CHAPTER 10 INFINITE SEQUENCES AND SERIES
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1 CHAPTER 10 INFINITE SEQUENCES AND SERIES 10.1 Sequeces 10.2 Ifiite Series 10.3 The Itegral Tests 10.4 Compariso Tests 10.5 The Ratio ad Root Tests 10.6 Alteratig Series: Absolute ad Coditioal Covergece 10.7 Power Series 10.8 Taylor ad MacLauri Series 10.1 Sequeces Defiitio A ifiite sequece or more simply a sequece is a uedig successio of umbers, called terms. It is uderstood that the terms have a defiite order, that is, there is a first term a 1, a secod term a 2, a third term a 3, ad so forth. Such a sequece would typically be writte as a 1, a 2, a 3, a 4,.a where the dots are used to idicate that the sequece cotiues idefiitely. Some specific example are 111 1, 2, 3, 4,.. 1,,,, , 4, 6, 8,.. 1, 1, 1, 1,. The umber a is called the th term or geeral term, of the sequece. Calculus & Aalytic Geometry II (MATF 144) 2
2 Example 10.1: I each part, fid the geeral term of the sequece. (a) ,,, (b) ,,, (c) ,,, (d) 1, 3, 5, 7,.. Calculus & Aalytic Geometry II (MATF 144) 3 Whe the geeral term of sequece is kow, there is o eed to write out the iitial terms, ad it is commo to write the geeral term eclosed i braces. Sequece ,,, Brace Notatio , 4, 8, ,,,...,( 1), , 3, 5, 7,..,2 1, 1 ( 1) Calculus & Aalytic Geometry II (MATF 144) 4
3 Covergece ad Divergece Sometimes the umbers i a sequece approach a sigle value as the idex icreases. This happes i the sequece 1, 1 2, 1 3, 1 4,,1, whose terms approach 0 as gets larger, ad i the sequece 0, 1 2, 2 3, 3 4, 4 5,,11, whose terms approach 1. O the other had, sequeces like 1, 2, 3,..,,. have terms that get larger tha ay umber as icreases, ad sequeces like 1,1,1,1.,1,. bouce back ad forth betwee 1 ad 1, ever covergig to a sigle value. Calculus & Aalytic Geometry II (MATF 144) 5 Example 10.2: Fid the limit of each of these sequeces. (a) 1 (b) ( 1) (c) 8 2 Calculus & Aalytic Geometry II (MATF 144) 6
4 Calculatig Limits of Sequeces Calculus & Aalytic Geometry II (MATF 144) 7 Example 10.3: Determie whether the sequece coverges or diverges. If it coverges, fid the limit. (a) 100 (b) 1 (c) (e) (b) (b) ( 1) Calculus & Aalytic Geometry II (MATF 144) 8
5 The Sadwich Theorem Example 10.4: Fid the limit of the sequece: si (a) 2 (b) 2 cos 3 (c) 1 ( 1) Calculus & Aalytic Geometry II (MATF 144) 9 Usig L Hôpital s rule. The ext theorem eables us to use L Hôpital s rule to fid the limits of some sequeces. It formalizes the coectio betwee lim a ad lim f( x) x. Example 10.5:(Evaluatig a limit usig L Hôpital s rule). (a) l lim (b) 2 lim 2 (c) lim 1/ (d) lim 1 1 Calculus & Aalytic Geometry II (MATF 144) 10
6 Commoly Occurig Limits The ext theorem gives some limits that arise frequetly. Calculus & Aalytic Geometry II (MATF 144) 11 Example 10.6: Fid the limit of each coverget sequece. l a (a) (b) a 3 (c) a 5 2 (d) a 1 3 (e) a (f) 3 a 55! Calculus & Aalytic Geometry II (MATF 144) 12
7 10.2 Ifiite Series Calculus & Aalytic Geometry II (MATF 144) 13 i) The th partial sum S of the series a is S a1 a2... a ii) The sequece of partial sums of the series a is S, S, S,... S, k Example 10.7: Give the series ( 1) (a) Fid S 1, S 2, S 3, S 4, S 5 ad S 6. (b) Fid S. (c) Show that the series coverges ad fid its sum. Calculus & Aalytic Geometry II (MATF 144) 14
8 Example 10.8: Give the series, 1 1 ( 1) 1 ( 1) 1 ( 1)... ( 1)... 1 (a) Fid S 1, S 2, S 3, S 4, S 5 ad S 6. (b) Fid S. (c) Show that the series diverges. Calculus & Aalytic Geometry II (MATF 144) 15 Geometric Series Example 10.9: Determie whether each of the followig geometric series coverges or diverges. If the series coverges, fid its sum. (a) (b) Calculus & Aalytic Geometry II (MATF 144) 16
9 Telescopig Series A telescopig series does ot have a set form, like the geometric series do. A telescopig series is ay series where early every term cacels with a precedig or followig term. For istace, the series Usig partial fractios, we fid that 1 2 k k k k k kk ( 1) k k1 Thus, the th partial sum of the give series ca be represeted as follows: Calculus & Aalytic Geometry II (MATF 144) 17 S k k 1 2 k1k k k The limit of the sequece of partial sums is so the series coverges, with sum S = lim S lim 1 1 Calculus & Aalytic Geometry II (MATF 144) 18
