 6 5... 5 5 6 5 6 7.... 9 7 8 8 6.... 5 7 9... 5. 6. S! (b) S (e) (f) 7. 9 8. 0 9. 5 0. 0 0. 7. I Exercises 8, verify that the ifiite series coverges... Use partial fractios. Use partial fractios. 5. 6. 7. 8. See www.calcchat.com for worked-out solutios to odd-umbered exercises..0.5.0 0.5 0 0 S 56789 0 7 8 9 0 5 0.9 0.9 0.8 0.79... 0 0.6 0.6 0.6 0.6... 0 Numerical, Graphical, ad Aalytic Aalysis I Exercises 9, (a) fid the sum of the series, (b) use a graphig utility to fid the idicated partial sum S ad complete the table, (c) use a graphig utility to graph the first 0 terms of the sequece of partial sums ad a horizotal lie represetig the sum, ad (d) explai the relatioship betwee the magitudes of the terms of the series ad the rate at which the sequece of partial sums approaches the sum of the series. 6 5 S 56789 5 6 7 8 9 5 6 7 8 9 5 0 0 50 00 (c) S (d) S S 5 6 7 8 9 6 5 5 6 7 8 9 9. 6 0.. 0.9.. 00.5. 0.85 5 I Exercises 5 50, fid the sum of the coverget series. 5. 6.
60_090.qxd //0 : PM Page 6 SECTION 9. Series ad Covergece 6 7. 8 8. 9. 0. 6 0 5 0.. 0 0. 0. 0.0 0.00.... 8 6 9 7 8... 5. 9... 6.... 7. 8. 0.7 0.9 0 9. 50. si 9 I Exercises 5 56, (a) write the repeatig decimal as a geometric series ad (b) write its sum as the ratio of two itegers. 5. 0. 5. 0.9 5. 0.8 5. 0.0 55. 0.075 56. 0.5 I Exercises 57 7, determie the covergece or divergece of the series. 57. 58. 0 0 59. 60. 6. 6. 6. 6. 0 0 65. 66..075 0 00 67. 68. l l 69. 70. k e 7. 7. l arcta Writig About Cocepts 7. State the defiitios of coverget ad diverget series. 7. Describe the differece betwee lim a 5 ad a 5. 75. Defie a geometric series, state whe it coverges, ad give the formula for the sum of a coverget geometric series. I Exercises 79 86, fid all values of x for which the series coverges. For these values of x, write the sum of the series as a fuctio of x. 79. 80. 8. x 8. 8. x 8. 0 85. 0 x 86. 87. (a) You delete a fiite umber of terms from a diverget series. Will the ew series still diverge? Explai your reasoig. (b) You add a fiite umber of terms to a coverget series. Will the ew series still coverge? Explai your reasoig. 88. Thik About It Cosider the formula Give x ad x, ca you coclude that either of the followig statemets is true? Explai your reasoig. (a)... (b) 8... I Exercises 89 ad 90, (a) fid the commo ratio of the geometric series, (b) write the fuctio that gives the sum of the series, ad (c) use a graphig utility to graph the fuctio ad the partial sums ad S 5. What do you otice? 89. x x 90. x x... I Exercises 9 ad 9, use a graphig utility to graph the fuctio. Idetify the horizotal asymptote of the graph ad determie its relatioship to the sum of the series. 9. Writig About Cocepts (cotiued) 76. State the th-term Test for Divergece. 77. Let a Discuss the covergece of a ad. a. 78. Explai ay differeces amog the followig series. (a) a (b) a (c) k a k x x x x x.... Fuctio S f x 0.5 x 0.5 k Series 0 9. f x 0.8 x 0.8 0 x 0 x x 0 x x 5 x x 8...
