CHAPTER 5 : SERIES. 5.2 The Sum of a Series Sum of Power of n Positive Integers Sum of Series of Partial Fraction Difference Method

Transcription

1 CHAPTER 5 : SERIES 5.1 Seies 5. The Sum of a Seies 5..1 Sum of Powe of Positive Iteges 5.. Sum of Seies of Patial Factio 5..3 Diffeece Method 5.3 Test of covegece Divegece Test 5.3. Itegal Test Ratio Test 5.4 Powe Seies 5.5 Taylo ad the Maclaui Seies 1

2 Review: Sequece What is a sequece? It is a set of umbes which ae witte i some paticula ode u, u,, u. 1 We sometimes wite u 1 fo the fist tem of the sequece, u fo the secod tem ad so o. We wite the th tem as u. Examples: 1, 3, 5, 9. fiite sequece 1,, 3, 4, 5,, fiite sequece 1, 1,, 3, 5, 8, - ifiite sequece

3 5.1 Seies Defiitio: The sum of the tems i the sequece is called a seies. Fo example, suppose we have the sequece u, u,, u. 1 The seies we obtai fom this is u u u 1. ad we wite S fo the sum of these tems. Fo example, let us coside the sequece of umbes The, S S S 1,, 3, 4, 5, 6,..., The diffeece betwee the sum of two cosecutive patial tems, S S 1, is the th tem of the seies. i.e u S S 1 If the sum of the tems eds afte a few tems, the the seies is called fiite seies. If the sum of the seies does ot ed, the the seies is called ifiite seies. 3

4 Example 1: Fid the 4 th tem ad 5 th tem of the sequece 1, 4, 7,. Hece, fid S 4 ad S 5 of the seies 1, 4, 7,. Example : The sum of the fist tems of the seies is give by 1 (5 S 11 ) 4 a) Fid the fist thee tems, ad b) The -th tem of the seies. 4

5 Summatio Notatio,. (ead as sigma) is used to epeset the sum of the seies. I geeal, S u u u u. (fiite) 1 S u u u u. (ifiite) 1 3 i1 i Example 3: Fid the -th tem of the followig seies. Hece, expess the seies usig otatio. a) ,to 10 tems. b) ,util 30 tems. 5

6 Befoe we poceed to the ext sub-topic, let us eview two impotat sequeces/pogessios, i.e. 1. Geometic Sequece. Aithmetic Sequece Geometic Sequece - GS is a sequece whee each ew tem afte the fist is obtaied by multiplyig the pecedig tem by a costat, called the commo atio. - If the fist tem of the sequece is a, the the GS is 3 a, a, a, a,... whee the -th tem is a 1. - E.g., 6, 18, 54,.. ( a, 3) 1, -, 4, -8 ( a1, ) Aithmetic Sequece - AS is a sequece whee each ew tem afte the fist is obtaied by addig a costat d, called the commo diffeece to the pecedig tem. - If the tem of the sequece is a, the the AS is a, a d, a d, a 3 d,... whee the -th tem is a ( 1) d. - E.g. 8, 5,, -1, -4,.. ( a 8, d 3) 6

7 5. Sum of Seies 5..1 Sum of Powe of Positive Iteges Example 4: Evaluate Example 5: ad Evaluate 1. Example 6: Fid the sum fo each of the followig seies: 4 6 (b) to 30 tems (a) 7

8 5.. Sum of Seies of Patial Factio I this sectio we shall discuss tems with patial factios such as We ae ot able to calculate the sum of the seies by usig the available fomula (so fa), but with the help of patial factio method, we ca solve the poblem. Example 7: Fid the sum of the fist tems of the seies The above poblem equies quite a log solutio. Howeve, i the ext sub-topic, we will see a diffeet appoach to solve the same poblem. We called the appoach, a diffeece method. 8

9 5..3 Diffeece method Let f( x ) be a fuctio of x ad the -th tem of the seies u is of the fom u f ( ) f ( 1), the f (1) f (0) f () f (1) f (3) f () f (4) f (3)... f ( ) f ( 1) f (0) f ( ) f ( ) f (0). To coclude, u f ( ) f ( 1) If u f f 1, the u f f O equivaletly If ku f f 1, the u f f whee k is a costat. 1 k Note: If we fail to expess u ito this fom, i.e. f f 1, the this method caot be used. 9

10 Example 8: Expess the -th tem of the seies ( )... as the diffeece of two fuctios of ad 1. Hece fid the sum of the fist tems of the seies. Solutio: Step 1: Fid the geeal fom of the -th tem: Step : Fom aothe sequece f() by addig oe moe facto to the ed of the geeal tem u : Step 3: Fid f( 1) : Step 4: Fom the diffeece: f ( ) f ( 1) Step 5: Fid the sum : 10

