Lecture Series (See Chapter i Boas) A basic ad very powerful (if pedestria, recall we are lazy AD smart) way to solve ay differetial (or itegral) equatio is via a series expasio of the correspodig solutio (or itegrad). (This is always possible i regios where the expasio exists.) Sice this process ca be carried out term-by-term, it is ideal for umerical solutios by computer to arbitrary, but specified, accuracy, which is especially useful i situatios where we have o other way to proceed. (This is also how your had calculator works.) Recall, i particular, the familiar Taylor series expasio of a fuctio about the poit x a (also called a Maclauri series for a 0) x a x a f x f a x a f a f a f a (.)!! where the primes sigify derivatives with respect to the variable x. We will retur to the subject of power series after we first cosider the properties of plai old umerical series. Cosider a ifiite sequece of umbers labeled by a idex a, a, a,, a, ;,,, (.) 3 We defie the series to be the correspodig sum, S a (.3). The idividual terms, a, are just umbers or are perhaps expressed as a fuctio of (ad ay other parameters), e.g., cx. This last expressio takes us back to the a power series expasio metioed above. I particular, this expressio, a cx, defies a geometric series where the ratio betwee subsequet terms is a sigle factor, x, i.e., the same factor for each pair of cotiguous terms. Do such series arise i physics? The aswer is yes, all the time! Cosider the simple example of droppig a ball from a height h. For a real ball the collisios with the floor result i the loss of eergy, i.e., the collisios are ielastic (some of the kietic eergy of the CM of the ball is coverted to heat, i.e., radom kietic eergy of the idividual molecules i the ball, each time the ball collides with the floor). If we defie the fractio of kietic eergy lost i each bouce to be x, the the height Physics 7 Lecture Autum 008
after oe bouce is xh (recall that the height of the bouce is determied by whe all the kietic eergy of the CM is coverted ito gravitatioal potetial eergy) E mgh mghx x. (.4) E mgh mgh 0 After bouces the height is h x h a. Such sequeces of umbers that progress by a power of a sigle factor correspod to geometric progressios. The geometric progressio is the simplest progressio. ow we ca ask how far does the ball travel before all kietic eergy is lost? (ote that, i priciple, this takes ifiite time.) I equatios we have D x h h h h h x x x h x h S x. (.5) I these expressios the ellipsis (the 3 dots, ) represet the implied missig terms. For a fairly elastic ball with x 45we fid 4 4 4 D h h h hs. (.6) 5 5 5 So the questio is, ca we give meaig (i.e., a umerical value) to this ifiite geometric series S 4 5 4 5? If the sum of the ifiite umber of terms exists, the that sum is the series, i.e., the value of the series. But how ca we tell if the sum exists? Eve if we kow that the sum exists, how do we evaluate it? ote that it would take a ifiite time for a computer to literally add up a ifiite umber of terms. We must be smarter ad use more powerful techiques as we will see below. First let's defie some termiology. If the sum exists (i.e., it is defied) ad is less tha ifiity, we say the series coverges. If the sum does ot exist (i.e., it is ot defied) or is ifiite, we say the series diverges. Physics 7 Lecture Autum 008
ow cosider how to evaluate the sum. Sice humas, like computers, do best cosiderig fiite thigs, cosider oly the first terms, 4 S a x x x x. (.7) 5 I a obvious otatio, this is called a partial sum. Sice this quatity is guarateed to be fiite (assumig a, ), we ca use ormal algebraic maipulatios (see below for the o-guarateed, ifiite case). Cosider the followig (true but umotivated) maipulatios, 3 xs x x x x S x x S x x x S x x x. (.8) ote that this last expressio implies the special ad very useful relatio (see below) 3 x x x. x (.9) ow cosider the limit x lim S lim x. x (.0) Evaluatig series is closely related to takig limits we must be able to do both! For the geeral case the possibilities are ) lim S exists ad is fiite, i which case the series is said to coverge ad S lim S ; Physics 7 Lecture 3 Autum 008
) lim S does ot exist, because either it exhibits a limit cycle, e.g., S 5 S 5 S 5 0 0 0,, or we have lim S ; i either case the series is said to diverge. It is ofte helpful to cosider the remaider R S S. The series coverges iff (if ad oly if) lim R 0. The series diverges for lim R 0, idepedet of the precise form of the ozero right-had-side. Aother way to express covergece is to say that, if you pick a small parameter,, I ca always fid a large iteger M such that R for all M. (ote that, if you are ot comfortable takig these limits,, you should practice doig so. It will be very useful below.) I our simple case above we have the followig possibilities (ote that x > 0 here, the fractio of eergy lost i each bouce is less tha 00%) x ) x, lim x 0 ad thus the series coverges, with S, x ) x, lim x ad thus the series diverges, 3) x, lim x, S, which is ill-defied, but usig the origial form we fid S ad the series diverges. Clearly, we must be careful i this last case. For our specific example we have 4 D h h 5 h 8h 9 h. 4 5 (.) The distace traveled is fiite, eve though there are a ifiite umber of steps. Of course, whe the height of the bouce is less tha a Agstrom there is little cotributio (ad the motio is difficult to detect). This series is said to coverge quickly. The cotributio from the bouces after bouce 00 is 00 4 4 4 9.6 0. D 00 h h S h (.) 0 5 5 5 Physics 7 Lecture 4 Autum 008
