Deck 7: Sequences and Series

Question 1

Find a possible formula for the general term of the given sequence, assume that the first term has index 1.{1, 4, 7, 10, 13, …}

A) {2n - 1}
B) {3n + 1}
C) {3n - 2}
D) $\{ 2 n + 1 \} _ { n = 0 } ^ { \infty }$

Answer 1

{3n - 2}

Question 2

Find a possible formula for the general term of the given sequence, assume that the first term has index 0. $\left\{ \frac { 1 } { 3 } , \frac { 2 } { 5 } , \frac { 3 } { 7 } , \frac { 4 } { 9 } , \ldots \right\}$

A) $\left\{ \frac { n - 1 } { 2 n + 1 } \right\} _ { n = 0 } ^ { \infty }$
B) $\left\{ \frac { 2 n - 1 } { 4 n - 1 } \right\}$
C) $\left\{ \frac { n - 1 } { 2 n + 3 } \right\} _ { n = 0 } ^ { \infty }$
D) $\left\{ \frac { n + 1 } { 2 n + 3 } \right\} _ { n = 0 } ^ { \infty }$

Answer 2

$\left\{ \frac { n + 1 } { 2 n + 3 } \right\} _ { n = 0 } ^ { \infty }$

Question 3

Find a possible formula for the general term of the given sequence, assume that the first term has index 1. $\left\{ \frac { 1 } { 5 } , \frac { 2 } { 25 } , \frac { 3 } { 125 } , \frac { 4 } { 625 } , \ldots \right\}$

A) $\left\{ \frac { n + 2 } { 5 ^ { n } } \right\}$
B) $\left\{ \frac { n } { 5 ^ { n } } \right\}$
C) $\left\{ \frac { n + 1 } { 5 ^ { n } } \right\}$
D) $\left\{ \frac { n ^ { 2 } } { 5 ^ { n } } \right\}$

Answer 3

$\left\{ \frac { n } { 5 ^ { n } } \right\}$

Question 4

Find a possible formula for the general term of the given sequence, assume that the first term has index 0. $\left\{ 3 , \frac { - 3 } { 2 } , \frac { 3 } { 4 } , \frac { - 3 } { 8 } , \frac { 3 } { 16 } , \ldots \right\}$

Question 5

Provide the first five terms of the sequence: $$\left\{ \frac { 1 - ( - 1 ) ^ { n } } { n + 1 } \right\}$$ Assume that the first term has index 1.

A) $\left\{ \frac { 1 } { 2 } , 0 , \frac { 1 } { 3 } , 0 , \frac { 1 } { 4 } , \ldots \right\}$
B) $\left\{ 0,1,0 , \frac { 1 } { 2 } , 0 , \ldots \right\}$
C) $\left\{ 1,0 , \frac { 1 } { 2 } , 0 , \frac { 1 } { 3 } , \ldots \right\}$
D) $\left\{ 1 , \frac { 1 } { 2 } , \frac { 1 } { 3 } , \frac { 1 } { 4 } , \frac { 1 } { 5 } , \ldots \right\}$

Question 6

Provide the first five terms of the sequence: $$\left\{ \frac { 2 n } { 3 n - 1 } \right\}$$ Assume that the first term has index 1.

A) $\left\{ 0,1 , \frac { 4 } { 5 } , \frac { 3 } { 4 } , \frac { 8 } { 11 } , \ldots \right\}$
B) $\left\{ 1 , \frac { 2 } { 3 } , \frac { 4 } { 5 } , \frac { 3 } { 4 } , \frac { 8 } { 11 } , \ldots \right\}$
C) $\left\{ 1 , \frac { 4 } { 5 } , \frac { 3 } { 4 } , \frac { 8 } { 11 } , \frac { 5 } { 7 } , \ldots \right\}$
D) $\left\{ 0,1 , \frac { 2 } { 3 } , \frac { 3 } { 5 } , \frac { 4 } { 7 } , \ldots \right\}$

Question 7

Provide the first five terms of the sequence $\left\{ \frac { \sin ( n x ) } { x ^ { n } + n ^ { 2 } } \right\}$ Assume that the first term has index 1.

