Question 1

Determine whether the given series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { n } { 8 ^ { n } }$

Accepted Answer

The answer of Determine whether the given series is convergent...

Question 2

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { 7 ^ { n } } { n ! n }$

Accepted Answer

The answer of Determine whether the series is convergent or...

Question 3

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { n \sqrt [ 6 ] { n } }$

Accepted Answer

absolutely...

Question 4

Find the radius of convergence and the interval of convergence of the power series. $\sum _ { n = 1 } ^ { \infty } \frac { 4 ^ { n } x ^ { n } } { n }$

Accepted Answer

The answer of Find the radius of convergence and the...

Question 5

Use the Integral Test to determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { 2 } { 7 n + 4 }$

Accepted Answer

The answer of Use the Integral Test to determine whether...

Question 6

Find the sum of the series. $\sum _ { n = 0 } ^ { \infty } \frac { 5 ^ { n } } { 6 ^ { n } n ! }$

Accepted Answer

The answer of Find the sum of the series. \(\sum...

Question 7

Determine which one of the p-series below is divergent.

Accepted Answer

A)    ∑n=1∞1n03\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 03 } }∑n=1∞​n031​ 
B)    ∑n=1∞n−4x\sum _ { n = 1 } ^ { \infty } n ^ { - 4 x }∑n=1∞​n−4x 
C)    ∑n=1∞1n3e\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 3 e } }∑n=1∞​n3e1​ 
D)    ∑n=1∞1n4\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 4 } }∑n=1∞​n41​

Question 8

Find the radius of convergence and the interval of convergence of the power series.  ∑n=2∞xnn(ln⁡n) 8\sum _ { n = 2 } ^ { \infty } \frac { x ^ { n } } { n ( \ln n )  ^ { 8 } }∑n=2∞​n(lnn) 8xn​

Accepted Answer

A)    R=0,I={0}R = 0 , I = \{ 0 \}R=0,I={0} 
B)    R=1,I=[−1,1]R = 1 , I = [ - 1,1 ]R=1,I=[−1,1] 
C)    R=1,I=(−1,1) R = 1 , I = ( - 1,1 ) R=1,I=(−1,1)  
D)    R=∞,I=(−∞,∞) R = \infty , I = ( - \infty , \infty ) R=∞,I=(−∞,∞)

Question 9

Determine whether the series converges or diverges. $\sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n } \ln n } { 9 ^ { n } }$

Accepted Answer

The answer of Determine whether the series converges or diverges....

Question 10

Determine whether the series converges or diverges. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { n + 4 }$

Accepted Answer

The answer of Determine whether the series converges or diverges....

Question 11

Determine whether the sequence defined by  an=sin⁡2n9na _ { n } = \frac { \sin 2 n } { 9 n }an​=9nsin2n​  converges or diverges. If it converges, find its limit.

Accepted Answer

A)   Diverges
B)   1
C)    29\frac { 2 } { 9 }92​ 
D)   0

Question 12

Use series to approximate the definite integral to within the indicated accuracy.  ∫005x2e−x2dx∣ error ∣<0.001\int _ { 0 } ^ { 05 } x ^ { 2 } e ^ { - x ^ { 2 } } d x \quad \mid 	ext { error } \mid < 0.001∫005​x2e−x2dx∣ error ∣<0.001

Accepted Answer

A)   0.0354
B)   0.0125
C)   0.0625
D)   0.1447
E)   0.2774

Question 13

Determine whether the series is convergent or divergent. If it is convergent, write its sum. Otherwise write divergent. $\sum _ { n = 1 } ^ { \infty } 5 \left( \frac { 3 } { 4 } \right) ^ { n - 1 }$

Accepted Answer

The answer of Determine whether the series is convergent or...

Question 14

Determine whether the series converges or diverges. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { \sqrt [ n ] { 6 n } }$

Accepted Answer

The answer of Determine whether the series converges or diverges....

Question 15

Find the radius of convergence and the interval of convergence of the power series.  ∑n=0∞xnn+2\sum _ { n = 0 } ^ { \infty } \frac { x ^ { n } } { n + 2 }∑n=0∞​n+2xn​

Accepted Answer

A)    R=2,I=[−2,2) R = 2 , I = [ - 2,2 ) R=2,I=[−2,2)  
B)    R=1,I=(−1,1) R = 1 , I = ( - 1,1 ) R=1,I=(−1,1)  
C)    R=1,I=[−1,1) R = 1 , I = [ - 1,1 ) R=1,I=[−1,1)  
D)    R=2,I=(−2,2) R = 2 , I = ( - 2,2 ) R=2,I=(−2,2)

Question 16

Determine which one of the p-series below is convergent.

