704 CHAPTER 11 Infinite Sequences and Series

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1 704 CHAPTER Ifiite Sequeces ad Series. (a) What is a sequece? (b) What does it mea to say that lim l ` a 8? (c) What does it mea to say that lim l ` a `?. (a) What is a coverget sequece? Give two examples. (b) What is a diverget sequece? Give two examples. 3 List the first five terms of the sequece. 3. a 4. a 5. a sd 5 6. a cos 7. a s d! 9. a, a 5a 3 0. a 6, a a. a, a a a. a, a, a a a 8. a sd! 3 8 Fid a formula for the geeral term a of the sequece, assumig that the patter of the first few terms cotiues. 3. 5, 4, 6, 8, 0, ,, 4, 6, 64, ,, 4 3, 8 9, 6 7, h5, 8,, 4, 7,...j 7. 5, 4 3, 9 4, 6 5, 5 6, h, 0,, 0,, 0,, 0,...j 9 Calculate, to four decimal places, the first te terms of the sequece ad use them to plot the graph of the sequece by had. Does the sequece appear to have a limit? If so, calculate it. If ot, explai why. 9. a a sd. a ( ). a Determie whether the sequece coverges or diverges. If it coverges, fid the limit. 3. a a a 6. a s0.86d 7. a a 3s s 9. a e ys 30. a a Î 4 S 3. a cos D 33. a s a sd s s 37. H s d!j 34. a e ysd 36. a sd s 38. H l l J 39. {si } 40. a ta 4. h e j 4. a ls d l 43. a cos 44. a s a sisyd 46. a cos 47. a S D 49. a ls d ls d 50. a sl d 5. a arctasl d 5. a s s h0,, 0, 0,, 0, 0, 0,,... j 54. 5, 3,, 4, 3, 5, 4, 6, a s Copyright 06 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s). Editorial review has deemed that ay suppressed cotet does ot materially affect the overall learig experiece. Cegage Learig reserves the right to remove additioal cotet at ay time if subsequet rights restrictios require it.

2 SECTION. Sequeces 705 ; 55. a! s3d 56. a! Use a graph of the sequece to decide whether the sequece is coverget or diverget. If the sequece is coverget, guess the value of the limit from the graph ad the prove your guess. (See the margi ote o page 699 for advice o graphig sequeces.) 57. a sd 58. a si 59. a arctas 4D 60. a s a cos 6. a 63. a? 3? 5?? s d!? 3? 5?? s d sd 64. (a) Determie whether the sequece defied as follows is coverget or diverget: a a 4 a for > (b) What happes if the first term is a? 65. If $000 is ivested at 6% iterest, compouded aually, the after years the ivestmet is worth a 000s.06d dollars. (a) Fid the first five terms of the sequece ha j. (b) Is the sequece coverget or diverget? Explai. 66. If you deposit $00 at the ed of every moth ito a accout that pays 3% iterest per year compouded mothly, the amout of iterest accumulated after moths is give by the sequece I 00S D (a) Fid the first six terms of the sequece. (b) How much iterest will you have eared after two years? 67. A fish farmer has 5000 catfish i his pod. The umber of catfish icreases by 8% per moth ad the farmer harvests 300 catfish per moth. (a) Show that the catfish populatio P after moths is give recursively by P.08P 300 P (b) How may catfish are i the pod after six moths? 68. Fid the first 40 terms of the sequece defied by a H a 3a if a is a eve umber if a is a odd umber ad a. Do the same if a 5. Make a cojecture about this type of sequece. 69. For what values of r is the sequece hr j coverget? 70. (a) If ha j is coverget, show that lim a lim l ` l ` a (b) A sequece ha j is defied by a ad a ys a d for >. Assumig that ha j is coverget, fid its limit. 7. Suppose you kow that ha j is a decreasig sequece ad all its terms lie betwee the umbers 5 ad 8. Explai why the sequece has a limit. What ca you say about the value of the limit? 7 78 Determie whether the sequece is icreasig, decreasig, or ot mootoic. Is the sequece bouded? 7. a cos 73. a a 75. a sd 76. a sd 77. a 3 e 78. a Fid the limit of the sequece 5s, ss, sss, A sequece ha j is give by a s, a s a. (a) By iductio or otherwise, show that ha j is icreasig ad bouded above by 3. Apply the Mootoic Sequece Theorem to show that lim l ` a exists. (b) Fid lim l ` a. 8. Show that the sequece defied by a a 3 a is icreasig ad a, 3 for all. Deduce that ha j is coverget ad fid its limit. 8. Show that the sequece defied by a a 3 a satisfies 0, a < ad is decreasig. Deduce that the sequece is coverget ad fid its limit. Copyright 06 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s). Editorial review has deemed that ay suppressed cotet does ot materially affect the overall learig experiece. Cegage Learig reserves the right to remove additioal cotet at ay time if subsequet rights restrictios require it.