Question 8

Provide the first five terms of the sequence $\left\{ \frac { \sqrt { n } } { n ^ { 2 } } \right\}$ Assume that the first term has index

Question 9

Find the least upper bound and the greatest lower bound of the sequence: $$\left\{ \frac { 2 } { n } \right\}$$

Question 10

Find the least upper bound and the greatest lower bound of the sequence: $$\left\{ \frac { 1 } { 3 } , \frac { 1 } { 2 } , \frac { 3 } { 5 } , \frac { 2 } { 3 } , \frac { 5 } { 7 } , \frac { 3 } { 4 } , \ldots \right\}$$

Question 11

Write the arithmetic sequence $\{ - 2,8,18,28 , \ldots \}$ in the form: $\{ c + d k \} _ { k = 0 } ^ { \infty }$

A) $\{ - 2 + 8 k \} _ { k = 0 } ^ { \infty }$
B) $\{ - 2 + 10 k \}$
C) $\{ - 2 + 10 k \} _ { k = 0 } ^ { \infty }$
D) none of these

Question 12

Write the arithmetic sequence $\{ 3,4.1,5.2,6.3,7.4,8.5 , \ldots \}$ in the form: $\{ c + d k \} _ { k = 0 } ^ { \infty }$

Question 13

Give the first five terms of the geometric sequence $\left\{ c r ^ { k } \right\} _ { k = 0 } ^ { \infty }$ with c = 5, and r = $1 / 2$

A) $\left\{ \frac { 5 } { 4 } , \frac { 5 } { 6 } , \frac { 5 } { 8 } , \frac { 1 } { 2 } , \frac { 5 } { 12 } , \ldots \right\}$
B) $\left\{ 5 , \frac { 5 } { 2 } , \frac { 5 } { 4 } , \frac { 5 } { 8 } , \frac { 5 } { 16 } , \ldots \right\}$
C) $\left\{ 5 , \frac { 5 } { 4 } , \frac { 5 } { 8 } , \frac { 5 } { 16 } , \frac { 5 } { 32 } , \ldots \right\}$
D) none of these

Question 14

Give the first five terms of the geometric sequence $\left\{ c r ^ { k } \right\} _ { k = 0 } ^ { \infty }$ with c = 3, and r = - $1 / 2$

Question 15

Use the difference test in Theorem 7.6 to analyze the monotonicity of the sequence $\left\{ k ^ { 2 } - k \right\}$ Assume that the first term has index 1.

A) Strictly decreasing
B) Eventually decreasing
C) Strictly increasing
D) Eventually increasing

Question 16

Use the difference test in Theorem 7.6 to analyze the monotonicity of the sequence $\left\{ \frac { k } { k + 3 } \right\}$

A) Strictly decreasing
B) Eventually decreasing
C) Eventually increasing
D) Strictly increasing

Question 17

Use the ratio test in Theorem 7.6 to analyze the monotonicity of the sequence $\left\{ \frac { 2 k ^ { 2 } } { k ! } \right\}$

A) Strictly increasing
B) Eventually increasing
C) Strictly decreasing
D) Eventually decreasing

Question 18

Use the derivative test in Theorem 7.6 to analyze the monotonicity of the sequence $\left\{ k ^ { 3 } - k \right\}$ Assume that the first term has index 1.

A) Strictly decreasing
B) Eventually decreasing
C) Strictly increasing
D) Eventually increasing

Question 19

Use the derivative test in Theorem 7.6 to analyze the monotonicity of the sequence $\left\{ \frac { \sqrt { k + 2 } } { k } \right\}$

A) Strictly increasing
B) Strictly decreasing
C) Eventually decreasing
D) Eventually increasing

Question 20

Use the derivative test in Theorem 7.6 to analyze the monotonicity of the sequence $\left\{ \frac { k ^ { 2 } } { e ^ { k } } \right\}$

A) Strictly decreasing
B) Strictly increasing
C) Eventually decreasing
D) Eventually increasing

Question 21

Determine whether the sequence $\left\{ \frac { k ^ { 2 } } { e ^ { k } } \right\}$ is monotonic or eventually monotonic and whether the sequence is bounded. If the sequence is convergent, find the limit.

