SalasSV_10_02_ex - 10.2 SEQUENCES OF REAL NUMBERS PROOF 593...

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Unformatted text preview: 10.2 SEQUENCES OF REAL NUMBERS PROOF 593 We will work with the function f (x ) = x . ex (See Example 2, Section 7.4.) Note that f (1) = 1/e = a1 , f (2) = 2/e2 = a2 , f (3) = 3/e3 = a3 , and so on. Differentiating f , we get f (x ) = ex − x ex 1−x = . 2x e ex Since f (x) < 0 for x > 1, f decreases on [1, ∞). Thus f (1) > f (2) > f (3) > · · · , that is, a1 > a2 > a3 > · · · . The sequence is decreasing. The first term a1 = 1/e is the least upper bound of the sequence. Since all the terms of the sequence are positive, 0 is a lower bound for the sequence. In Figure 10.2.3 we have sketched the graph of f (x) = x/ex and marked some points (n, an ). As the figure suggests, 0 is the greatest lower bound of the sequence. y l/e (1, a1) y= x ex (2, a2) (3, a3) 1 2 3 (4, a4) 4 5 x Figure 10.2.3 Example 5 The sequence an = n1/n decreases for n ≥ 3. PROOF We could compare an with an+1 directly, but it is easier to consider the function f (x) = x1/x instead. Since f (x) = e(1/x)ln x , we have f (x) = e(1/x) ln x d dx 1 ln x = x1/x x 1 − ln x . x2 For x > e, f (x) < 0. This shows that f decreases on [e, ∞). Since 3 > e, the function f decreases on [3, ∞), and the sequence decreases for n ≥ 3. EXERCISES 10.2 The first several terms of a sequence {an } are given. Assume that the pattern continues as indicated and find an explicit formula for an . 1. 2, 5, 8, 11, 14, . . . 3. 1, − 1 , 1 , − 1 , 1 , . . . 35 79 5. 2, 7. 1, 5 10 17 26 , , , ,... 2345 1 , 3, 1 , 5, 1 , . . . 2 4 6 Determine the boundedness and monotonicity of the sequence with an as indicated. 2 . n n + ( − 1)n 11. . n 9. 13. (0. 9)n . 10. ( − 1)n . n 2. 2, 0, 2, 0, 2, . . . 4. 1 , 3 , 7 , 15 , 31 , . . . 2 4 8 16 32 6. 8. 34 5 − 1 , 2 , − 16 , 25 , − 36 , . . . 49 1 1 1, 2, 1 , 4, 25 , 6, 49 , . . . 9 12. (1. 001)n . 14. n−1 . n 594 15. 17. 19. 21. 23. 25. CHAPTER 10 SEQUENCES; INDETERMINATE FORMS; IMPROPER INTEGRALS n2 . n+1 4n . √ 4n2 + 1 4n . n + 100 2 2n ln . n+1 (n + 1)2 . n2 1 4− . n 16. 18. n2 + 1. 53. a1 = 1; 54. a1 = 3; 55. a1 = 1, 56. a1 = 1, an+1 = an + · · · + a1 . an+1 = 4 − an . a2 = 3; a2 = 3; an+1 = 2an − an−1 , an+1 = 3an − 2n − 1, n ≥ 2. n ≥ 2. 2n . 4n + 1 n2 20. √ . n3 + 1 n+2 22. 10 √ . 3 n √ 24. (−1)n n. n+1 26. ln . n √ n+1 28. √ . n 1 1 30. − . 2n 2n + 3 32. (− 1 )n . 2 n+3 34. . ln (n + 3) 36. cos nπ . 38. 40. (−2)n . n10 1 − ( 1 )n 2 In Exercises 57–60, use mathematical induction to prove the following assertions for all n ≥ 1. 57. If a1 = 1 and an+1 = 2an + 1, then an = 2n − 1. 58. If a1 = 3 and an+1 = an + 5, then an = 5n − 2. n+1 n an , then an = n−1 . 59. If a1 = 1 and an+1 = 2n 2 1 1 60. If a1 = 1 and an+1 = an − , then an = . n(n + 1) n 61. Let r be a real number, r = 0. Define a sequence {Sn } by S1 = 1 S2 = 1 + r S3 = 1 + r + r 2 · · · Sn = 1 + r + r 2 + · · · + r n−1 · · · (a) Suppose r = 1. What is Sn for n = 1, 2, 3, . . .? (b) Suppose r = 1. Find a formula for Sn that does not involve adding up the powers of r . HINT: Calculate Sn − rSn . 1 62. Set an = , n = 1, 2, 3, . . ., and form the sequence n(n + 1) S 1 = a1 S 2 = a1 + a 2 S 3 = a1 + a 2 + a 3 · · · S n = a1 + a 2 + a 3 + · · · + a n · · · Find a formula for Sn , n = 1 , 2 , 3, . . . , that does not involve adding up the terms a1 , a2 , a3 , . . .. HINT: Use partial fractions to write 1/[k (k + 1)] as the sum of two fractions. 63. A ball is dropped from a height of 100 feet. Each time it hits the ground, it rebounds to 75% of its previous height. (a) Let Sn be the distance that the ball travels between the nth and (n + 1)st bounce, n = 1, 2, 3, . . . . Find a formula for Sn . √ 27. ( − 1)2n+1 n. 29. 