Question 1

Determine if the series converges or diverges; if the series converges, find its sum.
-$$\sum _ { n = 0 } ^ { \infty } \frac { \cos n \pi } { 7 ^ { n } }$$

Accepted Answer

A)  Converges; $\frac { 7 } { 6 }$ 
B)  Converges; $\frac { 1 } { 6 }$ 
C)  Converges; $\frac { 7 } { 8 }$ 
D)  Diverges 
A)  Converges; $\frac { 7 } { 6 }$ 
B)  Converges; $\frac { 1 } { 6 }$ 
C)  Converges; $\frac { 7 } { 8 }$ 
D)  Diverges

Question 2

Determine if the series converges or diverges; if the series converges, find its sum.
-$$\sum _ { n = 0 } ^ { \infty } \frac { n ! } { 900 ^ { n } }$$

Accepted Answer

A)  Converges; $\mathrm { e }$ 
B)  Converges; 1 
C)  converges; $\frac { 1 } { \mathrm { e } }$ 
D)  Diverges 
A)  Converges; $\mathrm { e }$ 
B)  Converges; 1 
C)  converges; $\frac { 1 } { \mathrm { e } }$ 
D)  Diverges

Question 3

Find a formula for the nth term of the sequence.
-$- 9 , - 8 , - 7 , - 6 , - 5$ (integers beginning with $- 9$ )

Accepted Answer

A)  $a _ { n } = n + 9$ 
B)  $a _ { n } = n - 8$ 
C)  $a _ { n } = n - 9$ 
D)  $a _ { n } = n - 10$ 
A)  $a _ { n } = n + 9$ 
B)  $a _ { n } = n - 8$ 
C)  $a _ { n } = n - 9$ 
D)  $a _ { n } = n - 10$

Question 4

Determine if the geometric series converges or diverges. If it converges, find its sum.
-$1 + ( - 9 ) + ( - 9 ) ^ { 2 } + ( - 9 ) ^ { 3 } + ( - 9 ) ^ { 4 } + \ldots$

Accepted Answer

A)  diverges 
B)  converges, 90 
C)  converges, 81 
D)  converges, 9 
A)  diverges 
B)  converges, 90 
C)  converges, 81 
D)  converges, 9

Question 5

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence.
-$$a _ { 1 } = 1 , a _ { 2 } = 3 , a _ { n + 2 } = a _ { n + 1 } - a _ { n }$$

Accepted Answer

A)  $1,3,2,1,0$ 
B)  $1 , - 1,2 , - 3,5$ 
C)  $1,3,2 , - 1 , - 3$ 
D)  $1 , - 3,4 , - 5,6$ 
A)  $1,3,2,1,0$ 
B)  $1 , - 1,2 , - 3,5$ 
C)  $1,3,2 , - 1 , - 3$ 
D)  $1 , - 3,4 , - 5,6$

Question 6

Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges.
-$$\frac { 9 } { 1 ^ { 2 } \cdot 2 ^ { 2 } } + \frac { 15 } { 2 ^ { 2 } \cdot 3 ^ { 2 } } + \frac { 21 } { 3 ^ { 2 } \cdot 4 ^ { 2 } } + \ldots + \frac { 3 ( 2 n + 1 ) } { n ^ { 2 } ( n + 1 ) ^ { 2 } } + \ldots$$

Accepted Answer

A)  $\frac { 3 n ( n + 2 ) } { ( n + 1 ) ^ { 2 } } ; 3$ 
B)  $\frac { 3 n ( n + 1 ) } { ( n + 2 ) ^ { 2 } } ; 3$ 
C)  $\frac { 3 n ^ { 2 } } { ( n + 1 ) ( n + 2 ) } ; 3$ 
D)  $\frac { 3 ( n + 1 ) ( n + 2 ) } { n ^ { 2 } } ; 3$ 
A)  $\frac { 3 n ( n + 2 ) } { ( n + 1 ) ^ { 2 } } ; 3$ 
B)  $\frac { 3 n ( n + 1 ) } { ( n + 2 ) ^ { 2 } } ; 3$ 
C)  $\frac { 3 n ^ { 2 } } { ( n + 1 ) ( n + 2 ) } ; 3$ 
D)  $\frac { 3 ( n + 1 ) ( n + 2 ) } { n ^ { 2 } } ; 3$

