Chapter 9 Review - Sequences and series calc 2 PDF

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Sequences and series calc 2...

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Name: ________________________ Class: ___________________ Date: __________

Chapter 9 Review Multiple Choice Identify the choice that best completes the statement or answers the question. ____

1. Match the sequence with its graph.

an 

6 n1

1

ID: A

Name: ________________________

ID: A

a.

d.

b.

e.

c.

2

Name: ________________________ ____

ID: A

2. Match the sequence with its graph.

an 

1n n!

a.

d.

b.

e.

c.

3

Name: ________________________ ____

ID: A

3. Match the sequence with its graph.

an 

4 1 n

n

a.

d.

b.

e.

c.

4

Name: ________________________

ID: A 

____

4. Identify the most appropriate test to be used to determine whether the series

 n1

15 1 n

n 1

converges or

diverges. a. b. c. d. e. ____

5. Determine the convergence or divergence of the series using any appropriate test from this chapter. Identify the test used. 

 n1

a. b. c. d. e. ____

Ratio Test  -Series Test Alternating Series Test T ele scoping Series Test Root Test

n

1 9 7n converges; I ntegr al Test converges; Ratio Test converges; Alterna tin gSeries Test diverges; Ratio Te st diverges; Int egralTe st

6. Determine the convergence or divergence of the series using any appropriate test from this chapter. Identify the test used. 

 n10

n1

a. b. c. d. e.

converges; Ratio Test both civerges; p-series and civerges; Integral Test civerges; p-series civerges; Integral Test converges; p-series

5

Name: ________________________

____

ID: A

12 and P1, a first-degree polynomial function whose value and slope x agree with the value and slope of f at x  8.

7. Use a graphing utility to graph f x 

3

a.

d.

b.

e.

c.

6

Name: ________________________

____

ID: A

  n x 8. Identify the graph of the first 10 terms of the sequence of partial sum of the series    for x  2. 3 n0 

a.

d.

b.

e.

c.

7

Name: ________________________

ID: A

Short Answer 1. Write the first five terms of the sequence.

 11  a n  1 n  2    n 2. Write the first three terms of the sequence.

an  1 

5 3  n n2 n

3. Graph the sequence a n  4 1 . 4. Write the next two apparent terms of the sequence 1, 

1 1 1 ,.... , , 27 3 9

5. Determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit.

  ln n 3    an  7n 6. Determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit.

an 

3n 4n

7. Write an expression for the nth term of the sequence 10, 22, 34, 46, .... 8. Write an expression for the nth term of the sequence

10 19 28 37 , , , ,.... 11 20 29 38

n    r  9. Consider the sequence A n  whose nth term is given by A n  P 1   where P is the principal, An is 12     the account balance in dollars after n months, and r is the interest rate compounded annually. Find the fourth term of the sequence if P  $9,000 and r  0.065. Round your answer to two decimal places.

8

Name: ________________________

ID: A

10. A deposit of $225 is made at the beginning of each month into an account at an annual interest rate of 3%   n compounded monthly. The balance in the account after n months is An  225 401  1.0025  1 . Find the   balance in the account after 4 years by computing the 48th term of the sequence. Round your answer to two decimal places. 11. A government program that currently costs taxpayers $2.5 billion per year is cut back by 30 percent per year. Write an expression for the amount budgeted for this program after n years. 12. A government program that currently costs taxpayers $3.5 billion per year is cut back by 30 percent per year. Compute the budget after the fifth year. Round your answer to the nearest integer.

1 13. If the rate of inflation is 3 % per year and the average price of a car is currently $40,000, the average price 2 n

after n years is Pn  $40,000 1.035 . Compute the average price after 6 years. Round your answer to two decimal places. 14. Write the first five terms of the sequence of partial sums.

5

5 5 5 5     4 9 16 25

15. Write the first five terms of the sequence of partial sums.

6 

36 216 1,296 7,776     7 343 2,401 49

16. Write the first five terms of the sequence of partial sums. 

 n1

5 6

n 1



17. True or false. The infinite series

 13nn 4 diverges.

n1

18. Find the sum of the convergent series. 

 n

 9 45 

n0

 

9

Name: ________________________

ID: A

19. Find the sum of the convergent series. 

 n0

 9  n 10    10 

20. Find the sum of the convergent series. 

 n1

9 n  9 n  11

21. Find the sum of the convergent series 8  1  

22. Find the sum of the convergent series

1 1    8 64

 1

 

n  0

6

n



1 7n

  .  