10 Example 10.10: I each case, express the th partial sums S i terms of ad determie whether the series coverges or diverges (a) 0 ( 1)( 2) (b) Calculus & Aalytic Geometry II (MATF 144) 19 Diverget Series Oe reaso that a series may fail to coverge is that its terms do t become small. For example, the series This series is diverges because the partial sums grow beyod every umber L. After = 1, the partial sums s is greater tha 2 Theorem above states that if a series coverges, the the limits of its th term a as is 0.Sometimes, it is possible for a series to become diverges. Calculus & Aalytic Geometry II (MATF 144) 20
11 So this theorem leads to a test for detectig the kid of divergece that occurred i some of the series. The th Term Test If, lim a 0 the further ivestigatio is ecessary to determie whether the series is coverge or diverge. 1 a Calculus & Aalytic Geometry II (MATF 144) 21 Example 10.11: Applyig the th Term Test (a) 12 1 (b) 1 2 (c) 1 ( 1) 1 Calculus & Aalytic Geometry II (MATF 144) 22
12 Combiig Series Calculus & Aalytic Geometry II (MATF 144) 23 Example 10.12: 7 2 (a) ( 1) (b) (c) Calculus & Aalytic Geometry II (MATF 144) 24
13 10.3 The Itegral Tests Calculus & Aalytic Geometry II (MATF 144) 25 Example 10.13: Determie whether the followig series coverge. (a) (b) (c) 0 l The p Series The Itegral Test is used to aalyze the covergece of a etire family of ifiite series 1 kow as the p series. For what values of the real umbers p does the p series p 1 coverge? Calculus & Aalytic Geometry II (MATF 144) 26
14 Calculus & Aalytic Geometry II (MATF 144) 27 Calculus & Aalytic Geometry II (MATF 144) 28
15 Example 10.14: Test each of the followig series for covergece. 1 1 (a) 10 1 (b) (c) 2 ( 1) 4 Calculus & Aalytic Geometry II (MATF 144) Compariso Tests We have see how to determie the covergece of geometric series, p series ad a few others. We ca test the covergece of may more series by comparig their terms to those of a series whose covergece is kow. Note:Let a, c ad d be series with positive terms. The series a coverges if it is smaller tha(domiated by) a kow coverget series c ad diverges if it is larger tha (domiates)a kow diverget series d. That is, smaller tha coverget is coverget, ad bigger tha diverget is diverget. Calculus & Aalytic Geometry II (MATF 144) 30
16 Example 10.15: Test the followig series for covergece. 1 (a) (b) l (c) 3 2 Calculus & Aalytic Geometry II (MATF 144) 31 Example 10.16: Test the followig series for covergece by usig the limit compariso test (a) (b) 1 (3 5) (c) 1 e 70 Calculus & Aalytic Geometry II (MATF 144) 32
17 10.5 The Ratio ad Root Tests Ituitively, a series of positive terms a coverges if ad oly if the sequece decrease rapidly toward 0. Oe way to measure the rate at which the sequece a decreasig is to examie the ratio followig theorem: a / a 1 a as grows large. This approach leads to the is Calculus & Aalytic Geometry II (MATF 144) 33 Note: You will fid the ratio test most useful with series ivolvig factorials or expoetials. Example 10.17: Use the ratio test to determie whether the followig series coverge or diverge. (a) 2 1! (b) 1! (c) e (d) 1 1 ( 1)( 2)! Calculus & Aalytic Geometry II (MATF 144) 34
18 The followig test is ofte useful if a cotais powers of. Calculus & Aalytic Geometry II (MATF 144) 35 Example 10.18: Determie whether the series is coverget or diverget. (a) 1 1 (l ) 2 (b) 2 k1 (c) 12 1 Calculus & Aalytic Geometry II (MATF 144) 36
19 Guidelies to Use the Covergece Test. Here is a reasoable course of actio whe testig a series of a positive terms 1 a for covergece. 1. Begi with the Divergece Test. If you show that lim a 0, the the series diverges ad your work is fiished. 2. Is the series a special series?recall the covergece properties for the followig series. Geometric Series: ar coverges if r 1 ad diverges for r 1. p series: 1 coverges for p p 1 ad diverges for p 1. Check also for a telescopig series. Calculus & Aalytic Geometry II (MATF 144) If the geeral th term of the series looks like a fuctio you ca itegrate, the try the Itegral Test. 4. If the geeral th term of the series ivolves, where a is a costat, the Ratio Test is advisable. Series with i a expoet may yield to the Root Test. 5. If the geeral th term of the series is a ratioal fuctio of (or a root or a ratioal fuctio), use the Direct Compariso or the Limit Compariso Test. Use the families of series give i Step 2 as a compariso series. Calculus & Aalytic Geometry II (MATF 144) 38
