Review Exercises for Chapter 9

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1 0_090R.qd //0 : PM Page CHAPTER 9 Ifiie Series I Eercises ad, wrie a epressio for he h erm of he sequece..,., 5, 0,,,, 0,... 7,... I Eercises, mach he sequece wih is graph. [The graphs are labeled (a), (b), (c), ad (d).] (a) (c) Review Eercises for Chaper 9 5 a a (b) (d). a. a 5. a 00.. a I Eercises 7 ad 8, use a graphig uiliy o graph he firs 0 erms of he sequece. Use he graph o make a iferece abou he covergece or divergece of he sequece. Verify your iferece aalyically ad, if he sequece coverges, fid is limi a 8. I Eercises 9, deermie he covergece or divergece of he sequece wih he give h erm. If he sequece coverges, fid is limi. ( b ad c are posiive real umbers.) 9. a 0. a. a. a l. a. a si a si 5. a. a b c 0 8 a a Compoud Ieres A deposi of $5000 is made i a accou ha ears 5% ieres compouded quarerly. The balace i he accou afer quarers is (a) Compue he firs eigh erms of he sequece A. (b) Fid he balace i he accou afer 0 years by compuig he 0h erm of he sequece. 8. Depreciaio A compay buys a machie for $0,000. Durig he e 5 years he machie will depreciae a a rae of 0% per year. (Tha is, a he ed of each year, he depreciaed value will be 70% of wha i was a he begiig of he year.) (a) Fid a formula for he h erm of he sequece ha gives he value V of he machie full years afer i was purchased. (b) Fid he depreciaed value of he machie a he ed of 5 full years. Numerical, Graphical, ad Aalyic Aalysis I Eercises 9, (a) use a graphig uiliy o fid he idicaed parial sum S k ad complee he able, ad (b) use a graphig uiliy o graph he firs 0 erms of he sequece of parial sums ! I Eercises, deermie he covergece or divergece of l I Eercises 7 0, fid he sum of he coverge series See for worked-ou soluios o odd-umbered eercises. A , k S k,,,....

2 0_090R.qd //0 : PM Page 89 REVIEW EXERCISES 89 I Eercises ad, (a) wrie he repeaig decimal as a geomeric series ad (b) wrie is sum as he raio of wo iegers Disace A ball is dropped from a heigh of 8 meers. Each ime i drops h meers, i rebouds 0.7h meers. Fid he oal disace raveled by he ball.. Salary You accep a job ha pays a salary of $,000 he firs year. Durig he e 9 years, you will receive a 5.5% raise each year. Wha would be your oal compesaio over he 0-year period? 5. Compoud Ieres A deposi of $00 is made a he ed of each moh for years i a accou ha pays % ieres, compouded coiuously. Deermie he balace i he accou a he ed of years.. Compoud Ieres A deposi of $00 is made a he ed of each moh for 0 years i a accou ha pays.5%, compouded mohly. Deermie he balace i he accou a he ed of 0 years. I Eercises 7 0, deermie he covergece or divergece of l I Eercises, deermie he covergece or divergece of I Eercises 5 8, deermie he covergece or divergece of I Eercises 9 5, deermie he covergece or divergece of 9. e 50.! e l Numerical, Graphical, ad Aalyic Aalysis I Eercises 5 ad 5, (a) verify ha he series coverges, (b) use a graphig uiliy o fid he idicaed parial sum S ad complee he able, (c) use a graphig uiliy o graph he firs 0 erms of he sequece of parial sums, ad (d) use he able o esimae he sum of S Wriig Use a graphig uiliy o complee he able for (a) p ad (b) p 5. Wrie a shor paragraph describig ad comparig he eries i he able. 5. Wriig You are old ha he erms of a posiive series appear o approach zero very slowly as approaches ifiiy. (I fac, a ) If you are give o oher iformaio, ca you coclude ha he series diverges? Suppor your aswer wih a eample. I Eercises 57 ad 58, fid he hird-degree Taylor polyomial ceered a c N N N 5 p d p f e, f a, c 0 c I Eercises 59, use a Taylor polyomial o approimae he fucio wih a error of less ha si cos0.75. l.75. e 0.5. A Taylor polyomial ceered a 0 will be used o approimae he cosie fucio. Fid he degree of he polyomial required o obai he desired accuracy over each ierval. Maimum Error Ierval 5 (a) , 0.5 (b) 0.00, (c) , 0.5 (d) 0.000,. Use a graphig uiliy o graph he cosie fucio ad he Taylor polyomials i Eercise.

