Question 1

Find the limit of the sequence if it converges; otherwise indicate divergence.
-$$a _ { n } = \frac { \ln ( n + 7 ) } { n ^ { 1 / n } }$$

Accepted Answer

A)  1 
B)  $\ln 7$ 
C)  0 
D)  Diverges 
A)  1 
B)  $\ln 7$ 
C)  0 
D)  Diverges

Question 2

Determine if the geometric series converges or diverges. If it converges, find its sum.
-$\frac { 1 } { 6 } + \left( \frac { 1 } { 6 } \right) ^ { 2 } + \left( \frac { 1 } { 6 } \right) ^ { 3 } + \left( \frac { 1 } { 6 } \right) ^ { 4 } + \left( \frac { 1 } { 6 } \right) ^ { 5 } + \ldots$

Accepted Answer

A)  converges, $\frac { 1 } { 5 }$ 
B)  diverges 
C)  converges, $\frac { 6 } { 5 }$ 
D)  converges, $\frac { 5 } { 6 }$ 
A)  converges, $\frac { 1 } { 5 }$ 
B)  diverges 
C)  converges, $\frac { 6 } { 5 }$ 
D)  converges, $\frac { 5 } { 6 }$

Question 3

Determine if the sequence is monotonic and if it is bounded.
-$$a _ { n } = \frac { 9 n + 1 } { n + 1 }$$

Accepted Answer

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

Question 4

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

Accepted Answer

A)  $\frac { 2187 } { 2 }$ 
B)  $\frac { 1 } { 4374 }$ 
C)  $\frac { 6561 } { 2 }$ 
D)  $\frac { 1 } { 13,122 }$ 
A)  $\frac { 2187 } { 2 }$ 
B)  $\frac { 1 } { 4374 }$ 
C)  $\frac { 6561 } { 2 }$ 
D)  $\frac { 1 } { 13,122 }$

Question 5

Determine if the series converges or diverges; if the series converges, find its sum.
-$$\sum _ { n = 0 } ^ { \infty } \left( 1 + \frac { - 3 } { n } \right) ^ { 3 n }$$

Accepted Answer

A)  Converges; $\mathrm { e } ^ { - 9 }$ 
B)  Converges; $\frac { 1 } { | - 3 | - 1 }$ 
C)  Converges; $\frac { 1 } { | - 3 | + 1 }$ 
D)  Diverges 
A)  Converges; $\mathrm { e } ^ { - 9 }$ 
B)  Converges; $\frac { 1 } { | - 3 | - 1 }$ 
C)  Converges; $\frac { 1 } { | - 3 | + 1 }$ 
D)  Diverges

Question 6

Find the sum of the series.
-$$\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \frac { 8 } { 7 ^ { n } }$$

Accepted Answer

A)  7 
B)  1 
C)  $\frac { 28 } { 3 }$ 
D)  $\frac { 4 } { 3 }$ 
A)  7 
B)  1 
C)  $\frac { 28 } { 3 }$ 
D)  $\frac { 4 } { 3 }$

Question 7

Determine if the series converges or diverges; if the series converges, find its sum.
-$$\sum _ { n = 0 } ^ { \infty } \sqrt { 10 }$$

Accepted Answer

A)  Converges; $1 - \sqrt { 10 }$ 
B)  Converges; $\sqrt { 10 } - 1$ 
C)  Converges; $\sqrt { 10 } + 1$ 
D)  Diverges 
A)  Converges; $1 - \sqrt { 10 }$ 
B)  Converges; $\sqrt { 10 } - 1$ 
C)  Converges; $\sqrt { 10 } + 1$ 
D)  Diverges

Question 8

Find the smallest value of N that will make the inequality hold for all n > N.
-$$a _ { n } = \frac { 2 ^ { n } } { n ! }$$

Accepted Answer

A)  L = 0, N = 8 
B)  L = ln 4, N = 8 
C)  L = ln 2, N = 6 
D)  diverges 
A)  L = 0, N = 8 
B)  L = ln 4, N = 8 
C)  L = ln 2, N = 6 
D)  diverges

Question 9

Determine if the sequence is monotonic and if it is bounded.
-$$\frac { \mathrm { n } ! } { 5 ^ { n } }$$

Accepted Answer

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

Question 10

Find the values of x for which the geometric series converges.
-$$\sum _ { n = 0 } ^ { \infty } \left( \frac { x - 10 } { 2 } \right) ^ { n }$$