20 10.6 Alteratig Series, Absolute ad Coditioal Covergece Alteratig Series We cosider alteratig series i which sigs strictly alterate, as i the series, The factor has the patter {,1, 1,1, 1, } ad provides the alteratig sigs. We prove the covergece of the alteratig series by applyig the Alteratig Series Test. Calculus & Aalytic Geometry II (MATF 144) 39 Example 10.19: (a) ( 1) (b) ( 1) (c) ( 1) Calculus & Aalytic Geometry II (MATF 144) 40
21 Absolute ad Coditioal Covergece The covergece test we have developed caot be applied to a series that has mixed terms or does ot strictly alterate. I such cases, it is ofte useful to apply the followig theorem. The series of absolute values, 1 ( 1) 2 is a example of a absolutely coverget series because the series 1 1 ( 1) is a coverget p series. I this case, removig the alteratig sigs i the series does ot affect its covergece. Calculus & Aalytic Geometry II (MATF 144) 41 O the other had, the coverget alteratig harmoic series property that the correspodig series of absolute values, 1 ( 1) ( 1) 1 1 has the does ot coverge. I this case, removig the alteratig sigs i the series does effect covergece. Calculus & Aalytic Geometry II (MATF 144) 42
22 Example 10.20: Determie whether the followig series diverge, coverge absolutely, or coverge coditioally. a) ( 1) 1 1 b) ( 1) si c) 1 d) ( 1) Calculus & Aalytic Geometry II (MATF 144) 43 Calculus & Aalytic Geometry II (MATF 144) 44
23 Calculus & Aalytic Geometry II (MATF 144) Power Series The most importat reaso for developig the theory i the previous sectios is to represet fuctios as power series that is, as series whose terms cotai powers of a variable x. The good way to become familiar with power series is to retur to geometric series. Recall that for a fixed umber r, 2 1 r 1 rr..., provided r 1 1 r 0 It s a small chage to replace the real umber r by the variable x. I doig so, the geometric series becomes a ew represetatio of a familiar fuctio: 2 1 x 1 xx..., provided x 1 1 x 0 Calculus & Aalytic Geometry II (MATF 144) 46
24 Covergece of Power Series We begi by establishig the termiology of power series. Calculus & Aalytic Geometry II (MATF 144) 47 Calculus & Aalytic Geometry II (MATF 144) 48
25 Example : Fid the iterval ad radius of covergece for each power series: x a) 0! Solutio: The ceter of the power series is 0 ad the terms of the series are. We test the series for absolute covergece usig Ratio Test: 1 x /( 1)! lim (Ratio Test) x /! x 1 x! lim (Ivert ad multiply) x ( 1)! 1 lim 0 (Simplify ad take the limit with x fixed) ( 1) Calculus & Aalytic Geometry II (MATF 144) 49 Notice that i takig the limit as, x is held fixed. Therefore, 0 for all values of x, which implies that the iterval of covergece of the power series is x ad the radius of covergece is R Calculus & Aalytic Geometry II (MATF 144) 50
26 ( 1) ( x 2) b) 0 4 Solutio: We test for absolute covergece usig the Root Test: ( 1) ( x 2) lim (Root Test) 4 x I this case, depeds o the value of x. For absolute covergece, x must satisfy x which implies that x 2 4. Usig stadard techiques for solvig iequalities, the solutio set is 4x24, or 2x 6. Thus, the iterval of covergece icludes ( 2, 6). Calculus & Aalytic Geometry II (MATF 144) 51 The Root Test does ot give iformatio about covergece at the edpoits, x = 2 ad x = 6, because at these poits, the Root Test results i 1. To test for covergece at the edpoits, we must substitute each edpoit ito the series ad carry out separate tests. At x = 2, the power series becomes ( 1) ( x 2) Substitute x 2 ad simplify 1 Diverges by th term test/divergece test The series clearly diverges at the left edpoit. At x = 6, the power series is ( 1) ( x 2) 4 1 Substitute x 6 ad simplify Diverges by th term test/divergece test Calculus & Aalytic Geometry II (MATF 144) 52
