Sequences and Series; Finite and Infinite

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1 Sequeces ad Series; Fiite ad Ifiite Calculus 2, Veritas Prep. We are about to do the coolest theorem i calculus. Not the most importat theorem, mid you that s the FTC the coolest. We are about to do the coolest, weirdest, most mid-blowig theorem i all of first-year calculus. I have stories about this theorem that I d be fired if I told you. We are about to do Taylor s theorem. Taylor s theorem, roughly speakig, says that everythig is a polyomial. Trig fuctios, expoetial fuctios, logarithms, ratioal fuctios, all sorts of weird stuff they ca all just be thought of as polyomials (albeit possibly ifiitely-log polyomials). Ad that is woderful ad weird, because polyomials are really ice fuctios. They re cotiuous everywhere they have o holes or asymptotes they have derivatives everywhere we ca plug ay real umber i they re ice ad predictable ad curvy. They re so simple! Remember how easy graphig them is, compared to graphig ratioal fuctios? All you do is factor them! Or thik about how easy it is to evaluate a polyomial at ay radom poit you just plug i the umber, do a lot of arithmetic, ad evetually get it! You ca t do that with trig fuctios (uless you re lucky ad you wat to plug i, say π/2): if f(x) = x 2 + 3x + 5, ad I wat to fid f(2), I just plug 2 i for x: f(2) = = 5 if f(x) = l(x), ad I wat to fid f(2), I just plug 2 i for x: f(2) = l(2) =...um. I do t kow how to work this out as a decimal. Ayway. Taylor s theorem says that everythig is a polyomial. For istace, sie is a polyomial albeit a ifiitely-log oe: si(x) = x x3 3! + x5 5! x7 7! + x9 9! So is e x : e x = + x + x2 2! + x3 3! + x4 4! + This stuff is really cool! But before we start talkig about it fully, there are some prelimiaries we eed to take care of. Namely: Taylor series are a type of series, so we should probably talk about those, ad series are made of idividual terms (a sequece of terms, added together), so we should probably talk about those, too. Sequeces A sequece, i essece, is just a ordered list of umbers. Here are some examples of sequeces:, 2, 3, 4, 5, 6, 2, 3, 4, (the atural umbers) (fractios!),,, 3, 3, 3, 3, 3, 4, 9, 6, 36 2, 4, 6, 8, x, x 2, x 3, x 4, (it s flippig back ad forth!) (kid of borig) (squares!) (eve umbers!) (a sequece of fuctios, rather tha umbers!) If you re a math perso, please do t rui the fu by shoutig out some idiotic commet. Look, I kow the cops are goig to break up this party evetually ad cite us for violatig uiform covergece, ad the i the morig we ll all wake up with deadly hagovers as a result of havig approximated, say, l( + x) far outside its radius of covergece... but for ow, let us just believe i the beauty ad the glory of EVERYTHING BEING POLYNOMIALS. Because it is beautiful. Tomorrow will come whe it will.

2 Note that these are all examples of ifiite sequeces, i.e., sequeces that could go o forever. Questio: I ve listed, for each sequece, the first term, the secod term, the third term, etc., etc.,... what will the th term look like?, 2, 3, 4, 5, 6,,, 2, 3, 4,,,,,,,, ( ), 3, 3, 3, 3, 3, 3,, 4, 9, 6, 36, 2, 2, 4, 6, 8,, 2, x, x 2, x 3, x 4,, x, Aother way of thikig about a sequece ad this is the way we d formally defie it is that it s a fuctio, but whereas basically all of the fuctios we deal with ca take ay real umber as iput, a sequece ca oly take a atural umber as iput. It s a fuctio whose domai is oly the atural umbers (but whose rage could be all real umbers). Here s a picture of what the sequece f() = / looks like: If I wat to fully describe a sequece, the rather tha list terms, it s probably a better idea to just give the formula for the th term: S = or f() = is the sequece, 2, 3, 4, 5, 6,, S = or f() = is the sequece, 2. 3, 4,,, S = ( ) or f() = 2 is the sequece,,,,, ( ), S = 3 or f() = 3 is the sequece 3, 3, 3, 3, 3, 3, S = 2 or f() = 2 is the sequece, 4, 9, 6, 36, 2, S = 2 or f() = 2 is the sequece 2, 4, 6, 8,, 2, S = x or f() = x is the sequece x, x 2, x 3, x 4,, x, Note that I m usig the letter rather tha x because, well, mathematicias ted to use the letter (or sometimes k, or, less ofte, m, i, ad j) to represet atural umbers, whereas x teds to be used to 2

