Question 1

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

Accepted Answer

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

Question 2

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

Accepted Answer

A)  750 
B)  731 
C)  737 
D)  733 
A)  750 
B)  731 
C)  737 
D)  733

Question 3

Provide an appropriate response.
-Let $f ( n ) = \sqrt { n ^ { 2 } + 4 } - \sqrt { n ^ { 2 } - 4 }$. What is $f ( n )$ approximately equal to as $n$ gets large? Hint: Compute various examples on your calculator.

Accepted Answer

A)  $\frac { 4 } { n ^ { 2 } }$ 
B)  $\frac { 4 } { \sqrt { n } }$ 
C)  $\frac { 4 } { 2 n }$ 
D)  $\frac { 4 } { n }$ 
A)  $\frac { 4 } { n ^ { 2 } }$ 
B)  $\frac { 4 } { \sqrt { n } }$ 
C)  $\frac { 4 } { 2 n }$ 
D)  $\frac { 4 } { n }$

Question 4

Provide an appropriate response.
-It can be shown that $\sqrt [ n ] { n ! } \approx \frac { n } { e }$ for large values of $n$. Find the smallest value of $N$ such that $\frac { \sqrt [ n ] { n ! } } { \frac { n } { e } } - 1  \mathrm { N }$.

Accepted Answer

A)  29 
B)  33 
C)  27 
D)  22 
A)  29 
B)  33 
C)  27 
D)  22

Question 5

Find the limit of the sequence if it converges; otherwise indicate divergence.
-$$a _ { n } = \frac { 7 n + 6 } { 2 + 1 \sqrt { n } }$$

Accepted Answer

A)  $\frac { 7 } { 2 }$ 
B)  7 
C)  6 
D)  Diverges 
A)  $\frac { 7 } { 2 }$ 
B)  7 
C)  6 
D)  Diverges

Question 6

Find a formula for the nth term of the sequence.
-$9 , - 9,9 , - 9,9$ (9's with alternating signs)

Accepted Answer

A)  $a _ { n } = 9 ( - 1 ) ^ { 2 n + 1 }$ 
B)  $a _ { n } = 9 ( - 1 ) ^ { n + 1 }$ 
C)  $a _ { n } = 9 ( - 1 ) ^ { 2 n - 1 }$ 
D)  $a _ { n } = 9 ( - 1 ) ^ { n }$ 
A)  $a _ { n } = 9 ( - 1 ) ^ { 2 n + 1 }$ 
B)  $a _ { n } = 9 ( - 1 ) ^ { n + 1 }$ 
C)  $a _ { n } = 9 ( - 1 ) ^ { 2 n - 1 }$ 
D)  $a _ { n } = 9 ( - 1 ) ^ { n }$

Question 7

Provide an appropriate response.
-It can be shown that $ \lim _{n \rightarrow \infty} \frac{1}{n^{c}}=0 $ for $ c>0 $ Find the smallest value of N such that $\mid\frac{1}{n^{c}}\mid$  N if

Accepted Answer

A)  11 
B)  13 
C)  10 
D)  12 
A)  11 
B)  13 
C)  10 
D)  12

Question 8

Find the values of x for which the geometric series converges.
-$$\sum _ { n = 0 } ^ { \infty } ( x - 6 ) ^ { n }$$

Accepted Answer

A)  $6 < x < 7$ 
B)  $5 < x < 6$ 
C)  $- 6 < x < 6$ 
D)  $5 < x < 7$ 
A)  $6 < x < 7$ 
B)  $5 < x < 6$ 
C)  $- 6 < x < 6$ 
D)  $5 < x < 7$

Question 9

Find the values of x for which the geometric series converges.
-$$\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \left( \frac { x - 8 } { 2 } \right) ^ { n }$$

Accepted Answer

A)  $- 10 < x < 10$ 
B)  $4 < x < 12$ 
C)  $6 < x < 10$ 
D)  $- 4 < x < 12$ 
A)  $- 10 < x < 10$ 
B)  $4 < x < 12$ 
C)  $6 < x < 10$ 
D)  $- 4 < x < 12$

Question 10

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

Accepted Answer

A)  converges; $\frac { 4 } { 3 }$ 
B)  converges; $\frac { 1 } { 2 }$ 
C)  converges; $\frac { 3 } { 2 }$ 
D)  diverges 
A)  converges; $\frac { 4 } { 3 }$ 
B)  converges; $\frac { 1 } { 2 }$ 
C)  converges; $\frac { 3 } { 2 }$ 
D)  diverges

