Math 1432 DAY 22. Dr. Melahat Almus.

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1 Math 43 DAY Dr. Melahat Almus If you me, please metio the course (43) i the subject lie. Bubble i PS ID ad Popper Number very carefully. If you make a bubblig mistake, your scatro ca t be saved i the system. I that case, you will ot get credit for the popper eve if you tured it i. Check your CASA accout for Quiz due dates. Do t miss ay olie quizzes! Be cosiderate of others i class. Respect your frieds ad do ot distract ayoe durig the lecture. Did you take practice test 3? Chapter 9 9. Sequeces Defiitio: GLB ad LUB of a sequece Least Upper Boud (LUB) Greatest Lower Boud (GLB) Smallest umber greater tha or equal to all the terms of the sequece. Largest umber less tha or equal to all the terms of the sequece. Math 43 Dr. Almus

2 Defiitio: Mootoe Sequeces The sequece {a} is said to be strictly icreasig if a < a+ for all, icreasig if a a+ for all, strictly decreasig if a > a+ for all, decreasig if a a+ for all. A sequece that satisfies ay of these coditios is called mootoic. Icreasig Sequece: a a a... a a... 3 Decreasig Sequece: a a a... a a... 3 How to test for mootoicity: a If for all, the the sequece is icreasig. a a If for all, the the sequece is decreasig. a Math 43 Dr. Almus

3 3 Math 43 Dr. Almus Example: Determie the mootoicity:

4 Sectio 9. Covergece of a Sequece Defiitio: Limit of a Sequece: We say that lima that N a L. L if for every 0, there exists a atural umber N such I this case, the sequece a is said to be coverget. Otherwise, the sequece is said to be diverget. OR: L is the limit of the sequece a if ad oly if for every positive umber epsilo, o matter how small, there is a atural umber N such that all terms of the sequece after a N stay withi epsilo uits of L. (That is, if the terms of the sequece approach a uique target umber L, we say the limit is L.) Example: a ; lim 4 Math 43 Dr. Almus

5 Example: lim Example: a ,,,,,,... ; lim a Example: a,,,,,,... ; lim a 5 Math 43 Dr. Almus

6 Let a be a sequece of real umbers. If the sequece has a fiite limit, we say it is coverget. If it does ot have a fiite limit, we say it is diverget. Possibilities: ) lima L where L is a fiite real umber. We say the sequece a coverges to L. lim a ; we say the sequece diverges to ifiity. ) 3) lima 4) a ; we say the sequece diverges to egative ifiity. lim : DNE because the sequece oscillates betwee two or more umbers. We say the sequece is diverget due to oscillatio. We worked o some simple examples ad observed the covergece. What if the geeral term of the sequece is more complicated? For example: a 5 We will work o more complicated cases i a bit; but first, some results: 6 Math 43 Dr. Almus

Theorem 9..4: Every coverget sequece is bouded. That is: If a is coverget, the it is bouded. If a is bouded, the it may be coverget OR diverget. IMPORTANT: If a is ot bouded, the it is diverget.

7 Theorem 9..4: Every coverget sequece is bouded. That is: If a is coverget, the it is bouded. If a is bouded, the it may be coverget OR diverget. IMPORTANT: If a is ot bouded, the it is diverget. (If a sequece is NOT bouded, you ca use this fact to prove that it is diverget!) Theorem 9..6: A bouded, icreasig sequece coverges to its least upper boud. A bouded, decreasig sequece coverges to its greatest lower boud. 7 Math 43 Dr. Almus

8 8 Math 43 Dr. Almus Example: Determie whether the sequece is coverget or diverget. If coverget, fid the limit.

9 9 Math 43 Dr. Almus For ratioal expressios of the form polyomial polyomial : 0 a lim b if the degree of deomiator is bigger. leadig coefficiet of umerator leadig coefficiet of deom. a lim b if the degree of top ad bottom are equal. a b if the degree of top is bigger (sequece is ot bouded, so diverget!) This trick works eve if there are radicals; compare the largest expoets: 3 3

10 For more complicated ratioal formulas, you ca compare how fast the top ad bottom are growig to decide whether the limit exists or ot. If the deomiator grows faster, the limit is 0. If the umerator grows faster, the sequece is diverget. Fastest to slowest growig expressios: e etc 3 >>! >> expoetials (,,..) >> polyomials(,,.) Fastest: The:! The: expoetials (, e,..) 3 The: polyomials(,, etc.) Examples: Determie if the sequece is coverget or diverget. If coverget, fid the limit.!!! 4!! 0 Math 43 Dr. Almus

11 FACT: For geometric expressios; For x, 0 x as. For x, x diverges Commo oscillatig examples: cos si Math 43 Dr. Almus

12 Some Importat Limits: lim e ad lim a b e ab lim / By Theorem 9..: lim a Llim l a l L lim a L lim si a si L lim a L lim cos a cos L For each real 0 For each real 0 lim si l lim x x, lim 0! / x, x 0 0 lim. Example: lim 0! /. Example: lim 5 Math 43 Dr. Almus

13 3 Math 43 Dr. Almus Example: 5 lim? POPPER # Choose (C) for coverget, (D) for diverget. Q# 5 Q# 5 Q# 3 Q# Q#! 5 Q#!

(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3

(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3 MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special