Accepted Answer

A)  $- 6 < x < 14$ 
B)  $- 12 < x < 12$ 
C)  $8 < x < 12$ 
D)  $6 < \mathrm { x } < 14$ 
A)  $- 6 < x < 14$ 
B)  $- 12 < x < 12$ 
C)  $8 < x < 12$ 
D)  $6 < \mathrm { x } < 14$

Question 11

Find the smallest value of N that will make the inequality hold for all n > N.
-At a plant that packages bottled spring water, the water is passed through a sequence of ion-exchange filters to reduce the sodium content prior to bottling. Each filter removes 74% of the sodium present in the water passing Through it. Determine the number of filters that must be used to reduce the sodium concentration from 22 Parts-per-million to 3.61 parts-per-million.

Accepted Answer

A)  8 filters 
B)  6 filters 
C)  7 filters 
D)  9 filters 
A)  8 filters 
B)  6 filters 
C)  7 filters 
D)  9 filters

Question 12

Determine if the series converges or diverges; if the series converges, find its sum.
-$$\sum _ { n = 0 } ^ { \infty } \frac { \sin \frac { ( n + 1 ) \pi } { 2 } } { 8 ^ { n } }$$

Accepted Answer

A)  Converges; $\frac { 1 } { 7 }$ 
B)  Converges; 8 
C)  Converges; $\frac { 8 } { 7 }$ 
D)  Diverges 
A)  Converges; $\frac { 1 } { 7 }$ 
B)  Converges; 8 
C)  Converges; $\frac { 8 } { 7 }$ 
D)  Diverges

Question 13

Determine if the series converges or diverges. If the series converges, find its sum.
-$ \sum_{n=1}^{\infty} \frac{7 n}{(2 n-1)^{2}(2 n+1)^{2}} $

Accepted Answer

A)  converges; $\frac{35}{6}$ 
B)  converges; $\frac{7}{8}$ 
C)  converges; $\frac{21}{8}$ 
D)  converges; $\frac{49}{6}$ 
A)  converges; $\frac{35}{6}$ 
B)  converges; $\frac{7}{8}$ 
C)  converges; $\frac{21}{8}$ 
D)  converges; $\frac{49}{6}$

Question 14

Find the sum of the geometric series for those x for which the series converges.
-$$\sum _ { n = 0 } ^ { \infty } 4 ^ { n } x ^ { n }$$

Accepted Answer

A)  $\frac { 1 } { 1 + 4 x }$ 
B)  $\frac { 4 } { 1 - 4 x }$ 
C)  $\frac { 4 } { 1 + 4 x }$ 
D)  $\frac { 1 } { 1 - 4 x }$ 
A)  $\frac { 1 } { 1 + 4 x }$ 
B)  $\frac { 4 } { 1 - 4 x }$ 
C)  $\frac { 4 } { 1 + 4 x }$ 
D)  $\frac { 1 } { 1 - 4 x }$

Question 15

Find the limit of the sequence if it converges; otherwise indicate divergence.
-$$a _ { n } = \frac { 5 + 3 n } { 3 + 2 n }$$

Accepted Answer

A)  1 
B)  32 
C)  53 
D)  Diverges 
A)  1 
B)  32 
C)  53 
D)  Diverges

Question 16

Find the values of x for which the geometric series converges.
-$$\sum _ { n = 0 } ^ { \infty } - 4 ^ { n } x ^ { n }$$

Accepted Answer

A)  $| x | < \frac { 1 } { 4 }$ 
B)  $| x | < 1$ 
C)  $| x | < 8$ 
D)  $| \mathrm { x } | < 4$ 
A)  $| x | < \frac { 1 } { 4 }$ 
B)  $| x | < 1$ 
C)  $| x | < 8$ 
D)  $| \mathrm { x } | < 4$

Question 17

Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges.
-$$\frac { 7 } { 2 \cdot 3 } + \frac { 7 } { 3 \cdot 4 } + \frac { 7 } { 4 \cdot 5 } + \ldots + \frac { 7 } { ( n + 1 ) ( n + 2 ) } + \ldots$$

