Infinite Sequences and Series

Transcription

1 -3 OJECTIVES Fid the it of the terms of a ifiite sequece. Fid the sum of a ifiite geometric series. Ifiite Sequeces ad Series ECONOMICS O Jauary 8, 999, Florida goveror Jeb ush proposed a tax cut that would allow the average family to keep a additioal $96. The margial propesity to cosume (MPC) is defied as the percetage of a dollar by which cosumptio icreases whe icome rises by a dollar. Suppose the MPC for households ad busiesses i 999 was 75%. What would be the total amout of moey spet i the ecoomy as a result of just oe family s tax savigs? This problem will be solved i Example 5. Real World A p plic atio Goveror Jeb ush Trasactio Expediture Terms of Sequece 96(0.75) 7 96(0.75) (0.75) (0.75) (0.75) (0.75) (0.75) (0.75) (0.75) ar Study the table at the left. Trasactio represets the iitial expediture of $96(0.75) or $7 by a family. The busiesses receivig this moey, Trasactio, would i tur sped 75%, ad so o. We ca write a geometric sequece to model this situatio with a 7 ad r Thus, the geometric sequece represetig this situatio is 7, 54, 40.50, 30.76,.78,. I theory, the sequece above ca have ifiitely may terms. Thus, it is called a ifiite sequece. As icreases, the terms of the sequece decrease ad get closer ad closer to zero. The terms of the modelig sequece will ever actually become zero; however, the terms approach zero as icreases without boud. Cosider the ifiite sequece,, 3, 4,,, 5 whose th term, a, is. Several terms of this sequece are graphed at the right. Notice that the terms approach a value of 0 as icreases. Zero is called the it of the terms i this sequece. a 6 O Chapter Sequeces ad Series

2 This it ca be expressed as follows. 0 is the symbol for ifiity. This is read the it of over, as approaches ifiity, equals zero. I fact, whe ay positive power of appears oly i the deomiator of a fractio ad approaches ifiity, the it equals zero. r 0, for r 0 If a geeral expressio for the th term of a sequece is kow, the it ca usually be foud by substitutig large values for. Cosider the followig ifiite geometric sequece. 7, 7 4, 7 7 7,, , This sequece ca be defied by the geeral expressio a 7 4. a a a Notice that as the value of icreases, the value for a appears to approach 0, suggestig Example Estimate the it of 9 5, 6 65, 4 7,, 7,. 3 7( 50) The 50th term is ( 50), or about ( 50) 7(00) The 00th term is, or about (00) 3(00) (500) The 500th term is 7, or about (500) 3(500) 7(000) The 000th term is, or (000) 3(000) Notice that as, the values appear to approach 3.5, suggestig Lesso -3 Ifiite Sequeces ad Series 775

3 For sequeces with more complicated geeral forms, applicatios of the followig it theorems, which we will preset without proof, ca make the it easier to fid. If the a exists, b exists, ad c is a costat, the the followig theorems are true. Limit of a Sum (a b ) a b Theorems for Limits Limit of a Differece Limit of a Product Limit of a Quotiet Limit of a Costat (a b ) a b a b a b a a, where b 0 b b c c, where c c for each The form of the expressio for the th term of a sequece ca ofte be altered to make the it easier to fid. Note that the Limit of a Sum theorem oly applies here because ad Example 3 each exist. Fid each it. a. ( 3 ) ( 3 ) or 3 Thus, the it is 3. b Rewrite as the sum of two fractios ad simplify. Limit of a Sum 0 ad 3 3 The highest power of i the expressio is. Divide each term i the umerator ad the deomiator by. Why does doig this produce a equivalet expressio? Simplify. 776 Chapter Sequeces ad Series

