44 Chapter 8 Discrete Mathematics: Fuctios o the Set of Natural Numbers (a) Take several odd, positive itegers for a ad write out eough terms of the 3N sequece to reach a repeatig loop (b) Show that ot every positive iteger reaches the sameloop(asappearstobethecaseforthe3n sequece) How maydifferet loops ca you fid? 76 Programs for computers or programmable calculators ofteuseaifthen istructio that termiates the programs if certai coditios hold but brach to other istructios otherwise Describe the kid of difficulties that we could coceivably ecouter if we programmed a computer to ru the 3N sequece ad prit out the terms util reachig Exercises 77 78 Compare Sequeces 77 Evaluate the first five terms of a ad b where a, b 4 6 3 3 8 4 4 Are the sequeces idetical? 78 (a) Evaluate the first six terms of a where a 9 6 Use your calculator to simplify each term Is a 3 for every? Explai (b) If b 3, are the sequeces a ad b idetical? Explai Exercises 79 80 Large Numbers 79 The factorial sequece! icreases very rapidly For istace, 0! 36 0 6 0! 4 0 8 50! 30 0 64 70! 0 00 To get some idea of how large these umbers are, look at 0! Simple computatio gives 0!,43,90,008,76,690,000 Now suppose a computer priter that operates at 00 characters per secod were to prit out a mauscript with 0! characters How log would it take the priter to do the job? 80 For the mauscript described i Exercise 79, suppose each page cotais about 4000 characters How thick would the mauscript be? The thickess of a ream of paper (500 pages) is approximately iches For compariso, the distace from the earth to the su is 93 millio miles Exercises 8 8 Fiboacci Sequece Use the defiitio o page 436 8 Show that f f f f for 8 Explore WriteoutthefirsttwelvetermsoftheFiboacci sequece Make a guess as to which terms are divisible by, by 3 8 GRAPHS AND CONVERGENCE It just came to me that I could use this techique, this theorem, i coectio with these curves i Hilbert space that I was dealig with ad get the aswer! It just came to me out of the blue oe day It has always struck me as so amazig Oe half of me had bee boucig aroud with this theorem a lot ad the other half had bee doig this problem, ad they had ever gotte together Adrew Gleaso the vast scope of moder mathematics I have i mid a expase swarmig with beaut y, worthy of beig surveyed from oe ed to the other ad studied eve i its smallest details: its valleys, streams, rocks, woods ad flowers Arthur Cayley Calculus is based o the study of limits At this poit, we use oly ituitive ideas of limits, but we ca use graphs ad their ed behavior to get strog feeligs about the existece or oexistece of limits of certai sequeces Without makig a precise defiitio, whe a sequece a has a limit L, wesaythat a coverges to L ad we write lim A a L Sice a sequece is a fuctio, we ca draw a graph; but we are oly iterested i those poits for which the x-pixel values are positive itegers I geeral, our graphs will be draw with a x-rage of 0, c, where c is the umber of pixel columsoourcalculator,adialmostallgraphswewillwattousedotmode
8 Graphs ad Covergece 443 TECHNOLOGY TIP Pixel colums The umber of pixel colums, c, for several calculators is as follows: Model cols Model cols TI-8 95 HP-38,48 30 TI-8 94 Casio fx-7700 94 TI-85 6 Casio fx-9700 6 To illustrate covergece, we look at the ed behavior for several sequeces i the first example [0, c] by [0, ] FIGURE EXAMPLE Covergece Use a graph to make a reasoable determiatio of the ed behavior for the sequece, ad the support your coclusio algebraically (a) a (b) b 3 (c) c 3 (a) Graphical Draw a graph of y x x usig 0, c 0, (see Techology Tip above) See Figure The right portio of the graph appears to be horizotal, but we kow that the calculator has oly so may pixels available Tracig alog the curve idicates that the y-coordiate is approachig as x icreases That is, it appears that lim a ; we say that the sequece a coverges A to Algebraic If we divide