Question 11

Find a formula for the nth term of the sequence.
-$0,2,2,2,0,2,2,2 ( 0,2,2,2$ repeated)

Accepted Answer

A)  $a _ { n } = 1 + ( - 1 ) ^ { n } ( n + 1 ) ( n + 2 ) / 6$ 
B)  $a _ { n } = 1 + ( - 1 ) ^ { n ( n + 1 ) / 2 }$ 
C)  $a _ { n } = 1 + ( - 1 ) ^ { ( n + 1 ) ( n + 2 ) / 2 }$ 
D)  $a _ { n } = 1 + ( - 1 ) ^ { n ( n - 1 ) / 2 }$ 
A)  $a _ { n } = 1 + ( - 1 ) ^ { n } ( n + 1 ) ( n + 2 ) / 6$ 
B)  $a _ { n } = 1 + ( - 1 ) ^ { n ( n + 1 ) / 2 }$ 
C)  $a _ { n } = 1 + ( - 1 ) ^ { ( n + 1 ) ( n + 2 ) / 2 }$ 
D)  $a _ { n } = 1 + ( - 1 ) ^ { n ( n - 1 ) / 2 }$

Question 12

Find the sum of the series.
-$$\sum _ { n = 0 } ^ { \infty } \left( \frac { 2 } { 9 ^ { n } } + \frac { 4 } { 5 ^ { n } } \right)$$

Accepted Answer

A)  $\frac { 29 } { 4 }$ 
B)  $- \frac { 23 } { 15 }$ 
C)  $\frac { 77 } { 15 }$ 
D)  $- \frac { 11 } { 4 }$ 
A)  $\frac { 29 } { 4 }$ 
B)  $- \frac { 23 } { 15 }$ 
C)  $\frac { 77 } { 15 }$ 
D)  $- \frac { 11 } { 4 }$

Question 13

Determine if the series converges or diverges; if the series converges, find its sum.
-$$\sum _ { n = 0 } ^ { \infty } \frac { 1 } { ( \sqrt { 6 } ) ^ { n } }$$

Accepted Answer

A)  Converges; $\frac { 6 - \sqrt { 6 } } { 5 }$ 
B)  Converges; $\frac { 6 + \sqrt { 6 } } { 5 }$ 
C)  Converges; $\frac { \sqrt { 6 } - 6 } { 5 }$ 
D)  Diverges 
A)  Converges; $\frac { 6 - \sqrt { 6 } } { 5 }$ 
B)  Converges; $\frac { 6 + \sqrt { 6 } } { 5 }$ 
C)  Converges; $\frac { \sqrt { 6 } - 6 } { 5 }$ 
D)  Diverges

Question 14

Find the sum of the series.
-$$\sum _ { n = 1 } ^ { \infty } \frac { 7 } { 3 ^ { n } }$$

Accepted Answer

A)  $\frac { 21 } { 2 }$ 
B)  $\frac { 21 } { 4 }$ 
C)  $\frac { 7 } { 2 }$ 
D)  $\frac { 7 } { 4 }$ 
A)  $\frac { 21 } { 2 }$ 
B)  $\frac { 21 } { 4 }$ 
C)  $\frac { 7 } { 2 }$ 
D)  $\frac { 7 } { 4 }$

Question 15

Find the sum of the geometric series for those x for which the series converges.
-$$\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } ( 8 x ) ^ { 2 n }$$

Accepted Answer

A)  $\frac { 1 } { 1 - 64 x ^ { 2 } }$ 
B)  $\frac { 64 } { 1 + 64 x ^ { 2 } }$ 
C)  $\frac { 8 } { 1 + 64 x ^ { 2 } }$ 
D)  $\frac { 1 } { 1 + 64 x ^ { 2 } }$ 
A)  $\frac { 1 } { 1 - 64 x ^ { 2 } }$ 
B)  $\frac { 64 } { 1 + 64 x ^ { 2 } }$ 
C)  $\frac { 8 } { 1 + 64 x ^ { 2 } }$ 
D)  $\frac { 1 } { 1 + 64 x ^ { 2 } }$

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

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

Provide an appropriate response. -Let $f ( n ) = \sqrt { n ^ { 2 } + 4 } - \sqrt { n ^ { 2 } - 4 }$ . What is $f ( n )$ approximately equal to as $n$ gets large? Hint: Compute various examples on your calculator.

Provide an appropriate response. -It can be shown that $\sqrt [ n ] { n ! } \approx \frac { n } { e }$ for large values of $n$ . Find the smallest value of $N$ such that $\frac { \sqrt [ n ] { n ! } } { \frac { n } { e } } - 1 < 10 ^ { - 1 }$ for all $\mathrm { n } > \mathrm { N }$ .

