Question 1

Write the first four elements of the sequence.
-sin (n$\pi$)

Accepted Answer

A)  1, 0, -1, 0 
B)  1, 1, 1, 1 
C)  0, 0, 0, 0 
D)  0, 1, 0, -1 
A)  1, 0, -1, 0 
B)  1, 1, 1, 1 
C)  0, 0, 0, 0 
D)  0, 1, 0, -1

Question 2

Find the limit of the sequence if it converges; otherwise indicate divergence.
-$$a _ { n } = \ln ( 9 n + 5 ) - \ln ( 4 n + 9 )$$

Accepted Answer

A)  $\ln 5$ 
B)  $\ln \left( \frac { 9 } { 4 } \right)$ 
C)  $\ln \left( \frac { 4 } { 9 } \right)$ 
D)  Diverges 
A)  $\ln 5$ 
B)  $\ln \left( \frac { 9 } { 4 } \right)$ 
C)  $\ln \left( \frac { 4 } { 9 } \right)$ 
D)  Diverges

Question 3

Find the smallest value of N that will make the inequality hold for all n > N.
-A company's annual revenue for the period since 2000 can be modeled by the function Rn = 2.11(1.06)n, where R is in millions of dollars and n = 0 corresponds to 2000. Assuming the model accurately predicts future
Revenue, find the year in which the revenue first exceeds $3.10 million.

Accepted Answer

A)  2007 
B)  2009 
C)  2006 
D)  2005 
A)  2007 
B)  2009 
C)  2006 
D)  2005

Question 4

Find the smallest value of N that will make the inequality hold for all n > N.
-$$| \sqrt [ n ] { 0.6 } - 1 | < 10 ^ { - 3 }$$

Accepted Answer

A)  507 
B)  511 
C)  509 
D)  514 
A)  507 
B)  511 
C)  509 
D)  514

Question 5

Find the sum of the series.
-$$\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } \frac { 8 } { 3 ^ { n } }$$

Accepted Answer

A)  4 
B)  12 
C)  6 
D)  2 
A)  4 
B)  12 
C)  6 
D)  2

Question 6

Find the smallest value of N that will make the inequality hold for all n > N.
-$$a _ { n } = ( 1 + 0.153 / n ) ^ { n }$$

Accepted Answer

A)  L ≈  1.1653, N = 6 
B)  L ≈  1.153, N = 6 
C)  L ≈  1.1653, N = 2 
D)  diverges 
A)  L ≈  1.1653, N = 6 
B)  L ≈  1.153, N = 6 
C)  L ≈  1.1653, N = 2 
D)  diverges

Question 7

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

Accepted Answer

A)  $\frac { 1 } { 3 } , \frac { 1 } { 3 } , \frac { 1 } { 3 } , \frac { 1 } { 3 }$ 
B)  $1 , \frac { 1 } { 3 } , \frac { 1 } { 9 } , \frac { 1 } { 27 }$ 
C)  $\frac { 1 } { 3 } , \frac { 1 } { 9 } , \frac { 1 } { 27 } , \frac { 1 } { 81 }$ 
D)  $\frac { 1 } { 3 } , - \frac { 1 } { 9 } , \frac { 1 } { 27 } , - \frac { 1 } { 81 }$ 
A)  $\frac { 1 } { 3 } , \frac { 1 } { 3 } , \frac { 1 } { 3 } , \frac { 1 } { 3 }$ 
B)  $1 , \frac { 1 } { 3 } , \frac { 1 } { 9 } , \frac { 1 } { 27 }$ 
C)  $\frac { 1 } { 3 } , \frac { 1 } { 9 } , \frac { 1 } { 27 } , \frac { 1 } { 81 }$ 
D)  $\frac { 1 } { 3 } , - \frac { 1 } { 9 } , \frac { 1 } { 27 } , - \frac { 1 } { 81 }$

Question 8

Assume that the sequence converges and find its limit.
-$$a _ { 1 } = 3 , a _ { n  + 1} = \frac { 72 } { 1 + a _ { n } }$$