4 Thus, the it is or 5 0 Apply it theorems. 5 5, 0, 4 4,, ad 0a Limits do ot exist for all ifiite sequeces. If the absolute value of the terms of a sequece becomes arbitrarily great or if the terms do ot approach a value, the sequece has o it. Example 3 illustrates both of these cases. Example 3 Fid each it. a Simplify. Note that 0 ad 5 5, but becomes icreasigly large as approaches ifiity. Therefore, the sequece has o it. b. ( ) 8 egi by rewritig ( ) as () 8. Now fid Divide the umerator ad deomiator by Simplify. Apply it theorems., 8 8, ad 0 Whe is eve, (). Whe is odd, (). Thus, the oddumbered terms of the sequece described by ( ) approach, ad the 8 8 eve-umbered terms approach. Therefore, the sequece has o it. 8 Lesso -3 Ifiite Sequeces ad Series 777

5 A ifiite series is the idicated sum of the terms of a ifiite sequece. Cosider the series Sice this is a geometric series, you ca fid the sum of the first 00 terms by usig the formula S a a r, r where r 5. S or Sice 5 00 is very close to 0, S 00 is early equal to. No matter how may 4 terms are added, the sum of the ifiite series will ever exceed, ad the 4 differece from 4 gets smaller as. Thus, is the sum of the ifiite series. 4 Sum of a Ifiite Series If S is the sum of the first terms of a series, ad S is a umber such that S S approaches zero as icreases without boud, the the sum of the ifiite series is S. S S If the sequece of partial sums S has a it, the the correspodig ifiite series has a sum, ad the th term a of the series approaches 0 as. If a 0, the series has o sum. If a 0, the series may or may ot have a sum. Recall that r meas r. The formula for the sum of the first terms of a geometric series ca be writte as follows. S a ( r ), r r Suppose ; that is, the umber of terms icreases without it. If r, r icreases without it as. However, whe r, r approaches 0 as a. Uder this coditio, the above formula for S approaches a value of. r Sum of a Ifiite Geometric Series The sum S of a ifiite geometric series for which r is give by a S. r 778 Chapter Sequeces ad Series

6 Example 4 Fid the sum of the series I the series, a ad r 7. Sice r, S a. r a S r 7 a, r or The sum of the series is I ecoomics, fidig the sum of a ifiite series is useful i determiig the overall effect of ecoomic treds. Example 5 Real World A p plic atio ECONOMICS Refer to the applicatio at the begiig of the lesso. What would be the total amout of moey spet i the ecoomy as a result of just oe family s tax savigs? For the geometric series modelig this situatio, a 7 ad r Sice r, the sum of the series is a equal to. r a S r 7 or Therefore, the total amout of moey spet is $88. You ca use what you kow about ifiite series to write repeatig decimals as fractios. The first step is to write the repeatig decimal as a ifiite geometric series. Example 6 Write 0.76 as a fractio ,0 00,000,000,000, I this series, a ad r (cotiued o the ext page) Lesso -3 Ifiite Sequeces ad Series 779

7 a S r or Thus, Check this with a calculator. C HECK FOR U NDERSTANDING Commuicatig Mathematics Read ad study the lesso to aswer each questio.. Cosider the sequece give by the geeral expressio a. a. Graph the first te terms of the sequece with the term umber o the x-axis ad the value of the term o the y-axis. b. Describe what happes to the value of a as icreases. c. Make a cojecture based o your observatio i part a as to the it of the sequece as approaches ifiity. d. Apply the techiques preseted i the lesso to evaluate. How does your aswer compare to your cojecture made i part c?. Cosider the ifiite geometric sequece give by the geeral expressio r. a. Determie the it of the sequece for r, r, r, r, ad 4 r 5. b. Write a geeral rule for the it of the sequece, placig restrictios o the value of r. 3. Give a example of a ifiite geometric series havig o sum. 4. You Decide Tyree ad Zota disagree o whether the ifiite sequece described by the geeral expressio 3 has a it. Tyree says that after 3 dividig by the highest-powered term, the expressio simplifies to, which has a it of as approaches ifiity. Zota says that the sequece has o it. Who is correct? Explai. Guided Practice Fid each it, or state that the it does ot exist ad explai your reasoig Write each repeatig decimal as a fractio Chapter Sequeces ad Series