3 0_090R.qd //0 : PM Page CHAPTER 9 Ifiie Series I Eercises 5 70, fid he ierval of covergece of he power series. (Be sure o iclude a check for covergece a he edpois of he ierval.) ! 70. I Eercises 7 ad 7, show ha he fucio represeed by he power series is a soluio of he differeial equaio I Eercises 7 ad 7, fid a geomeric power series ceered a 0 for he fucio Fid a power series for he derivaive of he fucio i Eercise Fid a power series for he iegral of he fucio i Eercise 7. I Eercises 77 ad 78, fid a fucio represeed by he series ad give he domai of he fucio y y! g h! y y y 0 y y y I Eercises 79 8, fid a power series for he fucio ceered a c. 79. f si, c 80. f cos, c 8. f, c f csc, c (firs hree erms) 8. f c 8. f,, c 85. g 5, c 0 8. h, c 0 I Eercises 87 9, fid he sum of he coverge series by usig a well-kow fucio. Ideify he fucio ad eplai how you obaied he sum ! Wriig Oe of he series i Eercises ad 9 coverges o is sum a a much lower rae ha he oher series. Which is i? Eplai why his series coverges so slowly. Use a graphig uiliy o illusrae he rae of covergece. 9. Use he biomial series o fid he Maclauri series for 95. Formig Maclauri Series Deermie he firs four erms of he Maclauri series for e (a) by usig he defiiio of he Maclauri series ad he formula for he coefficie of he h erm, a f 0!. (b) by replacig by i he series for e. (c) by muliplyig he series for e by iself, because e e e. 9. Formig Maclauri Series Follow he paer of Eercise 95 o fid he firs four erms of he series for si. (Hi: si si cos. ) I Eercises 97 00, fid he series represeaio of he fucio defied by he iegral f !!! si d. cos d l d e d I Eercises 0 ad 0, use a power series o fid he limi (if i eiss). Verify he resul by usig L Hôpial s Rule. arca 0. lim 0 arcsi 0. lim 0 5

0_090R.qd //0 : PM Page 9 P.S. Problem Solvig 9 P.S. Problem Solvig See www.calccha.com for worked-ou soluios o odd-umbered eercises.. The Caor se (Georg Caor, 85 98) is a subse of he ui ierval 0,.