3 represet real umbers. Of course, that s ot a ecessary covetio; it s just usual practice. The cotext of the problem should make it clear whether a variable rages over all possible real umbers (like the x i f(x) = x 2 ) or over just the atural umbers (like the i S = 2). Remember that N is the symbol for the atural umbers, whereas R is the symbol for real umbers. By the facy expressio i the above graph N, I mea that is a atural umber, or, more literally, is a member of the set of atural umbers, with meaig is a elemet/member of. The mai questio we ca ask about a sequece is: does it coverge or diverge? Meaig: as, does the sequece either: a) approach some sigle, fiite umber (coverge), or b) ot? (diverge) Ad as a secodary questio: if it does coverge, what does it coverge to? For example: The sequece S =, i.e., the sequece, 2, 3, 4, diverges, because as, the sequece just keeps gettig bigger ad bigger, i.e., S. The sequece S = / (i.e.,, 2, 3, 4, ) coverges, because as, S 0. So it coverges, ad moreover, it coverges to 0. The sequece S = ( ), i.e., the sequece, 2, 3, 4, 5, coverges, because as, S 0, just as i the last example. Note that, because of the ( ), it bouces aroud it keeps alteratig betwee beig positive ad egative. But it still approaches 0. The sequece S = ( ), i.e., the sequece,,,,, diverges, because as, the limit of S does t exist. It keeps alteratig betwee + ad. It ever settles dow. The sequece of fuctios S = x /, i.e., the sequece, x, x /2, x /3, x /4, coverges to x 0, or just. Series So, those are sequeces. They re kid of cool, but ot that cool, because they re easy to deal with. If we wat to fid whether a sequece coverges, we just take a limit! Not too difficult. But what if we add all the terms i a sequece together? For istace: rather tha havig the sequece:, 2, 3, 4, what if we have the series: This is a very differet creature. We kow that the sequece coverges, but we do t kow whether the series (i which we add everythig together) coverges. More formally, a series, or a sum, is whe we add the terms of a sequece together, rather tha simply listig them. They could be fiite (I add together a fiite umber of terms) or ifiite (I add together a ifiite umber of terms). Here s a example of a fiite series: There are three differet ways we ca write series: I could show each term explicitly: I could write it as a sigle umber: = 50 k=4 or I could write it usig Σ-otatio for sums: = 5k You ve see -otatio before, but as a quick remider, it works somethig like this: k=4 k= 5k = }{{} 5 + the k= term the k=2 term {}}{ }{{} the k=3 term k= the k=4 term 3 {}}{ 5 4 = = 50

4 Or, more geerally: k=b f(k) = f(a) + f(a + ) + f(a + 2) + + f(b 2) + f(b ) + f(b) k=a Obviously, if I have a ifiite series, I ca t write every term explicitly there are a ifiite umber of them! But I ca ask the same questio I ask of sequeces: give a series, does it coverge or diverge? For series, this is a much harder questio to aswer 2. With sequeces, we ca just take a limit; with series, we have just a smatterig of ad hoc methods, some of which might or might ot work for a give series. There are some obvious cosideratios first. Cosider the ifiite series: = = Obviously this diverges. We re addig bigger ad bigger umbers! The sum is gettig bigger ad bigger! So clearly, if each of the idividual terms i the series icrease, the series will diverge. I fact, eve if each of the terms stay the same, the series diverges: = 5 = = So we ca say this: if the idividual terms of a series do ot decrease (i.e., icrease or remai costat), the sequece will diverge. But just because the terms of a series decrease, it is ot always the case that the series coverges. For example, cosider the two series: The terms i both of these series go to 0. But (for reasos which you re about to see) oe of these series coverges, ad the other diverges. Oe last thig: do t get the terms sequece ad series cofused. I do it all the time. Do t follow my example: This is a sequece:, 2, 3, 4, 5, 6,,, This is a series: A Cool, Simple Series Cosider the series: a = + a + a 2 + a 3 + a 4 + for some umber a; for istace, if a = /2, this becomes the series: ( ) = Also ote that it s really oly a iterestig questio to ask about ifiite series, sice fiite series obviously coverge to whatever they add up to. 4

