Question 1

Find the Maclaurin series for $f ( x )$ using the definition of the Maclaurin series.
$$f ( x ) = x \cos ( 4 x )$$

Accepted Answer

The answer of Find the Maclaurin series for \(f (...

Question 2

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

Accepted Answer

The answer of \[\text { Use the power series for...

Question 3

Find the Maclaurin series for $f ( x )$ using the definition of the Maclaurin series.
$$f ( x ) = x \cos ( 4 x )$$

Accepted Answer

A)  $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } 4 ^ { n } x ^ { 2 n + 1 } } { ( 2 n ) ! }$ 
B) 
$\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } 4 ^ { 2 n } x ^ { 2 n + 1 } } { n ! }$ 
C) 
$\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } 4 ^ { 2 n } x ^ { 2 n } } { ( 2 n ) ! }$ 
D) 
$\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } 4 ^ { 2 n } x ^ { 2 n + 1 } } { ( 2 n ) ! }$ 
E) 
$\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } 4 ^ { 2 n } x ^ { 2 n + 1 } } { ( 2 n ) ! }$ 
A)  $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } 4 ^ { n } x ^ { 2 n + 1 } } { ( 2 n ) ! }$ 
B) 
$\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } 4 ^ { 2 n } x ^ { 2 n + 1 } } { n ! }$ 
C) 
$\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } 4 ^ { 2 n } x ^ { 2 n } } { ( 2 n ) ! }$ 
D) 
$\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } 4 ^ { 2 n } x ^ { 2 n + 1 } } { ( 2 n ) ! }$ 
E) 
$\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } 4 ^ { 2 n } x ^ { 2 n + 1 } } { ( 2 n ) ! }$

Question 4

Find the radius of convergence and the interval of convergence of the power series. $\backslash$ 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)  $R = 0 , l = \{ 0 \}$ 
D)  $R = 1 , I = [ - 1,1 ]$ 
A)  $R = \infty , I = ( - \infty , \infty )$ 
B)  $R = 1 , I = ( - 1,1 )$ 
C)  $R = 0 , l = \{ 0 \}$ 
D)  $R = 1 , I = [ - 1,1 ]$

Question 5

If $\$ 600$ is invested at $4 \%$ interest, compounded annually, then after $n$ years the investment is worth $a _ { n } = 600 ( 1.04 ) ^ { n }$ dollars. Find the size of investment after 7 years.

Accepted Answer

The answer of If $\$ 600$ is invested at \(4...

Question 6

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

The answer of Given the series \(\sum _ { m...

Question 7

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

Accepted Answer

conditiona

Question 8

Find the Maclaurin series for $f ( x )$ using the definition of a Maclaurin serires.
$$f ( x ) = ( 3 + x ) ^ { - 3 }$$

Accepted Answer

The answer of Find the Maclaurin series for \(f (...

Question 9

Determine whether the series is convergent or divergent by expressing $S _ { n }$ as a telescoping sum. If it is convergent, find its sum.
$$\sum _ { n = 2 } ^ { \infty } \frac { 5 } { n \left( n ^ { 2 } - 1 \right) }$$

Accepted Answer

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

Question 10

Use the binomial series to expand the function as a power series. Find the radius of convergence.
$$\frac { 1 } { ( 4 + x ) ^ { 5 } }$$

Accepted Answer

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

Question 11

Use series to approximate the definite integral to within the indicated accuracy.
$$\int _ { 0 } ^ { 0.5 } x ^ { 2 } e ^ { - x ^ { 2 } } d x \quad | e r r o r | < 0.001$$

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 12

Test the series for convergence or divergence.
$$\sum _ { k = 5 } ^ { \infty } \frac { 5 } { k ( \ln k ) ^ { 7 } }$$

Accepted Answer

The answer of Test the series for convergence or divergence.
\[\sum...

Question 13

Determine whether the given series converges or diverges. If it converges, find its sum.
$$\sum _ { n = 0 } ^ { \infty } \frac { 9 n ^ { 2 } + 3 } { 2 n ^ { 2 } + 5 }$$

Accepted Answer

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

Question 14

Determine whether the geometric series converges or diverges. If it converges, find its sum.
$$\sum _ { n = 0 } ^ { \infty } 3 ^ { n } 4 ^ { - n + 1 }$$

Accepted Answer

A)  12 
B)  Diverges 
C)  3 
D)  16 
A)  12 
B)  Diverges 
C)  3 
D)  16

Question 15

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

Accepted Answer

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

Question 16

Find an approximation of the sum of the series accurate to two decimal places.
$$\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { n ^ { 3 } }$$

Accepted Answer

The answer of Find an approximation of the sum of...