11 Tips: (A) If the geeal tem, u, of the seies is i "poduct" fom, you ca add oe moe facto to the ed of the geeal tem u, so as to fom a sequece f() ad the apply the diffeece method. (B) If the geeal tem, u, is i "quotiet" fom, you ca emove oe moe facto at the ed of the geeal tem u, so as to fom a sequece f() ad the apply the diffeece method. Example 9: By usig the diffeece method, fid the sum of the fist tems of the seies ( 1)( ) Example (10): Use the diffeece method; fid the sum of the seies

12 5.3 Test of Covegece Divegece Test If a coveges, the lima 0. Equivaletly, if lima diveges. 0, o lima does ot exist, the the seies is Example (11): Show that the seies diveges Example 1: Use Divegece Test to detemie whethe o ot. diveges 1l 1

13 5.3. The Itegal Test Suppose f is a cotiuous, positive, deceasig fuctio o 1, ad let a f. The the seies if ad oly if the impope itegal coveget. I othe wods (a) If (b) If 1 f x dx is coveget, the coveget. 1 f x dx is diveget, the 1 a f x dx is a is a is coveget is diveget. Note: Use this test whe f x is easy to itegate. 13

14 Example 13: Use the Itegal Test to detemie whethe the followig seies coveges o diveges. 14

15 5.3.3 Ratio Test Let a be a ifiite seies with positive tems ad let lim a a 1. a) If 0 1, the seies coveges. b) If 1, o, the seies diveges. c) If 1, the test is icoclusive. Example 14: Use the Ratio Test to detemie whethe the followig seies coveges o diveges. 15

16 5.4 Powe Seies Defiitio A powe seies about 0 x is a seies of the fom 3 ax a0 a1x ax a3x ax 0 A powe seies about x 0 a is a seies of the fom a x a a a x a a x a a x a ( ) 0 1( ) ( ) ( ) i which the cete a ad the coefficiets a, a, a, a, ae costats. 0 1, 16

17 5.4.1 Expasio of Expoet Fuctio The powe seies of the expoet fuctio ca be witte as x e x x x x! 3! 4! The expasio is tue fo all values of x. I geeal, Example (15): Give e x 1 x! ! 3! 4!! x 3 4 e 1 x x x x x Wite dow the fist five tems of the expasio of the followig fuctios (a) (b) e x x 1 e. Example (16): Wite dow the fist five tems o the expasio of the fuctio, 1 x e x i the fom of powe seies. 17

18 5.4. Expasio of Logaithmic Fuctio The expasio of logaithmic fuctio ca be witte as l(1 x) x x x x x x x The seies coveges fo 1 x 1 l 1 x is valid fo 1 x 1. By assumig x with x, we obtai l(1 x) x x x x x x x 6 7 Thus, this esult is tue fo 1 x 1o 1 x 1.. Thus the seies 18

19 Example (17): Wite dow the fist five tems of the expasio of the followig fuctios (a) l 1 3x (b) 3l 1x 1 3x Example (18): Fid the fist fou tems of the expasio of the fuctio, x x 3 1 l 1. 19

20 5.4.3 Expasio of Tigoometic Fuctio The powe seies fo tigoometic fuctios ca be witte as x x x x si x x 3! 5! 7! 9! x x x x cos x 1! 4! 6! 8! Both seies ae valid fo all values of x. Example (19): Give x x x x cos x 1! 4! 6! 8! Fid the expasio of cosx ad cos3x. Hece, by usig a appopiate tigoometic idetity fid the fist fou tems of the expasio of the followig fuctios: (a) (b) si 3 cos x x 0

21 5.5 Taylo ad the Maclaui Seies Example 0: Obtai the Taylo seies fo f ( x) 3x 6x 5 aoud the poit x 1. As: + 3(x-1) Example 1: Obtai Maclaui seies expasio fo the fist fou tems of x e ad five tems of si x. Hece, deduct that Maclaui seies fo x e si x is give by x x x x Example : Use Taylo s theoem to obtai a seies expasio of fist five deivatives fo cosx 3. Hece fid 0 cos6 coect to 4 dcp. As:

22 Example 3: If y lcos x, show that d y dy 1 0 dx dx Hece, by diffeetiatig the above expessio seveal times, obtai the Maclaui s seies of y lcos x i the ascedig 4 powe of x up to the tem cotaiig x. Solutio:

23 Fidig Limits with Taylo Seies ad Maclaui Seies. Example 4: x e 1 x Fid lim x0. x As: ½ Example 5: x Evaluate lim x0 cos x 4. 3x As: 1/36 Evaluatig Defiite Itegals with Taylo Seies ad Maclaui Seies. Example 6: Use the fist 6 tems of the Maclaui seies to appoximate the followig defiite itegal. a) 1 0 e x dx xcos( x ) dx b) As: 0.747,