ote that i (realistic) physics applicatios, where the ucertaity i the measuremets is always greater tha zero, we are typically also iterested i comparig to a theoretical calculatio, also with fiite accuracy, i.e., the value of for large. Returig to the more geeral expressio for the geometric series, we defie R S 0 ar a r S ar r a r r 0, (.3) which matches Eq. (.4) i Boas. For r, we ca take the limit as above to fid S a lim S. (.4) r It is clearly importat to be able to tell if a series coverges or ot. Eve though diverget series are sometimes useful (see below), we must be careful whe usig them! I particular, we caot reliably use the same algebraic maipulatios o full sums as we did o the partial sum. For example, we might cosider S 4 8 : x 0 S 4 8 S S, (.5) which is clearly osese (S = ). ext we observe the importace of sigs! Cosider the followig similar series (the Harmoic series), (.6) 3 Physics 7 Lecture 5 Autum 008
which diverges (as we demostrate below), ad, (.7) 3 which coverges. It is clearly essetial that we defie covergece tests that we ca geerally apply, icludig applyig to cases with alteratig sigs. Cosider agai the geeral expressio i Eq. (.3), a S a with all terms fiite,,. To this series/sum we ca apply the followig set of tests. 0) The Prelimiary Test or Divergece Test: if lim a 0, S diverges, if lim a 0, we must test further, i.e., this test is ecessary but ot sufficiet. For example, a, lim a 0 so we test further, but for a, lim a, or for a, lim a, we kow that the series diverges. ASIDE: I the curret discussio we may be worried about the sigs of the terms. If it is ot the case that all a 0, the we first cosider the absolute series S a. If S coverges, the we say S coverges absolutely. If the sum is ot absolutely coverget (i.e., S does ot coverge), S may still coverge coditioally, if there is eough cacellatio. We will discuss this case i more detail below. I ay case covergece always required that lim a 0. ) The Compariso Test: compare the series of iterest to a series whose covergece properties you already kow. This test has two parts. a) Assume we kow that the series a C c, with all c 0, coverges. If c for all, (with some large iteger), the S coverges absolutely. ote that the first - (fiite) terms caot affect covergece, oly the umerical value of S (if it covergeces). Physics 7 Lecture 6 Autum 008
b) Assume we kow that the series D d, with all d 0, diverges. If a d for all,, the S diverges. The series S may still coverge if there are varyig sigs ad eough cacellatio. ASIDE: If a c or a d, we lear othig from these comparisos. Cosider the example S, a,! 3 ad compare to the!! geometric series with x, c, which we kow coverges. We have S, 6 4 0 C. 4 8 6 3 Thus a c, 3 ad S coverges (very rapidly!). The dowside with the compariso test is that to use it we must first kow the covergece properties of may series. So let's develop some other tests. ) The Itegral Test: cosider the expressio ad with the itegrad treated as a (smooth) fuctio of. This will be useful for the case 0 a a for (some large ). The series coverges or diverges depedig o whether the itegral ad is fiite or ifiite (idepedet of the fiite lower limit). You ca see this from the followig figures. Physics 7 Lecture 7 Autum 008
.5 0.5 a.5 0.5 a 3 4 5 6 For the example (above) of the Harmoic series, a above figures, we lear that 3 4 5 6, which is illustrated i the d l. (.8) Thus the Harmoic series (with + sigs) diverges as oted earlier i Eq. (.6). The limitatio with this test is that we must be able to perform the itegral. 3) The Ratio Test: first defie the ratio of adjacet terms i the series ad the the asymptotic value of the ratio, a lim. (.9) a Sice for the geometric series is just the fixed ratio of terms, we lear from our aalysis above of the geometric series coupled with the compariso test that if, S coverges absolutely, may coverge, use further test., S diverges (.0) As familiar examples cosider Physics 7 Lecture 8 Autum 008
a a! 0 coverges,!!? (.) Clearly the case requires further study. I particular, we eed to specify ad uderstad the implicatios of how approaches as, which we ca aalyze a step at a time. To see what the possible behavior looks like we cosider some examples (ote this is a typical style of aalysis simply look at some examples!). Cosider first the Riema Zeta fuctio defied by the series p S p ad apply the itegral test. We have that (.) p d p, p I p p p, p l, p, (.3) where the special case p is just the Harmoic series. Thus the Zeta fuctio exists, i.e., the series coverges, for p (strictly for Re[ p] ) ad diverges otherwise. Back to the ratio test, we have for the Zeta fuctio expadig i powers of (thik about a Taylor series expasio of x p for small x) p p p p p. (.4) Thus we coclude, via the compariso test, that for a geeral series S, Physics 7 Lecture 9 Autum 008
