Show description

A nϩ1 ෇ 12 ͑a n ϩ 6͒ SOLUTION We begin by computing the first several terms: Mathematical induction is often used in dealing with recursive sequences. See page 87 for a discussion of the Principle of Mathematical Induction. 984375 These initial terms suggest that the sequence is increasing and the terms are approaching 6. To confirm that the sequence is increasing, we use mathematical induction to show that a nϩ1 Ͼ a n for all n ജ 1. This is true for n ෇ 1 because a 2 ෇ 4 Ͼ a 1. If we assume that it is true for n ෇ k, then we have a kϩ1 Ͼ a k a kϩ1 ϩ 6 Ͼ a k ϩ 6 so and 1 2 ͑a kϩ1 ϩ 6͒ Ͼ 2 ͑a k ϩ 6͒ 1 a kϩ2 Ͼ a kϩ1 Thus We have deduced that a nϩ1 Ͼ a n is true for n ෇ k ϩ 1.

4 Theorem nlϱ nlϱ n FIGURE 5 The sequence ͕ b n ͖ is squeezed between the sequences ͕ a n ͖ and ͕ c n ͖ . Խ Խ Խ Խ EXAMPLE 4 Find lim nlϱ n . 5: Divide numerator and denominator by the highest power of n that occurs in the denominator and then use the Limit Laws. lim nlϱ n ෇ lim nlϱ nϩ1 This shows that the guess we made earlier from Figures 1 and 2 was correct. ■ ■ ෇ nlϱ ln n . n 1ϩ 1 n 1 ෇1 1ϩ0 Here we used Equation 3 with r ෇ 1. 1 SEQUENCES ■ 561 SOLUTION Notice that both numerator and denominator approach infinity as n l ϱ.

C) What does it mean to say that lim n l ϱ a n ෇ ϱ? an ෇ n 2n ϩ 1 Does the sequence appear to have a limit? If so, find it. 25. ͕0, 1, 0, 0, 1, 0, 0, 0, 1, . . ͖ 26. a n ෇ ͑ln n͒ 2 n 27. a n ෇ ln͑2n 2 ϩ 1͒ Ϫ ln͑n 2 ϩ 1͒ 28. a n ෇ ͑Ϫ3͒n n! ■ 4. List the first nine terms of the sequence ͕cos͑n␲͞3͖͒. Does Find a formula for the general term a n of the sequence, assuming that the pattern of the first few terms continues. 6. 7. ͕2, 7, 12, 17, . ͖ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 9–28 ■ Determine whether the sequence converges or diverges.