Question 1

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 2

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 3

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 4

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 5

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

Accepted Answer

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

Question 6

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 7

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 8

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 9

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

Accepted Answer

A)  $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 7 } }$ 
B)  $\sum _ { n = 1 } ^ { \infty } n ^ { - 06 }$ 
C)  $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 08 } }$ 
D)  $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 3 / 4 } }$ 
A)  $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 7 } }$ 
B)  $\sum _ { n = 1 } ^ { \infty } n ^ { - 06 }$ 
C)  $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 08 } }$ 
D)  $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 3 / 4 } }$

Question 10

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

Accepted Answer

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

Question 11

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 12

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

Accepted Answer

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

Question 13

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 14

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 15

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 16

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

Accepted Answer

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

Question 17

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

Accepted Answer

A)  Diverges 
B)  1 
C)  $\frac { 2 } { 9 }$ 
D)  0 
A)  Diverges 
B)  1 
C)  $\frac { 2 } { 9 }$ 
D)  0

Question 18

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

Accepted Answer

A)  $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 03 } }$ 
B)  $\sum _ { n = 1 } ^ { \infty } n ^ { - 4 x }$ 
C)  $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 3 e } }$ 
D)  $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 4 } }$ 
A)  $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 03 } }$ 
B)  $\sum _ { n = 1 } ^ { \infty } n ^ { - 4 x }$ 
C)  $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 3 e } }$ 
D)  $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 4 } }$

Question 19

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 20

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...

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Final Exam

Filters

Exam 11: Infinite Sequences and Series

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​

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​

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

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​

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

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​

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​

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​

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

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​

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

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​

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​

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)

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

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​

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.

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

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​

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​

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Final Exam

Filters

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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