Math 2784 (or 2794W) University of Connecticut

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1 ORDERS OF GROWTH PAT SMITH Math 2784 (or 2794W) Uiversity of Coecticut Date: Mar. 2, 22.

2 ORDERS OF GROWTH. Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really wat to uderstad their behavior. It also helps you better grasp topics i calculus such as covergece of improper itegrals ad ifiite series. We wat to compare the growth of three differet kids of fuctios of x, as x : power fuctios for r > (such as x 3 or x = x /2 ), expoetial fuctios a x for a >, logarithmic fuctios log b x for b >. Some examples are plotted i Figure over the iterval [, ]. The relative sizes are quite differet for x ear ad for larger x. (Some coefficiets are icluded o 2 x ad x 2 to keep them from blowig up too quickly i the picture.) y y = 35 2x y = 5 x2 y = x y = log(x) x Figure. Graphs of several fuctios for x [, ]. All power fuctios, expoetial fuctios, ad logarithmic fuctios (as defied above) ted to as x. But these three classes of fuctios ted to at differet rates. The mai result we wat to focus o is the followig oe. It says e x grows faster tha ay power fuctio while log x grows slower tha ay power fuctio. (A power fuctio meas with r >, so /x 2 = x 2 does t cout.) log x Theorem.. For each r >, lim = ad lim x ex x =. This is illustrated i Figure 2. At first the fuctios are icreasig, but for larger x they ted to. After we prove Theorem. ad look at some cosequeces of it i Sectio 2, we will compare power, expoetial, ad log fuctios with the sequeces! ad ad evetually show that

3 2 PAT SMITH Figure 2. Graphs of x 3 /e x ad log(x)/x for x [, ]. betwee ay two fuctios with differet orders of growth we ca isert ifiitely may fuctios with differet orders of growth betwee them. 2. Proof of Theorem. ad some corollaries Proof. (of Theorem.) First we focus o the limit /e x. Whe r = this says x (2.) as x. ex This result follows from L Hopital s rule. To derive the geeral case from this special case, write ( ) x/r r (2.2) e x = rr e x/r. With r stayig fixed, as x also x/r, so (x/r)/e x/r by (2.) with x/r i place of x. The the right side of (2.2) teds to as x, so we re doe. Now we show (log x)/ as x. Writig y for log( ) = r log x, log x = y/r e y = r y e y. As x, also y, so (/r)(y/e y ) by (2.) with y i place of x. Thus (log x)/. p(x) Corollary 2.. For ay polyomial p(x), lim x e x =. Proof. By Theorem., for ay k > we have x k /e x as x. This is also true whe k =. Writig p(x) = a d x d + a d x d + + a x + a, we have p(x) e x x d = a d e x + a x d x d e x + + a e x + a Each x k /e x appearig here teds to as x, so p(x)/e x teds to as x. (log x) k Corollary 2.2. For ay r > ad k >, lim x =. Proof. Let y = log( ) = r log x, so (log x) k = yk /r k e y = r k yk e y. As x, also y. Therefore (/r k )(y k /e y ) by Theorem. (sice r k > ). We derived Corollary 2.2 from Theorem., but the argumet ca be reversed too. (So we could cosider Corollary 2.2 as the mai result ad Theorem. as a cosequece of it.) Take k = i e x.

