- OBJECTIVE Determie whether a series is coverget or diverget. Coverget ad Diverget Series HISTORY The Greek philosopher Zeo of Elea (c. 90 30 B.C.) proposed several perplexig riddles, or paradoxes. Oe of Zeo s paradoxes ivolves a race o a 00-meter track betwee the mythological Achilles ad a tortoise. Zeo claims that eve though Achilles ca ru twice as fast as the tortoise, if the tortoise is give a 0-meter head start, Achilles will ever catch him. Suppose Achilles rus 0 meters per secod ad the tortoise a remarkable 5 meters per secod. By the time Achilles has reached the 0-meter mark, the tortoise will be at 5 meters. By the time Achilles reaches the 5-meter mark, the tortoise will be at 7.5 meters, ad so o. Thus, Achilles is always behid the tortoise ad ever catches up. Real World A p plic atio Is Zeo correct? Let us look at the distace betwee Achilles ad the tortoise after specified amouts of time have passed. Notice that the distace betwee the two cotestats will be zero as approaches ifiity sice lim 0 0. To disprove Zeo s coclusio that Achilles will ever catch up to the tortoise, we must show that there is a time value for which this 0 differece ca be achieved. I other words, we eed to show that the ifiite series 8 has a sum, or limit. This problem will be solved i Example 5. Time (secods) Distace Apart (meters) 0 0 0 5 3 0.5 7 0.5 8 8 5 0 0.65 8 6.. 8 0 Startig with a time of secod, the partial sums of the time series form the sequece, 3, 7, 5,. As the umber of terms used for the partial sums 8 icreases, the value of the partial sums also icreases. If this sequece of partial sums approaches a limit, the related ifiite series is said to coverge. If this sequece of partial sums does ot have a limit, the the related ifiite series is said to diverge. 786 Chapter Sequeces ad Series
Coverget ad Diverget Series If a ifiite series has a sum, or limit, the series is coverget. If a series is ot coverget, it is diverget. Example There are may series that begi with the first few terms show i this example. I this chapter, always assume that the expressio for the geeral term is the simplest oe possible. Determie whether each arithmetic or geometric series is coverget or diverget. a. 8 6 This is a geometric series with r. Sice r, the series has a limit. Therefore, the series is coverget. b. 8 6 This is a geometric series with r. Sice r, the series has o limit. Therefore, the series is diverget. c. 0 8.5 7 5.5 This is a arithmetic series with d.5. Arithmetic series do ot have limits. Therefore, the series is diverget. Whe a series is either arithmetic or geometric, it is more difficult to determie whether the series is coverget or diverget. Several differet techiques ca be used. Oe test for covergece is the ratio test. This test ca oly be used whe all terms of a series are positive. The test depeds upo the ratio of cosecutive terms of a series, which must be expressed i geeral form. Let a ad a represet two cosecutive terms of a series of positive Ratio Test terms. Suppose lim a exists ad that r lim a a. The series is a coverget if r ad diverget if r. If r, the test provides o iformatio. The ratio test is especially useful whe the geeral form for the terms of a series cotais powers. Example Use the ratio test to determie whether each series is coverget or diverget. a. 3 8 6 First, fid a ad a. a ad a The use the ratio test. r lim (cotiued o the ext page) Lesso - Coverget ad Diverget Series 787
r lim Multiply by the reciprocal of the divisor. r lim r lim r lim lim Limit of a Product Divide by the highest power of ad the apply limit theorems. 0 r or Sice r, the series is coverget. b. 3 3 5 a r lim ad a or ( ) r lim r lim Divide by the highest power of ad apply limit theorems. 0 0 r or Sice r, the test provides o iformatio. 0 The ratio test is also useful whe the geeral form of the terms of a series cotais products of cosecutive itegers. Example 3 Use the ratio test to determie whether the series is coverget or diverget. 3 3 First fid the th term ad ( )th term. The, use the ratio test. a ad a ( ) r lim r lim ( ) ( ) Note that ( ) (). r lim or 0 Simplify ad apply limit theorems. Sice r, the series is coverget. 788 Chapter Sequeces ad Series