60_090.qxd //0 : PM Page 6 6 CHAPTER 9 Ifiite Series Writig I Exercises 9 ad 9, use a graphig utility to determie the first term that is less tha 0.000 i each of the coverget series. Note that the aswers are very differet. Explai how this will affect the rate at which the series coverges. 9. 9. 0.0, 8, 95. Marketig A electroic games maufacturer producig a ew product estimates the aual sales to be 8000 uits. Each year 0% of the uits that have bee sold will become ioperative. So, 8000 uits will be i use after year, 8000 0.98000 uits will be i use after years, ad so o. How may uits will be i use after years? 96. Depreciatio A compay buys a machie for $5,000 that depreciates at a rate of 0% per year. Fid a formula for the value of the machie after years. What is its value after 5 years? 97. Multiplier Effect The aual spedig by tourists i a resort city is $00 millio. Approximately 75% of that reveue is agai spet i the resort city, ad of that amout approximately 75% is agai spet i the same city, ad so o. Write the geometric series that gives the total amout of spedig geerated by the $00 millio ad fid the sum of the series. 98. Multiplier Effect Repeat Exercise 97 if the percet of the reveue that is spet agai i the city decreases to 60%. 99. Distace A ball is dropped from a height of 6 feet. Each time it drops h feet, it rebouds 0.8h feet. Fid the total distace traveled by the ball. 00. Time The ball i Exercise 99 takes the followig times for each fall. s 6t 6, s 6t 60.8, s 6t 60.8, s 6t 60.8, s 6t 60.8, Begiig with s, the ball takes the same amout of time to bouce up as it does to fall, ad so the total time elapsed before it comes to rest is give by t 0.9. Fid this total time. Probability I Exercises 0 ad 0, the radom variable represets the umber of uits of a product sold per day i a store. The probability distributio of is give by P. Fid the probability that two uits are sold i a give day [P] ad show that P P P.... P 0. 0. s 0 if t s 0 if t 0.9 s 0 if t 0.9 s 0 if t 0.9 s 0 if t 0.9 P 0. Probability A fair coi is tossed repeatedly. The probability that the first head occurs o the th toss is give by P, where. (a) Show that. (b) The expected umber of tosses required util the first head occurs i the experimet is give by Is this series geometric? (c) Use a computer algebra system to fid the sum i part (b). 0. Probability I a experimet, three people toss a fair coi oe at a time util oe of them tosses a head. Determie, for each perso, the probability that he or she tosses the first head. Verify that the sum of the three probabilities is. 05. Area The sides of a square are 6 iches i legth. A ew square is formed by coectig the midpoits of the sides of the origial square, ad two of the triagles outside the secod square are shaded (see figure). Determie the area of the shaded regios (a) if this process is cotiued five more times ad (b) if this patter of shadig is cotiued ifiitely. Figure for 05 Figure for 06 06. Legth A right triagle XYZ is show above where XY z ad X. Lie segmets are cotiually draw to be perpedicular to the triagle, as show i the figure. (a) Fid the total legth of the perpedicular lie segmets Yy x y x y... i terms of z ad. (b) If z ad fid the total legth of the perpedicular lie segmets. I Exercises 07 0, use the formula for the th partial sum of a geometric series i0. ar i a r r 07. Preset Value The wier of a $,000,000 sweepstakes will be paid $50,000 per year for 0 years. The moey ears 6% iterest per year. The preset value of the wiigs is 0 50,000.06.. 6 i. X θ z Y 6, y y Compute the preset value ad iterpret its meaig. x y y y 5 x x x x 5 Z
60_090.qxd //0 : PM Page 65 SECTION 9. 08. Sphereflake A sphereflake show below is a computergeerated fractal that was created by Eric Haies. The radius of the large sphere is. To the large sphere, ie spheres of radius are attached. To each of these, ie spheres of radius 9 are attached. This process is cotiued ifiitely. Prove that the sphereflake has a ifiite surface area. Series ad Covergece 65 5. 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) Use the regressio capabilities of a graphig utility to fid a model of the form a ce k,,, 5,..., for the data. Graphically compare the poits ad the model. (b) Use the data to fid the total sales for the 0-year period. (c) Approximate the total sales for the 0-year period usig the formula for the sum of a geometric series. Compare the result with that i part (b). Eric Haies 6. Salary You accept a job that pays a salary of $0,000 for the first year. Durig the ext 9 years you receive a % raise each year. What would be your total compesatio over the 0-year period? 