3 706 CHAPTER Ifiite Sequeces ad Series 83. (a) Fiboacci posed the followig problem: Suppose that rabbits live forever ad that every moth each pair produces a ew pair which becomes productive at age moths. If we start with oe ewbor pair, how may pairs of rabbits will we have i the th moth? Show that the aswer is f, where h f j is the Fiboacci sequece defied i Example 3(c). (b) Let a f yf ad show that a ya. Assumig that ha j is coverget, fid its limit. 84. (a) Let a a, a f sad, a 3 f sa d f s f sadd,..., a f sa d, where f is a cotiuous fuctio. If lim l ` a L, show that f sld L. (b) Illustrate part (a) by takig f sxd cos x, a, ad estimatig the value of L to five decimal places. ; 85. (a) Use a graph to guess the value of the limit 5 lim l `! (b) Use a graph of the sequece i part (a) to fid the smallest values of N that correspod to «0. ad «0.00 i Defiitio. 86. Use Defiitio directly to prove that lim l ` r 0 whe r,. 87. Prove Theorem 6. [Hit: Use either Defiitio or the Squeeze Theorem.] 88. Prove Theorem Prove that if lim l ` a 0 ad hb j is bouded, the lim l ` sa b d Let a S D. (a) Show that if 0 < a, b, the b a, s db b a (b) Deduce that b fs da bg, a. (c) Use a ys d ad b y i part (b) to show that ha j is icreasig. (d) Use a ad b ysd i part (b) to show that a, 4. (e) Use parts (c) ad (d) to show that a, 4 for all. (f) Use Theorem to show that lim l ` s yd exists. (The limit is e. See Equatio ) 9. Let a ad b be positive umbers with a. b. Let a be their arithmetic mea ad b their geometric mea: a a b Repeat this process so that, i geeral, a a b b sab b sa b (a) Use mathematical iductio to show that a. a. b. b (b) Deduce that both ha j ad hb j are coverget. (c) Show that lim l ` a lim l ` b. Gauss called the commo value of these limits the arithmetic-geometric mea of the umbers a ad b. 9. (a) Show that if lim l ` a L ad lim l ` a L, the ha j is coverget ad lim l ` a L. (b) If a ad a a fid the first eight terms of the sequece ha j. The use part (a) to show that lim l ` a s. This gives the cotiued fractio expasio s 93. The size of a udisturbed fish populatio has bee modeled by the formula p bp a p where p is the fish populatio after years ad a ad b are positive costats that deped o the species ad its eviromet. Suppose that the populatio i year 0 is p (a) Show that if h p j is coverget, the the oly possible values for its limit are 0 ad b a. (b) Show that p, sbyadp. (c) Use part (b) to show that if a. b, the lim l ` p 0; i other words, the populatio dies out. (d) Now assume that a, b. Show that if p 0, b a, the h p j is icreasig ad 0, p, b a. Show also that if p 0. b a, the h p j is decreasig ad p. b a. Deduce that if a, b, the lim l ` p b a. Copyright 06 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s). Editorial review has deemed that ay suppressed cotet does ot materially affect the overall learig experiece. Cegage Learig reserves the right to remove additioal cotet at ay time if subsequet rights restrictios require it.

