Question 1

Find the limit of the sequence if it converges; otherwise indicate divergence.
-$$a _ { n } = \frac { \tan ^ { - 1 } n } { \sqrt [ 7 ] { n } }$$

Accepted Answer

A)  $\frac { \pi } { 2 }$ 
B)  1 
C)  0 
D)  Diverges 
A)  $\frac { \pi } { 2 }$ 
B)  1 
C)  0 
D)  Diverges

Question 2

Find the sum of the series.
-$$\sum _ { n = 0 } ^ { \infty } \left( \frac { 1 } { 3 ^ { n } } + \frac { 1 } { 4 ^ { n } } \right)$$

Accepted Answer

A)  $\frac { 17 } { 6 }$ 
B)  $\frac { 31 } { 6 }$ 
C)  $\frac { 25 } { 6 }$ 
D)  $\frac { 23 } { 6 }$ 
A)  $\frac { 17 } { 6 }$ 
B)  $\frac { 31 } { 6 }$ 
C)  $\frac { 25 } { 6 }$ 
D)  $\frac { 23 } { 6 }$

Question 3

Find the limit of the sequence if it converges; otherwise indicate divergence.
-$$a _ { n } = \ln \left( 1 + \frac { 2 } { n } \right) ^ { n }$$

Accepted Answer

A)  0 
B)  $\ln 2$ 
C)  2 
D)  Diverges 
A)  0 
B)  $\ln 2$ 
C)  2 
D)  Diverges

Question 4

Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges.
-$$2 + \frac { 2 } { 7 } + \frac { 2 } { 49 } + \ldots + \frac { 2 } { 7 ^ { n - 1 } } + \ldots$$

Accepted Answer

A)  $\frac { 2 \left( 1 - \frac { 1 } { 7 ^ { n - 1 } } \right) } { 1 - \frac { 1 } { 7 } } ; \frac { 7 } { 4 }$ 
B)  $\frac { 2 \left( 1 - \frac { 1 } { 7 ^ { n } } \right) } { 1 - \frac { 1 } { 7 } } ; \frac { 7 } { 3 }$ 
C)  $\frac { 2 \left( 1 - \frac { 1 } { 7 ^ { n } } \right) } { 1 - \frac { 1 } { 7 } } ; \frac { 7 } { 4 }$ 
D)  $\frac { 2 \left( 1 - \frac { 1 } { 7 ^ { n - 1 } } \right) } { 1 - \frac { 1 } { 7 } } ; \frac { 7 } { 3 }$ 
A)  $\frac { 2 \left( 1 - \frac { 1 } { 7 ^ { n - 1 } } \right) } { 1 - \frac { 1 } { 7 } } ; \frac { 7 } { 4 }$ 
B)  $\frac { 2 \left( 1 - \frac { 1 } { 7 ^ { n } } \right) } { 1 - \frac { 1 } { 7 } } ; \frac { 7 } { 3 }$ 
C)  $\frac { 2 \left( 1 - \frac { 1 } { 7 ^ { n } } \right) } { 1 - \frac { 1 } { 7 } } ; \frac { 7 } { 4 }$ 
D)  $\frac { 2 \left( 1 - \frac { 1 } { 7 ^ { n - 1 } } \right) } { 1 - \frac { 1 } { 7 } } ; \frac { 7 } { 3 }$

Question 5

Use the nth-Term Test for divergence to show that the series is divergent, or state that the test is inconclusive.
-$$\sum _ { n = 1 } ^ { \infty } \cos \frac { 8 } { n }$$

Accepted Answer

A)  converges, 1 
B)  converges, 8 
C)  diverges 
D)  inconclusive 
A)  converges, 1 
B)  converges, 8 
C)  diverges 
D)  inconclusive

Question 6

Find the limit of the sequence if it converges; otherwise indicate divergence.
-$$a _ { n } = 7 + ( 0.3 ) ^ { n }$$

Accepted Answer

A)  8 
B)  7 
C)  7.3 
D)  Diverges 
A)  8 
B)  7 
C)  7.3 
D)  Diverges

Question 7

Assume that the sequence converges and find its limit.
-$a _ { 1 } = - 3 , a _ { n  + 1 }= \sqrt { 8 + 2 a _ { n } }$