the umerator ad deomiator of a by, weget a I this form, it should be clear that a A asa (b) Graphical ad Algebraic The graph of y 3x is a lie with slope 3 If we use a x-rage of 0, c, we eed a correspodigly large y-rage or the graph goes off scale almost immediately, but eve without a graph for this particular fuctio we kow the ed behavior of a lie The values of b cotiue to icrease without boud ad do ot approach ay umber We say that the sequece b diverges Ithiscase,asiChapter3wheworkigwith ratioal fuctios, we write lim A b (c) Graphical Graphig y x x x 3 i 0, c 3, 3 gives very differet lookig graphs i coected or dot modes Whichever you choose, make sure that each x-pixel coordiate is a iteger ad that you kow how to iterpret what the graph shows Tracig (i either mode) shows that as x icreases, the y-values jump back ad forth, with the positive values approachig ad the egative values approachig We coclude that c diverges because the c values do ot approach a sigle umber as A
444 Chapter 8 Discrete Mathematics: Fuctios o the Set of Natural Numbers Algebraic Disregardig the, the expressio lim A lim 3 A 3 3 does have a limit, It follows that c A,sothatwheis eve, c A ad whe is odd, c A Agai, c diverges because the c values do ot approach a sigle umber as A Sequeces Defied Recursively For sequeces defied recursively we caot eter fuctios for graphig as we did i Example Nevertheless, fuctioalitybuilt ito our graphig calculators makes it possible to ivestigate limits of some such sequeces quite easily Cosider the sequece a defied by a, a 3 a () We wat to calculate a umber of terms of the sequece without havig to go through all the steps of the recursive defiitio for each term We describe some optios i the followig Techology Tip, ad the look at additioal examples TECHNOLOGY TIP Calculatig recursively defied sequeces For algebraic operatio calculators (TI, Casio, ad HP 38), essetially all of the steps for evaluatig the sequece i Equatio () ca be hadled o the home scree by makig use of the machie capacity to store values Begi by storig the iitial value i the x-register: A X, ENT The compute ad store the ext value: 3 XAX, ENT The calculator displays the computed value, 5, which has bee stored Whe we press ENTER agai, the same computatio is repeated with the ew x-value ad displayed value,, is our a 3 Aswe ENTER repeatedly, the terms of the sequece are displayed It soo becomes clear that the terms are approachig a limit, 307756 O the HP 48, we ca write a simple program to accomplish the same thig Press (above the subtract key) to begi a program The type A X 3X A Num (above 57 EVAL )ad ENTER Whattheprogramdoesistotakethe umber o the stack, call it x, compute 3 x symbolically, covert the symbolic computatio to a umber The result is displayed o the stack so that the process ca be repeated To use the program, we eed to store it as a variable, so we type a ame, say RECR for recursive The ENTER ad 57 STO The ame RECR should appear o your 57 VAR meu Now eter o the stack, press the soft key beeath 57 RECR, ad the ew value appears By repeatedly pressig 57 RECR to chage, approachig 307756, the value cotiues
8 Graphs ad Covergece 445 Strategy: For (c), if the sequece coverges to a umber c, both a ad a approach c, leadig to the equatio c 3 c The solve for c EXAMPLE Nested square roots Sequece a is defied by a 3, a 3 a (a) Write out the first three terms i exact form (b) UsetheTechologyTip(page444)toapproximatethefirstfewtermsofthe sequece ad fid the apparet limit of the sequece (six decimal places) (c) Justify your coclusio i (b) algebraically (a) a 3, a 3 a 3 3, a 3 3 a 3 3 3 (b) Followig the Techology Tip, for all machies except the HP-48, we store 3 i the x-register, ad the eter (3 X) A X Repeatig the computatio gives a sequece begiig 7305, 7538, 74935, 9673, Afterseveral more terms, the sequece settles o a umber approximately equal to 30776 O the HP-48, we must chage