Question 1

Determine whether the sequence defined by $a _ { n } = 5 + 8 ( - 1 ) ^ { n }$ converges or diverges. If it converges, find its limit.

Accepted Answer

The answer of Determine whether the sequence defined by \(a...

Question 2

$$\text { For which positive integers } k \text { is the series } \sum _ { n = 1 } ^ { \infty } \frac { ( n ! ) ^ { 5 } } { ( k n ) ! } \text { convergent? }$$

Accepted Answer

The answer of \[\text { For which positive integers }...

Question 3

Determine whether the series is convergent or divergent.
$$\sum _ { n = 1 } ^ { \infty } \left( \frac { \ln \left( n ^ { 6 } \right) } { n } \right) ^ { n }$$

Accepted Answer

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

Question 4

Approximate the sum to the indicated accuracy.
$\sum _ { n = 1 } ^ { \infty } \frac { 4 ( - 1 ) ^ { n - 1 } } { n ^ { 7 } }$ (five decimal places)

Accepted Answer

The answer of Approximate the sum to the indicated accuracy.
\(\sum...

Question 5

Use the power series for $f ( x ) = \sqrt [ 3 ] { 5 + x }$ to estimate $\sqrt [ 3 ] { 5.07 }$ correct to four decimal places.

Accepted Answer

A)  $1.7179$ 
B)  $1.7189$ 
C)  $1.7195$ 
D)  $1.7156$ 
E)  $1.7200$ 
A)  $1.7179$ 
B)  $1.7189$ 
C)  $1.7195$ 
D)  $1.7156$ 
E)  $1.7200$

Question 6

Find the values of $p$ for which the series is convergent. Select the correct answer.
$$\sum _ { n - 2 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { \left( \ln \left( n ^ { 6 } \right) \right) ^ { p } }$$

Accepted Answer

A) $p > 1$ B) $p > 0$ C) $p < 0$ D) $p < 1$ A) $p > 1$ B) $p > 0$ C) $p < 0$ D) $p < 1$

Question 7

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 8

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

Accepted Answer

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

Question 9

How many terms of the series $\sum _ { m - 2 } ^ { \infty } \frac { 12 } { 6 m ( \ln m ) ^ { 2 } }$ would you need to add to find its sum to within $0.02 ?$

Accepted Answer

The answer of How many terms of the series \(\sum...

Question 10

Use the sum of the first 10 terms to approximate the sum of the series. Estimate the error.
$$\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 1 + 4 ^ { n } }$$

Accepted Answer

The answer of Use the sum of the first 10...

Question 11

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.
\[\sum...

Question 12

Given the series $\sum _ { m = 1 } ^ { \infty } \frac { 3 m } { 4 ^ { m } ( 3 m + 5 ) }$ estimate the error in using the partial sum $s _ { 8 }$ by comparison with the series $\sum _ { m = 9 } ^ { \infty } \frac { 1 } { 4 ^ { m } }$.

Accepted Answer

A)  $R _ { 8 } \leq 2.6130051$ 
B)  $R _ { 8 } \geq 0.0000052$ 
C)  $R _ { 8 } \leq 0.0000051$ 
D)  $R _ { 8 } \geq 0.0000051$ 
E)  $R _ { 8 } \leq 0.000005$ 
A)  $R _ { 8 } \leq 2.6130051$ 
B)  $R _ { 8 } \geq 0.0000052$ 
C)  $R _ { 8 } \leq 0.0000051$ 
D)  $R _ { 8 } \geq 0.0000051$ 
E)  $R _ { 8 } \leq 0.000005$

Question 13

For which positive integers $k$ is the series $\sum _ { n = 1 } ^ { \infty } \frac { ( n ! ) ^ { 5 } } { ( k n ) ! }$ convergent? Select the correct answer.

