Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece,, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet as a =, the secod elemet as a 2 =, the third elemet as a 3 = 2, etc.; so every positive iteger gets matched with a sequece elemet. We ca thik of the subscripts of the a as a way to refer to a specific sequece elemet. I particular, sice the subscripts cout the sequece elemets, it would t make sese to use a o-iteger subscript. Aother way to thik of a sequece is as a list where order matters; so a grocery list is ot a sequece, sice the order of items o the list is irrelevat, while a list of istructios for assemblig furiture certaily is a sequece, sice the order i which you complete the steps will make a differece i the outcome. There may be may differet ways to write the same sequece. If the patter is obvious, we could just write out the first few terms of the sequece, such as i this case we write, 4, 9, 6,...; {a } = {, 4, 9, 6,...}. Aother way to deote a sequece is to write a formula for the th elemet of the sequece. Usig the previous example, we ca deote the same sequece by specifyig that the th term of the sequece is a = 2. The a = 2 =, a 2 = 2 2 = 4, a 3 = 3 2 = 9, etc., so this is just aother way to write the origial sequece. Aother way to refer to the same sequece is by writig {a } = { 2 } =. Notice that the sequece {b } = { 2 } =3 is differet from {a } = { 2 } = ; the sequece {b } starts out 9, 6, 25,... ad misses the first few terms of {a }. Agai, a sequece is a fuctio, so that the order of the terms makes a differece. For istace, thik about the two sequeces below: {a } = {0, 9, 8, 7, 6, 5, 4,...} {b } = {9, 0, 8, 7, 6, 5, 4,...}. Eve though they oly differ i the first two places, the sequeces are distict. There are several differet ways to defie a sequece. We could use a fuctio defiitio, as we did above whe we set the th sequece elemet to be a = 2.
We ca also defie sequeces recursively. To build a recursive sequece, we choose some iitial terms, ad the the remaiig terms of the sequece are chose usig previous terms. For example, the extremely importat Fiboacci sequece is defied recursively. We choose the iitial terms f = ad f 2 =. To get the remaiig terms of the sequece, we set f = f + f 2 ; so the f 3 = f 2 + f = + = 2; f 4 = f 3 + f 2 = 2 + = 3. The first few terms of the sequece are writte out below: Fiboaci Sequece) {f } = {,, 2, 3, 5, 8, 3, 2, 34, 55,...}. May sequeces ivolve factorials; for ay positive iteger, we defie For cosistecy, we defie 0! =.! = ) 2) 3)... 3 2. The sequece {a } = {!} = ca be thought of as a recursive sequece; if we write a =, a 2 = 2 a, a 3 = 3 a 2..., we ca defie the sequece recursively usig the iitial coditio a = ; the remaiig terms are defied recursively by a = a. Example. Write a formula for the th term of the sequece begiig 4, 2 9, 3 6, 4 25,... Sice a = 4, a 2 = 2 9. etc., it appears that the umerator of the fractio matches up with the subscript of the a i. Similarly, it looks as if we ca get the deomiator of a i by squarig i +. The oly thig left to take care of is the egative sig. Sice the egatives appear o a i whe i is odd, the factor ) i should take care of the issue. So we guess that a = ) + ) 2. Let s check the first few terms: accordig to our work, a = ) 2) 2 = 4, 2
which matches up with the first term of the sequece. Let s check a 2 : a 2 = ) 2 2 3) 2 = 2 9. Similarly, a 3 = ) 3 3 4) 2 = 9 6. Whe ivestigatig the behavior of sequeces, it ofte helps to display the sequece graphically. For istace, the sequece, 2, 3, 4,..., whose terms are give by a = 2 ca be graphed as follows: 0 2 3 4 5 6 7 8 Notice that, sice a sequece has the itegers as its domai, the graph cosists of discrete poits as opposed to a coected lie or curve. Agai, this makes sese sice we do t defie, say, a.5th sequece elemet or did we assig outputs to ay o-itegers). We will be iterested i the ed behavior of sequeces, the way that {a } behaves for icreasigly large. A sequece ca coverge or diverge. Covergece simply meas that the sequece elemets a approach a fiite) umber as icreases without boud. Defiitio. The sequece {a } coverges to a limit L if the sequece elemets ca be made as close as we like to L by choosig large eough. We write a L or lim a = L. The graph of the sequece {a } = { } = idicates that a coverges to 0: 2 0 2 3 4 5 6 7 8 Of course, may sequeces do ot coverge; thus we record aother defiitio: Defiitio. The sequece {a } diverges if it does ot coverge to ay umber L. There are several differet ways that a sequece ca diverge. The sequece whose terms are give by a = diverges because the elemets icrease without boud: 3