the recursive part of the defiitio i our program RECR by pressig the tick-mark key ad the soft key uder RECR With RECR othestack,wepress57 EDIT ad go ito the program, replacig 3 X by the recursive part of our ew sequece, 3 X) Withtheewprogram,we eter 3 ad the repeat the soft key uder RECR, gettig the same sequece of terms (c) Followig the strategy, we wat to solve the equatio c 3 c for c Squarig both sides, we get the equatio c 3 c, orc c 3 0 By the quadratic formula, takig the positive sig (why ot?),wegetc 3 30775638, obviously the umber we were approximatig i part (b) EXAMPLE 3 Nested cube roots Repeat Example for the sequece c defied by c, 3 c 3 c (a) c, 3 c 3 c 3, 3 c 3 3 c 3 3 3 (b) O algebraic operatio machies, we store ^( 3) i the x-register, ad the eter ( X)^( 3) A X The sequece begis 599, 48754, 55797, 50575, 564, Thesequece settles o a umber approximately equal to 53797 O the HP-48, we eter ( X)^( 3) as the recursive part of the defiitio i RECR After eterig 3 o the stack, repeatig the soft key uder RECR gives the same sequece of terms (c) Sice it appears that the sequece coverges to a umber c, both c ad c must approach the same umber c, so the recursive portio of the defiitio gives a equatio which ca be cubed: c 3 c, or c 3 c 0 The cubic equatio is ot oe we ca solve i exact form coveietly, but by graphical methods, or by usig a solve routie, or by usig Newto s Method from Chapter 3, the oe real zero is approximately 53797
446 Chapter 8 Discrete Mathematics: Fuctios o the Set of Natural Numbers Cotiued Fractios Cotiued fractios is a topic studied i umber theory courses that has applicatios i may areas, icludig the programmig of routies for computers ad graphig calculators I the ext example we illustrate the cotiued fractio,, as a recursively defied sequece EXAMPLE 4 Cotiued fractios Sequece a is defied by a, a a (a) Write out the first four terms, first without simplifyig, ad the as a simple fractio (b) Approximate the first few terms of the sequece ad fid the apparet limit of the sequece (c) Justify your coclusio i (b) algebraically (a) a, a, a 3 a 3, a 4 a 3 3 3 5 3 The umbers i the umerator ad deomiator of the fractios remid us of the Fiboacci sequece f from page 436:,,, 3, 5, 8, 3, Thatis, a f f,a f 3 f,a 3 f 4 f 3,a 4 f 5 f 4, ad a reasoable guess is that a f f (b) For decimal approximatios, we use the Techology Tip, begiig with ad usig X A X for the recursio The sequece appears to settle dow o a umber c 68034 (c) If both a ad a approach c, the i the limit the recursio relatio becomes c c Multiplyig through by c leadstothequadraticequatioc c 0, 5 whose positive root is give by c 68034 5 The limit umber of the sequece,,iscalledthegolde Ratio, reflectig some aesthetic cosideratios of the aciet Greeks It is a umber that turs up i may diverse applicatios See Exercise 7 I the ext example, we see aother istace of a sequece that diverges eve though parts of the sequece, called subsequeces, coverge We had oe such sequece i Example, give by c 3 From the graph i dot mode, we saw that the sequece cosistig of the eve-umbered terms c, c 4, c 6, coverges to ; the odd-umbered terms form a subsequece that coverges to The same kid of behavior is possible with a recursively defied sequece
8 Graphs ad Covergece 447 EXAMPLE 5 Subsequeces Sequece a is defied by a 3, a a a Write out the first few terms Does the sequece have a limit? Describe some coverget subsequeces of the sequece Either by usig the Techology Tip (page 444) or by direct computatio, it is clear that the sequece begis 3, 6, 3, 6, 3, 6, 3, 6, Thesequece has o limit because the terms are ot gettig close to ay umber as A The subsequece of odd-umbered terms cotais oly 3, a 3,3,3,3,, which obviously coverges to 3 Similarly, the subsequece cosistig of eve-umbered terms a 6,6,6, coverges to 6 TECHNOLOGY TIP Calculatig with two-step recursios Ay graphig calculator ca be programmed to calculate