Whe the ratio test does ot determie if a series is coverget or diverget, other methods must be used. Example Determie whether the series is coverget or 3 5 diverget. Suppose the terms are grouped as follows. Begiig after the secod term, the umber of terms i each successive group is doubled. () 3 5 6 7 8 9 6 Notice that the first eclosed expressio is greater tha, ad the secod is equal to. Begiig with the third expressio, each sum of eclosed terms is greater tha. Sice there are a ulimited umber of such expressios, the sum of the series is ulimited. Thus, the series is diverget. A series ca be compared to other series that are kow to be coverget or diverget. The followig list of series ca be used for referece. Summary of Series for Referece. Coverget: a a r a r a r, r. Diverget: a a r a r a r, r 3. Diverget: a (a d ) (a d ) (a 3d ). Diverget: This series is kow as 3 5 the harmoic series. 5. Coverget:, p p 3 p p If a series has all positive terms, the compariso test ca be used to determie whether the series is coverget or diverget. Compariso Test A series of positive terms is coverget if, for, each term of the series is equal to or less tha the value of the correspodig term of some coverget series of positive terms. A series of positive terms is diverget if, for, each term of the series is equal to or greater tha the value of the correspodig term of some diverget series of positive terms. Example 5 Use the compariso test to determie whether the followig series are coverget or diverget. a. 5 7 9 The geeral term of this series is The geeral term of the diverget 3. series is. Sice for all,, the 3 5 3 series is also diverget. 5 7 9 Lesso - Coverget ad Diverget Series 789
b. 3 5 7 The geeral term of the series is. The geeral term of the ( ) coverget series is. Sice for 3 ( ) all, the series is also coverget. 3 5 7 With a better uderstadig of coverget ad diverget ifiite series, we are ow ready to tackle Zeo s paradox. Example 6 Real World A p plic atio HISTORY Refer to the applicatio at the begiig of the lesso. To disprove Zeo s coclusio that Achilles will ever catch up to the tortoise, we must show that the ifiite time series 0.5 0.5 has a limit. To show that the series 0.5 0.5 has a limit, we eed to show that the series is coverget. The geeral term of this series is. Try usig the ratio test for covergece of a series. a ad a r Sice r, the series coverges ad therefore has a sum. Thus, there is a time value for which the distace betwee Achilles ad the tortoise will be zero. You will determie how log it takes him to do so i Exercise 3. C HECK FOR U NDERSTANDING Commuicatig Mathematics Read ad study the lesso to aswer each questio.. a. Write a example, of a ifiite geometric series i which r. b. Determie the 5th, 50th, ad 00th terms of your series. c. Idetify the sum of the first 5, 50, ad 00 terms of your series. d. Explai why this type of ifiite geometric series does ot coverge.. Estimate the sum S of the series whose partial sums are graphed at the right. S 0 8 6 O 3 5 6 7 790 Chapter Sequeces ad Series
3. Cosider the ifiite series 3 3 3 33 3. a. Sketch a graph of the first eight partial sums of this series. b. Make a cojecture based o the graph foud i part a as to whether the series is coverget or diverget. c. Determie a geeral term for this series. d. Write a covicig argumet usig the geeral term foud i part c to support the cojecture you made i part b.. Math Joural Make a list of the methods preseted i this lesso ad i the previous lesso for determiig covergece or divergece of a ifiite series. Be sure to idicate ay restrictios o a method s use. The umber your list as to the order i which these methods should be applied. Guided Practice Use the ratio test to determie whether each series is coverget or diverget. 3 5. 3 7 5 6. 3 8 6 3 7. Use the compariso test to determie whether the series is 3 coverget or diverget. Determie whether each series is coverget or diverget. 5 3 7 8. 9. 6 8 6 3 0. 9. 3 3 3. Ecology A udergroud storage cotaier is leakig a toxic chemical. Oe year after the leak bega, the chemical had spread 500 meters from its source. After two years, the chemical had spread 900 meters more, ad by the ed of the third year, it had reached a additioal 50 meters. a. If this patter cotiues, how far will the spill have spread from its source after 0 years? b. Will the spill ever reach the grouds of a school located 000 meters away from the source? Explai. Practice A B E XERCISES Use the ratio test to determie whether each series is coverget or diverget. 3. 3 9. 7 8 5 8 0 5 8 6 5. 6. 3 3 3 5 3 5 7. 5 53 8. 5 3 3 5 3 9. Use the ratio test to determie whether the series 6 6 8 8 8 0 is coverget or diverget. 6 www.amc.glecoe.com/self_check_quiz Lesso - Coverget ad Diverget Series 79
