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11.1 Sequences Sequence A sequence is a list of numbers written in an explicit order. an a1, a2 , a3, ... , an , ... nth term Any real-valued function with domain a subset of the positive integers is a sequence. If the domain is finite, then the sequence is a finite sequence. In calculus, we will mostly be concerned with infinite sequences. Examples n an n2 an 1 n n 1 an 5 2 {1, 4,9,16, 25,...} {1, 1, 2, 3, 5, 8, 13, 21,...} The last example is a recursively defined sequence known as the Fibonacci Sequence. Limit and Convergence n • Let’s take a look at the sequence an n2 • What will happen as n gets large? • If a sequence {an} approach a number L as n approaches infinity, we will write lim an L n and say that the sequence converges to L. • If the limit of a sequence does not exist, then the sequence diverges. Example 2n 1 Does an converge? n 2n 1 2n 1 lim lim n n n n n 2n 1 lim lim n n n n 20 2 The sequence converges to 2. Graph the sequence. Properties of Limits • Same as limit laws for functions in chapter 2. • Theorem: Let f (x) be a function of a real variable such that lim f ( x) L x If {an} is a sequence such that f (n) = an for every positive integer n, then lim a L n n • Squeeze Theorem • Absolute Value Theorem: For the sequence {an}, if nlim a 0, then nlim a 0. n n Examples Determine the convergence of the following sequences. 3 an n sin( 2n) bn 1 n n 1 an 3n 2 bn (1) n! ln n an 3 n n 3 cn 5 n Monotonic Sequence • A sequence is called increasing if an an 1 for all n. • A sequence is called decreasing if an an 1 for all n. • It is called monotonic if it is either increasing or decreasing. Bounded Sequence • A sequence is bounded above if there is a number M such that an ≤ M for all n. • A sequence is bounded below if there is a number N such that N ≤ an for all n. • A sequence is a bounded sequence if it is bounded above and below. Theorem: Every bounded monotonic sequence is convergent. Examples Determine whether the sequence is bounded, monotonic and convergent. an n! 6 an 2 n n bn n 1 an (2) n an ( 1) 1 n 3 n cn 4 n 1000 A geometric sequence is a sequence in which the ratios between two consecutive terms are the same. That same ratio is called the common ratio. Example: 1, 2, 4, 8, 16, ... 102 , 101 , 1, 10, ... Geometric sequences can be defined recursively: or explicitly: r 2 101 r 2 10 10 an an 1 r an a1 r n 1 A sequence is defined recursively if there is a formula that relates an to previous terms. Example: b1 4 bn bn1 2 for all n 2 We find each term by looking at the term or terms before it: b1 4 b2 b1 2 6 b3 b2 2 8 b4 b3 2 10

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