A) Monotonic, bounded, converges to 1
B) Monotonic, bounded, converges to 0
C) Eventually monotonic, bounded, converges to 1
D) Eventually monotonic, bounded, converges to 0

Question 22

Determine whether the sequence $\left\{ 2 ^ { 1 / k } \right\}$ is monotonic or eventually monotonic and whether the sequence is bounded. If the sequence is convergent, find the limit.

A) Eventually monotonic, bounded, converges to 0
B) Monotonic, unbounded, diverges
C) Monotonic, bounded, converges to 0
D) Monotonic, bounded, converges to 1

Question 23

Determine whether the sequence $\{ \cos ( k \pi ) \}$ is monotonic or eventually monotonic and whether the sequence is bounded. If the sequence is convergent, find the limit.

A) Monotonic, bounded, converges to 1
B) Monotonic, unbounded, diverges
C) Not monotonic, bounded, diverges
D) Monotonic, bounded, converges to 0

Question 24

Determine whether the sequence $\left\{ \frac { k } { k ^ { 2 } + 3 k + 2 } \right\}$ is monotonic or eventually monotonic and whether the sequence is bounded. If the sequence is convergent, find the limit.

A) Monotonic, bounded, converges to 1
B) Monotonic, bounded, converges to 0
C) Eventually monotonic, bounded, converges to 0
D) Eventually monotonic, bounded, converges to 1

Question 25

Determine whether the sequence $\left\{ \frac { \cos k } { k } \right\}$ is monotonic or eventually monotonic and whether the sequence is bounded. If the sequence is convergent, find the limit.

A) Monotonic, bounded, converges to 0
B) Not monotonic, bounded, diverges
C) Not monotonic, bounded, converges to 1
D) Not monotonic, bounded, converges to 0

Question 26

Determine whether the sequence $\left\{ \cos \left( \frac { k \pi } { 2 } \right) \right\}$ is monotonic or eventually monotonic and whether the sequence is bounded. If the sequence is convergent, find the limit.

A) Monotonic, bounded, converges to 0
B) Monotonic, not bounded, diverges
C) Monotonic, bounded, converges to 1
D) Not monotonic, bounded, diverges

Question 27

Does the sequence $\left\{ \frac { \cos k } { 2 k } \right\}$ converge? If it does, give the limit.

A) Converges to 1
B) Diverges
C) Converges to -1
D) Converges to 0

Question 28

Does the sequence $\left\{ \tan \left( \frac { k \pi } { 2 } \right) \right\}$ converge? If it does, give the limit.

A) Converges to 1
B) Converges to 0
C) Diverges
D) Converges to -1

Question 29

Does the sequence $\left\{ \frac { \ln k } { 2 ^ { k } } \right\}$ converge? If it does, give the limit.

A) Converges to 1
B) Converges to 0
C) Converges to -1
D) Diverges

Question 30

Does the sequence $\left\{ \frac { \ln k } { 3 k ! } \right\}$ converge? If it does, give the limit.

A) Diverges
B) Converges to 1
C) Converges to -1
D) Converges to 0

Question 31

Does the sequence $\left\{ \frac { 5 ^ { k } } { k ! } \right\}$ converge? If it does, give the limit.

A) Diverges
B) Converges to 1
C) Converges to 0
D) Converges to -1

Question 32

Does the sequence $\left\{ \frac { k - e ^ { - k } } { e ^ { k } } \right\}$ converge? If it does, give the limit.

A) Converges to 1
B) Converges to 0
C) Converges to -1
D) Diverges

Question 33

Provide the first five terms of the series $\sum _ { n = 1 } ^ { \infty } \frac { n + 2 } { n ! }$

Question 34

Provide the first five terms of the series $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { 3 ( n ! ) }$

Question 35

Provide the first five terms of the series $\sum _ { n = 1 } ^ { \infty } \frac { 2 } { n ^ { 2 } + 1 }$

Question 36

Provide the first four terms of the sequence of partial sums for the series $\sum _ { n = 1 } ^ { \infty } \frac { n + 2 } { n ! }$

Question 37

Provide the first four terms of the sequence of partial sums for the series $\sum _ { n = 2 } ^ { \infty } ( 2 n + 3 )$

Question 38

Does the series $\sum _ { k = 0 } ^ { \infty } \frac { 4 } { 2 ^ { k } }$ converge or diverge? If it converges, then find the sum.