Accepted Answer

A)  $\frac { 7 n } { n + 2 } ; 7$ 
B)  $\frac { 7 n } { 2 ( n + 1 ) } ; \frac { 7 } { 2 }$ 
C)  $\frac { 7 n } { 2 ( n + 2 ) } ; \frac { 7 } { 2 }$ 
D)  $\frac { 7 n } { n + 1 } ; 7$ 
A)  $\frac { 7 n } { n + 2 } ; 7$ 
B)  $\frac { 7 n } { 2 ( n + 1 ) } ; \frac { 7 } { 2 }$ 
C)  $\frac { 7 n } { 2 ( n + 2 ) } ; \frac { 7 } { 2 }$ 
D)  $\frac { 7 n } { n + 1 } ; 7$

Question 18

Find the values of x for which the geometric series converges.
-$$\sum _ { \mathrm { n } = 0 } ^ { \infty } ( - 1 ) ^ { \mathrm { n } } ( 4 \mathrm { x } ) ^ { 2 \mathrm { n } }$$

Accepted Answer

A)  $| x | < 1$ 
B)  $| x | < 4$ 
C)  $| x | < \frac { 1 } { 4 }$ 
D)  $| x | < 8$ 
A)  $| x | < 1$ 
B)  $| x | < 4$ 
C)  $| x | < \frac { 1 } { 4 }$ 
D)  $| x | < 8$

Question 19

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

Accepted Answer

A)  $\mathrm { L } = 0 , \mathrm {~N} = 28$ 
B)  $\mathrm { L } = 1 , \mathrm {~N} = 28$ 
C)  $L = 1 , N = 6$ 
D)  diverges 
A)  $\mathrm { L } = 0 , \mathrm {~N} = 28$ 
B)  $\mathrm { L } = 1 , \mathrm {~N} = 28$ 
C)  $L = 1 , N = 6$ 
D)  diverges

Question 20

Assume that the sequence converges and find its limit.
-$$a _ { 1 } = 5 , a _ { n + 1} = \sqrt { 5 a _ { n } }$$

Accepted Answer

A)  $- 5$ 
B)  1 
C)  5 
D)  25 
A)  $- 5$ 
B)  1 
C)  5 
D)  25

Find the limit of the sequence if it converges; otherwise indicate divergence. - $a _ { n } = \frac { \ln ( n + 7 ) } { n ^ { 1 / n } }$

Determine if the geometric series converges or diverges. If it converges, find its sum. - $\frac { 1 } { 6 } + \left( \frac { 1 } { 6 } \right) ^ { 2 } + \left( \frac { 1 } { 6 } \right) ^ { 3 } + \left( \frac { 1 } { 6 } \right) ^ { 4 } + \left( \frac { 1 } { 6 } \right) ^ { 5 } + \ldots$

Determine if the sequence is monotonic and if it is bounded. - $a _ { n } = \frac { 9 n + 1 } { n + 1 }$

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

Determine if the series converges or diverges; if the series converges, find its sum. - $\sum _ { n = 0 } ^ { \infty } \left( 1 + \frac { - 3 } { n } \right) ^ { 3 n }$

Find the sum of the series. - $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \frac { 8 } { 7 ^ { n } }$

Determine if the series converges or diverges; if the series converges, find its sum. - $\sum _ { n = 0 } ^ { \infty } \sqrt { 10 }$

Find the smallest value of N that will make the inequality hold for all n > N. - $a _ { n } = \frac { 2 ^ { n } } { n ! }$

Determine if the sequence is monotonic and if it is bounded. - $\frac { \mathrm { n } ! } { 5 ^ { n } }$

Find the values of x for which the geometric series converges. - $\sum _ { n = 0 } ^ { \infty } \left( \frac { x - 10 } { 2 } \right) ^ { n }$

Determine if the series converges or diverges; if the series converges, find its sum. - $\sum _ { n = 0 } ^ { \infty } \frac { \sin \frac { ( n + 1 ) \pi } { 2 } } { 8 ^ { n } }$

Determine if the series converges or diverges. If the series converges, find its sum. - $\sum_{n=1}^{\infty} \frac{7 n}{(2 n-1)^{2}(2 n+1)^{2}}$

Find the sum of the geometric series for those x for which the series converges. - $\sum _ { n = 0 } ^ { \infty } 4 ^ { n } x ^ { n }$

Find the limit of the sequence if it converges; otherwise indicate divergence. - $a _ { n } = \frac { 5 + 3 n } { 3 + 2 n }$

Find the values of x for which the geometric series converges. - $\sum _ { n = 0 } ^ { \infty } - 4 ^ { n } x ^ { n }$

Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges. - $\frac { 7 } { 2 \cdot 3 } + \frac { 7 } { 3 \cdot 4 } + \frac { 7 } { 4 \cdot 5 } + \ldots + \frac { 7 } { ( n + 1 ) ( n + 2 ) } + \ldots$

Find the values of x for which the geometric series converges. - $\sum _ { \mathrm { n } = 0 } ^ { \infty } ( - 1 ) ^ { \mathrm { n } } ( 4 \mathrm { x } ) ^ { 2 \mathrm { n } }$

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

Assume that the sequence converges and find its limit. - $a _ { 1 } = 5 , a _ { n + 1} = \sqrt { 5 a _ { n } }$

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Exam 10: Infinite Sequences and Series

Find the limit of the sequence if it converges; otherwise indicate divergence. - an=ln⁡(n+7)n1/na _ { n } = \frac { \ln ( n + 7 ) } { n ^ { 1 / n } }an​=n1/nln(n+7)​

Determine if the sequence is monotonic and if it is bounded. - an=9n+1n+1a _ { n } = \frac { 9 n + 1 } { n + 1 }an​=n+19n+1​

Find the sum of the series. - ∑n=8∞13n\sum _ { n = 8 } ^ { \infty } \frac { 1 } { 3 ^ { n } }∑n=8∞​3n1​

Determine if the series converges or diverges; if the series converges, find its sum. - ∑n=0∞(1+−3n)3n\sum _ { n = 0 } ^ { \infty } \left( 1 + \frac { - 3 } { n } \right) ^ { 3 n }∑n=0∞​(1+n−3​)3n

Find the sum of the series. - ∑n=0∞(−1)n87n\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \frac { 8 } { 7 ^ { n } }∑n=0∞​(−1)n7n8​

Determine if the series converges or diverges; if the series converges, find its sum. - ∑n=0∞10\sum _ { n = 0 } ^ { \infty } \sqrt { 10 }∑n=0∞​10​

Find the smallest value of N that will make the inequality hold for all n > N. - an=2nn!a _ { n } = \frac { 2 ^ { n } } { n ! }an​=n!2n​

Determine if the sequence is monotonic and if it is bounded. - n!5n\frac { \mathrm { n } ! } { 5 ^ { n } }5nn!​

Find the values of x for which the geometric series converges. - ∑n=0∞(x−102)n\sum _ { n = 0 } ^ { \infty } \left( \frac { x - 10 } { 2 } \right) ^ { n }∑n=0∞​(2x−10​)n

Determine if the series converges or diverges; if the series converges, find its sum. - ∑n=0∞sin⁡(n+1)π28n\sum _ { n = 0 } ^ { \infty } \frac { \sin \frac { ( n + 1 ) \pi } { 2 } } { 8 ^ { n } }∑n=0∞​8nsin2(n+1)π​​

Determine if the series converges or diverges. If the series converges, find its sum. - ∑n=1∞7n(2n−1)2(2n+1)2 \sum_{n=1}^{\infty} \frac{7 n}{(2 n-1)^{2}(2 n+1)^{2}} ∑n=1∞​(2n−1)2(2n+1)27n​

Find the sum of the geometric series for those x for which the series converges. - ∑n=0∞4nxn\sum _ { n = 0 } ^ { \infty } 4 ^ { n } x ^ { n }∑n=0∞​4nxn

Find the limit of the sequence if it converges; otherwise indicate divergence. - an=5+3n3+2na _ { n } = \frac { 5 + 3 n } { 3 + 2 n }an​=3+2n5+3n​

Find the values of x for which the geometric series converges. - ∑n=0∞−4nxn\sum _ { n = 0 } ^ { \infty } - 4 ^ { n } x ^ { n }∑n=0∞​−4nxn

Find the values of x for which the geometric series converges. - ∑n=0∞(−1)n(4x)2n\sum _ { \mathrm { n } = 0 } ^ { \infty } ( - 1 ) ^ { \mathrm { n } } ( 4 \mathrm { x } ) ^ { 2 \mathrm { n } }∑n=0∞​(−1)n(4x)2n

Find the smallest value of N that will make the inequality hold for all n > N. - an=ntan⁡1na _ { n } = n \tan \frac { 1 } { n }an​=ntann1​

Assume that the sequence converges and find its limit. - a1=5,an+1=5ana _ { 1 } = 5 , a _ { n + 1} = \sqrt { 5 a _ { n } }a1​=5,an+1​=5an​​