Accepted Answer

A)  $k \geq 5$ 
B)  $k \leq 0$ 
C)  $k \geq 0$ 
D)  $k \geq 1$ 
E)  $k \leq - 5$ 
A)  $k \geq 5$ 
B)  $k \leq 0$ 
C)  $k \geq 0$ 
D)  $k \geq 1$ 
E)  $k \leq - 5$

Question 14

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

Accepted Answer

The answer of Determine whether the series converges or diverges.
\[\sum...

Question 15

Find a power series representation for the function and determine the radius of convergence.
$$f ( x ) = \arctan \left( \frac { x } { 3 } \right)$$

Accepted Answer

The answer of Find a power series representation for the...

Question 16

Find the radius of convergence and the interval of convergence of the power series.
$$\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( x - 8 ) ^ { n } } { \sqrt { n } }$$

Accepted Answer

A)  $R = 1 , l = [ 7,9 )$ 
B)  $R = 1 , I = ( 7,9 ]$ 
C)  $R = 8 , l = [ - 8,8 )$ 
D)  $R = 8 , I = ( - 8,8 )$ 
A)  $R = 1 , l = [ 7,9 )$ 
B)  $R = 1 , I = ( 7,9 ]$ 
C)  $R = 8 , l = [ - 8,8 )$ 
D)  $R = 8 , I = ( - 8,8 )$

Question 17

Find the radius of convergence and the interval of convergence of the power series.
$$\sum _ { n = 0 } ^ { \infty } \frac { ( 7 x ) ^ { n } } { n ! }$$

Accepted Answer

A)  $R = 7 , L = ( - 7,7 )$ 
B)  $R = 0 , I = \{ 0 \}$ 
C)  $R = 7 , I = [ - 7,7 ]$ 
D)  $R = \infty , I = ( - \infty , \infty )$ 
A)  $R = 7 , L = ( - 7,7 )$ 
B)  $R = 0 , I = \{ 0 \}$ 
C)  $R = 7 , I = [ - 7,7 ]$ 
D)  $R = \infty , I = ( - \infty , \infty )$

Question 18

Use the binomial series to expand the function as a power series. Find the radius of convergence
$$\sqrt [ 4 ] { 1 + x ^ { 6 } }$$

Accepted Answer

The answer of Use the binomial series to expand the...

Question 19

Find the radius of convergence and the interval of convergence of the power series. ]
Select the correct answer.
$$\sum _ { n = 1 } ^ { \infty } \frac { 3 \cdot 6 \cdot 9 \cdot \cdots \cdot 3 n } { 4 \cdot 7 \cdot 10 \cdots \cdot ( 3 n + 1 ) } x ^ { 2 n + 1 }$$

Accepted Answer

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

Question 20

Find the radius of convergence and the interval of convergence of the power series.
$$\sum _ { n = 0 } ^ { \infty } \left( \frac { n x } { 6 } \right) ^ { n }$$

Accepted Answer

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

Determine whether the sequence defined by $a _ { n } = 5 + 8 ( - 1 ) ^ { n }$ converges or diverges. If it converges, find its limit.

$\text { For which positive integers } k \text { is the series } \sum _ { n = 1 } ^ { \infty } \frac { ( n ! ) ^ { 5 } } { ( k n ) ! } \text { convergent? }$

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \left( \frac { \ln \left( n ^ { 6 } \right) } { n } \right) ^ { n }$

Approximate the sum to the indicated accuracy. $\sum _ { n = 1 } ^ { \infty } \frac { 4 ( - 1 ) ^ { n - 1 } } { n ^ { 7 } }$ (five decimal places)

Use the power series for $f ( x ) = \sqrt [ 3 ] { 5 + x }$ to estimate $\sqrt [ 3 ] { 5.07 }$ correct to four decimal places.