Question 7

Find the values of x for which the geometric series converges.
-$$\sum _ { n = 0 } ^ { \infty } ( 7 x + 1 ) ^ { n }$$

Accepted Answer

A)  $0 < x < \frac { 1 } { 7 }$ 
B)  $- \frac { 1 } { 7 } < x < \frac { 1 } { 7 }$ 
C)  $- \frac { 2 } { 7 } < x < 0$ 
D)  $0 < x < \frac { 2 } { 7 }$ 
A)  $0 < x < \frac { 1 } { 7 }$ 
B)  $- \frac { 1 } { 7 } < x < \frac { 1 } { 7 }$ 
C)  $- \frac { 2 } { 7 } < x < 0$ 
D)  $0 < x < \frac { 2 } { 7 }$

Question 8

Find the limit of the sequence if it converges; otherwise indicate divergence.
-$$a _ { n } = \left( \frac { 4 } { n } \right) ^ { 4 / n }$$

Accepted Answer

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

Question 9

Find a formula for the nth term of the sequence.
-$0 , \frac { 2 } { 3 } , 0 , \frac { 2 } { 3 } , 0$ (alternating 0 's and $\frac { 2 } { 3 }$ 's)

Accepted Answer

A)  $a _ { n } = \frac { 1 + ( - 1 ) ^ { n + 1 } } { 2 + ( - 1 ) ^ { n + 1 } }$ 
B)  $a _ { n } = \frac { 1 - ( - 1 ) ^ { n } } { 2 + ( - 1 ) ^ { n } }$ 
C)  $a _ { n } = \frac { 1 + ( - 1 ) ^ { n + 1 } } { 2 - ( - 1 ) ^ { n + 1 } }$ 
D)  $a _ { n } = \frac { 1 + ( - 1 ) ^ { n } } { 2 + ( - 1 ) ^ { n } }$ 
A)  $a _ { n } = \frac { 1 + ( - 1 ) ^ { n + 1 } } { 2 + ( - 1 ) ^ { n + 1 } }$ 
B)  $a _ { n } = \frac { 1 - ( - 1 ) ^ { n } } { 2 + ( - 1 ) ^ { n } }$ 
C)  $a _ { n } = \frac { 1 + ( - 1 ) ^ { n + 1 } } { 2 - ( - 1 ) ^ { n + 1 } }$ 
D)  $a _ { n } = \frac { 1 + ( - 1 ) ^ { n } } { 2 + ( - 1 ) ^ { n } }$

Question 10

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

Accepted Answer

A)  Converges; $\frac { 1 } { 2 }$ 
B)  Converges; 2 
C)  Converges; 1 
D)  Diverges 
A)  Converges; $\frac { 1 } { 2 }$ 
B)  Converges; 2 
C)  Converges; 1 
D)  Diverges

Question 11

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

Accepted Answer

A)  $- 10 < x < - 8$ 
B)  $- 10 < x < - 9$ 
C)  $- 9 < \mathrm { x } < 9$ 
D)  $- 9 < \mathrm { x } < - 8$ 
A)  $- 10 < x < - 8$ 
B)  $- 10 < x < - 9$ 
C)  $- 9 < \mathrm { x } < 9$ 
D)  $- 9 < \mathrm { x } < - 8$

Question 12

Determine whether the nonincreasing sequence converges or diverges.
-$$a _ { 1 } = 1 , a _ { n + 1 } = 2 a _ { n } - 8$$

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}=5, a_{n+1}=(-1)^{n} a_{n} $

Accepted Answer

A)  $- 5,5 , - 5,5 , - 5$ 
B)  $5,5 , - 5 , - 5,5$ 
C)  $5 , - 5 , - 5,5,5$ 
D)  $5 , - 5,5 , - 5,5$ 
A)  $- 5,5 , - 5,5 , - 5$ 
B)  $5,5 , - 5 , - 5,5$ 
C)  $5 , - 5 , - 5,5,5$ 
D)  $5 , - 5,5 , - 5,5$

Question 14

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence.
-$$\mathrm { a } _ { 1 } = 4 , \mathrm { a } _ { \mathrm { n } + 1 } = - \mathrm { a } _ { \mathrm { n } }$$