There is a startlig result about this series: it coverges to a : a = + a + a 2 + a 3 + a 4 + = a So, for istace, i the case of a = /2, o matter how may terms we add, the series simply gets closer ad

5 There is a startlig result about this series: it coverges to a : a = + a + a 2 + a 3 + a 4 + = a So, for istace, i the case of a = /2, o matter how may terms we add, the series simply gets closer ad closer to 2: ( ) = = /2 = /2 = 2 Let s prove this. Imagie, for the sake of coveiece, we call this series S: S = + a + a 2 + a 3 + a 4 + The: what happes if we multiply both sides of this equatio by a? We ll have: as = a + a 2 + a 3 + a 4 + a 5 What if we cosider the quatity S as? This must be just: S {}}{{}}{ S as = ( + a + a 2 + a 3 + a 4 + ) ( a + a 2 + a 3 + a 4 + a 5 ) But if I combie those two paretheses, most early everythig will cacel out we ve got a a, ad a a; we ve got a a 2, ad a a 2, etc. The oly thig that will be left is : S as = as But we ca easily solve this for S: S( a) = S = a Ta-da! Now we ca deal with ay series of the form a. I fact, we have quite a bit of power, because most of our tests for covergece oly tell us whether a certai series coverges they do t tell us what such series coverge to. But this test does. For istace, if we have the series: which is really just the series: ( ) 2 3 we kow, usig this formula we just proved, that this series must coverge, ad it must coverge to 3: ( ) 2 = 3 2/3 = /3 = 3 Ufortuately (or fortuately, depedig o how you look at it), there s a pretty substatial caveat. Namely: what if we have the series 2 : (2) =

6 Does this series coverge? Certaily ot! It just keeps gettig bigger ad bigger! It goes to! But accordig to our theorem, it does coverge: (2) = 2 = This series is ot. You caot keep addig powers of two ad get. That is utterly ridiculous. You will keep gettig bigger ad bigger umbers. The problem is pervasive cosider a = : () = = So really, we should specify that i order for this series to coverge, a must be less tha. Ad, actually, we should say that a has to be betwee ad, because umbers below give us the same problem: ( ) = + = does t coverge! keeps flippig betwee 0 ad ( 2) = = does t coverge! gets bigger ad bigger, but with alteratig sigs So we should restate our theorem a little bit more precisely: a =, but oly if < a < a This type of series, by the way, is kow as a geometric series. Not that the ame really meas much, but other people use the ame, so you should probably kow it. Aother Type of Series Here s aother series: what if I have somethig like: or, writte i -otatio: = = (This is called the harmoic series, by the way.) Does this coverge? It might, sice the terms get smaller. It does t obviously diverge. But how do we deal with it? Allow me to suggest this: did the last series remid you at all of a itegral? a improper itegral? remember how weird it was that sometimes we have itegrals to that coverge? i.e., shapes that are ifiitely log but have fiite area? like, for example: x 2 = Ad if we have a series, it s kid of like addig up areas! Thik about it like this: remember that the sequece f() = / looks like this (just a buch of poits): 6

7 but if we tur each of these poits ito the top of rectagles, ad the fid the areas of all the rectagles, that s like the series : = So I guess we could make the aalogy: a series is to a sequece as a itegral is to a smooth curve. Ayway, what if we simultaeously cosider the fuctio f() = /, where ca be ay real umber (ot just a atural umber), ad the fid the area udereath that curve? 7

8 Clearly, there will be slightly less area udereath the curve / tha there is i the boxes (i.e., i the series ). Put differetly, we kow that: d < But we kow how to work out that itegral! We kow: d = [l()] = = l( ) l() = So the area udereath the curve / is ifiite... but the area udereath the boxes is greater tha the area uder the curve, meaig the boxes must have ifiite area, too, ad sice the boxes are the series, the series must be ifiite, too, ad thus the series diverges! so d < = < = = = I fact, we ca state this as a more geeral priciple: if if a a f() d diverges, the f() d coverges, the =a =a f() also diverges f() also coverges We could have use this test to figure out that the geometric series (our previous example) coverges: we 8

