Please #55,59,73,79

27-62 Determine whether the sequence converges or diverges.

If it converges, find the limit.

78-84 Determine whether the sequence is increasing, decreas ing, or not monotonic. Is the sequence bounded? 78. an = cos n 1 - n 79. an = 80. an 2n + 3 2+ n 81. an = n(-1)" 82. an = 2 + (-1)" n 83. an = 3 - 2ne-" 84. a, = n' - 3n + 3An' - 3n 4n' - 3n 29. an = n cosn 2n- + 1 30. an = 67. an = 2n + 1 1 + n2 31. an = n' - 2n 32. an = 2 + (0.86)" 1 . 3 . 5 . . . . . (2n - 1) 68. an = n! 33. an = 3"7-n 34. an = 3 vn 69. an = 1 . 3 . 5 . . . . . (2n - 1) Vn + 2 (2n)' 35. an = e-1/vn 36. an = 1 + 9" 70. (a) Determine whether the sequence defined as follows is 1 + 4n2 convergent or divergent: 37. an = 1 + n2 38. an = cos n + 1 a = 1 an+1 = 4 - an for n > 1 (b) What happens if the first term is a, = 2? 39. an = 40. an = e2n/(n+2) Vn' + 4n 71. If $1000 is invested at 6% interest, compounded annually, then after n years the investment is worth a,, = 1000(1.06)" dollars. 41. an, = (-1)" 42. an, =- (-1)"t 'n 2vn n + n (a) Find the first five terms of the sequence {an}. (b) Is the sequence convergent or divergent? Explain. 43. (2n - 1)! (2n + 1)! 44. In n In(2n) 72. If you deposit $100 at the end of every month into an account that pays 3% interest per year compounded 45. { sinn } 46. an = tan 'n monthly, the amount of interest accumulated after n months n is given by the sequence 47. {n'e -" } 48. an = In(n + 1) - Inn In = 100 1.0025" - 1 49. an = cos'n 0.0025 50. an = "/21+3n (a) Find the first six terms of the sequence. 51. an = n sin(1) 52. an = 2 " cos nT (b) How much interest will you have earned after two years? 53 . an (1 + 2 ) 73. A fish farmer has 5000 catfish in his pond. The number of 54. an = n'/ catfish increases by 8% per month and the farmer harvests 300 catfish per month. 55. an = In(2n? + 1) - In(n? + 1) (a) Show that the catfish population P, after n months is (In n)2 given recursively by 56. an, = Pn = 1.08 PM-1 - 300 Po = 5000 57. an = arctan(In n) (b) Find the number of catfish in the pond after six months. 58. an = n - Vn + 1vn+ 3 74. Find the first 40 terms of the sequence defined by 59. {0, 1, 0, 0, 1, 0, 0, 0, 1,...} an+1 = if an is an even number 60. { t, 3, 2, 4, 3, 3. 4. b.... } 3an + 1 if an is an odd number 61. an = 62. a, = (-3)" and a1 = 11. Do the same if a1 = 25. Make a conjecture n! about this type of sequence. 75. For what values of r is the sequence {nr"} convergent? 63-69 Use a graph of the sequence to decide whether the 76. (a) If {a,, } is convergent, show that sequence is convergent or divergent. If the sequence is conver- gent, guess the value of the limit from the graph and then prove lim an+1 = lim a, your guess. (b) A sequence {a, } is defined by a1 = 1 and 63. an = (-1)" 64. an = sinn anti = 1/(1 + an) for n > 1. Assuming that {an } is n + 1 n convergent, find its limit. 65. an = arctan 66. an = V3" + 5" 77. Suppose you know that {a,, } is a decreasing sequence and all its terms lie between the numbers 5 and 8. Explain why