Accepted Answer

A)  4 
B)  8 
C)  2 
D)  3 
A)  4 
B)  8 
C)  2 
D)  3

Question 9

Determine if the sequence is monotonic and if it is bounded.
-$$a _ { n } = \frac { ( n + 2 ) ! } { ( n + 1 ) ! }$$

Accepted Answer

A)  increasing; bounded 
B)  not monotonic; unbounded 
C)  nondecreasing; bounded 
D)  nondecreasing; unbounded 
A)  increasing; bounded 
B)  not monotonic; unbounded 
C)  nondecreasing; bounded 
D)  nondecreasing; unbounded

Question 10

Find the limit of the sequence if it converges; otherwise indicate divergence.
-$$a _ { n } = \sqrt [ n ] { 4 n }$$

Accepted Answer

A)  $\ln 4$ 
B)  1 
C)  0 
D)  Diverges 
A)  $\ln 4$ 
B)  1 
C)  0 
D)  Diverges

Question 11

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

Accepted Answer

A)  converges; $\frac { \pi } { 2 }$ 
B)  converges; $\pi$ 
C)  diverges 
D)  converges; $- \frac { \pi } { 2 }$ 
A)  converges; $\frac { \pi } { 2 }$ 
B)  converges; $\pi$ 
C)  diverges 
D)  converges; $- \frac { \pi } { 2 }$

Question 12

Find the smallest value of N that will make the inequality hold for all n > N.
-$$a _ { n } = 1291 ^ { 1 / n }$$

Accepted Answer

A)  L = 0, N = 1301 
B)  L = 1, N = 720 
C)  L = 1, N = 1301 
D)  diverges 
A)  L = 0, N = 1301 
B)  L = 1, N = 720 
C)  L = 1, N = 1301 
D)  diverges

Question 13

Determine if the series converges or diverges; if the series converges, find its sum.
-$$\sum _ { n = 1 } ^ { \infty } \ln \frac { 8 } { n }$$

Accepted Answer

A)  Converges; $\ln \frac { 1 } { 8 }$ 
B)  Converges; 1 
C)  Converges; $\ln 8$ 
D)  Diverges 
A)  Converges; $\ln \frac { 1 } { 8 }$ 
B)  Converges; 1 
C)  Converges; $\ln 8$ 
D)  Diverges

Question 14

Find the smallest value of N that will make the inequality hold for all n > N.
-$$a _ { n } = \cos n$$

Accepted Answer

A)  $\mathrm { L } = 1 , \mathrm {~N} = 394$ 
B)  $\mathrm { L } = 0 , \mathrm {~N} = 237$ 
C)  $\mathrm { L } = 0 , \mathrm {~N} = 394$ 
D)  diverges 
A)  $\mathrm { L } = 1 , \mathrm {~N} = 394$ 
B)  $\mathrm { L } = 0 , \mathrm {~N} = 237$ 
C)  $\mathrm { L } = 0 , \mathrm {~N} = 394$ 
D)  diverges

Question 15

Find the smallest value of N that will make the inequality hold for all n > N.
-$$| \sqrt [ n ] { 0.8 } - 1 | < 10 ^ { - 2 }$$

Accepted Answer

A)  27 
B)  23 
C)  25 
D)  20 
A)  27 
B)  23 
C)  25 
D)  20

Question 16

Determine if the sequence is monotonic and if it is bounded.
-$$a _ { n } = \frac { 4 ^ { n } } { ( 4 n ) ! }$$

Accepted Answer

A)  not monotonic; bounded 
B)  nondecreasing; bounded 
C)  increasing; unbounded 
D)  decreasing; bounded 
A)  not monotonic; bounded 
B)  nondecreasing; bounded 
C)  increasing; unbounded 
D)  decreasing; bounded

Question 17

Use the nth-Term Test for divergence to show that the series is divergent, or state that the test is inconclusive.
-$$\sum _ { n = 1 } ^ { \infty } \ln \frac { 10 } { n }$$