8 Fid the sum of each ifiite series, or state that the sum does ot exist ad explai your reasoig Etertaimet Pete s Pirate Ride operates like the bob of a pedulum. O its logest swig, the ship travels through a arc 75 meters log. Each successive swig is two-fifths the legth of the precedig swig. If the ride is allowed to cotiue without itervetio, what is the total distace the ship will travel before comig to rest? 75 m E XERCISES Practice Fid each it, or state that the it does ot exist ad explai your reasoig. A (3 4)( ) ( ) 4 3. Fid the it of the sequece described by the geeral expressio 5 ( ), or state that the it does ot exist. Explai your reasoig. Write each repeatig decimal as a fractio Explai why the sum of the series exists. The fid the sum. Fid the sum of each series, or state that the sum does ot exist ad explai your reasoig C Lesso -3 Ifiite Sequeces ad Series 78

9 Applicatios ad Problem Solvig Real World A p plic atio 40. Physics A basketball is dropped from a height of 35 meters ad bouces 5 of the distace after each fall. a. Fid the first five terms of the ifiite series represetig the vertical distace traveled by the ball. b. What is the total vertical distace the ball travels before comig to rest? (Hit: Rewrite the series foud i part a as the sum of two ifiite geometric series.) 4. Critical Thikig Cosider the sequece whose th term is described by. a. Explai why. b. Fid. 4. Egieerig Fracisco desigs a toy with a rotary flywheel that rotates at a maximum speed of 70 revolutios per miute. Suppose the flywheel is operatig at its maximum speed for oe miute ad the the power supply to the toy is tured off. Each subsequet miute thereafter, the flywheel rotates two-fifths as may times as i the precedig miute. How may complete revolutios will the flywheel make before comig to a stop? 43. Critical Thikig Does cos exist? Explai. 44. Medicie A certai drug developed to fight cacer has a half-life of about hours i the bloodstream. The drug is formulated to be admiistered i doses of D milligrams every 6 hours. The amout of each dose has yet to be determied. a. What fractio of the first dose will be left i the bloodstream before the secod dose is admiistered? b. Write a geeral expressio for the geometric series that models the umber of milligrams of drug left i the bloodstream after the th dose. c. About what amout of medicie is preset i the bloodstream for large values of? d. A level of more tha 350 milligrams of this drug i the bloodstream is cosidered toxic. Fid the largest possible dose that ca be give repeatedly over a log period of time without harmig the patiet. 78 Chapter Sequeces ad Series 45. Geometry If the midpoits of a square are joied by straight lies, the ew figure will also be a square. a. If the origial square has a perimeter of 0 feet, fid the perimeter of the ew square. (Hit: Use the Pythagorea Theorem.) b. If this process is cotiued to form a sequece of ested squares, what will be the sum of the perimeters of all the squares?

