Question 1

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

Accepted Answer

A)  L = 0, N = 7 
B)  L = ln 4, N = 8 
C)  L = ln 2, N = 6 
D)  diverges 
A)  L = 0, N = 7 
B)  L = ln 4, N = 8 
C)  L = ln 2, N = 6 
D)  diverges

Question 2

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

Accepted Answer

A)  2 
B)  0 
C)  1 
D)  Diverges 
A)  2 
B)  0 
C)  1 
D)  Diverges

Question 3

Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges.
-$$5 - \frac { 5 } { 7 } + \frac { 5 } { 49 } - \frac { 5 } { 343 } + \ldots + ( - 1 ) ^ { n - 1 } \frac { 5 } { 7 ^ { n - 1 } } + \ldots$$

Accepted Answer

A)  $\frac { 5 \left( 1 - \frac { 1 } { ( - 7 ) ^ { n - 1 } } \right) } { 1 + \frac { 1 } { 7 } } ; \frac { 35 } { 6 }$ 
B)  $\frac { 5 \left( 1 - \frac { 1 } { ( - 7 ) ^ { n } } \right) } { 1 + \frac { 1 } { 7 } } ; \frac { 35 } { 6 }$ 
C)  $\frac { 5 \left( 1 - \frac { 1 } { ( - 7 ) ^ { n - 1 } } \right) } { 1 + \frac { 1 } { 7 } } ; \frac { 35 } { 8 }$ 
D)  $\frac { 5 \left( 1 - \frac { 1 } { ( - 7 ) ^ { n } } \right) } { 1 + \frac { 1 } { 7 } } ; \frac { 35 } { 8 }$ 
A)  $\frac { 5 \left( 1 - \frac { 1 } { ( - 7 ) ^ { n - 1 } } \right) } { 1 + \frac { 1 } { 7 } } ; \frac { 35 } { 6 }$ 
B)  $\frac { 5 \left( 1 - \frac { 1 } { ( - 7 ) ^ { n } } \right) } { 1 + \frac { 1 } { 7 } } ; \frac { 35 } { 6 }$ 
C)  $\frac { 5 \left( 1 - \frac { 1 } { ( - 7 ) ^ { n - 1 } } \right) } { 1 + \frac { 1 } { 7 } } ; \frac { 35 } { 8 }$ 
D)  $\frac { 5 \left( 1 - \frac { 1 } { ( - 7 ) ^ { n } } \right) } { 1 + \frac { 1 } { 7 } } ; \frac { 35 } { 8 }$

Question 4

Find the sum of the geometric series for those x for which the series converges.
-$$\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \left( \frac { x - 6 } { 10 } \right) ^ { n }$$

Accepted Answer

A)  $\frac { 10 } { - 4 + x }$ 
B)  $\frac { 10 } { 4 - x }$ 
C)  $\frac { 10 } { - 4 - x }$ 
D)  $\frac { 10 } { 4 + x }$ 
A)  $\frac { 10 } { - 4 + x }$ 
B)  $\frac { 10 } { 4 - x }$ 
C)  $\frac { 10 } { - 4 - x }$ 
D)  $\frac { 10 } { 4 + x }$

Question 5

Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges.
-$$\frac { 8 } { 1 \cdot 3 } + \frac { 8 } { 2 \cdot 4 } + \frac { 8 } { 3 \cdot 5 } + \ldots + \frac { 8 } { n ( n + 2 ) } + \ldots$$

Accepted Answer

A)  $\frac { 8 n ( 3 n + 4 ) } { 4 n ( n + 1 ) } ; 6$ 
B)  $\frac { 8 n ( 3 n + 5 ) } { 4 ( n + 1 ) ( n + 2 ) } ; 6$ 
C)  $\frac { 8 n ( 3 n + 5 ) } { 4 n ( n + 2 ) } ; 6$ 
D)  $\frac { 8 n ( 3 n + 4 ) } { 4 n ( n + 2 ) } ; 6$ 
A)  $\frac { 8 n ( 3 n + 4 ) } { 4 n ( n + 1 ) } ; 6$ 
B)  $\frac { 8 n ( 3 n + 5 ) } { 4 ( n + 1 ) ( n + 2 ) } ; 6$ 
C)  $\frac { 8 n ( 3 n + 5 ) } { 4 n ( n + 2 ) } ; 6$ 
D)  $\frac { 8 n ( 3 n + 4 ) } { 4 n ( n + 2 ) } ; 6$