9 had the series 0 a, where 0 < a <, ad we kow: [ a d = l(a) a 0 ] = l(a) a l(a) a0 = l(a) 0 l(a) =, which is fiite l(a) so sice 0 a d coverges, 0 a must coverge, too. The disadvatage is that usig this test, we kow that it coverges, but we do t kow what it coverges to. It does t coverge to, because a) we l(a) already kow it coverges to /( a), ad clearly those two thigs are ot equal, ad b) this is because a itegral is ot precisely the same as a sum there s extra area i the boxes that the smooth curve of the itegral misses (look at the previous graph the area of the itegral ad of the sum are similar, but ot idetical). Plus, this test wo t always work. What if we have a series like? We ca t cosider!! d, because we have o idea how to atidifferetiate!. Likewise with, say, 0, sice we ca t atid- ( ) ifferetiate that, either. (Our rule for a x dx oly works whe a > 0, because otherwise the fuctio is really really bizarre ad somewhat itractable.) Aother Test For Covergece/Divergece Here s aother cool series: Ay idea what its formula is? No? It s a factorial! It s /!: =! = So the obvious questio we ca ask about this series, like about every other series, is: does it coverge? Each of the terms are gettig smaller, so it might. But we ca t come up with a ice little algebraic formula for its sum, like we could for a geometric series. We ca t treat it like a itegral ad see whether the itegral coverges, because we do t kow how to atidifferetiate! ad it s ot just that we do t kow how to, either, but that the very idea of atidifferetiatig (or differetiatig) such a fuctio makes o sese, because! is t a smooth curve like x 2 or e x. (What is 3.5!? Is it ? Is it 3 2? Does the questio ot eve make sese?) What we ll have to do is compare it to a series that we do kow coverges. I the last example, we made this argumet:. we kow that the itegral 2. we kow that the series 3. therefore, = = diverges. d diverges; is greater tha 9 d; 0

10 We ca make a similar argumet with /!. Here s what we kow: we kow that! is a skyrocketig fuctio it goes up really, really fast. Compare it, for example, with x 2 (or 2 ) ad 2 : 2 2! , It takes! slightly loger to get goig it s ot i first place util = 4 but after that, it zooms up. Which meas, of course, that /! is zoomig dow, zoomig dow to 0 far faster tha / 2 or /2. BUT WAIT. We kow that /2 coverges. That s just a geometric series: AND we kow that 2 =. ( ) = 2. /2 = /2 = 2 so the we kow that 2 >, at least after = 4! =4 2 > (Note how I chaged the s to start at 4. Not a big deal, though. I ll address it more i a momet.) Sice the /2 series coverges, ad sice /! is less tha /2, the /! series must coverge, too: =4 2 }{{} this coverges > =4! =4! }{{} so this must, too Of course, our origial questio was ot about /! whe starts at 4; our questio was about /! whe starts at. But this is t a cocer. If I just add the first few terms back i, I ll just be addig some fiite umber, ad that wo t all-of-sudde make the series diverge. Put more formally: the = term! = {}}{ }{{} the =2 term! = =4 the =3 term {}}{ 6 + We kow that the series from = 4 coverges, so addig 0/6 wo t make it diverge. It will still coverge, eve if we add 0/6. The sum from = will coverge to a slightly differet poit, yes (a poit 0/6! =4! 0