Accepted Answer

A)  $- 9$ 
B)  $- 3$ 
C)  8 
D)  4 
A)  $- 9$ 
B)  $- 3$ 
C)  8 
D)  4

Question 8

Express the number as the ratio of two integers.
-$0.592592 \ldots$

Accepted Answer

A)  $\frac { 592 } { 99 }$ 
B)  $\frac { 160 } { 27 }$ 
C)  $\frac { 5920 } { 99 }$ 
D)  $\frac { 16 } { 27 }$ 
A)  $\frac { 592 } { 99 }$ 
B)  $\frac { 160 } { 27 }$ 
C)  $\frac { 5920 } { 99 }$ 
D)  $\frac { 16 } { 27 }$

Question 9

Find the limit of the sequence if it converges; otherwise indicate divergence.
-$$a _ { n } = 1 + ( - 1 ) ^ { n } + ( - 1 ) ^ { n ( n + 1 ) }$$

Accepted Answer

A)  1 
B)  3 
C)  0 
D)  Diverges 
A)  1 
B)  3 
C)  0 
D)  Diverges

Question 10

Write the first four elements of the sequence.
-$\frac { \ln ( n + 1 ) } { n ^ { 3 } }$

Accepted Answer

A)  $0 , \frac { \ln 2 } { 27 } , \frac { \ln 3 } { 64 } , \frac { \ln 4 } { 81 }$ 
B)  $0 , \frac { \ln 2 } { 8 } , \frac { \ln 3 } { 27 } , \frac { \ln 4 } { 64 }$ 
C)  $\ln 2 , \frac { \ln 3 } { 8 } , \frac { \ln 4 } { 27 } , \frac { \ln 5 } { 64 }$ 
D)  $\frac { \ln 2 } { 8 } , \frac { \ln 3 } { 27 } , \frac { \ln 4 } { 64 } , \frac { \ln 5 } { 81 }$ 
A)  $0 , \frac { \ln 2 } { 27 } , \frac { \ln 3 } { 64 } , \frac { \ln 4 } { 81 }$ 
B)  $0 , \frac { \ln 2 } { 8 } , \frac { \ln 3 } { 27 } , \frac { \ln 4 } { 64 }$ 
C)  $\ln 2 , \frac { \ln 3 } { 8 } , \frac { \ln 4 } { 27 } , \frac { \ln 5 } { 64 }$ 
D)  $\frac { \ln 2 } { 8 } , \frac { \ln 3 } { 27 } , \frac { \ln 4 } { 64 } , \frac { \ln 5 } { 81 }$

Question 11

Find the smallest value of N that will make the inequality hold for all n > N.
-$$\frac { n ^ { 2 } } { 2 ^ { n } } < 10 ^ { - 3 }$$

Accepted Answer

A)  17 
B)  20 
C)  19 
D)  18 
A)  17 
B)  20 
C)  19 
D)  18

Question 12

Determine whether the nonincreasing sequence converges or diverges.
-$$a _ { n } = \frac { 1 + n \cdot 4 ^ { n } } { 4 ^ { n } }$$

Accepted Answer

A)  Converges 
B)  Diverges 
A)  Converges 
B)  Diverges

Question 13

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence.
-$$a _ { 1 } = 1 , a _ { n + 1 } = a _ { n } + 6$$

Accepted Answer

A)  $7,13,19,25,31$ 
B)  $1,7,13,19,25$ 
C)  $1,7,13,19,25,31$ 
D)  $1,6,36,216,1296,7776$ 
A)  $7,13,19,25,31$ 
B)  $1,7,13,19,25$ 
C)  $1,7,13,19,25,31$ 
D)  $1,6,36,216,1296,7776$

Question 14

Find the limit of the sequence if it converges; otherwise indicate divergence.
-$$a _ { n } = ( - 1 ) ^ { n } \frac { 8 } { n }$$

Accepted Answer

A)  8 
B)  ±8 
C)  0 
D)  Diverges 
A)  8 
B)  ±8 
C)  0 
D)  Diverges

Question 15

Find the sum of the geometric series for those x for which the series converges.
-$$\sum _ { n = 0 } ^ { \infty } ( x - 2 ) ^ { n }$$