23. Write the repeating decimal 0. 75 as a geometric series. 

24. True or false. The series

 n1

6n  1 is convergent. 12n  1



25.

True or false. The series 

20 is divergent. n n  0 10

26. Determine the convergence or divergence of the series. 



5 n n03

27. Find all values of x for which the series





n

n0





 9 x 9 8 

10

converges.

Name: ________________________

ID: A

28. Find all values of x for which the series converges. For these values of x, write the sum of the series as a function of x. 

 n0

 x  8  n  10   10 

29. Suppose an electronic games manufacturer producing a new product estimates the annual sales to be 4,000 units. Each year 20% of the units that have been sold will become inoperative. So, 4,000 units will be in use   after 1 year,  4,000  0.80 4,000  units will be in use after 2 years, and so on. How many units will be in   use after n years? 30. Suppose the annual spending by tourists in a resort city is $100 million. Approximately 90% of that revenue is again spent in the resort city, and of that amount approximately 90% is again spent in the same city, and so on. Write the expression that gives the total amount of spending generated by the $100 million after n years. 31. Suppose the annual spending by tourists in a resort city is $100 million. Approximately 75% of that revenue is again spent in the resort city, and of that amount approximately 75% is again spent in the same city, and so 

on. Summing all of this spending indefinitely, leads to the geometric series

 100  0.75i . Find the sum of

i0

this series. 32. Suppose a ball is dropped from a height of 16 feet. Each time it drops h feet, it rebounds 0.85h feet. Find the total distance traveled by the ball. Round your answer to two decimal places. 33. Suppose the winner of a $4,000,000 sweepstakes will be paid $100,000 per year for 40 years, starting a year 40

from now. The money earns 5% interest per year. The present value of the winnings is

 n1

 1  n  . 100,000   1.05 

Compute the present value using the formula for the nth partial sum of a geometric series. Round your answer to two decimal places. 34. Suppose you go to work at a company that pays $0.08 for the first day, $0.16 for the second day, $0.32 for the third day, and so on. If the daily wage keeps doubling, what would your total income be for working 29 days? Round your answer to two decimal places. 35. Use the Integral Test to determine the convergence or divergence of the series. 

 n1

2 9n  2

11

Name: ________________________

ID: A

36. Use the Integral Test to determine the convergence or divergence of the series. 

 ne



n 2

n1

37. True or false: The series

ln2 ln3 ln4 ln5 ln6         converges. 4 10 8 6 12

38. True or false: The series

1 2 3 n  2       converges. 3 6 11 n 2 

39. True or false: The series

 n1

1 converges. n5

40. Use the Integral Test to determine the convergence or divergence of the series. 

 lnn n 2

n

10

41. Use the Integral Test to determine the convergence or divergence of the series. 

10 n  2 n lnn





42. True or false: The series

 n1

1 6n  7

3

converges.



43. True or false: The series

6n diverges. 4n 2  1 n1



44. Use Theorem 9.11 to determine the convergence or divergence of the series. 

 n1

2 n

5 9

12

Name: ________________________

ID: A

45. Use Theorem 9.11 to determine the convergence or divergence of the series.

1

1 3



2

2

1 3

3

 2

1 3

 2

4

1 3

5

2



46. Use Theorem 9.11 to determine the convergence or divergence of the series. 

1



n1n

0.86





47. Sketch the graph of the sequence of partial sum of the series

n 1



1



48. Find the positive values of p for which the series

n2 



49. Find the positive values of p for which the series

n1

n(ln n )

p

2 4

n

. 3

converges.

 p 3n 8  n 2  converges.  

50. Determine the convergence or divergence of the series. 

4 8 n  n n1



51. Determine the convergence or divergence of the series. 

8

1

 n1

n

1.15

52. Determine the convergence or divergence of the series. 

   n0

1   4 

n



53. Use the Direct Comparison Test (if possible) to determine whether the series  n5

diverges.

13

1 2

7n  4

converges or

Name: ________________________

ID: A 

54. Use the Direct Comparison Test to determine the convergence or divergence of the series 

n  1 7n

1 2

9

.

55. Use the Direct Comparison Test (if possible) to determine whether the series 

 n9n

1 5 6

8 

2n converges or n n 1 8 1

56. Use the Direct Comparison Test (if possible) to determine whether the series  diverges.

57. Use the Direct Comparison Test to determine the convergence or divergence of the series 

8n . n n 09 7 

58. Use the Direct Comparison Test to determine the convergence or divergence of the series 



n15

4

1 n 1 

e

59. Use the Direct Comparison Test to determine the convergence or divergence of the series

n0

60. Use the Limit Comparison Test to determine the convergence or divergence of the series 

9n . 2 9n 2 n1 



61. Use the Limit Comparison Test (if possible) to determine whether the series  n1

2 9

n2 9

.