4 SECTION. Series 75 3 EXAMPLE Fid the sum of the series S s d D. SOLUTION The series oy is a geometric series with a ad r, so I Example 8 we foud that s d So, by Theorem 8, the give series is coverget ad S D 3 s d 3 s d 3? 4 NOTE 4 A fiite umber of terms does t affect the covergece or divergece of a series. For istace, suppose that we were able to show that the series is coverget. Sice it follows that the etire series o ǹ ys 3 d is coverget. Similarly, if it is kow that the series o ǹ N a coverges, the the full series is also coverget. a o N a a N. (a) What is the differece betwee a sequece ad a series? (b) What is a coverget series? What is a diverget series?. Explai what it meas to say that o ǹ a Calculate the sum of the series o ǹ a whose partial sums are give. 3. s 3s0.8d 4. s Calculate the first eight terms of the sequece of partial sums correct to four decimal places. Does it appear that the series is coverget or diverget? s 3 ; 7. si sd 8.! 9 4 Fid at least 0 partial sums of the series. Graph both the sequece of terms ad the sequece of partial sums o the same scree. Does it appear that the series is coverget or diverget? If it is coverget, fid the sum. If it is diverget, explai why. 9. s5d 0. cos. s Copyright 06 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s). Editorial review has deemed that ay suppressed cotet does ot materially affect the overall learig experiece. Cegage Learig reserves the right to remove additioal cotet at ay time if subsequet rights restrictios require it.

5 76 CHAPTER Ifiite Sequeces ad Series Ssi si D 5. Let a 3. (a) Determie whether ha j is coverget. (b) Determie whether o ǹ a is coverget. 6. (a) Explai the differece betwee o a i ad o a j i j (b) Explai the differece betwee o a i ad o a j i i 7 6 Determie whether the geometric series is coverget or diverget. If it is coverget, fid its sum s0.73d 5. s3d sd e ? Determie whether the series is coverget or diverget. If it is coverget, fid its sum k 30. k k k fs0.d s0.6d g e e 35. ssi 00d k k ( 3) ls 38. D (s ) k k arcta 4. S e s dd 40. S 3 5 D e Determie whether the series is coverget or diverget by expressig s as a telescopig sum (as i Ex am ple 8). If it is coverget, fid its sum s 3d 47. se y e ysd d 44. l 46. 4S s D s 49. Let x (a) Do you thik that x, or x? (b) Sum a geometric series to fid the value of x. (c) How may decimal represetatios does the umber have? (d) Which umbers have more tha oe decimal represetatio? 50. A sequece of terms is defied by a Calculate o ǹ a. a s5 da 5 56 Express the umber as a ratio of itegers Fid the values of x for which the series coverges. Fid the sum of the series for those values of x. 57. s5d x 58. sx d sx d s4d sx 5d x 63. e x 0 6. si x 0 3 Copyright 06 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s). Editorial review has deemed that ay suppressed cotet does ot materially affect the overall learig experiece. Cegage Learig reserves the right to remove additioal cotet at ay time if subsequet rights restrictios require it.

