CHAPTER 1 SEQUENCES AND INFINITE SERIES

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1 CHAPTER SEQUENCES AND INFINITE SERIES

2 SEQUENCES AND INFINITE SERIES (0 meetigs) Sequeces ad limit of a sequece Mootoic ad bouded sequece Ifiite series of costat terms Ifiite series of positive terms Alteratig series Power series Differetiatio ad Itegratio of Power Series Taylor ad Maclauri series

3 CHAPTER OBJECTIVE At the ed of the chapter, you should be able to:.determie if a give sequece is coverget or diverget. 2.Determie if a give series is coverget or diverget. 3.Differetiate/itegrate a ifiite series.

4 CHAPTER OBJECTIVE At the ed of the chapter, you should be able to: 4. Fid the iterval ad radius of covergece of a give series. 5. Write the Maclauri/Taylor series expasio of a fuctio.

5 . Sequeces A sequece of real umbers 2 is a fuctio that assigs to each positive iteger a a umber. a, a,..., a,... DOMAIN: N Some books use Domai: W The umbers i the rage are called the elemets or terms of the sequece.

6 . Sequeces NOTATIONS: a a f

7 What s ext i the sequece? ,,,, , 3, 8, 5, 24, 35, 48,, 2, 3, 5, 8, 3, 2

8 NOTE Fiboacci sequece:,, 2, 3, 5, 8, 3, Iterative/Recursive Relatio (differece equatio): Recurrece formula f + 2 = f + + f, where f() =, f(2) = Geeral term (solutio to the differece equatio): Explicit formula f = f(+) FYI: lim f() = ϕ = =.68 (golde ratio)

9 OUR INTEREST IN SEQUENCES: Behavior of f as Let lim f L.

10 OUR INTEREST IN SEQUENCES: Some idicators of existece of limit: icreasig or decreasig mootoicity is ot ecessary bouded boudedess is ecessary but ot sufficiet

11 Example. 2 f Let. f

12 f

13 Example 2. Let g. g

14 g

15 Example 3. Let h e. h e 2 e 3 e 4 e 5 e 6 e 7 e

16 h e 2 e 3 e 4 e 5 e 6 e 7 e

17 Example 4. Let j. j

18 j

19 Example 5. Try this i MS Excel x where x 3.9x x assume x [0,] Chaos!

20 The Limit of a Sequece The limit of a sequece f is the real umber L if for ay there exists a umber is a atural umber ad if, f L the. 0 N 0, however small, N such that if We write: lim f L

21 Cosider lim = Example For ay real umber ε > 0, take N = ε We eed to fid this If > N, the > ε ε > ε > ε >. Illustratio: Suppose ε = 0.0 N = 0.0 = 00 Hece, for all > 00, the distace tha 0.0. is less

22 Theorem. lim If ad is defied for every x f x L positive iteger the. e Recall: lim 0 f lim f L Note that f x e x is defied for x e x every positive iteger ad lim 0.

23 Defiitio. If i lim f L, L exists, The the sequece is said to be coverget. Otherwise it is diverget.

24 Which of the ff sequeces is/are coverget? ta Arc cos 2! 0 3 2!

25 NOTE Speed of fuctios, rakig: - costat (e.g. 0) - logarithmic (e.g. log, log( 2 )) - fractioal power (e.g. sqrt()) - liear (e.g., 5+0) - logliear (e.g. log, log!) - quadratic (e.g. 2, ) - cubic (e.g. 3, ) - higher degree polyomials (FYI: 2 log is as fast as polyomials) - expoetial (e.g., 2,. 2, ) where base> - factorial (e.g.!, 2!+3) Ca you still remember how to get horizotal asymptote? Use LHR!

26 .2 Mootoic ad Bouded Sequeces Mootoe Covergece Theorem (MCT) for Sequeces. A bouded mootoic sequece is coverget. Whe are sequeces mootoic? bouded?

27 .2 Mootoic ad Bouded Sequeces Defiitios. A sequece is mootoic if it is either icreasig or decreasig for all. A sequece A sequece a a is mootoe icreasig if a a N, is mootoe decreasig if a a N,

28 How do we determie if a sequece is mootoic or ot?. Observe. a 2. Obtai. The Compare result to (oe). f a x 2. Fid. ' a

29 Defiitios. A sequece is bouded if it has both a upper boud ad a lower boud. A real umber l is a lower boud of the sequece if l a, N A lower boud is the greatest lower boud (glb) of the sequece if l lower boud. g l g for all

30 Defiitios. A real umber u is a upper boud of the sequece if u a, N A upper boud (lub) of the sequece if u boud. v is the least upper boud u v for all upper

31 Example. Let f x 5x 2x 5 2 Sice, f ' 2 x 2 f ' x 0 x f is decreasig. Now,. 4x 5 0 f has 0 as a lower boud (5/2 2 is the glb) ad 3 as a upper boud. Thus, the sequece is mootoic ad bouded.

32 Example 2. Let a! 0! 0 a! 0 Now, a a! 0 0! That is, a a Thus, the sequece is mootoic (icreasig).

33 ! Example 2. 0! 0 0 Note that.! 0 has 0 as a lower boud (/0 is the glb) but has o upper boud. Thus, the sequece is ubouded.

34 Example 3. Recall: a Thus, the sequece is bouded but is either icreasig or decreasig.

35 Example 4. Let a 3 3 2! 2! a 3 3! Now, a 3 3! a 2! That is, a a Thus, the sequece is mootoic (decreasig).

36 Example ! Thus, the sequece is bouded. Note that ! has 0 as a lower boud ad has ½ as a upper boud. 3 2!

37 REMARKS: A bouded mootoe decreasig sequece coverges to its greatest lower boud. Similarly, a bouded mootoe icreasig sequece coverges to its least upper boud.

38 Example (MCT is ot applicable but has a limit): Let j. j

39 j

40 REMARKS: Relaxig MCT: It is ot ecessary that the sequece be mootoic iitially, oly that they be mootoic from some poit o, that is, for >K. Two evetually similar sequeces have the same limit.

SOLUTIONS TO EXAM 3. Solution: Note that this defines two convergent geometric series with respective radii r 1 = 2/5 < 1 and r 2 = 1/5 < 1.

SOLUTIONS TO EXAM 3. Solution: Note that this defines two convergent geometric series with respective radii r 1 = 2/5 < 1 and r 2 = 1/5 < 1. SOLUTIONS TO EXAM 3 Problem Fid the sum of the followig series 2 + ( ) 5 5 2 5 3 25 2 2 This series diverges Solutio: Note that this defies two coverget geometric series with respective radii r 2/5 < ad