11 greater tha what the sum from = 4 coverges to), but sice we do t kow what the sum from = 4 coverges to i the first place, this does t really chage our situatio. The poit is: =! coverges. Ayway, the poit is, this is aother method we ca use to determie whether series coverge or diverge: we ca compare them to a series we kow somethig about. We might have a series that s always greater tha a series which diverges; therefore, we kow it diverges. We might have a series that s always smaller tha a series which coverges; therefore, we kow it coverges. (If we have a series that s always greater tha a series which coverges; we kow othig, sice the series could either coverge or diverge (the latter if it s too much greater tha the series that coverges). If we have a series that s always less tha a series that diverges, we kow othig, sice the series could either coverge or diverge (the latter if it s ot smaller eough tha the series which diverges).) Here is a simpler form of the two argumets we could make: ARGUMENT, VERY BRIEFLY: We have a series that s always smaller tha a series which coverges; therefore, we kow it coverges. ARGUMENT, SLIGHTLY MORE DETAILED:. we kow that series B coverges; 2. the terms of series B are always greater tha the terms of series A; 3. therefore series B is greater tha series A; 4. put differetly, series A is less tha series B; 5. therefore, sice series B coverges, series A also coverges. ARGUMENT, VERY BRIEFLY: We have a series that s always greater tha a series which diverges; therefore, we kow it diverges. ARGUMENT, SLIGHTLY MORE DETAILED:. we kow that series B diverges to ; 2. the terms of series B are always less tha the terms of series A; 3. therefore series B is less tha series A; 4. put differetly, series A is greater tha series B; 5. therefore, sice series B diverges, series A also diverges. We could use this sort of argumet for all sorts of differet series for example, we could have a series that we could t take a itegral of, because we do t kow how to atidifferetiate it, but we could compare it to oe we did kow how to atidifferetiate.

12 Fid a formula for the th term of the sequece: Problems.,,,, 2.,,,,, 3., 4, 9, 6, 25 4., 4, 9, 6, 25, 5. 0, 3, 8, 5, 24, 6. 3,,, 0,, 7., 5, 9, 3, , 6, 0, 4, 8, 9. 2, 3, 5, 7,, 3, 7, 9, 23, 29 For each of the followig sequeces, write out the first few terms, ad the determie whether it coverges or diverges. If it coverges, what does it coverge to? 0. S = ( ) ( S = ) 2. f() = (! 27. S = 2 ) ( ) 2 2. f() = ( ) S = 2 + ( ) 4. S = 2 + (0.) 28. f() = ( )+ 2 ( 29. S = ) 2 5. S = ( ) 6. f() = ( ) 7. S = + ( ) 8. S = f() = S = f() = f() = S = S = + ( ) ( 25. f() = ( ) ) 30. f() = si() 3. f() = si2 () S = + ( π 33. f() = si 2 + ) 34. f() = l() l( + ) 35. f() = f() = f() = l( + ) 38. f() = l() l(2) 39. f() = 8 / 40. f() = (0.03) / 2

13 Write out each of the followig sums (give i Σ-otatio): k=2 k=5 ( ) ( ) k 4. (3k + ) 43. k! k=3 42. k=3 (2 k ) 44. k=4 k=2 3 k Write each of the followig sums i Σ-otatio f(x ) x + f(x 2 ) x f(x ) x Which of the followig series coverge, ad which diverge? Why? For those that coverge, ca you determie what they coverge to? 49. ( ) cos(π) 70. = ( ( ) 5 = = = = ( 2 ) l() l(/) ) ( ) ( 2 ) = cos(π) 5 e e 2 ( ) 5 k 2 k 2 2 k 4 l k k ( ) + = = 2 3! 000 x ( ) x ( ) x 2 = l( k) k k k ( ) l + 3

14 = = = ( + ) + ( ) 2 / 2 /(+) ( ) l( + 2) l( + ) k k 2 + 3k + 2 e ke k2 k 2 2 k3 ( 2 ) 89. For what values of p does the series k p coverge? for what values of p does it diverge? 90. The followig is a sketch of a fractal, costructed by takig a square, drawig a square iside of it usig the midpoits as corers, drawig a square iside of that usig the midpoits as corers, ad so forth, ad ifiitum. Fid a) the total area eclosed by all the squares, ad b) the total perimeter of all the squares. 9. This is a famous fractal: Helge vo Koch s sowflake curve, kow geerally as the Koch curve or Koch sowflake. It is draw by takig a equilateral triagle, the pastig o three equilateral triagles i the ceterthird of each side, the pastig o equilateral triagles ito the ceter-third of the sides of the triagles costructed i the previous step, ad so forth. Show that a) the total area eclosed by the Koch curve is fiite (what is it?), but that b) its perimeter is ifiite. (Sorry the sketch is t better; go olie if you wat a clearer picture (ad aimatios!).) 4

Chapter 6: Numerical Series

Chapter 6: Numerical Series 327 Chapter 6 Overview: Sequeces ad Numerical Series I most texts, the topic of sequeces ad series appears, at first, to be a side topic. There are almost o derivatives or itegrals