Accepted Answer

A)  $\frac { 1 } { 3 + x }$ 
B)  $\frac { 1 } { 3 - x }$ 
C)  $\frac { 1 } { - 1 + x }$ 
D)  $\frac { 1 } { - 1 - x }$ 
A)  $\frac { 1 } { 3 + x }$ 
B)  $\frac { 1 } { 3 - x }$ 
C)  $\frac { 1 } { - 1 + x }$ 
D)  $\frac { 1 } { - 1 - x }$

Question 16

Determine if the series converges or diverges. If the series converges, find its sum.
-$$\sum _ { n = 1 } ^ { \infty } \frac { 6 } { n ( n + 3 ) }$$

Accepted Answer

A)  diverges 
B)  converges; 5 
C)  converges; $\frac { 7 } { 3 }$ 
D)  converges; $\frac { 11 } { 3 }$ 
A)  diverges 
B)  converges; 5 
C)  converges; $\frac { 7 } { 3 }$ 
D)  converges; $\frac { 11 } { 3 }$

Question 17

Determine if the series converges or diverges. If the series converges, find its sum.
-$$\sum _ { n = 1 } ^ { \infty } \left( \frac { 1 } { n ^ { 3 / 2 } } - \frac { 1 } { ( n + 1 ) ^ { 3 / 2 } } \right)$$

Accepted Answer

A)  converges; $\frac { 1 } { 6 \sqrt { 6 } }$ 
B)  converges; $\frac { 1 } { 3 \sqrt { 3 } }$ 
C)  diverges 
D)  converges; 1 
A)  converges; $\frac { 1 } { 6 \sqrt { 6 } }$ 
B)  converges; $\frac { 1 } { 3 \sqrt { 3 } }$ 
C)  diverges 
D)  converges; 1

Question 18

Write the first four elements of the sequence.
-$$\left( 1 + \frac { 1 } { n } \right) ^ { n }$$

Accepted Answer

A)  $0,2 , \frac { 9 } { 4 } , \frac { 64 } { 27 }$ 
B)  $0,1 , \frac { 9 } { 4 } , \frac { 64 } { 27 }$ 
C)  $1 , \frac { 9 } { 4 } , \frac { 64 } { 27 } , \frac { 625 } { 64 }$ 
D)  $2 , \frac { 9 } { 4 } , \frac { 64 } { 27 } , \frac { 625 } { 256 }$ 
A)  $0,2 , \frac { 9 } { 4 } , \frac { 64 } { 27 }$ 
B)  $0,1 , \frac { 9 } { 4 } , \frac { 64 } { 27 }$ 
C)  $1 , \frac { 9 } { 4 } , \frac { 64 } { 27 } , \frac { 625 } { 64 }$ 
D)  $2 , \frac { 9 } { 4 } , \frac { 64 } { 27 } , \frac { 625 } { 256 }$

Question 19

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence.
-$$a _ { 1 } = 1 , a _ { n + 1 } = \frac { a _ { n } } { n + 6 }$$

Accepted Answer

A)  $1 , \frac { 1 } { 7 } , \frac { 7 } { 8 } , \frac { 8 } { 9 } , \frac { 9 } { 10 }$ 
B)  $1 , \frac { 1 } { 7 } , 56 , \frac { 1 } { 504 } , 5040$ 
C)  $1 , \frac { 1 } { 7 } , \frac { 1 } { 56 } , \frac { 1 } { 504 } , \frac { 1 } { 5040 }$ 
D)  $1 , \frac { 1 } { 7 } , \frac { 1 } { 8 } , \frac { 1 } { 9 } , \frac { 1 } { 10 }$ 
A)  $1 , \frac { 1 } { 7 } , \frac { 7 } { 8 } , \frac { 8 } { 9 } , \frac { 9 } { 10 }$ 
B)  $1 , \frac { 1 } { 7 } , 56 , \frac { 1 } { 504 } , 5040$ 
C)  $1 , \frac { 1 } { 7 } , \frac { 1 } { 56 } , \frac { 1 } { 504 } , \frac { 1 } { 5040 }$ 
D)  $1 , \frac { 1 } { 7 } , \frac { 1 } { 8 } , \frac { 1 } { 9 } , \frac { 1 } { 10 }$

Question 20

Find the values of x for which the geometric series converges.
-$$\sum _ { \mathrm { n } = 0 } ^ { \infty } \frac { ( - 1 ) ^ { \mathrm { n } } } { 5 } \frac { 1 } { ( 8 + \sin \mathrm { x } ) ^ { \mathrm { n } } }$$