2n − 1 . 2n π 31. sin . n+1 33. (1. 2)−n . 35. 1 1 − . n n+1 ln (n + 2) 37. . n+2 3n . 39. (n + 1)2 . ( 1 )n 2 41. Show that the sequence an = 5n /n! decreases for n ≥ 5. Is the sequence nonincreasing? 42. Let M be a positive integer. Show that an = M n /n! decreases for n ≥ M . 43. Show that, if 0 < c < d , then the sequence an = (cn + d n )1/n is bounded and monotonic. 44. Show that linear combinations and products of bounded sequences are bounded. Sequences can be defined recursively: one or more terms are given explicitly; the remaining ones are then defined in terms of their predecessors. In Exercises 45–56, give the first six terms of the sequence and then give the nth term. 45. a1 = 1; 46. a1 = 1; 47. a1 = 1; 48. a1 = 1; 49. a1 = 1; 50. a1 = 1; 51. a1 = 1; 52. a1 = 1; an+1 = an+1 1 an . n+1 = an + 3n(n + 1) + 1. 1 2 an+1 = 1 (an + 1). 2 an+1 = an + 1. an+1 = an + 2. n an+1 = an . n+1 an+1 = an + 2n + 1. an+1 = 2an + 1. 10.3 LIMIT OF A SEQUENCE 595 (b) Let Tn be the time that the ball is in the air between the nth and (n + 1)st bounce, n = 1, 2, 3, . . . . Find a formula for Tn . 64. Suppose that the number of bacteria in a culture is growing exponentially (see Section 7.6) and that the number doubles every 12 hours. Find a formula for the number Pn of bacteria in the culture after n hours, given that there are 500 bacteria initially. c 68. Let {an } be the sequence defined recursively by setting a1 = 1; an+1 = 3a n , n = 1, 2, 3, . . . . (a) Show by induction that {an } is an increasing sequence. (b) Show by induction that {an } is bounded above. (c) Calculate a2 , a3 , a4 , . . . , a15 . Estimate the least upper bound of the sequence. c 65. Let {an } be the sequence defined by setting an = √ n2 + n − n, n = 1, 2, 3, . . . . Use a CAS to determine whether {an } is increasing, nondecreasing, decreasing, nonincreasing, or none of these. c 66. Repeat Exercise 65 with the sequence {an } defined recursively by setting a1 = 100; an+1 = 2 + an , n = 1, 2, 3, . . . . c 69. Let pn denote the nth prime number (see Exercise 34, Section 10.1). Define the sequence {an } by setting an = pn , n ln n n = 2, 3, 4, . . . . (a) Use a CAS to investigate an for large n. (b) Is {an } a bounded sequence? (c) Does it appear that lim an exists? If so, what is the limit? c 70. Let {an } be the sequence defined recursively by setting a1 = 1, a2 = 3, and an+2 = 1 + an+1 , n = 1, 2, 3, . . . . an n→∞ c 67. Let {an } be the sequence defined recursively by setting a1 = 1; an+1 = 1 + √ an , n = 1, 2, 3, . . . . (a) Show by induction that {an } is an increasing sequence. (b) Show by induction that {an } is bounded above. (c) Calculate a2 , a3 , a4 , . . . , a15 . Estimate the least upper bound of the sequence. (a) Use a CAS to investigate the behavior of the sequence. Is the sequence monotone? Is it bounded? If so, give the least upper bound and the greatest lower bound. (b) Experiment with other “initial” values a1 and a2 . 10.3 LIMIT OF A SEQUENCE You have seen the limit process applied in various settings. The limit process applied to sequences is exactly what you would expect. DEFINITION 10.3.1 LIMIT OF A SEQUENCE n→∞ lim an = L if for each > 0, there exists a positive integer K such that if n ≥ K, then |an − L| < . Example 1 PROOF Let 4n − 1 = 4. n→∞ n > 0. We must show that there exists an integer K such that lim if n ≥ K, then 4n − 1 −4 < . n Note that 4n − 1 1 4n − 1 − 4n −1 −4 = = =. n n n n ...
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