Chapter 2 Sequeces I 2. Itroductio A sequece is a list of umbers i a defiite order so that we kow which umber is i the first place, which umber is i the secod place ad, for ay atural umber, we kow which umber is i the th place. All the sequeces i this course are ifiite ad cotai oly real umbers. For example:,2,3,4,5,...,,,,,..., 2, 3, 4, 5,... si,si2,si3,si4,... I geeral we deote a sequece by: a = a,a 2,a 3,a 4,... Notice that for each atural umber,, there is a term a i the sequece; thus a sequece ca be thought of as a fuctio a : N R give by a = a. Sequeces, like may fuctios, ca be plotted o a graph. Let s deote the first three sequeces above by a, b ad c, so the th terms are give by: Iitially Sometimes you will see a 0 as the iitial term of a sequece. We will see later that, as far as covergece is cocered, it does t matter where you start the sequece. a = ; b = ; c =. Figure 2. shows roughly what the graphs look like. Aother represetatio is obtaied by simply labellig the poits of the sequece o the real lie, see figure 2.2. These pictures show types of behaviour that a sequece might have. The sequece a goes to ifiity, the sequece b jumps back ad forth betwee - ad, ad the sequece c coverges to 0. I this chapter we will decide how to give each of these phrases a precise meaig. Sie Time What do you thik the fourth sequece, si, looks like whe you plot it o the real lie? 5
6 CHAPTER 2. SEQUENCES I a b c Figure 2.: Graphig sequeces as fuctios N R. a a2 d a a3 b c bl br d c3 c2 c Figure 2.2: Number lie represetatios of the sequeces i figure 2.. Exercise Write dow a formula for the th term of each of the sequeces below. The plot the sequece i each of the two ways described above..,3,5,7,9,... 2. 2, 4, 8, 6, 32,... 3. 0, 2,0, 2,0, 2,... 4. 2, 2 3, 3 4, 4 5, 5 6,... Labour Savers Note that: strictly icreasig = icreasig ad ot decreasig strictly decreasig = decreasig ad ot icreasig icreasig = mootoic decreasig = mootoic. 2.2 Icreasig ad Decreasig Sequeces Defiitio A sequece a is: strictly icreasig if, for all, a < a + ; icreasig if, for all, a a + ; strictly decreasig if, for all, a > a + ; decreasig if, for all, a a + ; mootoic if it is icreasig or decreasig or both; o-mootoic if it is either icreasig or decreasig. Example Recallthesequecesa,b adc,givebya =,b = ad c =. We see that:. for all, a = < + = a +, therefore a is strictly icreasig; 2. b = < = b 2, b 2 = > = b 3, therefore b is either icreasig or decreasig, i.e. o-mootoic; 3. for all, c = > + = c +, therefore c is strictly decreasig.
2.3. BOUNDED SEQUENCES 7 a b c d e f Figure 2.3: Which sequeces are mootoic? Exercise 2 Test whether each of the sequeces defied below has ay of the followig properties: icreasig; strictly icreasig; decreasig; strictly decreasig; o-mootoic. [A graph of the sequece may help you to decide, but use the formal defiitios i your proof.]. a = 2. a 2 =,a 2 = 3. a = 4. a = 2 5. a = + 6. a = si Be Dotty Whe you are graphig your sequeces, remember ot to joi the dots. Sequeces are fuctios defied o the atural umbers oly. Hit: I part 5, try usig the idetity a b = a2 b 2 a+b. 2.3 Bouded Sequeces Defiitio A sequece a is: bouded above if there exists U such that, for all, a U; U is a upper boud for a ; bouded below if there exists L such that, for all, a L; L is a lower boud for a ; bouded if it is both bouded above ad bouded below. Example. The sequece is bouded sice 0 <. 2. The sequece is bouded below but is ot bouded above because for each value C there exists a umber such that > C. Boudless Bouds If U is a upper boud the so is ay umber greater tha U. If L is a lower boud the so is ay umber less tha L. Bouds are ot uique.
