CHAPTER 10 INFINITE SEQUENCES AND SERIES

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,,,,... 2 3 4 2, 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

Example 10.1: I each part, fid the geeral term of the sequece. (a) 1 2 3 4,,,... 2 3 4 5 (b) 1 1 1 1,,,... 2 4 8 16 (c) 1 2 3 4,,,... 2 3 4 5 (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 1 2 3 4,,,...... 2 3 4 5 1 Brace Notatio 1 1 1 1 1 1 1 2, 4, 8, 16... 2... 1 2 1 1 2 3 4 1,,,...,( 1),... 2 3 4 5 1 1, 3, 5, 7,..,2 1, 1 ( 1) 1 2 1 1 1 Calculus & Aalytic Geometry II (MATF 144) 4

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

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) 4 3 1 4 2 5 2 1 (e) 2 2 57 3 (b) 5 3 2 4 2 7 3 (b) ( 1) 1 2 1 Calculus & Aalytic Geometry II (MATF 144) 8

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

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

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,... 1 2 3 k Example 10.7: Give the series 1 1 1 1...... 12 23 34 ( 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

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) 1 3 0 72 (b) 1 3 0 5 Calculus & Aalytic Geometry II (MATF 144) 16

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 1 1 1 1 1 2 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 1 1 1 k k 1 2 k1k k k1 1 1 1 1 1 1 1 1.. 2 2 3 3 4 1 1 1 1 1 1 1 1 1.. 2 2 3 4 1 1 1 1 The limit of the sequece of partial sums is so the series coverges, with sum S = 1. 1 1 lim S lim 1 1 Calculus & Aalytic Geometry II (MATF 144) 18

Example 10.10: I each case, express the th partial sums S i terms of ad determie whether the series coverges or diverges. 1 1 1 (a) 0 ( 1)( 2) (b) 0 2 1 2 3 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 1 2 2 14 9...... This series is diverges because the partial sums grow beyod every umber L. After = 1, the partial sums s. 2 149... 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

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

Combiig Series Calculus & Aalytic Geometry II (MATF 144) 23 Example 10.12: 7 2 (a) ( 1) 1 3 1 1 1 (b) 1 1 2 6 1 (c) 1 1 1 2 3 2 Calculus & Aalytic Geometry II (MATF 144) 24

10.3 The Itegral Tests Calculus & Aalytic Geometry II (MATF 144) 25 Example 10.13: Determie whether the followig series coverge. (a) 2 1 1 (b) 1 1 2 5 (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

Calculus & Aalytic Geometry II (MATF 144) 27 Calculus & Aalytic Geometry II (MATF 144) 28

Example 10.14: Test each of the followig series for covergece. 1 1 (a) 10 1 (b) 1 3 1 (c) 2 ( 1) 4 Calculus & Aalytic Geometry II (MATF 144) 29 10.4 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

Example 10.15: Test the followig series for covergece. 1 (a) 1 3 1 (b) 2 1 1 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. 1 3 2 100 (a) 1 2 5 (b) 1 (3 5) (c) 1 e 70 Calculus & Aalytic Geometry II (MATF 144) 32

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

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

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) 37 3. 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

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 1 1 1 1 1 1 1 1 1 1 1 1 1... 2 3 4 5 6 7 8 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) 2 1 1 (b) ( 1) 1 1 2 1 2 2 4 3 (c) ( 1) 1 4 3 Calculus & Aalytic Geometry II (MATF 144) 40

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) 1 2 2 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 ( 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

Example 10.20: Determie whether the followig series diverge, coverge absolutely, or coverge coditioally. a) ( 1) 1 1 b) ( 1) 3 1 1 si c) 1 d) ( 1) 1 1 2 Calculus & Aalytic Geometry II (MATF 144) 43 Calculus & Aalytic Geometry II (MATF 144) 44

Calculus & Aalytic Geometry II (MATF 144) 45 10.7 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

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

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

( 1) ( x 2) b) 0 4 Solutio: We test for absolute covergece usig the Root Test: ( 1) ( x 2) lim (Root Test) 4 x 2 1 4 I this case, depeds o the value of x. For absolute covergece, x must satisfy x 2 1 4 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) 4 0 4 04 0 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 4 4 0 0 0 1 Diverges by th term test/divergece test Calculus & Aalytic Geometry II (MATF 144) 52

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

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

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 0 2 3 0 1 2 3 fx ( ) ax a axax ax... ax... for every x i the iterval of covergece. If r < x < r, the (i) (ii) 1 2 3 2 1 1 1 f ( x) a 2a x 3 a x... a x... a x 2 3 1 x x x x a 0 ( ) 0 1 2...... 2 3 1 1 1 f tdt ax a a a x 1 Calculus & Aalytic Geometry II (MATF 144) 58

Example 10.25: Fid a fuctio f that is represeted by the power series 2 3 1 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

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 0 2 3 4 0 1 2 3 4 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, 2 3 1 1 2 3 4 1 f() x a 2a x3a x 4 a x... a x 2 3 4 2 2 2 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, 3 4 3 ( k f ) ( x) ( 1)( 2)...( k1) a x k k 3 Calculus & Aalytic Geometry II (MATF 144) 62

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 0 1 2 3 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

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