Accepted Answer

A)  $- 4,4 , - 4,4 , - 4$ 
B)  $4,0 , - 4 , - 8 , - 12$ 
C)  $4 , - 16,64 , - 256,1024$ 
D)  $4 , - 4,4 , - 4,4$ 
A)  $- 4,4 , - 4,4 , - 4$ 
B)  $4,0 , - 4 , - 8 , - 12$ 
C)  $4 , - 16,64 , - 256,1024$ 
D)  $4 , - 4,4 , - 4,4$

Question 15

Find the smallest value of N that will make the inequality hold for all n > N.
-$$| \sqrt [ n ] { 2 n } - 1 | < 10 ^ { - 3 }$$

Accepted Answer

A)  9907 
B)  9891 
C)  9883 
D)  9899 
A)  9907 
B)  9891 
C)  9883 
D)  9899

Question 16

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

Accepted Answer

A)  L = 0, N = 8 
B)  L = 0, N = 10 
C)  L = 0, N = 13 
D)  diverges 
A)  L = 0, N = 8 
B)  L = 0, N = 10 
C)  L = 0, N = 13 
D)  diverges

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.
-$\frac { 2 } { 1 \cdot 2 } + \frac { 2 } { 2 \cdot 3 } + \frac { 2 } { 3 \cdot 4 } + \ldots + \frac { 2 } { n ( n + 1 ) } + \ldots$

Accepted Answer

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

Question 18

Express the number as the ratio of two integers.
-$0.616161 \ldots$

Accepted Answer

A)  $\frac { 61 } { 999 }$ 
B)  $\frac { 61 } { 99 }$ 
C)  $\frac { 610 } { 999 }$ 
D)  $\frac { 610 } { 99 }$ 
A)  $\frac { 61 } { 999 }$ 
B)  $\frac { 61 } { 99 }$ 
C)  $\frac { 610 } { 999 }$ 
D)  $\frac { 610 } { 99 }$

Question 19

Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges.
-$7 + 14 + 28 + \ldots + 7 \cdot 2 ^ { n - 1 } + \ldots$

Accepted Answer

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

Question 20

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence.
-$a _ { 1 } = 1 , a _ { n + 1 } = a _ {{ n } ^ { 3 }}$

Accepted Answer

A)  $1,3,9,27,81,243$ 
B)  $1,3,9,27,81$ 
C)  $1,1,1,1,1$ 
D)  $1,4,7,10,13$ 
A)  $1,3,9,27,81,243$ 
B)  $1,3,9,27,81$ 
C)  $1,1,1,1,1$ 
D)  $1,4,7,10,13$

Determine if the series converges or diverges; if the series converges, find its sum. - $\sum _ { n = 0 } ^ { \infty } \frac { \cos n \pi } { 7 ^ { n } }$

Determine if the series converges or diverges; if the series converges, find its sum. - $\sum _ { n = 0 } ^ { \infty } \frac { n ! } { 900 ^ { n } }$

Find a formula for the nth term of the sequence. - $- 9 , - 8 , - 7 , - 6 , - 5$ (integers beginning with $- 9$ )

Determine if the geometric series converges or diverges. If it converges, find its sum. - $1 + ( - 9 ) + ( - 9 ) ^ { 2 } + ( - 9 ) ^ { 3 } + ( - 9 ) ^ { 4 } + \ldots$

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence. - $a _ { 1 } = 1 , a _ { 2 } = 3 , a _ { n + 2 } = a _ { n + 1 } - a _ { n }$

Find the values of x for which the geometric series converges. - $\sum _ { n = 0 } ^ { \infty } ( 7 x + 1 ) ^ { n }$

Find the limit of the sequence if it converges; otherwise indicate divergence. - $a _ { n } = \left( \frac { 4 } { n } \right) ^ { 4 / n }$

Find a formula for the nth term of the sequence. - $0 , \frac { 2 } { 3 } , 0 , \frac { 2 } { 3 } , 0$ (alternating 0 's and $\frac { 2 } { 3 }$ 's)

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

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

Determine whether the nonincreasing sequence converges or diverges. - $a _ { 1 } = 1 , a _ { n + 1 } = 2 a _ { n } - 8$

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence. - $a_{1}=5, a_{n+1}=(-1)^{n} a_{n}$

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence. - $\mathrm { a } _ { 1 } = 4 , \mathrm { a } _ { \mathrm { n } + 1 } = - \mathrm { a } _ { \mathrm { n } }$