27 This series also diverges at the right edpoit. Therefore, the iterval of covergece is ( 2, 6), excludig the edpoits ad the radius of covergece is R =4. (Whe the covergece set is the etire x axis). x Show that the power series 1! coverges for all x. Calculus & Aalytic Geometry II (MATF 144) 53 Example 10.21: (Covergece oly at the poit x=0). x Show that the power series 1! coverges oly whe x = 0. Calculus & Aalytic Geometry II (MATF 144) 54
28 Example 10.22: (Covergece set is a bouded iterval). Fid the covergece set for the power series 1 x. Calculus & Aalytic Geometry II (MATF 144) 55 ax Accordig to the Theorem 11.23, the set of umbers for which the power series 0 coverges is a iterval cetered at x =0. We call this the iterval of covergece of the power series. If this iterval has legth 2R, the R is called the radius of covergece of the series. If the series has radius of covergece R =0, ad if it coverges for all x, we say that R. Calculus & Aalytic Geometry II (MATF 144) 56
29 Example 10.23: Fid the iterval of covergece for the power series 0 What is the radius of the covergece? 2 x. Example 10.24: ( x 1) Fid the iterval of covergece of the power series 0 3. Calculus & Aalytic Geometry II (MATF 144) 57 Term by term Differetiatio ad Itegratio. ax Suppose that a power series 0 has a radius of covergece r>0, ad let f be defied by fx ( ) ax a axax ax... ax... for every x i the iterval of covergece. If r < x < r, the (i) (ii) f ( x) a 2a x 3 a x... a x... a x x x x x a 0 ( ) f tdt ax a a a x 1 Calculus & Aalytic Geometry II (MATF 144) 58
30 Example 10.25: Fid a fuctio f that is represeted by the power series xx x... ( 1) x... Example 10.26: 1 Fid a power series represetatio for 2 (1 x) if x 1. Calculus & Aalytic Geometry II (MATF 144) 59 Example 10.27: Fid a power series represetatio for l(1 x) if x 1. Example 10.28: Fid a power series represetatio for ta 1 x. Calculus & Aalytic Geometry II (MATF 144) 60
31 10.8 Taylor ad MacLauri Series I the previous lecture, we cosidered power series represetatio for several special fuctios, icludig those where f(x) has the form 1 (1 x), l(1 x) ad ta provided x is suitably restricted. We ow wish to cosider the followig questio. If a fuctio f has a power series represetatio what is the form of Suppose that, a? fx () 0 0 ax fx () a( ) or xc fx ( ) ax a axax ax ax... 1 x, Calculus & Aalytic Geometry II (MATF 144) 61 ad the radius of covergece of the series is r > 0. A power series represetatio for f() x may be obtaied by differetiatig each term of the series for f(x). We may the fid a series for f () x by differetiatig the terms of the series for f () x. Series for f () x, f (4) () x ad so o, ca be foud i similar fashio. Thus, f() x a 2a x3a x 4 a x... a x f( x) 2 a (32) a x(4 3) a x... ( 1) a x f( x) (32) a (4 32) a x... ( 1)( 2) a x ad for every positive iteger k, ( k f ) ( x) ( 1)( 2)...( k1) a x k k 3 Calculus & Aalytic Geometry II (MATF 144) 62
32 Each series obtaied by differetiatio has the same radius of covergece r as the series for f(x). Substitutig 0 for x i each of these represetatios, we obtai f(0) a, f(0) a, f(0) 2 a, f(0) (3 2) a ad for every positive iteger k, ( k f ) ( x) k( k1)( k 2)...(1) a k If we let k =, the ( f ) () x! a Solvig the precedig equatios for a 0, a1, a2,..., we see that f(0) f(0) a0 f(0), a1 f(0), a2, a3 2 (3 2) Ad, i geeral, a ( f ) (0)! Calculus & Aalytic Geometry II (MATF 144) 63 MacLauri Series for f(x) If a fuctio f has a power series represetatio fx () 0 ax with radius of covergece r > 0, the () k f (0) exists for every positive iteger k ad a ( f ) (0)!. Thus, ( ) f(0) 2 f (0) fx ( ) f(0) f(0) x x... x... 2!! 0 ( ) f (0) x! Calculus & Aalytic Geometry II (MATF 144) 64
33 Taylor Series for f(x) If a fuctio f has a power series represetatio fx () a( xc) 0 with radius of covergece r > 0, the f ( k ) () c exists for every positive iteger k ad a ( ) () f c!. Thus, f() c 2 fx () fc () f()( c xc) ( xc)... 2! 0 ( ) f (0) ( ) xc...! ( ) f () c ( x c)! Calculus & Aalytic Geometry II (MATF 144) 65 Example 10.29: Fid the MacLauri Series for f(x) = cos x. Example 10.30: Fid the Taylor Series for f(x) = l x at c =1. Prepared by: P.Suriawati Sahari Calculus & Aalytic Geometry II (MATF 144) 66
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