Accepted Answer

A)  inconclusive 
B)  diverges 
C)  converges; $\ln \frac { 1 } { 10 }$ 
D)  converges; $\ln 10$ 
A)  inconclusive 
B)  diverges 
C)  converges; $\ln \frac { 1 } { 10 }$ 
D)  converges; $\ln 10$

Question 18

Find the sum of the series.
-$$\sum _ { n = 1 } ^ { \infty } \left( \frac { 1 } { 3 ^ { n } } - \frac { 1 } { 6 ^ { n } } \right)$$

Accepted Answer

A)  $\frac { 3 } { 10 }$ 
B)  $- \frac { 3 } { 10 }$ 
C)  $- \frac { 9 } { 10 }$ 
D)  $\frac { 9 } { 10 }$ 
A)  $\frac { 3 } { 10 }$ 
B)  $- \frac { 3 } { 10 }$ 
C)  $- \frac { 9 } { 10 }$ 
D)  $\frac { 9 } { 10 }$

Question 19

Find the sum of the geometric series for those x for which the series converges.
-$$\sum _ { n = 0 } ^ { \infty } - 8 ^ { n } x ^ { n }$$

Accepted Answer

A)  $\frac { 8 } { 1 + 8 x }$ 
B)  $\frac { 8 } { 1 - 8 x }$ 
C)  $\frac { 1 } { 1 - 8 x }$ 
D)  $\frac { 1 } { 1 + 8 x }$ 
A)  $\frac { 8 } { 1 + 8 x }$ 
B)  $\frac { 8 } { 1 - 8 x }$ 
C)  $\frac { 1 } { 1 - 8 x }$ 
D)  $\frac { 1 } { 1 + 8 x }$

Question 20

Determine whether the nonincreasing sequence converges or diverges.
-$$a _ { n } = \frac { 1 + \sqrt { 3 n } } { n }$$

Accepted Answer

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

Write the first four elements of the sequence. -sin (n $\pi$ )

Find the limit of the sequence if it converges; otherwise indicate divergence. - $a _ { n } = \ln ( 9 n + 5 ) - \ln ( 4 n + 9 )$

Find the smallest value of N that will make the inequality hold for all n > N. - $| \sqrt [ n ] { 0.6 } - 1 | < 10 ^ { - 3 }$

Find the sum of the series. - $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } \frac { 8 } { 3 ^ { n } }$

Find the smallest value of N that will make the inequality hold for all n > N. - $a _ { n } = ( 1 + 0.153 / n ) ^ { n }$

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

Assume that the sequence converges and find its limit. - $a _ { 1 } = 3 , a _ { n + 1} = \frac { 72 } { 1 + a _ { n } }$

Determine if the sequence is monotonic and if it is bounded. - $a _ { n } = \frac { ( n + 2 ) ! } { ( n + 1 ) ! }$

Find the limit of the sequence if it converges; otherwise indicate divergence. - $a _ { n } = \sqrt [ n ] { 4 n }$

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

Find the smallest value of N that will make the inequality hold for all n > N. - $a _ { n } = 1291 ^ { 1 / n }$

Determine if the series converges or diverges; if the series converges, find its sum. - $\sum _ { n = 1 } ^ { \infty } \ln \frac { 8 } { n }$

Find the smallest value of N that will make the inequality hold for all n > N. - $a _ { n } = \cos n$

Find the smallest value of N that will make the inequality hold for all n > N. - $| \sqrt [ n ] { 0.8 } - 1 | < 10 ^ { - 2 }$

Determine if the sequence is monotonic and if it is bounded. - $a _ { n } = \frac { 4 ^ { n } } { ( 4 n ) ! }$

Use the nth-Term Test for divergence to show that the series is divergent, or state that the test is inconclusive. - $\sum _ { n = 1 } ^ { \infty } \ln \frac { 10 } { n }$