We say that this sequece diverges to ifiity, ad write a. Similarly, if the sequece elemets decrease without boud, as i the followig graph, we say that the sequece diverges to egative ifiity, a : A sequece may diverge eve if its elemets do ot to icrease or decrease a great deal. For istace, the diverget sequece { ) } = is graphed below: This sequece diverges because it ever approaches a specific umber, but simply bouces back ad forth betwee ad. We would like to have methods for determiig whether a sequece coverges or diverges; i additio, if a sequece coverges, we would like to determie its limit. Fortuately, it turs out that there are may similarities betwee sequeces ad fuctios. Earlier we looked at the sequece {a } = { } =. Although this is differet from the fuctio fx) = x we ca evaluate f for o-itegers!), it is similar eough that we ca use what we kow about the fuctio to help us determie whether or ot {a } coverges, ad eve to fid the sequece s limit. Notice that fx) ad a match up wheever x is a iteger: 4
This leads us to the followig theorem: Theorem. Suppose that {a } is a sequece ad fx) is a cotiuous fuctio so that f) = a wheever is a iteger. If lim fx) exists, the lim a exists ad x lim fx) = lim a. x We ca ow cofirm our suspicio that {a } = { } = coverges to 0: sice fx) = x is a fuctio so that f) = a whe is a iteger ad lim fx) = 0, the x as well. lim a = 0 Because of the correspodece betwee fuctios ad sequeces, we ca borrow a few rules about fuctio limits to use i the cotext of sequeces. Theorem. If {a } ad {b } are sequeces so that {a } coverges to a real umber L ad {b } coverges to a real umber M, the. Sums/differeces) lim a ± b m ) = L ± M 2. Products) lim a b = L M 3. Quotiets) lim a b ) = L, provided that M 0 M 4. Costat multiples) lim ca = cl, where c is ay costat 5. Cotiuous fuctios) If fx) is cotiuous, the lim fa ) = f ) lim a = fl). The first rule says that we ca create a ew sequece by addig the correspodig terms of {a } ad {b }; if both the origial sequeces coverge, so will the ew oe. The remaiig rules have similar iterpretatios. Notice that the rules tell us othig about the idicated limits if either of {a } or {b } diverge. 5
The similarities betwee ormal fuctios ad sequeces gives us aother tool: L Hopital s Rule. For example, we ca use L Hopital s Rule to determie if the sequece whose th term is a = l le + ) coverges. Sice fx) = l x lex + ) is a fuctio that matches up with a wheever x is a iteger, we check the limit of f: lim x ) x l x lex + ) = lim l x ex )) x = l lim x ex LR = l lim x e ) = l e ) = l e = l e =. Sice fx) approaches, the sequece a coverges to. Yet aother helpful tool that we gai from our study of limits of fuctios is the Squeeze Theorem: Theorem. Let {a }, {b }, ad {c } be sequeces of real umbers. a b c for all > N, ad if If there is a N so that the lim a = lim c = L, lim b = L. Example. Determie if the sequece whose th term is give by a = si 3 coverges or diverges. Sice si, we kow that 3 real umbers, we calculate lim x x 3 = 0 ad So the origial sequece coverges to 0 as well. si 3 3. Viewig 3 ad 3 as fuctios o the lim x x 3 = 0. O a related ote, we record the followig useful theorem: 6
Theorem. If lim a = 0, the lim a = 0 as well. I other words, if a sequece has both positive ad egative terms, ad if makig all of the egative terms positive meas that this ew sequece coverges to 0, the the origial sequece does as well. Some sequeces are defied i a way that makes them difficult to work with as fuctios. For istace, cosider the sequece whose th term is a = 2!. Sice we do ot kow how to calculate limits of fuctios ivolvig factorials, the correspodig sequece could be hard to work with. Fortuately, the followig theorem ca help us evaluate the limits of several differet types of difficult sequeces. Theorem. The sequeces below coverge to the idicated limits:. lim = 2. lim b = for ay costat b > 0) 3. lim b = 0 if < b < ) 4. lim + b = e ) b for ay costat b) b 5. lim! = 0 for ay costat b) Example. Determie if the sequeces whose th term coverges or diverges. a =. This sequece ca be rewritte:. =.), which coverges to by the secod rule. Example. Determie if the sequeces whose th term is b = 0 ) coverges or diverges. 7