Find the limit of the sequence if it converges; otherwise indicate divergence. - $a _ { n } = \frac { 7 n + 6 } { 2 + 1 \sqrt { n } }$

Find a formula for the nth term of the sequence. - $9 , - 9,9 , - 9,9$ (9's with alternating signs)

Provide an appropriate response. -It can be shown that $\lim _{n \rightarrow \infty} \frac{1}{n^{c}}=0$ for $c>0$ Find the smallest value of N such that $\mid\frac{1}{n^{c}}\mid$ < $\varepsilon$ for all n> N if

Find the values of x for which the geometric series converges. - $\sum _ { n = 0 } ^ { \infty } ( x - 6 ) ^ { n }$

Find the values of x for which the geometric series converges. - $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \left( \frac { x - 8 } { 2 } \right) ^ { n }$

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

Find a formula for the nth term of the sequence. - $0,2,2,2,0,2,2,2 ( 0,2,2,2$ repeated)

Find the sum of the series. - $\sum _ { n = 0 } ^ { \infty } \left( \frac { 2 } { 9 ^ { n } } + \frac { 4 } { 5 ^ { n } } \right)$

Determine if the series converges or diverges; if the series converges, find its sum. - $\sum _ { n = 0 } ^ { \infty } \frac { 1 } { ( \sqrt { 6 } ) ^ { n } }$

Find the sum of the series. - $\sum _ { n = 1 } ^ { \infty } \frac { 7 } { 3 ^ { n } }$

Find the sum of the geometric series for those x for which the series converges. - $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } ( 8 x ) ^ { 2 n }$

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Exam 10: Infinite Sequences and Series

Determine if the sequence is monotonic and if it is bounded. - an=4n4na _ { n } = \frac { 4 ^ { n } } { 4 n }an​=4n4n​

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

Provide an appropriate response. -Let f(n)=n2+4−n2−4f ( n ) = \sqrt { n ^ { 2 } + 4 } - \sqrt { n ^ { 2 } - 4 }f(n)=n2+4​−n2−4​ . What is f(n)f ( n )f(n) approximately equal to as nnn gets large? Hint: Compute various examples on your calculator.

Find the limit of the sequence if it converges; otherwise indicate divergence. - an=7n+62+1na _ { n } = \frac { 7 n + 6 } { 2 + 1 \sqrt { n } }an​=2+1n​7n+6​

Find a formula for the nth term of the sequence. - 9,−9,9,−9,99 , - 9,9 , - 9,99,−9,9,−9,9 (9's with alternating signs)

Provide an appropriate response. -It can be shown that lim⁡n→∞1nc=0 \lim _{n \rightarrow \infty} \frac{1}{n^{c}}=0 limn→∞​nc1​=0 for c>0 c>0 c>0 Find the smallest value of N such that ∣1nc∣\mid\frac{1}{n^{c}}\mid∣nc1​∣ < ε\varepsilonε for all n> N if

Find the values of x for which the geometric series converges. - ∑n=0∞(x−6)n\sum _ { n = 0 } ^ { \infty } ( x - 6 ) ^ { n }∑n=0∞​(x−6)n

Find the values of x for which the geometric series converges. - ∑n=0∞(−1)n(x−82)n\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \left( \frac { x - 8 } { 2 } \right) ^ { n }∑n=0∞​(−1)n(2x−8​)n

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

Find a formula for the nth term of the sequence. - 0,2,2,2,0,2,2,2(0,2,2,20,2,2,2,0,2,2,2 ( 0,2,2,20,2,2,2,0,2,2,2(0,2,2,2 repeated)

Find the sum of the series. - ∑n=0∞(29n+45n)\sum _ { n = 0 } ^ { \infty } \left( \frac { 2 } { 9 ^ { n } } + \frac { 4 } { 5 ^ { n } } \right)∑n=0∞​(9n2​+5n4​)

Determine if the series converges or diverges; if the series converges, find its sum. - ∑n=0∞1(6)n\sum _ { n = 0 } ^ { \infty } \frac { 1 } { ( \sqrt { 6 } ) ^ { n } }∑n=0∞​(6​)n1​

Find the sum of the series. - ∑n=1∞73n\sum _ { n = 1 } ^ { \infty } \frac { 7 } { 3 ^ { n } }∑n=1∞​3n7​

Find the sum of the geometric series for those x for which the series converges. - ∑n=0∞(−1)n(8x)2n\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } ( 8 x ) ^ { 2 n }∑n=0∞​(−1)n(8x)2n