more ivolved recursively defied sequeces, but geeral programmig is ot our focus i this text To lear about programmig o your calculator, cosult your istructio maual The Texas Istrumet TI-8 ad TI-85, HP 38, ad the Casio fx7700 ad fx9700, allow us to hadle two-step recursively defied sequeces, such as the Fiboacci sequece, directly o the home scree, as described below The Fiboacci sequece (page 436) is defied by f, f, f f f, O the home scree, we store the iitial values ad their sum: A A: A B: A B A C ad ENTER (The colo is located above the decimal poit o TI ad HP 38, ad o the PRGM meu o the Casio) The display shows f 3 as We must reassig values for the ext step: B A A:C A B: A B A C Now ENTER adwesee3asf 4, ad we ca repeat for as may terms as desired We revisit this problem i matrix form i Exercise 70 of Sectio 96 Equatios of the Form F(x) x Whe lookig for the roots of a equatio it is sometimes possible to isolate a x, writig the equatio i the form F x x Uder certai circumstaces it is possible to use a iterative process to approximate a root of such a equatio to great accuracy Basically, the solutio is foud as the limit of a recursive sequece We begi with a iitial approximatio a ad defie a F a Determiig the coditios uder which such a iteratio coverges requires calculus, but we illustrate the procedure i Example 6
448 Chapter 8 Discrete Mathematics: Fuctios o the Set of Natural Numbers EXAMPLE 6 Solvig a equatio (a) Approximate the root of the equatio x cos x 0fromagraph (b) Write the equatio i the form f x x ad use the approximatio from part (a) as a ad let a f a Iterate to approximate the limit L of the sequece to 8 decimal places ad verify that L satisfies the origial equatio (a) From a graph, we ca see that there is a root of the equatio ear 05 (b) The equatio is equivalet to x cos x,sowetakef x cos x ad defie the sequece by a 05, a cos a Followig the Techology Tip for recursive sequeces, we store 05 A X, ENT The follow with (COS X) A X, ENT ad iterate, gettig a sequece begiig 0438798, 045639, Thesequece settles quickly o the umber L 0450836, ad whe we substitute L for x i the expressio x cos x, we get a umber very ear 0, as desired EXERCISES 8 Check Your Uderstadig It will be helpful to use the Techology Tip (page 444) to get the first several terms of a Exercises 5 True or False Give reasos Use sequece a defied by a ad a a Every term of a is less tha or equal to The sequece is decreasig; that is a a for every 3 The eve-umbered terms are greater tha the oddumbered terms 4 The subsequece cosistig of the odd-umbered terms, a, a 3, a 5,,isdecreasig 5 The subsequece cosistig of the eve-umbered terms, a, a 4, a 6,,isicreasig Exercises 6 0 Fill i the blak so that the resultig statemet is true Exercises 6 8 Sequece a is defied by a ad a a 4 6 The smallest iteger greater tha a 5 is 7 The umber of terms of a betwee 8 ad 0 is 8 The sum of the first 5 terms is Exercises 9 0 Sequece b is defied by b ad b b 9 b 5 0 The smallest prime umber that is greater tha b 5 is Develop Mastery Exercises 0 Does it Coverge? Use a graph to help you determie whether or ot the sequece appears to coverge Explai a 5 a 3 a 3 4 a 5 a 3 6 a 7 a 9 a 64 6 0 a 5 8 a
8 Graphs ad Covergece 449 Exercises 6 Sequeces Defied Recursively Use the Techology Tip (page 444) (a) Give the first three terms of a (b) The sequece coverges to a umber c Use your calculator to get a six-decimal-place approximatio for c (c) Justify your aswer algebraically See Examples ad 3 a 3, a 3 a a 5, a 5 a 3 a 7, a 7 a 4 a 5, 3 a 5 3 a 5 a 6, 3 a 6 3 a 6 a 5, 4 a 5 4 a Exercises 7 Cotiued Fractios (a) Fid the first three terms of c (b) Fid a 6-decimal-place approximatio for the limit to which c appears to coverge (c) Justify algebraically See Example 4 7 c 3, c 3 c 8 c 5, c 5 c 9 c 3, c 3 c 0 c, c c c 3, c 3 c c, c c Exercises 3 4 Repeatig Terms Sequece a is defied recursively (a) Give the first six terms Does