Find the values of $p$ for which the series is convergent. Select the correct answer. $\sum _ { n - 2 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { \left( \ln \left( n ^ { 6 } \right) \right) ^ { p } }$

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

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

How many terms of the series $\sum _ { m - 2 } ^ { \infty } \frac { 12 } { 6 m ( \ln m ) ^ { 2 } }$ would you need to add to find its sum to within $0.02 ?$

Use the sum of the first 10 terms to approximate the sum of the series. Estimate the error. $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 1 + 4 ^ { n } }$

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

Given the series $\sum _ { m = 1 } ^ { \infty } \frac { 3 m } { 4 ^ { m } ( 3 m + 5 ) }$ estimate the error in using the partial sum $s _ { 8 }$ by comparison with the series $\sum _ { m = 9 } ^ { \infty } \frac { 1 } { 4 ^ { m } }$ .

For which positive integers $k$ is the series $\sum _ { n = 1 } ^ { \infty } \frac { ( n ! ) ^ { 5 } } { ( k n ) ! }$ convergent? Select the correct answer.

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

Find a power series representation for the function and determine the radius of convergence. $f ( x ) = \arctan \left( \frac { x } { 3 } \right)$

Find the radius of convergence and the interval of convergence of the power series. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( x - 8 ) ^ { n } } { \sqrt { n } }$

Find the radius of convergence and the interval of convergence of the power series. $\sum _ { n = 0 } ^ { \infty } \frac { ( 7 x ) ^ { n } } { n ! }$

Use the binomial series to expand the function as a power series. Find the radius of convergence $\sqrt [ 4 ] { 1 + x ^ { 6 } }$

Find the radius of convergence and the interval of convergence of the power series. ] Select the correct answer. $\sum _ { n = 1 } ^ { \infty } \frac { 3 \cdot 6 \cdot 9 \cdot \cdots \cdot 3 n } { 4 \cdot 7 \cdot 10 \cdots \cdot ( 3 n + 1 ) } x ^ { 2 n + 1 }$

Find the radius of convergence and the interval of convergence of the power series. $\sum _ { n = 0 } ^ { \infty } \left( \frac { n x } { 6 } \right) ^ { n }$

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Exam 11: Infinite Sequences and Series

Determine whether the sequence defined by an=5+8(−1)na _ { n } = 5 + 8 ( - 1 ) ^ { n }an​=5+8(−1)n converges or diverges. If it converges, find its limit.

Determine whether the series is convergent or divergent. ∑n=1∞(ln⁡(n6)n)n\sum _ { n = 1 } ^ { \infty } \left( \frac { \ln \left( n ^ { 6 } \right) } { n } \right) ^ { n }∑n=1∞​(nln(n6)​)n

Approximate the sum to the indicated accuracy. ∑n=1∞4(−1)n−1n7\sum _ { n = 1 } ^ { \infty } \frac { 4 ( - 1 ) ^ { n - 1 } } { n ^ { 7 } }∑n=1∞​n74(−1)n−1​ (five decimal places)

Use the power series for f(x)=5+x3f ( x ) = \sqrt [ 3 ] { 5 + x }f(x)=35+x​ to estimate 5.073\sqrt [ 3 ] { 5.07 }35.07​ correct to four decimal places.

Find the values of ppp for which the series is convergent. Select the correct answer. ∑n−2∞(−1)n(ln⁡(n6))p\sum _ { n - 2 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { \left( \ln \left( n ^ { 6 } \right) \right) ^ { p } }∑n−2∞​(ln(n6))p(−1)n​

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​

Determine whether the series is convergent or divergent. ∑n=1∞9nn!n\sum _ { n = 1 } ^ { \infty } \frac { 9 ^ { n } } { n ! n }∑n=1∞​n!n9n​

How many terms of the series ∑m−2∞126m(ln⁡m)2\sum _ { m - 2 } ^ { \infty } \frac { 12 } { 6 m ( \ln m ) ^ { 2 } }∑m−2∞​6m(lnm)212​ would you need to add to find its sum to within 0.02?0.02 ?0.02?

