# Sequences I. Chapter Introduction

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1 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

2 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, , 4, 8, 6, 32, , 2,0, 2,0, 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.

3 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.

4 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 si 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

5 2.4. SEQUENCES TENDING TO INFINITY 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 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 =

8 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.

9 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 /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 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 = 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

11 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 [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.

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