10 46. Techology Sice the mid-980s, the umber of computers i schools has steadily icreased. The graph below shows the correspodig declie i the studet-computer ratio. 60 Studets Per Computer i U.S. Public Schools Studets Per Computer Aother publicatio states that the average umber of studets per computer i U.S. public schools ca be estimated by the sequece model a ( ), for,, 3,, with the school year correspodig to. a. Fid the first te terms of the model. Roud your aswers to the earest teth. b. Use the model to estimate the average umber of studets havig to share a computer durig the school year. How does this estimate compare to the actual data give i the graph? c. Make a predictio as to the average umber of studets per computer for the school year. d. Does this sequece approach a it? If so, what is the it? e. Realistically, will the studet computer ratio ever reach this it? Explai. Year Source: QED's Techology i Public Schools, 6th Editio Mixed Review 47. The first term of a geometric sequece is 3, ad the commo ratio is. Fid 3 the ext four terms of the sequece. (Lesso -) 48. Fid the 6th term of the arithmetic sequece for which a.5 ad d 0.5. (Lesso -) 49. Name the coordiates of the ceter, foci, ad vertices, ad the equatio of the asymptotes of the hyperbola that has the equatio x 4y x 6y 6. (Lesso 0-4) 50. Graph r 6 cos 3. (Lesso 9-) 5. Navigatio A ship leavig port sails for 5 miles i a directio 0 orth of due east. Fid the magitude of the vertical ad horizotal compoets. (Lesso 8-) 5. Use a half-agle idetity to fid the exact value of cos.5. (Lesso 7-4) 53. Graph y cos x o the iterval 80 x 360. (Lesso 6-3) 54. List all possible ratioal zeros of the fuctio f(x) 8x 3 3x. (Lesso 4-4) 55. SAT/ACT Practice If a 4b 6, ad b is a positive iteger, the a could be divisible by all of the followig EXCEPT A 4 C 5 D 6 E 7 Extra Practice See p. A49. Lesso -3 Ifiite Sequeces ad Series 783

11 GRAPHING CALCULATOR EXPLORATION -3 Cotiued Fractios A Extesio of Lesso -3 OJECTIVE Explore sequeces geerated by cotiued fractios. A expressio of the followig form is called a cotiued fractio. a a b a 3 b b 3 a4 y usig oly a fiite umber of decks ad values of a ad b that follow regular patters, you ca ofte obtai a sequece of terms that approaches a it, which ca be represeted by a simple expressio. For example, if all of the umbers a ad b are equal to, the the cotiued fractio gives rise to the followig sequece.,,,, The golde ratio is closely related to the Fiboaci sequece, which you will lear about i Lesso -7. It ca be show that the terms of this sequece approach the it umber is ofte called the golde ratio. Now cosider the followig more geeral sequece. A, A, A, A, A A A A A A 5. This To help you visualize what this sequece represets, suppose A 5. The sequece becomes 5, 5 6, 5, 5, 5 5 or 5,, , , A calculator approximatio of this sequece is 5, 5., , 5,959593,. Graphig Calculator Programs To dowload this graphig calculator program, visit our website at glecoe.com Each term of the sequece is the sum of A ad the reciprocal of the previous term. The program at the right calculates the value of the th term of the above sequece for 3 ad a specified value of A. Whe you ru the program it will ask you to iput values for A ad N. PROGRAM: CFRAC : Prompt A : Disp INPUT TERM : Disp NUMER N, N 3 : Prompt N : K :A /A C : Lbl :A /C C :K K : If K N : The: Goto : Else: Disp C 784 Chapter Sequeces ad Series

12 TRY THESE Eter the program ito your calculator ad use it for the exercises that follow.. What output is give whe the program is executed for A ad N 0?. With A, determie the least value of N ecessary to obtai a output 5 that agrees with the calculator s ie decimal approximatio of. 3. Use algebra to show that the cotiued fractio has a 5 value of. (Hit: If x, the x. Solve x this last equatio for x.) 4. Fid the exact value of Execute the program with A 3 ad N 40. How does this output compare to the decimal approximatio of the expressio foud i Exercise 4? 6. Fid a radical expressio for A. A A 7. Write a modified versio of the program that calculates the th term of the followig sequece for 3. A, A A, A, A, A A A A A 8. Choose several positive iteger values for A ad ad compare the program output with the decimal approximatio of A for several values of, for 3. Describe your observatios. 9. Use algebra to show that for A 0 ad 0, A has a A A value of A. Hit: If x A, the x A A. A A A A WHAT DO YOU THINK? 0. If you execute the origial program for A ad N 0 ad the execute it for A ad N 0, how will the two outputs compare?. What values ca you use for A ad i the program for Exercise 7 i order to approximate 5? Lesso -3 Cotiued Fractios 785