This sequece matches the form of the fourth rule: lim 0 ) = lim + 0 ) = e 0. Thus the sequece coverges. Example. Determie if the sequeces whose th term is c = ) 0 ) coverges or diverges. This sequece is more challegig; it does ot match up precisely with ay of the give rules, so we will eed to thik about it differetly. Notice that the factor of ) will chage the sig of the terms back ad forth. Let s thik about the sequece i two parts whe is odd ad whe is eve. If we could restrict to be odd, the lim ) 0 ) = lim 0 ) = e 0. O the other had, if we oly cosidered eve, the lim ) 0 ) = lim 0 ) = e 0. Sice the limits do ot match up, the sequece is jumpig back ad forth betwee two differet umbers, ad ever approaches a sigle umber. Thus the sequece diverges. The Nodecreasig Sequece Theorem The last tool that we will add to our collectio of theorems o sequece covergece is oly applicable to certai types of sequeces. We eed a ew defiitio here: Defiitio. A sequece {a } is odecreasig if a i a i+ for all i. I other words, the elemets of the sequece may stay the same or get larger, but they may ot become ay smaller as the sequece progresses. As a example, the sequece { } { {a } = = + 2, 2 3, 3 } 4,... = is odecreasig, as is the costat sequece {a } = {} =. 8
As with ay other type of sequece, a odecreasig sequece may either coverge or diverge. If a odecreasig sequece diverges, the sice the sequece elemets ca ever decrease, the sequece itself must diverge to ifiity: O the other had, a odecreasig sequece may coverge to a limit L: I this case, sice the sequece is odecreasig ad coverges to L, it is clear that the sequece elemets ca ever be larger tha L. This leads to a defiitio that is applicable to all sequeces, ot just odecreasig sequeces: Defiitio. A sequece {a } is bouded from above if there is some umber M so that a M for all. We say that M is a upper boud of {a }. If o umber less tha M is a upper boud for {a }, we call M the least upper boud of {a }. If the umber M is a upper boud of a sequece, the the sequece ca ever cross the horizotal) lie y = M, ad the least upper boud is the lie with the smallest y value. For example, i the graphic below, the sequece elemets ca be made as close as we like to the lie y = 2, but they will ever cross the lie. Thus 2 is the least upper boud for the sequece: The sequece {a } = {2, 4, 6, 8,...} has o upper boud. However the sequece {a } = {, 2, 3, 4,...} has may upper bouds; for istace o sequece elemet is ever greater tha 2, or π, or, so all of these are upper bouds. However, the least upper boud of the sequece is ; o smaller umber ca be a upper boud sice a =. 9
A geeral sequece that has a upper boud may or may ot coverge. For istace, the sequece {a } = {,,,,...} has least upper boud, but does ot coverge: However, if we kow that a sequece is odecreasig ad has a upper boud M, we automatically kow that the sequece coverges: the sequece elemets ca ot decrease, or ca they become larger tha M. I particular, if the odecreasig sequece has least upper boud L, the the sequece coverges to L: Theorem. The Nodecreasig Sequece Theorem A odecreasig sequece of real umbers coverges if ad oly if it is bouded above. If the odecreasig sequece coverges, the it coverges to its least upper boud. Example. Does the sequece whose th term is a = + coverge or diverge? We otice that we ca apply the Nodecreasig Sequece Theorem to this problem, sice + < + + 2. If we ca fid a upper boud for the sequece, the it coverges. Now we kow that + < 0
sice < + + as for all > 0. I additio, o smaller umber ca be a upper boud, because we ca make close to as we like. By the Nodecreasig Sequece Theorem, we see that {a } coverges to.