6 SECTION. Series 77 CAS 64. We have see that the harmoic series is a diverget series whose terms approach 0. Show that ls D is aother series with this property Use the partial fractio commad o your CAS to fid a coveiet expressio for the partial sum, ad the use this expressio to fid the sum of the series. Check your aswer by usig the CAS to sum the series directly s d If the th partial sum of a series o ǹ a is s fid a ad o ǹ a. 68. If the th partial sum of a series oǹ a is s 3, fid a ad o ǹ a. 69. A doctor prescribes a 00-mg atibiotic tablet to be take every eight hours. Just before each tablet is take, 0% of the drug remais i the body. (a) How much of the drug is i the body just after the secod tablet is take? After the third tablet? (b) If Q is the quatity of the atibiotic i the body just after the th tablet is take, fid a equatio that expresses Q i terms of Q. (c) What quatity of the atibiotic remais i the body i the log ru? 70. A patiet is ijected with a drug every hours. Immediately before each ijectio the cocetratio of the drug has bee reduced by 90% ad the ew dose icreases the cocetratio by.5 mgyl. (a) What is the cocetratio after three doses? (b) If C is the cocetratio after the th dose, fid a formula for C as a fuctio of. (c) What is the limitig value of the cocetratio? 7. A patiet takes 50 mg of a drug at the same time every day. Just before each tablet is take, 5% of the drug remais i the body. (a) What quatity of the drug is i the body after the third tablet? After the th tablet? (b) What quatity of the drug remais i the body i the log ru? 7. After ijectio of a dose D of isuli, the cocetratio of isuli i a patiet s system decays expoetially ad so it ca be writte as De at, where t represets time i hours ad a is a positive costat. (a) If a dose D is ijected every T hours, write a expressio for the sum of the residual cocetratios just before the s dst ijectio. ; (b) Determie the limitig pre-ijectio cocetratio. (c) If the cocetratio of isuli must always remai at or above a critical value C, determie a miimal dosage D i terms of C, a, ad T. 73. Whe moey is spet o goods ad services, those who receive the moey also sped some of it. The people receivig some of the twice-spet moey will sped some of that, ad so o. Ecoomists call this chai reactio the multiplier effect. I a hypothetical isolated commuity, the local govermet begis the process by spedig D dollars. Suppose that each recipiet of spet moey speds 00c% ad saves 00s% of the moey that he or she receives. The val ues c ad s are called the margial propesity to cosume ad the margial propesity to save ad, of course, c s. (a) Let S be the total spedig that has bee geerated after trasactios. Fid a equatio for S. (b) Show that lim l ` S kd, where k ys. The umber k is called the multiplier. What is the multiplier if the margial propesity to cosume is 80%? Note: The federal govermet uses this priciple to justify deficit spedig. Baks use this priciple to justify led ig a large percetage of the moey that they receive i deposits. 74. A certai ball has the property that each time it falls from a height h oto a hard, level surface, it rebouds to a height rh, where 0, r,. Suppose that the ball is dropped from a iitial height of H meters. (a) Assumig that the ball cotiues to bouce idefiitely, fid the total distace that it travels. (b) Calculate the total time that the ball travels. (Use the fact that the ball falls tt meters i t secods.) (c) Suppose that each time the ball strikes the surface with velocity v it rebouds with velocity kv, where 0, k,. How log will it take for the ball to come to rest? 75. Fid the value of c if s cd 76. Fid the value of c such that e c I Example 9 we showed that the harmoic series is diverget. Here we outlie aother method, makig use of the fact that e x. x for ay x. 0. (See Exercise ) If s is the th partial sum of the harmoic series, show that e s.. Why does this imply that the harmoic series is diverget? 78. Graph the curves y x, 0 < x <, for 0,,, 3, 4,... o a commo scree. By fidig the areas betwee successive curves, give a geometric demostratio of the fact, show i Example 8, that s d Copyright 06 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s). Editorial review has deemed that ay suppressed cotet does ot materially affect the overall learig experiece. Cegage Learig reserves the right to remove additioal cotet at ay time if subsequet rights restrictios require it.