Find the smallest value of N that will make the inequality hold for all n > N. - $| \sqrt [ n ] { 2 n } - 1 | < 10 ^ { - 3 }$

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

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

Express the number as the ratio of two integers. - $0.616161 \ldots$

Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges. - $7 + 14 + 28 + \ldots + 7 \cdot 2 ^ { n - 1 } + \ldots$

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence. - $a _ { 1 } = 1 , a _ { n + 1 } = a _ {{ n } ^ { 3 }}$

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Exam 10: Infinite Sequences and Series

Determine if the series converges or diverges; if the series converges, find its sum. - ∑n=0∞cos⁡nπ7n\sum _ { n = 0 } ^ { \infty } \frac { \cos n \pi } { 7 ^ { n } }∑n=0∞​7ncosnπ​

Determine if the series converges or diverges; if the series converges, find its sum. - ∑n=0∞n!900n\sum _ { n = 0 } ^ { \infty } \frac { n ! } { 900 ^ { n } }∑n=0∞​900nn!​

Find a formula for the nth term of the sequence. - −9,−8,−7,−6,−5- 9 , - 8 , - 7 , - 6 , - 5−9,−8,−7,−6,−5 (integers beginning with −9- 9−9 )

Determine if the geometric series converges or diverges. If it converges, find its sum. - 1+(−9)+(−9)2+(−9)3+(−9)4+…1 + ( - 9 ) + ( - 9 ) ^ { 2 } + ( - 9 ) ^ { 3 } + ( - 9 ) ^ { 4 } + \ldots1+(−9)+(−9)2+(−9)3+(−9)4+…

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence. - a1=1,a2=3,an+2=an+1−ana _ { 1 } = 1 , a _ { 2 } = 3 , a _ { n + 2 } = a _ { n + 1 } - a _ { n }a1​=1,a2​=3,an+2​=an+1​−an​

Find the values of x for which the geometric series converges. - ∑n=0∞(7x+1)n\sum _ { n = 0 } ^ { \infty } ( 7 x + 1 ) ^ { n }∑n=0∞​(7x+1)n

Find the limit of the sequence if it converges; otherwise indicate divergence. - an=(4n)4/na _ { n } = \left( \frac { 4 } { n } \right) ^ { 4 / n }an​=(n4​)4/n

Find a formula for the nth term of the sequence. - 0,23,0,23,00 , \frac { 2 } { 3 } , 0 , \frac { 2 } { 3 } , 00,32​,0,32​,0 (alternating 0 's and 23\frac { 2 } { 3 }32​ 's)

Determine if the series converges or diverges; if the series converges, find its sum. - ∑n=1∞(−1)n−143n\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } \frac { 4 } { 3 ^ { n } }∑n=1∞​(−1)n−13n4​

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

Determine whether the nonincreasing sequence converges or diverges. - a1=1,an+1=2an−8a _ { 1 } = 1 , a _ { n + 1 } = 2 a _ { n } - 8a1​=1,an+1​=2an​−8

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence. - a1=5,an+1=(−1)nan a_{1}=5, a_{n+1}=(-1)^{n} a_{n} a1​=5,an+1​=(−1)nan​

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence. - a1=4,an+1=−an\mathrm { a } _ { 1 } = 4 , \mathrm { a } _ { \mathrm { n } + 1 } = - \mathrm { a } _ { \mathrm { n } }a1​=4,an+1​=−an​

Find the smallest value of N that will make the inequality hold for all n > N. - ∣2nn−1∣<10−3| \sqrt [ n ] { 2 n } - 1 | < 10 ^ { - 3 }∣n2n​−1∣<10−3

Find the smallest value of N that will make the inequality hold for all n > N. - an=cos⁡nn2a _ { n } = \frac { \cos n } { n ^ { 2 } }an​=n2cosn​

Express the number as the ratio of two integers. - 0.616161…0.616161 \ldots0.616161…

Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges. - 7+14+28+…+7⋅2n−1+…7 + 14 + 28 + \ldots + 7 \cdot 2 ^ { n - 1 } + \ldots7+14+28+…+7⋅2n−1+…

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=an3a _ { 1 } = 1 , a _ { n + 1 } = a _ {{ n } ^ { 3 }}a1​=1,an+1​=an3​