3n  1 . n n 1 8 1

62. Use the Limit Comparison Test to determine the convergence or divergence of the series 



4n  1 converges or n n 15 6

63. Use the Limit Comparison Test (if possible) to determine whether the series  diverges.

14

n

6

.

Name: ________________________

ID: A 

2n 2  7 converges or 7 n  1 7n  3n  2

64. Use the Limit Comparison Test (if possible) to determine whether the series  diverges.



9

65. Use the Limit Comparison Test to determine the convergence or divergence of the series  n1

n2  6

n



66. Use the Limit Comparison Test to determine the convergence or divergence of the series  2sin n1

.

1 . n



1 . n 6 n 6

67. Use the Limit Comparison Test to determine the convergence or divergence of the series 

7

68. Use the Direct Comparison Test to determine the convergence or divergence of the series 

 n1

7n .  2 2   n  7  

69. Use the polynomial test to determine whether the series

5 4 3 2 1  . . . converges or diverges.    20 23 28 35 44 

1 converges or diverges. n 1n 9

70. Use the polynomial test to determine whether the series 

71. Determine the convergence or divergence of the series

72. Determine the convergence or divergence of the series 

73. Consider the series

 n1

1 6n  1 

74. True or false: The series

 n1

2

5

1 1 1 1 . . ..    550 1100 1650 2200 1 1 1 1 ....    251 258 277 314

2 . The sum of the series is  / 6 Find the sum of the series



 n5

n 1

1 n converges. 6n  4

15

1 2

6n  1

.

Name: ________________________

ID: A 

75. True or false: The series

 n1

1

n

diverges.

9n



76. True or false: The series

 sec n   converges. n1 

77. True or false: The series

 n1

n

1 converges . 6n  3! 

78. Determine whether the series

 n1 

79. Determine whether the series

 n1 

80. Determine whether the series

 n2

n 1

1 n 2

1

n

n3 /8

converges absolutely, converges conditionally, or diverges.

n

1 converges conditionally or absolutely, or diverges. ln2n



81.

converges conditionally or absolutely, or diverges.

Determine whether the series  n0

cos n  converges conditionally or absolutely, or diverges. n3

82. Approximate the sum of the series by using the first six terms. 

 n0

n

1 4 n!

83. Approximate the sum of the series by using the first six terms. 

 n1

1 n

n 1

2

3

84. Determine the minimal number of terms required to approximate the sum of the series with an error of less than 0.007. 

 n1

1

n 1

n3

16

Name: ________________________

ID: A

85. Determine the minimal number of terms required to approximate the sum of the series with an error of less than 0.004. 

 n0

1 n!

n

86. Determine the minimal number of terms required to approximate the sum of the series with an error of less than 0.008. 

 n1

1

n 1

2n3  1

87. Determine the minimal number of terms required to approximate the sum of the series with an error of less than 0.005. 

 n0

n

1 (2n)!

88. Approximate the sum of the series by using the first six terms. 

 n1

n 1

1 3 ln(n  1) 

89. Use the Ratio Test to determine the convergence or divergence of the series

 n!n .

n0

90. Use the Ratio Test to determine the convergence or divergence of the series. 



n





 n 103 

n1

91. Use the Ratio Test to determine the convergence or divergence of the series. 

n6 n n  1 10



92. Use the Ratio Test to determine the convergence or divergence of the series.



 n1

1

 7 n   2  

n 1  

n2

17

8

Name: ________________________

ID: A 

93. Use the Ratio Test to determine the convergence or divergence of the series

 n0

n

8n

1 7 . 7n  1!



94. Use the Root Test to determine the convergence or divergence of the series



1 . n 16 n1

95. Use the Root Test to determine the convergence or divergence of the series. 



n

  3n4n 1

n1



96. Use the Root Test to determine the convergence or divergence of the series. 

 7n  1  n     4n  1   n1



97. Use the Root Test to determine the convergence or divergence of the series.

n   7n 2  1      2 10n    1 n1  





98. Use the Root Test to determine the convergence or divergence of the series

 e 3n. n1



99. Determine the convergence or divergence of the series

 n1

7 1 n

n 1

using any appropr...