Find the sum of the series. - $\sum _ { n = 1 } ^ { \infty } \left( \frac { 1 } { 3 ^ { n } } - \frac { 1 } { 6 ^ { n } } \right)$

Find the sum of the geometric series for those x for which the series converges. - $\sum _ { n = 0 } ^ { \infty } - 8 ^ { n } x ^ { n }$

Determine whether the nonincreasing sequence converges or diverges. - $a _ { n } = \frac { 1 + \sqrt { 3 n } } { n }$

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Exam 10: Infinite Sequences and Series

Write the first four elements of the sequence. -sin (n π\piπ )

Find the limit of the sequence if it converges; otherwise indicate divergence. - an=ln⁡(9n+5)−ln⁡(4n+9)a _ { n } = \ln ( 9 n + 5 ) - \ln ( 4 n + 9 )an​=ln(9n+5)−ln(4n+9)

Find the smallest value of N that will make the inequality hold for all n > N. - ∣0.6n−1∣<10−3| \sqrt [ n ] { 0.6 } - 1 | < 10 ^ { - 3 }∣n0.6​−1∣<10−3

Find the sum of the series. - ∑n=1∞(−1)n−183n\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } \frac { 8 } { 3 ^ { n } }∑n=1∞​(−1)n−13n8​

Find the smallest value of N that will make the inequality hold for all n > N. - an=(1+0.153/n)na _ { n } = ( 1 + 0.153 / n ) ^ { n }an​=(1+0.153/n)n

Write the first four elements of the sequence. - (13)n\left( \frac { 1 } { 3 } \right) ^ { n }(31​)n

Assume that the sequence converges and find its limit. - a1=3,an+1=721+ana _ { 1 } = 3 , a _ { n + 1} = \frac { 72 } { 1 + a _ { n } }a1​=3,an+1​=1+an​72​

Determine if the sequence is monotonic and if it is bounded. - an=(n+2)!(n+1)!a _ { n } = \frac { ( n + 2 ) ! } { ( n + 1 ) ! }an​=(n+1)!(n+2)!​

Find the limit of the sequence if it converges; otherwise indicate divergence. - an=4nna _ { n } = \sqrt [ n ] { 4 n }an​=n4n​

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

Find the smallest value of N that will make the inequality hold for all n > N. - an=12911/na _ { n } = 1291 ^ { 1 / n }an​=12911/n

Determine if the series converges or diverges; if the series converges, find its sum. - ∑n=1∞ln⁡8n\sum _ { n = 1 } ^ { \infty } \ln \frac { 8 } { n }∑n=1∞​lnn8​

Find the smallest value of N that will make the inequality hold for all n > N. - an=cos⁡na _ { n } = \cos nan​=cosn

Find the smallest value of N that will make the inequality hold for all n > N. - ∣0.8n−1∣<10−2| \sqrt [ n ] { 0.8 } - 1 | < 10 ^ { - 2 }∣n0.8​−1∣<10−2

Determine if the sequence is monotonic and if it is bounded. - an=4n(4n)!a _ { n } = \frac { 4 ^ { n } } { ( 4 n ) ! }an​=(4n)!4n​

Use the nth-Term Test for divergence to show that the series is divergent, or state that the test is inconclusive. - ∑n=1∞ln⁡10n\sum _ { n = 1 } ^ { \infty } \ln \frac { 10 } { n }∑n=1∞​lnn10​

Find the sum of the series. - ∑n=1∞(13n−16n)\sum _ { n = 1 } ^ { \infty } \left( \frac { 1 } { 3 ^ { n } } - \frac { 1 } { 6 ^ { n } } \right)∑n=1∞​(3n1​−6n1​)

Find the sum of the geometric series for those x for which the series converges. - ∑n=0∞−8nxn\sum _ { n = 0 } ^ { \infty } - 8 ^ { n } x ^ { n }∑n=0∞​−8nxn

Determine whether the nonincreasing sequence converges or diverges. - an=1+3nna _ { n } = \frac { 1 + \sqrt { 3 n } } { n }an​=n1+3n​​