Use the sum of the first 10 terms to approximate the sum of the series. Estimate the error. ∑n=1∞11+4n\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 1 + 4 ^ { n } }∑n=1∞​1+4n1​

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​

For which positive integers kkk is the series ∑n=1∞(n!)5(kn)!\sum _ { n = 1 } ^ { \infty } \frac { ( n ! ) ^ { 5 } } { ( k n ) ! }∑n=1∞​(kn)!(n!)5​ convergent? Select the correct answer.

Determine whether the series converges or diverges. ∑n=1∞(−1)nnsin⁡(π9n)\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } n \sin \left( \frac { \pi } { 9 n } \right)∑n=1∞​(−1)nnsin(9nπ​)

Find a power series representation for the function and determine the radius of convergence. f(x)=arctan⁡(x3)f ( x ) = \arctan \left( \frac { x } { 3 } \right)f(x)=arctan(3x​)

Find the radius of convergence and the interval of convergence of the power series. ∑n=1∞(−1)n(x−8)nn\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( x - 8 ) ^ { n } } { \sqrt { n } }∑n=1∞​n​(−1)n(x−8)n​

Find the radius of convergence and the interval of convergence of the power series. ∑n=0∞(7x)nn!\sum _ { n = 0 } ^ { \infty } \frac { ( 7 x ) ^ { n } } { n ! }∑n=0∞​n!(7x)n​

Use the binomial series to expand the function as a power series. Find the radius of convergence 1+x64\sqrt [ 4 ] { 1 + x ^ { 6 } }41+x6​

Find the radius of convergence and the interval of convergence of the power series. ∑n=0∞(nx6)n\sum _ { n = 0 } ^ { \infty } \left( \frac { n x } { 6 } \right) ^ { n }∑n=0∞​(6nx​)n

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

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

Filters

Determine whether the sequence defined by $a _ { n } = 5 + 8 ( - 1 ) ^ { n }$ converges or diverges. If it converges, find its limit.

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \left( \frac { \ln \left( n ^ { 6 } \right) } { n } \right) ^ { n }$

Approximate the sum to the indicated accuracy. $\sum _ { n = 1 } ^ { \infty } \frac { 4 ( - 1 ) ^ { n - 1 } } { n ^ { 7 } }$ (five decimal places)

Use the power series for $f ( x ) = \sqrt [ 3 ] { 5 + x }$ to estimate $\sqrt [ 3 ] { 5.07 }$ correct to four decimal places.

Find the values of $p$ for which the series is convergent. Select the correct answer. $\sum _ { n - 2 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { \left( \ln \left( n ^ { 6 } \right) \right) ^ { p } }$

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

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

How many terms of the series $\sum _ { m - 2 } ^ { \infty } \frac { 12 } { 6 m ( \ln m ) ^ { 2 } }$ would you need to add to find its sum to within $0.02 ?$

Use the sum of the first 10 terms to approximate the sum of the series. Estimate the error. $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 1 + 4 ^ { n } }$

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

For which positive integers $k$ is the series $\sum _ { n = 1 } ^ { \infty } \frac { ( n ! ) ^ { 5 } } { ( k n ) ! }$ convergent? Select the correct answer.

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

Find a power series representation for the function and determine the radius of convergence. $f ( x ) = \arctan \left( \frac { x } { 3 } \right)$

Find the radius of convergence and the interval of convergence of the power series. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( x - 8 ) ^ { n } } { \sqrt { n } }$

Find the radius of convergence and the interval of convergence of the power series. $\sum _ { n = 0 } ^ { \infty } \frac { ( 7 x ) ^ { n } } { n ! }$

Use the binomial series to expand the function as a power series. Find the radius of convergence $\sqrt [ 4 ] { 1 + x ^ { 6 } }$

Find the radius of convergence and the interval of convergence of the power series. $\sum _ { n = 0 } ^ { \infty } \left( \frac { n x } { 6 } \right) ^ { n }$