Question 6

Find the sum of the geometric series for those x for which the series converges.
-$$\sum _ { n = 0 } ^ { \infty } ( 9 x + 1 ) ^ { n }$$

Accepted Answer

A)  $- \frac { 1 } { 9 x }$ 
B)  $\frac { 1 } { 9 x }$ 
C)  $\frac { 1 } { 1 - 9 x }$ 
D)  $\frac { 1 } { 1 + 9 x }$ 
A)  $- \frac { 1 } { 9 x }$ 
B)  $\frac { 1 } { 9 x }$ 
C)  $\frac { 1 } { 1 - 9 x }$ 
D)  $\frac { 1 } { 1 + 9 x }$

Question 7

Find the smallest value of N that will make the inequality hold for all n > N.
-$$\frac { 5 ^ { n } } { n ! } < 10 ^ { - 2 }$$

Accepted Answer

A)  16 
B)  13 
C)  15 
D)  14 
A)  16 
B)  13 
C)  15 
D)  14

Question 8

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

Question 9

Find the limit of the sequence if it converges; otherwise indicate divergence.
-$$a _ { n } = \frac { 8 + 5 n + 5 n ^ { 4 } } { 7 n ^ { 4 } - 1 n ^ { 3 } - 7 }$$

Accepted Answer

A)  $- \frac { 5 } { 7 }$ 
B)  $\frac { 8 } { 7 }$ 
C)  $\frac { 5 } { 7 }$ 
D)  Diverges 
A)  $- \frac { 5 } { 7 }$ 
B)  $\frac { 8 } { 7 }$ 
C)  $\frac { 5 } { 7 }$ 
D)  Diverges

Question 10

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

Accepted Answer

A)  $\frac { 2 n } { 2 ( n + 1 ) } ; 1$ 
B)  $\frac { 2 n } { n + 2 } ; 2$ 
C)  $\frac { 4 n } { 2 n + 1 } ; 2$ 
D)  $\frac { 2 n } { 2 n + 1 } ; 1$ 
A)  $\frac { 2 n } { 2 ( n + 1 ) } ; 1$ 
B)  $\frac { 2 n } { n + 2 } ; 2$ 
C)  $\frac { 4 n } { 2 n + 1 } ; 2$ 
D)  $\frac { 2 n } { 2 n + 1 } ; 1$

Question 11

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

Accepted Answer

A)  17 
B)  15 
C)  16 
D)  18 
A)  17 
B)  15 
C)  16 
D)  18

Question 12

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence.
-$$a _ { 1 } = 2 , a _ { n + 1 } = \frac { ( - 1 ) ^ { n + 1 } } { a _ { n } }$$

Accepted Answer

A)  $2 , \frac { 1 } { 2 } , - 2 , - \frac { 1 } { 2 } , 2$ 
B)  $2 , - \frac { 1 } { 2 } , 2 , - \frac { 1 } { 2 } , 2$ 
C)  $- 2 , \frac { 1 } { 2 } , 2 , - \frac { 1 } { 2 } , - 2$ 
D)  $2 , - \frac { 1 } { 2 } , - 2 , \frac { 1 } { 2 } , 2$ 
A)  $2 , \frac { 1 } { 2 } , - 2 , - \frac { 1 } { 2 } , 2$ 
B)  $2 , - \frac { 1 } { 2 } , 2 , - \frac { 1 } { 2 } , 2$ 
C)  $- 2 , \frac { 1 } { 2 } , 2 , - \frac { 1 } { 2 } , - 2$ 
D)  $2 , - \frac { 1 } { 2 } , - 2 , \frac { 1 } { 2 } , 2$

Question 13

Provide an appropriate response.
-A sequence of rational numbers $\left\{ r _ { n } \right\}$ is defined by $r _ { 1 } = \frac { 1 } { 1 }$, and if $r _ { n } = \frac { a } { b }$ then $r _ { n } + 1 = \frac { a + 5 b } { a + b }$. Find $\lim _ { n \rightarrow } r _ { n }$. Hint: Compute the square of several terms of the sequence on a calculator.