4 ORDERS OF GROWTH 3 Corollary 2.2 to get the log part of Theorem. ad use the chage of variables y = e x i /e x to get the expoetial part of Theorem. from Corollary 2.2. Specifically, whe y = e x (log y)r =, ex y ad as x we have y = e x, so by Corollary 2.2 we get (log y) r /y. /e x as x. (log x) k Corollary 2.3. For ay ocostat polyomial p(x) ad positive k, lim x p(x) =. Therefore Proof. For large x, p(x) sice ozero polyomials have oly a fiite umber of roots. Write p(x) = a d x d + a d x d + + a x + a, where d > ad a d. The (log x) k p(x) = (log x)k x d a d + a d /x + + a /x d. As x, the first factor teds to by Corollary 2.2 while the secod factor teds to /a d, so the product teds to. Corollary 2.4. As x, x /x. Proof. The logarithm of x /x is log(x /x ) = (log x)/x, which teds to as x. Expoetiatig, x /x = e (log x)/x e =. Replacig e x with a x for ay a > ad log x with log b x for ay b > leads to completely aalogous results. Theorem 2.5. Fieal umbers a > ad b >. For ay r > ad iteger k >, lim x a x =, lim (log b x) k x =. Proof. To deduce this theorem from the earlier results, write a x = e (log a)x ad log b x = (log x)/(log b). The umbers log a ad log b are positive. The, for istace, if we set y = (log a)x, a x = xr e (log a)x = y r (log a) r e y. Whe x, also y sice log a >, so the behavior of /a x follows from that of y r /e y usig Theorem.. Sice log b x = (log x)/(log b) is a costat multiple of log x, carryig over the results o log x to log b x is just a matter of rescalig. For istace, if we set y = log x, so log b x = y/ log b, the (log b x) k = yk /(log b) k e ry = (log b) k y k (e r ) y. As x, also y, so the expoetial fuctio (e r ) y domiates over the power fuctio y k : y k /(e r ) y. Therefore (log b x) k / as x. For ay ocostat polyomial p(x), it follows from Theorem 2.5 that lim x p(x) (log =, lim b x) k ax x p(x) i the same way we proved Corollaries 2. ad 2.3. =

5 4 PAT SMITH 3. Growth of basic sequeces We wat to compare the growth of five kids of sequeces: power sequeces r for r > :, 2 r, 3 r, 4 r, 5 r,... expoetial sequeces a for a > : a, a 2, a 3, a 4, a 5,... log sequeces log b for b > :, log b 2, log b 3, log b 4, log b 5,...!:, 2, 6, 24, 2,... :, 4, 27, 256, 325,... The first three sequeces are just the fuctios we have already treated, except the real variable x has bee replaced by a iteger variable. That is, we are lookig at those old fuctios at iteger values of x ow. Some otatio to covey domiatig rates of growth will be coveiet. For two sequeces x ad y, write x y to mea x /y as. I other words, x grows substatially slower tha y (if it just grew at half the rate, for istace, the x /y would be aroud /2 rather tha ted to ). For istace, 2 ad. (The otatio is take from [, Chap. 9], which has a whole chapter o orders of growth.) Remark 3.. The otatio x y does ot mea x < y for all. Maybe some iitial terms i the x sequece are larger tha the correspodig oes i the y sequece, but this will evetually stop ad the log term growth of y domiates. For istace, 2 eve though 2 < for all small. Ideed, the ratio 2 = teds to as, but the ratio is ot small util gets quite large. Theorem. tells us that (3.) log r e for ay r >. By Theorem 2.5, we ca say more geerally that (3.2) log b r a for ay a > ad b >. How do the sequeces! ad fit ito (3.2)? They belog o the right, as follows. Theorem 3.2. For ay a >, a!. Equivaletly, a lim!! =, lim =. Proof. To compare a ad!, we use Euler s itegral formula for!:! = x e x dx. (This formula ca be proved by iductio o usig itegratio by parts.) The itegral has for a lower boud the same itegral carried out just over [, ]:! > x e x dx.