p S coverges for p if as, (.5) S diverges for p while for p we must work harder still. ext cosider the similar series S s for which the itegral test tells us that, s (.6) l, s, m, s d dm s ml s (.7) l i.e., the form of the fial itegral (after the chage of variable) is just as for the Zeta fuctio. O the other had the ratio test for this series has the form (thik about how to perform the idicated expasios for large ) s l l l s, l s (.8) where we saw from the itegral test that this correspods to a coverget sum iff (if ad oly if) s. We ca summarize these isights i the followig geeral result for the ratio test. Keep the largest terms as ad defie the followig sequece of ever more detailed results i this limit. Physics 7 Lecture 0 Autum 008
, S diverges If,, S coverges p p, S diverges else if ad as, p, S coverges s else if, pad as l s, S diverges, s, S coverges etc. (.9) As a example cosider agai our fried the Harmoic series a (.) we have. From Eq., (.30) a where the last step used Eq. (.9), but with x x (a useful result i its ow right), 3 4 x x x x x. x (.3) I Eq. (.30) we see that the coefficiet of -/ is correspodig to p = ad that the coefficiet of l is zero (sice the ext term is ), s = 0. Hece by the rules of Eq. (.9) the Harmoic series diverges as advertised. 4) The Combied Test: combie test ) ad 3) (ad )), agai i two steps. a) Pick a coverget series C c, c 0 ad take the ratio to (the absolute value of ) terms i S a, a if lim, the S is absolutely coverget. (.3) c Physics 7 Lecture Autum 008
This meas that the terms i S, for large, are proportioal to those i C ad both series coverge if oe does. I priciple, we do t care if this proportioality costat is large, while i test ) we were really checkig for proportioality costat. b) Pick a diverget series D d, d 0, ad agai cosider the ratio, a if a 0 ad lim 0, the S diverges. (.33) d Agai we must be careful i the case of alteratig sigs (see below). This last test allows us to simplify the series we are cosiderig by choosig a simpler compariso series with the same asymptotic behavior, which we illustrate with some examples. Cosider the series defied by a 5 3, 3. 4 7 (.34) For large we have a. 3 (.35) 4 So we do t eed to cosider the full complexity of a, but we ca istead cosider the simpler series give by B b,. Both the itegral test ad the ratio test tell us that B coverges, b d, p. (.36) Physics 7 Lecture Autum 008
Hece we use B as the coverget compariso series ad coclude that S is coverget, a b. (.37) As aother example cosider 3 a, 5 5 l 3 l e e 3 a 5 5. (.38) So try the compariso series d 3 5, which we kow diverges from the ratio test, 5 3 3 5 3 3. 5 (.39) So the origial series diverges sice 5 3 a 3 3 0. 5 3 d 3 5 5 (.40) ow let s retur to cosider the case of a series with terms alteratig i sig, which does ot coverge absolutely. It ca still coverge, i which case we say that it exhibits coditioal covergece. A iterestig example is the alteratig sig versio of the Harmoic series, a. The absolute value versio of this series, the usual Harmoic series, we kow to diverge. Such a situatio is addressed by out last test. 5) The Alteratig Sig (ot Absolutely Coverget) Test: a series with terms that alterate i sig, which is ot absolutely coverget, is coditioally coverget if (ad oly if) the terms systematically shrik i magitude ad asymptotically vaish, Physics 7 Lecture 3 Autum 008
a a, lim a 0. (.4) The trick with coditioally coverget series is determiig the value to which they coverge. It is easy to be misled o this issue, especially if oe just looks at the series. The aswer ca deped o how the series is arraged. ote that the test i Eq. (.4) depeds o the order of the terms. From our kowledge of the (Maclauri) series expasio of the logarithm we have 3 4 x x x l x x, (.4) 3 4 which we could obtai directly by itegratig Eq. (.3). Thus we kow that the (stadard order) alteratig sig Harmoic series coverges to l 0.69347. (.43) O the other had, we could arrage the terms i the series i the followig explicitly misleadig fashio (the right-had-side is the ruig sum), 3 5.5333.0333 7 9 3 5.58 4.78 7 9 3 5.544 (.44) which could be iterpreted to mea covergece to.5 if the series is defied i this way. The lesso is that you have to be very careful with ifiite series, especially whe the sigs alterate. I the later case we are effectively subtractig two diverget series i order to obtai a coditioally coverget series. (I the case Physics 7 Lecture 4 Autum 008
above we have the two series A ad B 0 Clearly the order i which we combie the terms ca matter., S A B.) We close this discussio with a summary of useful facts about ifiite series, which follow from the above cosideratios. ) Covergece of a series is ot affected by overall multiplicatio (i.e., multiplyig every term) by a costat, or by chagig a fiite umber of terms. ) Two coverget series, A a, B b, ca be added or subtracted term by term to obtai aother coverget series, C a b A B, i.e., coverget series ca be treated like ordiary umbers. (This is ot true of diverget series.) 3) The terms i a absolutely coverget series ca be re-arraged without fear of chagig the covergece or the value of the series. Coditioally coverget series deped o the order of the terms, both for the questio of covergece ad for the value of the series (i.e., the sum). Physics 7 Lecture 5 Autum 008