A) Converges, sum is 4
B) Converges, sum is 8
C) Diverges
D) Converges, sum is 2

Question 39

Does the series $\sum _ { k = 2 } ^ { \infty } \frac { 3 ^ { k } } { 2 }$ converge or diverge? If it converges, then find the sum.

A) Converges, sum is 3
B) Converges, sum is 3/2
C) Diverges
D) Converges, sum is 1

Question 40

Does the series $\sum _ { k = 2 } ^ { \infty } \frac { ( - 1 ) ^ { k } } { 2 ^ { k + 1 } }$ converge or diverge? If it converges, then find the sum.

A) Converges, sum is $1 / 2$
B) Converges, sum is $1 / 4$
C) Converges, sum is 1/12
D) Diverges

Question 41

Does the series $\sum _ { k = 2 } ^ { \infty } \left( - \frac { 2 } { 5 } \right) ^ { k }$ converge or diverge? If it converges, then find the sum.

A) Converges, sum is 2/5
B) Converges, sum is 4/25
C) Diverges
D) Converges, sum is 4/35

Question 42

Does the series $\sum _ { k = 0 } ^ { \infty } \frac { 3 ^ { k + 1 } } { 4 ^ { k } }$ converge or diverge? If it converges, then find the sum.

A) Converges, sum is $3 / 4$
B) Converges, sum is 3/16
C) Converges, sum is 12
D) Diverges

Question 43

Does the series $\sum _ { k = 1 } ^ { \infty } \left( \frac { 1 } { k } - \frac { 1 } { k + 3 } \right)$ converge or diverge? If it converges, then find the sum.

A) Diverges
B) Converges, sum is 5/6
C) Converges, sum is 11/6
D) Converges, sum is 1/3

Question 44

Does the series $\sum _ { k = 1 } ^ { \infty } \frac { 4 } { k ^ { 2 } + k }$ converge or diverge? If it converges, then find the sum.

A) Diverges
B) Converges, sum is 2
C) Converges, sum is 8
D) Converges, sum is 4

Question 45

Does the series $\sum _ { k = 1 } ^ { \infty } \frac { 5 } { k ^ { 2 } + 3 k + 2 }$ converge or diverge? If it converges, then find the sum.

A) Diverges
B) Converges, sum is 5/2
C) Converges, sum is 5/3
D) Converges, sum is 2/3

Question 46

Use divergence test to analyze the series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n }$

Question 47

Use divergence test to analyze the series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 } }$

Question 48

Use divergence test to analyze the series $\sum _ { n = 1 } ^ { \infty } \frac { n } { 4 n + 3 }$

Question 49

Use integral test to determine whether the series $\sum _ { n = 1 } ^ { \infty } \frac { n } { e ^ { n } }$ converges or diverges.

Question 50

Use integral test to determine whether the series $\sum _ { n = 2 } ^ { \infty } \frac { 1 } { n \ln n }$ converges or diverges.

Question 51

Use integral test to determine whether the series $\sum _ { n = 1 } ^ { \infty } \frac { 2 + n } { n ^ { 2 } }$ converges or diverges.

Question 52

Use either divergence or integral test to determine whether the series $\sum _ { k = 1 } ^ { \infty } \frac { k } { k ^ { 2 } + 1 }$ converges or diverges.

Question 53

Use either divergence or integral test to determine whether the series $\sum _ { k = 1 } ^ { \infty } \frac { 2 } { 3 k + 1 }$ converges or diverges.

Question 54

Use either divergence or integral test to determine whether the series $\sum _ { k = 1 } ^ { \infty } \frac { \ln k } { 2 k }$ converges or diverges.

Question 55

Use either divergence or integral test to determine whether the series $\sum _ { k = 1 } ^ { \infty } \frac { 2 } { k ^ { 2 } }$ converges or diverges.

Question 56

Use one of the comparison tests to determine whether the series $\sum _ { k = 1 } ^ { \infty } \frac { 2 + \ln k } { k }$ converges or diverges.

Question 57

Use one of the comparison tests to determine whether the series $\sum _ { k = 0 } ^ { \infty } \frac { 2 k ^ { 2 } + 1 } { k ^ { 3 } + k ^ { 2 } + 3 }$ converges or diverges.