8 CHAPTER 2. SEQUENCES I a b c U L L U Figure 2.4: Sequeces bouded above, below ad both. Bouds for Mootoic Sequeces Each icreasig sequece a is bouded below by a. Each decreasig sequece a is bouded above by a. Exercise 3 Decide whether each of the sequeces defied below is bouded above, bouded below, bouded. If it is oe of these thigs the explai why. Idetify upper ad lower bouds i the cases where they exist. Note that, for a positive real umber x, x, deotes the positive square root of x.. 2. 4. si 5. 3. a = + 6. Exercise 4. A sequece a is kow to be icreasig. a Might it have a upper boud? b Might it have a lower boud? c Must it have a upper boud? d Must is have a lower boud? Give a umerical example to illustrate each possibility or impossibility. 2. If a sequece is ot bouded above, must it cotai a a positive term, b a ifiite umber of positive terms? 2.4 Sequeces Tedig to Ifiity We say a sequece teds to ifiity if its terms evetually exceed ay umber we choose. Defiitio A sequece a teds to ifiity if, for every C > 0, there exists a atural umber N such that a > C for all > N. We will use three differet ways to write that a sequece a teds to ifiity, a, a, as ad lim a =. Example
2.4. SEQUENCES TENDING TO INFINITY 9. 3. Let C > 0. We wat to fid N such that if > N the 3 > C. Note that 3 > C > 3C. So choose N 3C. If > N the 3 > N 3 C. I the margi draw a graph of the sequece ad illustrate the positios of C ad N. 2. 2. Let C > 0. We wat to fid N such that if > N the 2 > C. Note that 2 > C > log 2 C. So choose N log 2 C. If > N the 2 > 2 N 2 log 2 C = C. Exercise 5 Whe does the sequece evetually exceed 2, 2 ad 000? The prove that teds to ifiity. Exercise 6 Select values of C to demostrate that the followig sequeces do ot ted to ifiity..,, 2,, 3,...,,,... 2., 2, 3, 4,...,,... 3., 2,, 2,...,, 2,... Is Ifiity a Number? We have ot defied ifiity to be ay sort of umber - i fact, we have ot defied ifiity at all. We have side-stepped ay eed for this by defiig the phrase teds to ifiity as a self-cotaied uit. Exercise 7 Thik of examples to show that:. a icreasig sequece eed ot ted to ifiity; 2. a sequece that teds to ifiity eed ot be icreasig; 3. a sequece with o upper boud eed ot ted to ifiity. Theorem Let a ad b be two sequeces such that b a for all. If a the b. Proof. Suppose C > 0. We kow that there exists N such that a > C wheever > N. Hece b a > C wheever > N. Example We kow that 2 ad, hece 2. Defiitio A sequece a teds to mius ifiity if, for every C < 0, there exists a umber N such that a < C for all > N. The correspodig three ways to write that a teds to mius ifiity are a,a, as ad lim a =
20 CHAPTER 2. SEQUENCES I Figure 2.5: Does this look like a ull sequece? Example You ca show that a if ad oly if a. Hece,, 2 ad all ted to mius ifiity. Theorem Suppose a ad b. The a + b, a b, ca whe c > 0 ad ca whe c < 0. Proof. We ll just do the first part here. Suppose a ad b. Let C > 0. Sice a ad C/2 > 0 there exists a atural umber N such that a > C/2 wheever > N. Also, sice b ad C/2 > 0 there exists a atural umber N 2 such that b > C/2 wheever > N 2. Now let N = max{n,n 2 }. Suppose > N. The This gives that > N ad > N 2 so that a > C/2 ad b > C/2. a +b > C/2+C/2 = C. This is exactly what it meas to say that a +b. Try doig the other parts i your portfolio. [Hit: for the secod part use C istead of C/2 i a proof similar to the above.] 2.5 Null Sequeces If someoe asked you whether the sequece, 2, 3, 4, 5,...,,... Is Zero Allowed? We are goig to allow zeros to appear i sequeces that ted to zero ad ot let their presece bother us. We are eve goig to say that the sequece 0,0,0,0,0,... tedstozero, youmightdrawagraphlikefigure2.5adtheprobablyaswer yes. After a little thought you might go o to say that the sequeces ad also ted to zero.,0, 2,0, 3,,0, 4,0,...,,0,..., 2, 3, 4, 5,...,,... teds to zero.