Use the compariso test to determie whether each series is coverget or diverget. 0.. 6 9 8 65 3. 5 5 5 3. 3 3 6. Use the compariso test to determie whether the series 3 5 9 7 is coverget or diverget. Determie whether each series is coverget or diverget. 3 9 5. 5 7 6. 3 8 3 3 5 C 7. 8. 5 5 5 3 3 5 9. 3 5 7 30. 3 6 3 8 6 3 Applicatios ad Problem Solvig Real World A p plic atio 3. Ecoomics The MagicSoft software compay has a proposal to the city coucil of Alva, Florida, to relocate there. The proposal claims that the compay will geerate $3.3 millio for the local ecoomy by the $ millio i salaries that will be paid. The city coucil estimates that 70% of the salaries will be spet i the local commuity, ad 70% of that moey will agai be spet i the commuity, ad so o. a. Accordig to the city coucil s estimates, is the claim made by MagicSoft accurate? Explai. b. What is the correct estimate of the amout geerated to the local ecoomy? 79 Chapter Sequeces ad Series 3. Critical Thikig Give a example of a series a a a 3 a that diverges, but whe its terms are squared, the resultig series a a a 3 a coverges. 33. Cellular Growth Leticia Cox is a biochemist. She is testig two differet types of drugs that iduce cell growth. She has selected two cultures of 000 cells each. To culture A, she admiisters a drug that raises the umber of cells by 00 each day ad every day thereafter. Culture B gets a drug that icreases cell growth by 8% each day ad everyday thereafter. a. Assumig o cells die, how may cells will have grow i each culture by the ed of the seveth day? b. At the ed of oe moth s time, which drug will prove to be more effective i promotig cell growth? Explai. 3. Critical Thikig Refer to Example 6 of this lesso. The sequece of partial sums, S, S, S 3,, S,, for the time series is, 3, 7, 5,. 8 a. Fid a geeral expressio for the th term of this sequece. b. To determie how log it takes for Achilles to catch-up to the tortoise, fid the sum of the ifiite time series. (Hit: Recall from the defiitio of the sum S of a ifiite series that lim S S.)
35. Clocks The hour ad miute hads of a clock travel aroud its face at differet speeds, but at certai times of the day, the two hads coicide. I additio to oo ad midight, the hads also coicide at times occurrig betwee the hours. Accordig the figure at the right, it is :00. a. Whe the miute had poits to, what fractio of the distace betwee ad 5 will the hour had have traveled? b. Whe the miute had reaches the hour had s ew positio, what additioal fractio will the hour had have traveled? c. List the ext two terms of this series represetig the distace traveled by the hour had as the miute had chases its positio. d. At what time betwee the hours of ad 5 o clock will the two hads coicide? 0 9 8 7 6 5 3 Mixed Review 36. Evaluate lim 5. (Lesso -3) 3 37. Fid the ith term of the geometric sequece,,,. (Lesso -) 38. Form a arithmetic sequece that has five arithmetic meas betwee ad 9. (Lesso -) 39. Solve 5.9 e 0.075t (Lesso -6) 0. Navigatio A submarie soar is trackig a ship. The path of the ship is recorded as 6 r cos ( 30 ). Fid the liear equatio of the path of the ship. (Lesso 9-). Fid a ordered pair that represets AB for A(8, 3) ad B(5, ). (Lesso 8-). SAT/ACT Practice How may umbers from to 00 iclusive are equal to the cube of a iteger? A oe B two C three D four E five MID-CHAPTER QUIZ. Fid the 9th term i the sequece for which a ad d. (Lesso -). Fid S 0 for the arithmetic series for which a ad d 6. (Lesso -) 3. Form a sequece that has two geometric meas betwee 56 ad 89. (Lesso -). Fid the sum of the first eight terms of the series 3 6. (Lesso -) 5. Fid lim 5 or explai why the limit does ot exist. (Lesso -3) 6. Recreatio A bugee jumper rebouds 55% of the height jumped. If a bugee jump is made usig a cord that stretches 50 feet, fid the total distace traveled by the jumper before comig to rest. (Lesso -3) 7. Fid the sum of the followig series.. (Lesso -3) 5 50 500 Determie whether each series is coverget or diverget. (Lesso -) 6 8. 0 00 000 0,000 6 9. 5 5 5 0. Fiace Ms. Fuetes ivests $500 quarterly (Jauary, April, July, ad October ) i a retiremet accout that pays a APR of % compouded quarterly. Iterest for each quarter is posted o the last day of the quarter. Determie the value of her ivestmet at the ed of the year. (Lesso -) Extra Practice See p. A9. Lesso - Coverget ad Diverget Series 793