888 CHAPTER 12 Sequences; Induction; the Binomial Theorem Examining a Divergent Sequence Discuss the sequence { } { } = s 2 . n n Solution EXAMPLE 2 The terms of this sequence are … 2, 4, 8, 16, 32, 64, Since each successive term is twice the preceding term, the terms of the sequence are increasing without bound as n gets larger and larger. Since there is no number L that the terms get closer to, conclude that the sequence is divergent. As n gets larger and larger, the value of n 1 gets closer to zero so + n n 2 1 gets closer to 2. A graphing utility can be used to draw the same conclusion. See Table 9. Note that as n gets large, the graphing utility rounds the sequence value to 2 even though the actual value is not 2. In calculus, this idea is expressed by saying that the sequence { } sn converges, and limit notation is used to write = + = →∞ →∞ s n n lim lim 2 1 2 n n n In general, the notation = →∞ s L lim n n is used to mean that as n gets larger and larger, the terms of the sequence { } sn get closer and closer to the number L. In such a case, the sequence is said to converge to L, and L is called the limit of the sequence. Finding the Limit of a Sequence Find the limit of each sequence: (a) { } { } = − a n n 6 4 n 2 2 (b) { } { } = + − + + b n n n n 3 5 2 6 10 8 n 3 2 3 Solution EXAMPLE 1 (a) Divide both the numerator and the denominator by n .2 Then − = − n n n 6 4 6 1 4 2 2 2 As n gets larger and larger, the value of n 4 2 gets closer to 0. Conclude that − = →∞ n n lim 6 4 6 n 2 2 (b) Divide both the numerator and the denominator by n .3 Then + − + + = + − + + n n n n n n n n 3 5 2 6 10 8 3 5 2 6 10 8 3 2 3 3 2 3 As n gets larger and larger, the values of n n n 5 , 2 , 10 , 3 2 and n 8 3 each get closer and closer to 0. Conclude that + − + + = = →∞ n n n n lim 3 5 2 6 10 8 3 6 1 2 n 3 2 3 For the sequence of Example 2, since the terms of the sequence are increasing without bound, we write =∞ →∞ s lim n n A sequence can be divergent in other ways, as illustrated in the next example. Table 9 Some sequences do not converge. If a sequence does not converge, it is said to be divergent.
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