2.5. NULL SEQUENCES 2 We wat to develop a precise defiitio of what it meas for a sequece to ted to zero. As a first step, otice that for each of the sequeces above, every positive umber is evetually a upper boud for the sequece ad every egative umber is evetually a lower boud. So the sequece is gettig squashed closer to zero the further alog you go. Exercise 8. Use the sequeces below which are ot ull to demostrate the iadequacy of the followig attempts to defie a ull sequece. a A sequece i which each term is strictly less tha its predecessor. b A sequece i which each term is strictly less tha its predecessor while remaiig positive. c A sequece i which, for sufficietly large, each term is less tha some small positive umber. d A sequece i which, for sufficietly large, the absolute value of each term is less tha some small positive umber. e A sequece with arbitrarily small terms. I. 2,, 0,, 2, 3, 4,...,,... II. 3 2, 2, 4 3, 5 4, 6 5,..., +,... III. 2,, 0,, 0., 0., 0.,..., 0.,... IV. 2,, 0, 0., 0.0, 0.00, 0.0, 0.00,..., 0.0, 0.00,... V., 2,, 4,, 8,... 2. Examie the sequece, 2, 3, 4, 5,...,,... a Beyod what stage i the sequece are the terms betwee 0. ad 0.? b Beyod what stage i the sequece are the terms betwee 0.0 ad 0.0? c Beyod what stage i the sequece are the terms betwee 0.00 ad 0.00? d Beyod what stage i the sequece are the are the terms betwee ε ad ε, where ε is a give positive umber? You oticed i Exercise 8 2. that for every value of ε, o matter how tiy, the sequece was evetually sadwiched betwee ε ad ε i.e. withi ε of zero. We use this observatio to create our defiitio. See figure 2.6 Defiitio A sequece a teds to zero if, for each ε > 0, there exists a atural umber N such that a < ε for all > N. εrror. The choice of ε, the Greek e, is to stad for error, where the terms of a sequece are thought of as sucessive attempts to hit the target of 0. Make Like a Elephat This defiitio is the trickiest we ve had so far. Eve if you do t uderstad it yet Memorise It! I fact, memorise all the other defiitios while you re at it.
22 CHAPTER 2. SEQUENCES I 0 e -e N Figure 2.6: Null sequeces; first choose ε, the fid N. Archimedea Property Oe property of the real umbers that we do t ofte give much thought to is this: Give ay real umber x there is a iteger N such that N > x. Where have we used this fact? The three ways to write a sequece teds to zero are, a 0, a 0, as, ad lim a = 0. We also say a coverges to zero, or a is a ull sequece. Example The sequece a = teds to zero. Let ε > 0. We wat to fid N such that if > N, the a = < ε. Note that < ε > ε. So choose a atural umber N ε. If > N, the a = < N ε. Exercise 9 Prove that the sequece teds to zero. Exercise 0 Prove that the sequece,,,,,,,... does ot ted to zero. Fid a value of ε for which there is o correspodig N. Lemma If a ad a 0 for ay, the a 0. Exercise Prove this lemma. Exercise 2 Thik of a example to show that the opposite statemet, if a 0 the a, is false, eve if a 0 for all.