a coverge? You may wish to use the Techology Tip (b) Determie a 60 ad the sum of the first sixty terms 3 a, a a 4 a, a a Exercises 5 6 Recogizig a Patter Fid the first four terms of a Make a geeralizatio ad justify algebraically 5 (a) a, a 05 a a (b) a, a 05 a 4 a (c) a 3, a 05 a 9 a (d) a 4, a 05 a 6 a 6 (a) a, a a a (b) a, (c) a 3, a 5a a a 0a a (d) a 4, a 7a a 7 Fiboacci Related For a 3, a 3 a, (a) Write the first six terms as simple fractios (b) Guess a relatioship betwee a ad the Fiboacci sequece 8 For a l, a l a, (a) use the Techology Tip to fid a six-decimal-place approximatio to the limit c to which a coverges (b) Show that c isarootofe x x 0 5 Exercises 9 30 Golde Ratio, Give that a coverges to the umber c, use a algebraic approach to verify that c is the umber give The use the Techology Tip to get a calculator check 9 a, a ; c is the square root of the a golde ratio 30 a 3, a ; c istheegativeofthegolde a ratio Exercises 3 3 Explore The recursive formula for a is give alog with differet values of a I each case use the Techology Tip to get the first three terms ad the limit (six decimal places) to which the sequece coverges Try other values of a Describe the role that a plays 3 a a, (a) a (b) a 8 (c) a 4 3 a, a (a) a (b) a 3 (c) a 5 Exercises 33 36 Subsequeces (a) Does a coverge? (b) Fid subsequeces of a that coverge See Example 5 33 a 34 a 35 a si 36 a cos
450 Chapter 8 Discrete Mathematics: Fuctios o the Set of Natural Numbers Exercises 37 40 Your Choice Give a sequece a of your choice that meets the give coditios 37 a 0ifis odd, a 0ifis eve 38 The sequece a does ot coverge, but the subsequece of odd-umbered terms (all of which are greater tha ) coverges to, while the subsequece of eveumbered terms (all of which are less tha ) coverges to 39 For every, 0 a, ad a coverges to (a) ; (b) 05 40 For all odd, 0 a, ad for all eve, a, ad a coverges to Exercises4 43 Repeatig Terms (a) Givethefirstfour terms of a (b) What is a 47? a 7? (c) Fid the sum of thefirsttwetyterms (d) Explai the repeatig behavior of a (Hit for Exercise 4: Cosider f x x x, ad show that f x f x What is f f x?) 4 a 6, a a a 4 a, a 3a a 3 43 a 4, a 3a a 3 Exercises 44 50 Roots of f~x! x Follow the istructios for Example 6 for the give equatio 44 cos x x 45 5x cosx 0 46 x e x 47 x cos x 4 48 x cos 4 x 49 x l 4 x 4 50 x 4 e x Exercises 5 5 Roots of x x 0 Fid the limit L of a to eight decimal places Show that L is a root of the equatio x x 0 5 a 50, a 0 l a l 5 a, a 0a 53 Explore I the recursive formula for Exercise 30, a a, may differet iitial values give sequeces that coverge to the same value There are iitial values that do ot work, however We obviously caot use a 0 because a would the be udefied, ad we caot use a umber as a iitial value that would lead to 0 For example, solvig a 0, we get a If we were to try a, we would get a 0ad the a 3 would be udefied (a) Solve ad a fid aother iadmissible value for a (b) Fid a sequece i exact form of iadmissible iitial value umbers (c) Show that a 3 5 is iadmissible by computig the first few terms i exact form (d) Try a 06 ad compute the first few terms by usig the Techology Tip (page 444) Explai the differece i results from part (c) 54 The sequece a is give by a x x, where 3 i x (a) Use DeMoivre s theorem to evaluate x x, the show that a cos( 0 ) (b) Write out the first six terms of the sequece ad fid their sum (c) What is the sum of the first 00 terms? Exercises 55 60 Sequece a coverges to a umber L (a) Use the Techology Tip to approximate L (b) Use algebra to fid the exact value of L 55 a a a 4 a for 56 a a a 3 a for 57 a a a a for 58 a a a for 59 a 6 a 6 a for 60 a a 4 a for Exercises 6 6 Sequeces a ad b coverge; sequece c diverges (a) Fid approximatios (6 decimal places) for the limits of a ad b (b) Fid subsequeces of c that coverge ad approximate their limits 6 a, a 4 a ; b, b 4 b c, c 4 c 6 a 4, a 4 3 a ; b 4, b 4 3 b c 4, c 4 3 c