Accepted Answer

A)    ∑n=1∞1n7\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 7 } }∑n=1∞​n71​ 
B)    ∑n=1∞n−06\sum _ { n = 1 } ^ { \infty } n ^ { - 06 }∑n=1∞​n−06 
C)    ∑n=1∞1n08\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 08 } }∑n=1∞​n081​ 
D)    ∑n=1∞1n3/4\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 3 / 4 } }∑n=1∞​n3/41​

Question 17

Determine whether the sequence converges or diverges. If it converges, find the limit. $a _ { n } = 2 e ^ { 4 n / ( n + 2 ) }$

Accepted Answer

The answer of Determine whether the sequence converges or diverges....

Question 18

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 2 } + 9 } { n ^ { 7 } + n }$

Accepted Answer

The answer of Determine whether the series is convergent or...

Question 19

Use the Integral Test to determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 8 n + 2 }$

Accepted Answer

The answer of Use the Integral Test to determine whether...

Question 20

Find the radius of convergence and the interval of convergence of the power series.  ∑n=1∞(−1) n−1(x−6) nn⋅5n\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 )  ^ { n - 1 } ( x - 6 )  ^ { n } } { n \cdot 5 ^ { n } }∑n=1∞​n⋅5n(−1) n−1(x−6) n​

Accepted Answer

A)    R=6,I=(−1,11]R = 6 , I = ( - 1,11 ]R=6,I=(−1,11] 
B)    R=5,I=(1,11]R = 5 , I = ( 1,11 ]R=5,I=(1,11] 
C)    R=15,I=[−15,15) R = \frac { 1 } { 5 } , I = \left[ - \frac { 1 } { 5 } , \frac { 1 } { 5 } \right) R=51​,I=[−51​,51​)  
D)    R=5,I=[−5,5) R = 5 , I = [ - 5,5 ) R=5,I=[−5,5)

Exam 11: Infinite Sequences and Series

Determine whether the series converges or diverges. ∑n=1∞(−1)nn+4\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { n + 4 }∑n=1∞​n+4(−1)n​

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Determine whether the series is convergent or divergent. If it is convergent, write its sum. Otherwise write divergent. ∑n=1∞5(34)n−1\sum _ { n = 1 } ^ { \infty } 5 \left( \frac { 3 } { 4 } \right) ^ { n - 1 }∑n=1∞​5(43​)n−1

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Determine which one of the p-series below is convergent.

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Determine which one of the p-series below is divergent.

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Determine whether the given series is convergent or divergent. ∑n=1∞n8n\sum _ { n = 1 } ^ { \infty } \frac { n } { 8 ^ { n } }∑n=1∞​8nn​

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Use series to approximate the definite integral to within the indicated accuracy. ∫005x2e−x2dx∣ error ∣<0.001\int _ { 0 } ^ { 05 } x ^ { 2 } e ^ { - x ^ { 2 } } d x \quad \mid \text { error } \mid < 0.001∫005​x2e−x2dx∣ error ∣<0.001

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Determine whether the series is convergent or divergent. ∑n=1∞n2+9n7+n\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 2 } + 9 } { n ^ { 7 } + n }∑n=1∞​n7+nn2+9​

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Use the Integral Test to determine whether the series is convergent or divergent. ∑n=1∞27n+4\sum _ { n = 1 } ^ { \infty } \frac { 2 } { 7 n + 4 }∑n=1∞​7n+42​

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Determine whether the series is convergent or divergent. ∑n=1∞7nn!n\sum _ { n = 1 } ^ { \infty } \frac { 7 ^ { n } } { n ! n }∑n=1∞​n!n7n​

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Find the radius of convergence and the interval of convergence of the power series. ∑n=2∞xnn(ln⁡n) 8\sum _ { n = 2 } ^ { \infty } \frac { x ^ { n } } { n ( \ln n ) ^ { 8 } }∑n=2∞​n(lnn) 8xn​

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Find the radius of convergence and the interval of convergence of the power series. ∑n=0∞xnn+2\sum _ { n = 0 } ^ { \infty } \frac { x ^ { n } } { n + 2 }∑n=0∞​n+2xn​

