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Section 8.1: Sequences Definition: A sequence is an ordered set of real numbers that has a one-to-one correspondence with the positive integers. a1 , a2 , a3 , a4 , ... , an , ... • Often denoted by {an}. • Ex. {2n} = 21, 22, 23, … , 2n, … • The expression for the nth term is called a closed-form definition for the sequence. Definition: A sequence can also be defined as a function where the domain is restricted to the positive integers. In this case f(n) = an. f(1), f(2), f(3), ... , f(n), ... The points (1, f(1)), (2, f(2)), (3, f(3)), … can be discretely graphed on the coordinate plane. Example 1 List the first 4 terms and the tenth term of each sequence. 3 4 5 11 a. n 1 2, , , and a n! 2 n n 1 b. (1) 3n 1 2 6 24 10 10! 1 4 9 16 100 ,- , ,and a10 2 5 8 11 29 A recursively defined sequence will state the first term, a1, and then define ak+1 in terms of the preceding term ak. Ex. Find the first four terms of the sequence defined recursively by a1 = 2 and ak+1 = 3ak for k ≥ 1. Can you find a closed form definition for this series? 2, 6, 18, 54 . Yes, an = 2(3)n-1 (geometric) Definition: A sequence {an} has the limit L, or converges to L, if for every ε > 0 there exists a positive number N such that |an – L | < ε whenever n > N. Denoted: lim an L n If such a number L does not exist, the sequence has no limit, or diverges. Example 2 Let {an} = a. Find 1 n1 2 lim an n lim n 1 2 n 1 0 b. Let ε = .001. Find some integer N such that for all n > N an is less than ε units from the limit above. N 10 Properties of Sequences Let {an} be a sequence and f(n) = an. If f(x) exits for every real number x ≥ 1, then 1) If lim f ( x) L , then lim f ( n) L x n 2) If lim f ( x) , then lim f ( n) Also, x For any real number r, n r 0 if r 1 . 1) lim n 2) lim r n if r 1 n n Example 3 Determine whether the sequence converges or diverges. If it converges, find the limit. a. (1) n1 diverges b. c. 5n 2n e 2 converges to 0 2 n 3 converges to 2 More Properties of Sequences Sandwich Theorem for Sequences: If {an}, {bn}, and {cn} are sequences and an ≤ bn ≤ cn for every n and if lim an L lim cn , then lim bn L n n n and Theorem 8.8: Let {an} be a sequence. If lim an 0 , then lim an 0 n n Example 4 Determine whether the sequence converges or diverges. If it converges, find the limit. a. (1)n 1 1 converges to 0 b. cos 2 n n 3 n converges to 0