CHAPTER SEQUENCES AND INFINITE SERIES
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
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.
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.
. 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.
. Sequeces NOTATIONS: a a f
What s ext i the sequece? 3 35 357,,,, 2 24 246 2468 3579 24680 0, 3, 8, 5, 24, 35, 48,, 2, 3, 5, 8, 3, 2
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 = 5 + 5 0 + 5 2 + 5 5 0 5 2 f(+) FYI: lim f() = ϕ = + 5 2 =.68 (golde ratio)
OUR INTEREST IN SEQUENCES: Behavior of f as Let lim f L.
OUR INTEREST IN SEQUENCES: Some idicators of existece of limit: icreasig or decreasig mootoicity is ot ecessary bouded boudedess is ecessary but ot sufficiet
Example. 2 f Let. f 2 3 4 5 6 7 0 3 8 5 24 35 48
f 2 3 4 5 6 7 0 3 8 5 24 35 48 0
Example 2. Let g. g 2 3 4 5 6 7
g 2 3 4 5 6 7
Example 3. Let h e. h 2 3 4 5 6 7 e 2 e 3 e 4 e 5 e 6 e 7 e
h 2 3 4 5 6 7 e 2 e 3 e 4 e 5 e 6 e 7 e
Example 4. Let j. j 2 3 4 5 6 7 2 3 4 5 6 7
2 3 4 5 6 7 j 2 3 4 5 6 7
Example 5. Try this i MS Excel x where x 3.9x x assume x [0,] 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0. 0 0 0 20 30 40 50 Chaos!
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
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
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.
Defiitio. If i lim f L, L exists, The the sequece is said to be coverget. Otherwise it is diverget.
Which of the ff sequeces is/are coverget? 3 4 2 ta Arc 7 3 4 cos 2! 0 3 2!
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, 7 2 +9) - cubic (e.g. 3, 8 3 +5+2) - 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!
.2 Mootoic ad Bouded Sequeces Mootoe Covergece Theorem (MCT) for Sequeces. A bouded mootoic sequece is coverget. Whe are sequeces mootoic? bouded?
.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,
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
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
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
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.
Example 2. Let a! 0! 0 a! 0 Now, a a! 0 0! That is, a a Thus, the sequece is mootoic (icreasig).
! 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.
Example 3. Recall: a 2 3 4 5 6 7 Thus, the sequece is bouded but is either icreasig or decreasig.
Example 4. Let a 3 3 2! 2! a 3 3! Now, a 3 3! a 2! 3 3 3 That is, a a Thus, the sequece is mootoic (decreasig).
Example 4. 3 2! Thus, the sequece is bouded. Note that. 3 0 2! has 0 as a lower boud ad has ½ as a upper boud. 3 2!
REMARKS: A bouded mootoe decreasig sequece coverges to its greatest lower boud. Similarly, a bouded mootoe icreasig sequece coverges to its least upper boud.
Example (MCT is ot applicable but has a limit): Let j. j 2 3 4 5 6 7 2 3 4 5 6 7
2 3 4 5 6 7 j 2 3 4 5 6 7
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.