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Determine whether the series converges or diverges. ∑n=2∞(−1)nln⁡n9n\sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n } \ln n } { 9 ^ { n } }∑n=2∞​9n(−1)nlnn​

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Use the Integral Test to determine whether the series is convergent or divergent. ∑n=1∞18n+2\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 8 n + 2 }∑n=1∞​8n+21​

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Determine whether the series is absolutely convergent, conditionally convergent, or divergent. ∑n=1∞(−1)nnn6\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { n \sqrt [ 6 ] { n } }∑n=1∞​n6n​(−1)n​

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Determine whether the series converges or diverges. ∑n=1∞(−1)n6nn\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { \sqrt [ n ] { 6 n } }∑n=1∞​n6n​(−1)n​

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Find the sum of the series. ∑n=0∞5n6nn!\sum _ { n = 0 } ^ { \infty } \frac { 5 ^ { n } } { 6 ^ { n } n ! }∑n=0∞​6nn!5n​

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Determine whether the sequence converges or diverges. If it converges, find the limit. an=2e4n/(n+2)a _ { n } = 2 e ^ { 4 n / ( n + 2 ) }an​=2e4n/(n+2)

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Find the radius of convergence and the interval of convergence of the power series. ∑n=1∞4nxnn\sum _ { n = 1 } ^ { \infty } \frac { 4 ^ { n } x ^ { n } } { n }∑n=1∞​n4nxn​

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Find the radius of convergence and the interval of convergence of the power series. ∑n=1∞(−1) n−1(x−6) nn⋅5n\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } ( x - 6 ) ^ { n } } { n \cdot 5 ^ { n } }∑n=1∞​n⋅5n(−1) n−1(x−6) n​

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Determine whether the sequence defined by an=sin⁡2n9na _ { n } = \frac { \sin 2 n } { 9 n }an​=9nsin2n​ converges or diverges. If it converges, find its limit.

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Determine whether the series converges or diverges. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { n + 4 }$

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Determine whether the series is convergent or divergent. If it is convergent, write its sum. Otherwise write divergent. $\sum _ { n = 1 } ^ { \infty } 5 \left( \frac { 3 } { 4 } \right) ^ { n - 1 }$

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Correct Answer
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Correct Answer
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Determine whether the given series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { n } { 8 ^ { n } }$

Correct Answer
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Use series to approximate the definite integral to within the indicated accuracy. $\int _ { 0 } ^ { 05 } x ^ { 2 } e ^ { - x ^ { 2 } } d x \quad \mid \text { error } \mid < 0.001$

Correct Answer
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Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 2 } + 9 } { n ^ { 7 } + n }$

Correct Answer
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Use the Integral Test to determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { 2 } { 7 n + 4 }$

Correct Answer
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Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { 7 ^ { n } } { n ! n }$

Correct Answer
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Find the radius of convergence and the interval of convergence of the power series. $\sum _ { n = 2 } ^ { \infty } \frac { x ^ { n } } { n ( \ln n ) ^ { 8 } }$

Correct Answer
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Find the radius of convergence and the interval of convergence of the power series. $\sum _ { n = 0 } ^ { \infty } \frac { x ^ { n } } { n + 2 }$

Correct Answer
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Determine whether the series converges or diverges. $\sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n } \ln n } { 9 ^ { n } }$

Correct Answer
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Use the Integral Test to determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 8 n + 2 }$

Correct Answer
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Determine whether the series is absolutely convergent, conditionally convergent, or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { n \sqrt [ 6 ] { n } }$

Correct Answer
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absolutely...
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Determine whether the series converges or diverges. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { \sqrt [ n ] { 6 n } }$

Correct Answer
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Find the sum of the series. $\sum _ { n = 0 } ^ { \infty } \frac { 5 ^ { n } } { 6 ^ { n } n ! }$

Correct Answer
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Determine whether the sequence converges or diverges. If it converges, find the limit. $a _ { n } = 2 e ^ { 4 n / ( n + 2 ) }$

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Find the radius of convergence and the interval of convergence of the power series. $\sum _ { n = 1 } ^ { \infty } \frac { 4 ^ { n } x ^ { n } } { n }$

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Find the radius of convergence and the interval of convergence of the power series. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } ( x - 6 ) ^ { n } } { n \cdot 5 ^ { n } }$

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Determine whether the sequence defined by $a _ { n } = \frac { \sin 2 n } { 9 n }$ converges or diverges. If it converges, find its limit.

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