Question 58

Use one of the comparison tests to determine whether the series $\sum _ { k = 1 } ^ { \infty } \frac { 1 + \ln k } { k ^ { 2 } }$ converges or diverges.

Question 59

Use one of the comparison tests to determine whether the series $\sum _ { k = 1 } ^ { \infty } \frac { \sin ^ { 2 } k } { k ^ { 2 } }$ converges or diverges.

Question 60

Use one of the comparison tests to determine whether the series $\sum _ { k = 1 } ^ { \infty } \frac { 2 } { 1 + k ^ { 2 } }$ converges or diverges.

Question 61

Use one of the comparison tests to determine whether the series $\sum _ { k = 1 } ^ { \infty } \frac { 4 } { 1 + 3 k }$ converges or diverges.

Question 62

Use one of the comparison tests to determine whether the series $\sum _ { k = 1 } ^ { \infty } \sin \left( \frac { 2 } { k } \right)$ converges or diverges.

Question 63

Use one of the comparison tests to determine whether the series $\sum _ { k = 1 } ^ { \infty } \frac { \sqrt { k } } { k ^ { 2 } + 1 }$ converges or diverges.

Question 64

Use one of the comparison tests to determine whether the series $\sum _ { k = 1 } ^ { \infty } \frac { \ln k } { 2 k }$ converges or diverges.

Question 65

Use ratio test to determine whether the series $\sum _ { k = 0 } ^ { \infty } \frac { 3 ^ { k } } { k ! }$ converges or diverges.

Question 66

Use ratio test to determine whether the series $\sum _ { k = 1 } ^ { \infty } \frac { 2 ^ { k } } { k ^ { 5 } }$ converges or diverges.

Question 67

Use ratio test to determine whether the series $\sum _ { k = 1 } ^ { \infty } \frac { k ! } { ( 3 k ) ! }$ converges or diverges.

Question 68

Use ratio test to determine whether the series $\sum _ { k = 1 } ^ { \infty } \frac { 2 ^ { 2 k - 1 } } { ( 2 k - 1 ) ! }$ converges or diverges.

Question 69

Use ratio test to determine whether the series $\sum _ { k = 1 } ^ { \infty } \frac { ( 2 k ) ! } { ( k + 1 ) ! }$ converges or diverges.

Question 70

Use root test to determine whether the series $\sum _ { k = 1 } ^ { \infty } \frac { 3 ^ { k } } { k ^ { 2 } }$ converges or diverges.

Question 71

Use root test to determine whether the series $\sum _ { k = 2 } ^ { \infty } \left( \frac { 2 } { \ln k } \right) ^ { k }$ converges or diverges.

Question 72

Use root test to determine whether the series $\sum _ { k = 1 } ^ { \infty } \left( \frac { 1 + 3 k } { 4 k - 1 } \right) ^ { k }$ converges or diverges.

Question 73

Use root test to determine whether the series $\sum _ { k = 1 } ^ { \infty } \left( \frac { 1 + 5 k } { 3 k - 1 } \right) ^ { k }$ converges or diverges.

Question 74

Use root test to determine whether the series $\sum _ { k = 1 } ^ { \infty } \frac { 2 ^ { k } } { k ^ { k } }$ converges or diverges.

Question 75

Use any convergence test to determine whether the series $\sum _ { k = 1 } ^ { \infty } \left( \frac { 1 + k } { 2 k + 1 } \right) ^ { k }$ converges or diverges.

Question 76

Use any convergence test to determine whether the series $\sum _ { k = 1 } ^ { \infty } \frac { k ^ { 2 } } { 3 ^ { k } }$ converges or diverges.

Question 77

Use any convergence test to determine whether the series $\sum _ { k = 1 } ^ { \infty } \frac { 3 } { 2 + k }$ converges or diverges.

Question 78

Use any convergence test to determine whether the series $\sum _ { k = 1 } ^ { \infty } \frac { 2 } { k ( 1 + k ) }$ converges or diverges.

Question 79

Use any convergence test to determine whether the series $\sum _ { k = 1 } ^ { \infty } \frac { k } { 1 + 2 k ^ { 3 } }$ converges or diverges.

Question 80

Use any convergence test to determine whether the series $\sum _ { k = 1 } ^ { \infty } \frac { 3 ^ { k } } { ( k + 1 ) ! }$ converges or diverges.