Accepted Answer

A)  diverges for all $x$ 
B)  $- \infty < x < \infty$ 
C)  $\{ x \mid x$ is not a multiple of $ \pi \rangle$ 
D)  $\{ x \mid x$ is not a multiple of $2 \pi \rangle$ 
A)  diverges for all $x$ 
B)  $- \infty < x < \infty$ 
C)  $\{ x \mid x$ is not a multiple of $ \pi \rangle$ 
D)  $\{ x \mid x$ is not a multiple of $2 \pi \rangle$

Find the limit of the sequence if it converges; otherwise indicate divergence. - $a _ { n } = \frac { \tan ^ { - 1 } n } { \sqrt [ 7 ] { n } }$

Find the sum of the series. - $\sum _ { n = 0 } ^ { \infty } \left( \frac { 1 } { 3 ^ { n } } + \frac { 1 } { 4 ^ { n } } \right)$

Find the limit of the sequence if it converges; otherwise indicate divergence. - $a _ { n } = \ln \left( 1 + \frac { 2 } { n } \right) ^ { n }$

Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges. - $2 + \frac { 2 } { 7 } + \frac { 2 } { 49 } + \ldots + \frac { 2 } { 7 ^ { n - 1 } } + \ldots$

Use the nth-Term Test for divergence to show that the series is divergent, or state that the test is inconclusive. - $\sum _ { n = 1 } ^ { \infty } \cos \frac { 8 } { n }$

Find the limit of the sequence if it converges; otherwise indicate divergence. - $a _ { n } = 7 + ( 0.3 ) ^ { n }$

Assume that the sequence converges and find its limit. - $a _ { 1 } = - 3 , a _ { n + 1 }= \sqrt { 8 + 2 a _ { n } }$

Express the number as the ratio of two integers. - $0.592592 \ldots$

Find the limit of the sequence if it converges; otherwise indicate divergence. - $a _ { n } = 1 + ( - 1 ) ^ { n } + ( - 1 ) ^ { n ( n + 1 ) }$

Write the first four elements of the sequence. - $\frac { \ln ( n + 1 ) } { n ^ { 3 } }$

Find the smallest value of N that will make the inequality hold for all n > N. - $\frac { n ^ { 2 } } { 2 ^ { n } } < 10 ^ { - 3 }$

Determine whether the nonincreasing sequence converges or diverges. - $a _ { n } = \frac { 1 + n \cdot 4 ^ { n } } { 4 ^ { n } }$

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence. - $a _ { 1 } = 1 , a _ { n + 1 } = a _ { n } + 6$

Find the limit of the sequence if it converges; otherwise indicate divergence. - $a _ { n } = ( - 1 ) ^ { n } \frac { 8 } { n }$

Find the sum of the geometric series for those x for which the series converges. - $\sum _ { n = 0 } ^ { \infty } ( x - 2 ) ^ { n }$

Determine if the series converges or diverges. If the series converges, find its sum. - $\sum _ { n = 1 } ^ { \infty } \frac { 6 } { n ( n + 3 ) }$

Determine if the series converges or diverges. If the series converges, find its sum. - $\sum _ { n = 1 } ^ { \infty } \left( \frac { 1 } { n ^ { 3 / 2 } } - \frac { 1 } { ( n + 1 ) ^ { 3 / 2 } } \right)$

Write the first four elements of the sequence. - $\left( 1 + \frac { 1 } { n } \right) ^ { n }$

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence. - $a _ { 1 } = 1 , a _ { n + 1 } = \frac { a _ { n } } { n + 6 }$

Find the values of x for which the geometric series converges. - $\sum _ { \mathrm { n } = 0 } ^ { \infty } \frac { ( - 1 ) ^ { \mathrm { n } } } { 5 } \frac { 1 } { ( 8 + \sin \mathrm { x } ) ^ { \mathrm { n } } }$

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Exam 10: Infinite Sequences and Series

Find the limit of the sequence if it converges; otherwise indicate divergence. - an=tan⁡−1nn7a _ { n } = \frac { \tan ^ { - 1 } n } { \sqrt [ 7 ] { n } }an​=7n​tan−1n​