6 ORDERS OF GROWTH 5 O the iterval [, ], e x has its smallest value at the right ed: e x e. Therefore x e x x e o [, ]. Itegratig both sides of this iequality from x = to x = gives ( Therefore! > e ) +, so x e x dx = e x e dx x dx = + e + ( ) = e +. a! < a ( ae ) + (/e) (/( + )) =. This fial expressio is a upper boud o a /!. How does it behave as? For large, ae/ /2, so (ae/) (/2). Therefore (ae/). Sice the other factor ( + )/ teds to, we see our upper boud o a /! teds to, so a /! as. To show the other part of the theorem, that!/ as, we will get a upper boud o! ad divide the upper boud by. Write e x as e x/2 e x/2 i Euler s factorial itegral:! = x e x dx = (x e x/2 )e x/2 dx. The fuctio x e x/2 drops off to as x. Where does it have its maximum value? The derivative is x e x/2 ( x/2) (check this), so x e x/2 vaishes at x = 2. Checkig the sigs of the derivative to the left ad right of x = 2, we see x e x/2 has a maximum value at x = 2, where the value is (2) e. Therefore x e x/2 (2) e for all x >, so Dividig throughout by gives! = (x e x/2 )e x/2 dx (2) e e x/2 dx = (2) e e x/2 dx = (2) e 2. ( )! 2 2. e Sice 2 < e, the right side teds to, so!/ as. Remark 3.3. I the proof we showed ( ) (! 2 + ). + e e The true order of magitude of! is (/e) 2π, by Stirlig s formula [2, pp. 6 23]. The fact that a! is ituitively reasoable, for the followig reaso: each of these expressios (a,!, ad ) is a product of umbers, but the ature of these umbers is differet.

7 6 PAT SMITH I a, all umbers are the same value a, which is idepedet of : a = a } a {{ a}. times I!, the umbers are the itegers from to :! = 2 3 ( ). Sice the terms i this product keep growig, while the terms i a stay the same, it makes sese that! grows faster tha a (at least oce gets larger tha a). I, all umbers equal : = } {{ }. times Sice all the terms i this product equal, while i! the terms are the umbers from to, it is plausible that grows a lot faster tha!. To summarize our results o sequeces, we combie (3.2) ad Theorem 3.2: log b r a! Here a >, b >, ad r > (ot just r >!). All sequeces here ted to as, but the rates of growth are all differet: ay sequece which comes to the left of aother sequece o this list grows at a substatially smaller rate, i the sese that the ratio teds to. For example, ca we fid a (atural) sequece whose growth is itermediate betwee ad r for every r >? That is, we wat to fid a sigle sequece of umbers x such that x r for every r >. Oe choice is x = log. Ideed, log = log, so log, ad for ay r > log r = log r, which teds to sice r > ad log grows slower tha ay power fuctio (with a positive expoet) by Theorem.. Usig powers of log, we ca write dow ifiitely may sequeces with differet rates of growth betwee ad every sequece r for r > : log (log ) 2 (log ) 3 (log ) k r, where k rus through the positive itegers. Is it possible to isert ifiitely may sequeces with differet rates of growth betwee ay two sequeces with differet rates of growth? Theorem 3.4. If x y, there are sequeces {z () }, {z (2) }, {z (3) },... such that x z () z (2) z (3) y. Proof. Sice x /y, for large the ratio x /y is small. Specifically, < x /y < for large. For small positive umbers, takig roots makes them larger but less tha : < a < = < a < a < 3 a < < k a < <. Sice x /y < for large, this presets us with the iequalities < x x x x < < y y 3 < < y k < < y for large ad k =, 2, 3,.... Multiply through by y : (3.3) < x < x y < x /3 y 2/3 < < x /k y /k < < y.

8 ORDERS OF GROWTH 7 For k < l, the ratio of the k-th root sequece to the l-th root sequece is x /k y /k ( ) /k /l x x /l y /l =. y Sice /k /l >, this ratio teds to as. Therefore (3.3) leads to ifiitely may sequeces with growth itermediate betwee {x } ad {y }, amely the sequeces z (k) = x /k y /k for k = 2, 3, 4,... : (3.4) x x y x /3 y 2/3 x /k (If you wat to label the first sequece with k =, set z (k) y /k y. = x /(k+) y /(k+) for k =, 2, 3,....) The differece betwee (3.3) ad (3.4) is that (3.3) is a set of iequalities which is valid for large (amely large eough to have x /y < ), while (3.4) is a statemet about rates of growth betwee differet sequeces: it makes o sese to ask if (3.4) is true at a particular value of, ay more tha it would make sese to ask if the limit relatio Refereces + is true at = 45. [] D. E. Kuth, R. Graham, O. Patashik, Cocrete Mathematics, Addiso-Wesley, 989. [2] S. Lag, Udergraduate Aalysis, 2d ed., Spriger-Verlag, 997.