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Sequences (11/13/13) A sequence is an infinite list of real numbers: {a1, a2, a3, a4, a5 ….} = {an}. (Order counts!) A sequence can be described by – – listing some initial terms, hence establishing a pattern, or giving its general formula. Examples: {1, 4, 9, 16,…} = formula? {n / 2n} = what initial elements? Clicker Question 1 What are the first 4 terms of {(-1)n+1 / n} ? – A. {1, 1/2, 1/3, 1/4,…} – B. {1, -1/2, 1/3, -1/4,…} – C. {-1, 1/2, -1/3, 1/4,…} – D. {-1, -1/2, -1/3, -1/4,…} – E. {1, -2, 3, -4,…} Clicker Question 2 What is a formula for the sequence {2/3, 4/9, 6/27, 8/81,…} ? – A. {2n / 3n} – B. {2n / 3n} – C. {2n / 3n} – D. {2n / 3n} – E. {(n + 1)/3n} Convergent and Divergent Sequences A sequence {an} converges to L (a real number) if, given any positive distance from L, we can go far enough out in the sequence so that every term from there out is within that given distance from L. We write limnan = L or {an} L . If {an} does not converge, we say it diverges. Examples Do these sequences converge or diverge? If converge, to what? (Btw, sound familiar?) {1, 1/4, 1/9, 1/16, …} {(-1)n+1(1/n2)} {(-1)n+1} {1, 1+1/4, 1+1/4+1/9, 1+1/4+1/9+1/16, …} {1, 1+1/2, 1+1/2+1/3, 1+1/2+1/3+1/4, …} Clicker Question 3 {1, 1+1/2, 1+1/2+1/4, 1+1/2+1/4+1/8,…} – A. converges to 1. – B. converges to 2. – C. converges to some number greater than 2. – D. diverges Monotone & Bounded Sequences If a1 < a2 < a3 < …, {an} is called increasing. Likewise decreasing. In either case, {an} is called monotone. If an < M (for some M and for all n), {an} is called bounded above. Likewise bounded below. If a sequence is monotone, we can simply say bounded. Monotone Convergence Theorem: If {an} is both monotone and bounded, then it is convergent. Example: {1, 1+1/4, 1+1/4+1/9, 1+1/4+1/9+1/16, …} is bounded above by 1 + 1 1 2 dx = ?? x Hence it must converge, but to what?? For Friday Read Section 11.1 as needed (long section). In that section, please do Exercises 1, 2, 3-17 odd, 23, 25, 27, 31, 33, 35, 43, 49, 53, 55, 65, 73, 75, 77. (Again, stay calm, most of these are quickies!)