2.6. ARITHMETIC OF NULL SEQUENCES 23 Lemma Absolute Value Rule a 0 if ad oly if a 0. Proof. The absolute value of a is just a, i.e. a = a. So a 0 iff for each ε > 0 there exists a atural umber N such that a < ε wheever > N. But, by defiitio, this is exactly what a 0 meas. Example We showed before that 0. Now =. Hece 0. Theorem Sadwich Theorem for Null Sequeces Suppose a 0. If b a the b 0. Example. Clearly 0 +. Therefore + 0. 2. 0 3/2. Therefore 3/2 0. Exercise 3 Prove that the followig sequeces are ull usig the result above. Idicate what ull sequece you are usig to make your Sadwich. si. 2. + 2.6 Arithmetic of Null Sequeces Theorem Suppose a 0 ad b 0. The for all umbers c ad d: Examples ca +db 0 Sum Rule for Null Sequeces; a b 0 Product Rule for Null Sequeces. 2 = 0 Product Rule 2 5 2 = 2 5 2 0 Sum Rule The Sum Rule ad Product Rule are hardly surprisig. If they failed we would surely have the wrog defiitio of a ull sequece. So provig them carefully acts as a test to see if our defiitio is workig. Exercise 4
24 CHAPTER 2. SEQUENCES I. If a is a ull sequece ad c is a costat umber, prove that c a is a ull sequece. [Hit: Cosider the cases c 0 ad c = 0 i tur]. 2. Deduce that 0 is a ull sequece. 3. Suppose that a ad b are both ull sequeces, ad ε > 0 is give. a Must there be a N such that a < 2 ε whe > N? b Must there be a N 2 such that b < 2 ε whe > N 2? c Is there a N 0 such that whe > N 0 both > N ad > N 2? d If > N 0 must a +b < ε? You have proved that the termwise sum of two ull sequeces is ull. e If the sequece c is also ull, what about a +b +c? What about the sum of k ull sequeces? Exercise 5 Suppose a ad b are both ull sequeces, ad ε > 0 is give.. Must there be a N such that a < ε whe > N? 2. Must there be a N 2 such that b < whe > N 2? 3. Is there a N 0 such that whe > N 0 both > N ad > N 2? 4. If > N 0 must a b < ε? You have proved that the termwise product of two ull sequeces is ull. 5. If the sequece c is also ull, what about a b c? What about the product of k ull sequeces? Example To show that 2 +2+3 is a ull sequece, ote that 2 +2+3 3 = 3 + 2 + 3 2. We kow that 0 so 3 ad 2 are ull by the Product 3 Rule. It follows that is ull by the Sum Rule. 2 +2+3 3 2.7 * Applicatio - Estimatig π * Recall Archimedes method for approximatig π: A ad a are the areas of the circumscribed ad iscribed regular sided polygo to a circle of radius. Archimedes used the formulae to estimate π. a 2 = a A A 2 = 2A a 2 A +a 2 Exercise 6 Why is the sequece a 4,a 8,a 6,a 32,... icreasig? Why are all the values betwee 2 ad π? What similar statemets ca you make about the sequece A 4,A 8,A 6,A 32,...?
2.7. * APPLICATION - ESTIMATING π * 25 Usig Archimedes formulae we see that A 2 a 2 = 2A a 2 A +a 2 a 2 = A a 2 a 2 2 A +a 2 a 2 Exercise 7 Explai why = A a 2 A +a 2 a 2 = A a A A +a 2 a 2 A = A +a 2 A + A a a a 2 A A +a 2 A + a is ever larger tha 0.4. [Hit: use the bouds from the previous questio.] Hece show that the error A a i calculatig π reduces by at least 0.4 whe replacig by 2. Show that by calculatig A 2 0 ad a 2 0 we ca estimate π to withi 0.004. [Hit: recall that a π A.] Check Your Progress By the ed of this chapter you should be able to: Explai the term sequece ad give a rage of examples. Plot sequeces i two differet ways. Test whether a sequece is strictly icreasig, strictly decreasig, mootoic, bouded above or bouded below - ad formally state the meaig of each of these terms. Test whether a sequece teds to ifiity ad formally state what that meas. Test whether a sequece teds to zero ad formally state what that meas. Apply the Sadwich Theorem for Null Sequeces. Provethatifa adb areullsequecesthesoare a, ca +db ad a b.