Find the sum of the series. - ∑n=0∞(13n+14n)\sum _ { n = 0 } ^ { \infty } \left( \frac { 1 } { 3 ^ { n } } + \frac { 1 } { 4 ^ { n } } \right)∑n=0∞​(3n1​+4n1​)

Find the limit of the sequence if it converges; otherwise indicate divergence. - an=ln⁡(1+2n)na _ { n } = \ln \left( 1 + \frac { 2 } { n } \right) ^ { n }an​=ln(1+n2​)n

Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges. - 2+27+249+…+27n−1+…2 + \frac { 2 } { 7 } + \frac { 2 } { 49 } + \ldots + \frac { 2 } { 7 ^ { n - 1 } } + \ldots2+72​+492​+…+7n−12​+…

Use the nth-Term Test for divergence to show that the series is divergent, or state that the test is inconclusive. - ∑n=1∞cos⁡8n\sum _ { n = 1 } ^ { \infty } \cos \frac { 8 } { n }∑n=1∞​cosn8​

Find the limit of the sequence if it converges; otherwise indicate divergence. - an=7+(0.3)na _ { n } = 7 + ( 0.3 ) ^ { n }an​=7+(0.3)n

Assume that the sequence converges and find its limit. - a1=−3,an+1=8+2ana _ { 1 } = - 3 , a _ { n + 1 }= \sqrt { 8 + 2 a _ { n } }a1​=−3,an+1​=8+2an​​

Express the number as the ratio of two integers. - 0.592592…0.592592 \ldots0.592592…

Find the limit of the sequence if it converges; otherwise indicate divergence. - an=1+(−1)n+(−1)n(n+1)a _ { n } = 1 + ( - 1 ) ^ { n } + ( - 1 ) ^ { n ( n + 1 ) }an​=1+(−1)n+(−1)n(n+1)

Write the first four elements of the sequence. - ln⁡(n+1)n3\frac { \ln ( n + 1 ) } { n ^ { 3 } }n3ln(n+1)​

Find the smallest value of N that will make the inequality hold for all n > N. - n22n<10−3\frac { n ^ { 2 } } { 2 ^ { n } } < 10 ^ { - 3 }2nn2​<10−3

Determine whether the nonincreasing sequence converges or diverges. - an=1+n⋅4n4na _ { n } = \frac { 1 + n \cdot 4 ^ { n } } { 4 ^ { n } }an​=4n1+n⋅4n​

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence. - a1=1,an+1=an+6a _ { 1 } = 1 , a _ { n + 1 } = a _ { n } + 6a1​=1,an+1​=an​+6

Find the limit of the sequence if it converges; otherwise indicate divergence. - an=(−1)n8na _ { n } = ( - 1 ) ^ { n } \frac { 8 } { n }an​=(−1)nn8​

Find the sum of the geometric series for those x for which the series converges. - ∑n=0∞(x−2)n\sum _ { n = 0 } ^ { \infty } ( x - 2 ) ^ { n }∑n=0∞​(x−2)n

Determine if the series converges or diverges. If the series converges, find its sum. - ∑n=1∞6n(n+3)\sum _ { n = 1 } ^ { \infty } \frac { 6 } { n ( n + 3 ) }∑n=1∞​n(n+3)6​

Determine if the series converges or diverges. If the series converges, find its sum. - ∑n=1∞(1n3/2−1(n+1)3/2)\sum _ { n = 1 } ^ { \infty } \left( \frac { 1 } { n ^ { 3 / 2 } } - \frac { 1 } { ( n + 1 ) ^ { 3 / 2 } } \right)∑n=1∞​(n3/21​−(n+1)3/21​)

Write the first four elements of the sequence. - (1+1n)n\left( 1 + \frac { 1 } { n } \right) ^ { n }(1+n1​)n

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence. - a1=1,an+1=ann+6a _ { 1 } = 1 , a _ { n + 1 } = \frac { a _ { n } } { n + 6 }a1​=1,an+1​=n+6an​​

Find the values of x for which the geometric series converges. - ∑n=0∞(−1)n51(8+sin⁡x)n\sum _ { \mathrm { n } = 0 } ^ { \infty } \frac { ( - 1 ) ^ { \mathrm { n } } } { 5 } \frac { 1 } { ( 8 + \sin \mathrm { x } ) ^ { \mathrm { n } } }∑n=0∞​5(−1)n​(8+sinx)n1​