7 78 CHAPTER Ifiite Sequeces ad Series 79. The figure shows two circles C ad D of radius that touch at P. The lie T is a commo taget lie; C is the circle that touches C, D, ad T; C is the circle that touches C, D, ad C ; C 3 is the circle that touches C, D, ad C. This procedure ca be cotiued idefiitely ad produces a ifiite sequece of circles hc j. Fid a expressio for the diameter of C ad thus provide aother geometric demostratio of Example 8. C C C 80. A right triagle ABC is give with /A ad AC b. CD is draw perpedicular to AB, DE is draw perpedicular to BC, EF AB, ad this process is cotiued idefi itely, as show i the figure. Fid the total legth of all the perpediculars P C CD DE EF FG i terms of b ad. B 8. What is wrog with the followig calculatio? H s d s d s d F G D E C A b D T 84. If o a is diverget ad c ± 0, show that o ca is diverget. 85. If o a is coverget ad o b is diverget, show that the series o sa b d is diverget. [Hit: Argue by cotradictio.] 86. If o a ad o b are both diverget, is o sa b d ecessarily diverget? 87. Suppose that a series o a has positive terms ad its partial sums s satisfy the iequality s < 000 for all. Explai why o a must be coverget. 88. The Fiboacci sequece was defied i Sectio. by the equatios f, f, f f f > 3 Show that each of the followig statemets is true. (a) f f f f f f (b) f f (c) f f f 89. The Cator set, amed after the Germa mathematicia Georg Cator (845 98), is costructed as follows. We start with the closed iterval [0, ] ad remove the ope iterval ( 3, 3 ). That leaves the two itervals f0, 3 g ad f 3, g ad we remove the ope middle third of each. Four itervals remai ad agai we remove the ope middle third of each of them. We cotiue this procedure idefiitely, at each step removig the ope middle third of every iterval that remais from the precedig step. The Cator set cosists of the umbers that remai i [0, ] after all those itervals have bee removed. (a) Show that the total legth of all the itervals that are removed is. Despite that, the Cator set cotais ifiitely may umbers. Give examples of some umbers i the Cator set. (b) The Sierpiski carpet is a two-dimesioal couterpart of the Cator set. It is costructed by removig the ceter oe-ith of a square of side, the removig the ceters of the eight smaller remaiig squares, ad so o. (The figure shows the first three steps of the costructio.) Show that the sum of the areas of the removed squares is. This implies that the Sierpiski carpet has area 0. s d s d s d (Guido Ubaldus thought that this proved the existece of God because somethig has bee created out of othig. ) 8. Suppose that o ǹ a sa ± 0d is kow to be a coverget series. Prove that oǹ ya is a diverget series. 83. Prove part (i) of Theorem (a) A sequece ha j is defied recursively by the equatio a sa ad for > 3, where a ad a ca be ay real umbers. Experimet with various values of a ad a ad use your calculator to guess the limit of the sequece. Copyright 06 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s). Editorial review has deemed that ay suppressed cotet does ot materially affect the overall learig experiece. Cegage Learig reserves the right to remove additioal cotet at ay time if subsequet rights restrictios require it.

8 SECTION.3 The Itegral Test ad Estimates of Sums 79 (b) Fid lim l ` a i terms of a ad a by expressig a a i terms of a a ad summig a series. 9. Cosider the series o ǹ ys d!. (a) Fid the partial sums s, s, s 3, ad s 4. Do you recogize the deomiators? Use the patter to guess a formula for s. (b) Use mathematical iductio to prove your guess. (c) Show that the give ifiite series is coverget, ad fid its sum. 9. I the figure at the right there are ifiitely may circles approachig the vertices of a equilateral triagle, each circle touchig other circles ad sides of the triagle. If the triagle has sides of legth, fid the total area occupied by the circles. s o i i I geeral, it is difficult to fid the exact sum of a series. We were able to accomplish this for geometric series ad the series o yfs dg because i each of those cases we could fid a simple formula for the th partial sum s. But usually it is t easy to discover such a formula. Therefore, i the ext few sectios, we develop several tests that eable us to determie whether a series is coverget or diverget without explicitly fidig its sum. (I some cases, however, our methods will eable us to fid good esti mates of the sum.) Our first test ivolves improper itegrals. We begi by ivestigatig the series whose terms are the reciprocals of the squares of the positive itegers: There s o simple formula for the sum s of the first terms, but the computer-geerated table of approximate values give i the margi suggests that the partial sums are approachig a umber ear.64 as l ` ad so it looks as if the series is coverget. We ca cofirm this impressio with a geometric argumet. Figure shows the curve y yx ad rectagles that lie below the curve. The base of each rectagle is a iterval of legth ; the height is equal to the value of the fuctio y yx at the right edpoit of the iterval. y y= x FIGURE area= 3@ area= 4@ area= 5@ Copyright 06 Cegage Learig. All Rights Reserved. May ot be copied, scaed, or duplicated, i whole or i part. Due to electroic rights, some third party cotet may be suppressed from the ebook ad/or echapter(s). Editorial review has deemed that ay suppressed cotet does ot materially affect the overall learig experiece. Cegage Learig reserves the right to remove additioal cotet at ay time if subsequet rights restrictios require it.