4 0_090R.qd //0 : PM Page 9 P.S. Problem Solvig 9 P.S. Problem Solvig See for worked-ou soluios o odd-umbered eercises.. The Caor se (Georg Caor, 85 98) is a subse of he ui ierval 0,. To cosruc he Caor se, firs remove he middle hird, of he ierval, leavig wo lie segmes. For he secod sep, remove he middle hird of each of he wo remaiig segmes, leavig four lie segmes. Coiue his procedure idefiiely, as show i he figure. The Caor se cosiss of all umbers i he ui ierval 0, ha sill remai.. Ideical blocks of ui legh are sacked o op of each oher a he edge of a able. The ceer of graviy of he op block mus lie over he block below i, he ceer of graviy of he op wo blocks mus lie over he block below hem, ad so o (see figure). 0 0 The Grager Collecio (a) Fid he oal legh of all he lie segmes ha are removed. (b) Wrie dow hree umbers ha are i he Caor se. (c) Le C deoe he oal legh of he remaiig lie segmes afer seps. Fid lim C.. I ca be show ha GEORG CANTOR (85 98) Caor was a Germa mahemaicia kow for his work o he developme of se heory, which is he basis of moder mahemaical aalysis. This heory eeds o he cocep of ifiie (or rasfiie) umbers. [see Eample (b), Secio 9.]. Use his fac o show ha 8.. Le T be a equilaeral riagle wih sides of legh. Le a be he umber of circles ha ca be packed ighly i rows iside he riagle. For eample, a, a, ad a, as show i he figure. Le A be he combied area of he a circles. Fid lim A. (a) If here are hree blocks, show ha i is possible o sack hem so ha he lef edge of he op block eeds ui beyod he edge of he able. (b) Is i possible o sack he blocks so ha he righ edge of he op block eeds beyod he edge of he able? (c) How far beyod he able ca he blocks be sacked? 5. (a) Cosider he power series a 5... i which he coefficies a,,,,,,,... are periodic of period p. Fid he radius of covergece ad he sum of his power series. (b) Cosider a power series a i which he coefficies are periodic, a p a p ad a > 0. Fid he radius of covergece ad he sum of his power series.. For wha values of he posiive cosas a ad b does he followig series coverge absoluely? For wha values does i coverge codiioally? a b a b a 5 b a 7 b (a) Fid a power series for he fucio f e ceered a 0. Use his represeaio o fid he sum of he ifiie series!. (b) Differeiae he power series for f e. Use he resul o fid he sum of he ifiie series.!

5 0_090R.qd //0 : PM Page 9 9 CHAPTER 9 Ifiie Series 8. Fid f (0 if f e. (Hi: Do o calculae derivaives.) 9. The graph of he fucio f, 0 si, > 0 is show below. Use he Aleraig Series Tes o show ha he improper iegral f d coverges. y. Derive each ideiy usig he appropriae geomeric series. (a) Cosider a idealized populaio wih he characerisic ha each member of he populaio produces oe offsprig a he ed of every ime period. Each member has a life spa of hree ime periods ad he populaio begis wih 0 ewbor members. The followig able shows he populaio durig he firs five ime periods. (b) Time Period Age Bracke π π π π Toal (a) Prove ha coverges if ad oly if p >. l d p (b) Deermie he covergece or divergece of he series l.. (a) Cosider he followig sequece of umbers defied recursively. a a a a a Wrie he decimal approimaios for he firs si erms of his sequece. Prove ha he sequece coverges ad fid is limi. (b) Cosider he followig sequece defied recursively by a a ad a a a, where a >. a, a a, a a a,... Prove ha his sequece coverges ad fid is limi.. Le a be a sequece of posiive umbers saisfyig lim a r > 0. Prove ha he series a r L < r, coverges.. Cosider he ifiie series. (a) Fid he firs five erms of he sequece of parial sums. (b) Show ha he Raio Tes is icoclusive for his series. (c) Use he Roo Tes o es for he covergece or divergece of his series. The sequece for he oal populaio has he propery ha S S S S, Fid he oal populaio durig each of he e five ime periods.. Imagie you are sackig a ifiie umber of spheres of decreasig radii o op of each oher, as show i he figure. The radii of he spheres are meer, meer, meer, ec. The spheres are made of a maerial ha weighs ewo per cubic meer. (a) How high is his ifiie sack of spheres? (b) Wha is he oal surface area of all he spheres i he sack? (c) Show ha he weigh of he sack is fiie. 7. (a) Deermie he covergece or divergece of he series.... (b) Deermie he covergece or divergece of he series si si. >. m m m

In exercises 1 and 2, (a) write the repeating decimal as a geometric series and (b) write its sum as the ratio of two integers _

In exercises 1 and 2, (a) write the repeating decimal as a geometric series and (b) write its sum as the ratio of two integers _ Chapter 9 Curve I eercises ad, (a) write the repeatig decimal as a geometric series ad (b) write its sum as the ratio of two itegers _.9.976 Distace A ball is dropped from a height of 8 meters. Each time