Accepted Answer

A)  2 
B)  $\sqrt { 6 }$ 
C)  $\sqrt { 10 }$ 
D)  $\sqrt { 5 }$ 
A)  2 
B)  $\sqrt { 6 }$ 
C)  $\sqrt { 10 }$ 
D)  $\sqrt { 5 }$

Question 14

Determine if the sequence is monotonic and if it is bounded.
-$$a _ { n } = \frac { 4 ^ { n } } { 7 ^ { n } n ! }$$

Accepted Answer

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

Question 15

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

Accepted Answer

A)  $\mathrm { L } = \ln 2 , \mathrm {~N} = 394$ 
B)  $L = 1 , N = 652$ 
C)  $\mathrm { L } = 1 , \mathrm {~N} = 394$ 
D)  diverges 
A)  $\mathrm { L } = \ln 2 , \mathrm {~N} = 394$ 
B)  $L = 1 , N = 652$ 
C)  $\mathrm { L } = 1 , \mathrm {~N} = 394$ 
D)  diverges

Question 16

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

Accepted Answer

A)  31 
B)  30 
C)  20 
D)  21 
A)  31 
B)  30 
C)  20 
D)  21

Question 17

Determine whether the nonincreasing sequence converges or diverges.
-$$a _ { n } = \frac { 8 ^ { n + 1 } + 7 ^ { n } } { n \cdot 8 ^ { n } }$$

Accepted Answer

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

Question 18

Find a formula for the nth term of the sequence.
-$8,10,12,14,16$ (every other integer starting with 8 )

Accepted Answer

A)  $a _ { n } = n + 14$ 
B)  $a _ { n } = 2 n + 7$ 
C)  $a _ { n } = n + 15$ 
D)  $a _ { n } = 2 n + 6$ 
A)  $a _ { n } = n + 14$ 
B)  $a _ { n } = 2 n + 7$ 
C)  $a _ { n } = n + 15$ 
D)  $a _ { n } = 2 n + 6$

Question 19

Determine if the series converges or diverges. If the series converges, find its sum.
-$$\sum _ { n = 1 } ^ { \infty } \left( \frac { 1 } { \ln ( n + 3 ) } - \frac { 1 } { \ln ( n + 4 ) } \right)$$

Accepted Answer

A)  converges; $\frac { 1 } { \ln 3 }$ 
B)  diverges 
C)  converges; $\ln 4$ 
D)  converges; $\frac { 1 } { \ln 4 }$ 
A)  converges; $\frac { 1 } { \ln 3 }$ 
B)  diverges 
C)  converges; $\ln 4$ 
D)  converges; $\frac { 1 } { \ln 4 }$

Question 20

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

Accepted Answer

A)  $- 1,1 , \frac { 3 } { 5 } , \frac { 1 } { 2 }$ 
B)  $\frac { 1 } { 3 } , \frac { 1 } { 2 } , \frac { 3 } { 4 } , \frac { 2 } { 3 }$ 
C)  $0 , \frac { 1 } { 3 } , \frac { 1 } { 2 } , \frac { 3 } { 4 }$ 
D)  $1 , \frac { 3 } { 5 } , \frac { 1 } { 2 } , \frac { 5 } { 11 }$ 
A)  $- 1,1 , \frac { 3 } { 5 } , \frac { 1 } { 2 }$ 
B)  $\frac { 1 } { 3 } , \frac { 1 } { 2 } , \frac { 3 } { 4 } , \frac { 2 } { 3 }$ 
C)  $0 , \frac { 1 } { 3 } , \frac { 1 } { 2 } , \frac { 3 } { 4 }$ 
D)  $1 , \frac { 3 } { 5 } , \frac { 1 } { 2 } , \frac { 5 } { 11 }$

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

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

Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges. - $5 - \frac { 5 } { 7 } + \frac { 5 } { 49 } - \frac { 5 } { 343 } + \ldots + ( - 1 ) ^ { n - 1 } \frac { 5 } { 7 ^ { n - 1 } } + \ldots$

Find the sum of the geometric series for those x for which the series converges. - $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \left( \frac { x - 6 } { 10 } \right) ^ { n }$

Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges. - $\frac { 8 } { 1 \cdot 3 } + \frac { 8 } { 2 \cdot 4 } + \frac { 8 } { 3 \cdot 5 } + \ldots + \frac { 8 } { n ( n + 2 ) } + \ldots$

Find the sum of the geometric series for those x for which the series converges. - $\sum _ { n = 0 } ^ { \infty } ( 9 x + 1 ) ^ { n }$

Find the smallest value of N that will make the inequality hold for all n > N. - $\frac { 5 ^ { n } } { n ! } < 10 ^ { - 2 }$

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

Find the limit of the sequence if it converges; otherwise indicate divergence. - $a _ { n } = \frac { 8 + 5 n + 5 n ^ { 4 } } { 7 n ^ { 4 } - 1 n ^ { 3 } - 7 }$

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

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

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence. - $a _ { 1 } = 2 , a _ { n + 1 } = \frac { ( - 1 ) ^ { n + 1 } } { a _ { n } }$

Determine if the sequence is monotonic and if it is bounded. - $a _ { n } = \frac { 4 ^ { n } } { 7 ^ { n } n ! }$

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

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

Determine whether the nonincreasing sequence converges or diverges. - $a _ { n } = \frac { 8 ^ { n + 1 } + 7 ^ { n } } { n \cdot 8 ^ { n } }$

Find a formula for the nth term of the sequence. - $8,10,12,14,16$ (every other integer starting with 8 )

Determine if the series converges or diverges. If the series converges, find its sum. - $\sum _ { n = 1 } ^ { \infty } \left( \frac { 1 } { \ln ( n + 3 ) } - \frac { 1 } { \ln ( n + 4 ) } \right)$

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

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Exam 10: Infinite Sequences and Series

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

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

Find the sum of the geometric series for those x for which the series converges. - ∑n=0∞(−1)n(x−610)n\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \left( \frac { x - 6 } { 10 } \right) ^ { n }∑n=0∞​(−1)n(10x−6​)n

Find the sum of the geometric series for those x for which the series converges. - ∑n=0∞(9x+1)n\sum _ { n = 0 } ^ { \infty } ( 9 x + 1 ) ^ { n }∑n=0∞​(9x+1)n

Find the smallest value of N that will make the inequality hold for all n > N. - 5nn!<10−2\frac { 5 ^ { n } } { n ! } < 10 ^ { - 2 }n!5n​<10−2

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

Find the limit of the sequence if it converges; otherwise indicate divergence. - an=8+5n+5n47n4−1n3−7a _ { n } = \frac { 8 + 5 n + 5 n ^ { 4 } } { 7 n ^ { 4 } - 1 n ^ { 3 } - 7 }an​=7n4−1n3−78+5n+5n4​

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

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence. - a1=2,an+1=(−1)n+1ana _ { 1 } = 2 , a _ { n + 1 } = \frac { ( - 1 ) ^ { n + 1 } } { a _ { n } }a1​=2,an+1​=an​(−1)n+1​

Determine if the sequence is monotonic and if it is bounded. - an=4n7nn!a _ { n } = \frac { 4 ^ { n } } { 7 ^ { n } n ! }an​=7nn!4n​

Find the smallest value of N that will make the inequality hold for all n > N. - an=n+1na _ { n } = \sqrt [ n ] { n + 1 }an​=nn+1​

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

Determine whether the nonincreasing sequence converges or diverges. - an=8n+1+7nn⋅8na _ { n } = \frac { 8 ^ { n + 1 } + 7 ^ { n } } { n \cdot 8 ^ { n } }an​=n⋅8n8n+1+7n​

Find a formula for the nth term of the sequence. - 8,10,12,14,168,10,12,14,168,10,12,14,16 (every other integer starting with 8 )

Determine if the series converges or diverges. If the series converges, find its sum. - ∑n=1∞(1ln⁡(n+3)−1ln⁡(n+4))\sum _ { n = 1 } ^ { \infty } \left( \frac { 1 } { \ln ( n + 3 ) } - \frac { 1 } { \ln ( n + 4 ) } \right)∑n=1∞​(ln(n+3)1​−ln(n+4)1​)

Write the first four elements of the sequence. - n+13n−1\frac { n + 1 } { 3 n - 1 }3n−1n+1​