09. Salary You go to work at a compay that pays $0.0 for the first day, $0.0 for the secod day, $0.0 for the third day, ad so o. If the daily wage keeps doublig, what would your total icome be for workig (a) 9 days, (b) 0 days, ad (c) days? 0. Auities Whe a employee receives a paycheck at the ed of each moth, P dollars is ivested i a retiremet accout. These deposits are made each moth for t years ad the accout ears iterest at the aual percetage rate r. If the iterest is compouded mothly, the amout A i the accout at the ed of t years is A P P P r r r... P r t t. Pe r Pe t r P e rt. e r Verify the formulas for the sums give above. Auities I Exercises, cosider makig mothly deposits of P dollars i a savigs accout at a aual iterest rate r. Use the results of Exercise 0 to fid the balace A after t years if the iterest is compouded (a) mothly ad (b) cotiuously.. P $50, r %, t 0 years. P $75, r 5%, t 5 years. P $00, r %, t 0 years. P $0, r 6%, t 50 years a 7. If lim a 0, the a 8. If coverges. L, the a L a0. 0 9. If r <, the ar a. r 000 diverges. 0. The series. 0.75 0.79999..... Every decimal with a repeatig patter of digits is a ratioal umber. If the iterest is compouded cotiuously, the amout A i the accout after t years is A P Pe r True or False? I Exercises 7, determie whether the statemet is true or false. If it is false, explai why or give a example that shows it is false.. Show that the series a ca be writte i the telescopig form c S c S where S0 0 ad S is the th partial sum.. Let a be a coverget series, ad let RN an an... be the remaider of the series after the first N terms. Prove that lim RN 0. N 5. Fid two diverget series a ad b such that a b coverges. 6. Give two ifiite series a ad b such that a coverges ad b diverges, prove that a b diverges. 7. Suppose that a diverges ad c is a ozero costat. Prove that ca diverges.
60_090.qxd //0 : PM Page 66 66 CHAPTER 9 Ifiite Series 8. If coverges where is ozero, show that a a diverges. 9. The Fiboacci sequece is defied recursively by a a a, where a ad a. a Sectio Project: Cator s Disappearig Table (a) Show that a a a a a a. (b) Show that. 0 a a 0. Fid the values of x for which the ifiite series x x x x x 5 x 6... coverges. What is the sum whe the series coverges?. Prove that for r r r >. r... r,. Writig The figure below represets a iformal way of showig that Explai how the figure implies this <. coclusio. The followig procedure shows how to make a table disappear by removig oly half of the table! (a) Origial table has a legth of L. L (b) Remove of the table cetered at the midpoit. Each remaiig piece has a legth that is less tha L. 7 6 5 (c) Remove 8 of the table by takig sectios of legth 6 L from the ceters of each of the two remaiig pieces. Now, you have removed 8 of the table. Each remaiig piece has a legth that is less tha L. FOR FURTHER INFORMATION For more o this exercise, see the article Covergece with Pictures by P.J. Rippo i America Mathematical Mothly.. Writig Read the article The Expoetial-Decay Law Applied to Medical Dosages by Gerald M. Armstrog ad Calvi P. Midgley i Mathematics Teacher. (To view this article, go to the website www.matharticles.com.) The write a paragraph o how a geometric sequece ca be used to fid the total amout of a drug that remais i a patiet s system after equal doses have bee admiistered (at equal time itervals). (d) Remove 6 of the table by takig sectios of legth 6 L from the ceters of each of the four remaiig pieces. Now, you have removed 8 6 of the table. Each remaiig piece has a legth that is less tha 8 L. Putam Exam Challege. Write as a ratioal umber. k k k k k 5. Let f be the sum of the first terms of the sequece 0,,,,,,,,..., where the th term is give by a,, 6 k if is eve if is odd. Show that if x ad y are positive itegers ad x > y the xy fx y fx y. These problems were composed by the Committee o the Putam Prize Competitio. The Mathematical Associatio of America. All rights reserved. Will cotiuig this process cause the table to disappear, eve though you have oly removed half of the table? Why? FOR FURTHER INFORMATION Read the article Cator s Disappearig Table by Larry E. Kop i The College Mathematics Joural. To view this article, go to the website www.matharticles.com.