Question 17

$$\text { The terms of a series are defined recursively by the equations } a _ { 1 } = 6 , a _ { n + 1 } = \frac { 7 n + 1 } { 6 n + 3 } a _ { n } \text {. }$$ $$\text { Determine whether } \sum a _ { n } \text { converges or diverges. }$$

Accepted Answer

The answer of \[\text { The terms of a series...

Question 18

$$\text { Express the number } 0 . \overline { 81 } \text { as a ratio of integers. }$$

Accepted Answer

The answer of \[\text { Express the number } 0...

Question 19

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

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 20

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...

Find the Maclaurin series for $f ( x )$ using the definition of the Maclaurin series. $f ( x ) = x \cos ( 4 x )$

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

Find the Maclaurin series for $f ( x )$ using the definition of the Maclaurin series. $f ( x ) = x \cos ( 4 x )$

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

If $\$ 600$ is invested at $4 \%$ interest, compounded annually, then after $n$ years the investment is worth $a _ { n } = 600 ( 1.04 ) ^ { n }$ dollars. Find the size of investment after 7 years.

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 $\sum _ { m - 9 } ^ { \infty } \frac { 1 } { 4 ^ { m } }$ .

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

Find the Maclaurin series for $f ( x )$ using the definition of a Maclaurin serires. $f ( x ) = ( 3 + x ) ^ { - 3 }$

Determine whether the series is convergent or divergent by expressing $S _ { n }$ as a telescoping sum. If it is convergent, find its sum. $\sum _ { n = 2 } ^ { \infty } \frac { 5 } { n \left( n ^ { 2 } - 1 \right) }$

Use the binomial series to expand the function as a power series. Find the radius of convergence. $\frac { 1 } { ( 4 + x ) ^ { 5 } }$

Use series to approximate the definite integral to within the indicated accuracy. $\int _ { 0 } ^ { 0.5 } x ^ { 2 } e ^ { - x ^ { 2 } } d x \quad | e r r o r | < 0.001$

Test the series for convergence or divergence. $\sum _ { k = 5 } ^ { \infty } \frac { 5 } { k ( \ln k ) ^ { 7 } }$

Determine whether the given series converges or diverges. If it converges, find its sum. $\sum _ { n = 0 } ^ { \infty } \frac { 9 n ^ { 2 } + 3 } { 2 n ^ { 2 } + 5 }$

Determine whether the geometric series converges or diverges. If it converges, find its sum. $\sum _ { n = 0 } ^ { \infty } 3 ^ { n } 4 ^ { - n + 1 }$

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

Find an approximation of the sum of the series accurate to two decimal places. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { n ^ { 3 } }$

$\text { The terms of a series are defined recursively by the equations } a _ { 1 } = 6 , a _ { n + 1 } = \frac { 7 n + 1 } { 6 n + 3 } a _ { n } \text {. }$ $\text { Determine whether } \sum a _ { n } \text { converges or diverges. }$

$\text { Express the number } 0 . \overline { 81 } \text { as a ratio of integers. }$

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

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

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

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Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

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Exam 11: Infinite Sequences and Series

Find the Maclaurin series for f(x)f ( x )f(x) using the definition of the Maclaurin series. f(x)=xcos⁡(4x)f ( x ) = x \cos ( 4 x )f(x)=xcos(4x)

Find the Maclaurin series for f(x)f ( x )f(x) using the definition of the Maclaurin series. f(x)=xcos⁡(4x)f ( x ) = x \cos ( 4 x )f(x)=xcos(4x)

If $600\$ 600$600 is invested at 4%4 \%4% interest, compounded annually, then after nnn years the investment is worth an=600(1.04)na _ { n } = 600 ( 1.04 ) ^ { n }an​=600(1.04)n dollars. Find the size of investment after 7 years.

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. ∑n=1∞(−1)n−1n6\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } } { \sqrt [ 6 ] { n } }∑n=1∞​6n​(−1)n−1​

Find the Maclaurin series for f(x)f ( x )f(x) using the definition of a Maclaurin serires. f(x)=(3+x)−3f ( x ) = ( 3 + x ) ^ { - 3 }f(x)=(3+x)−3

Determine whether the series is convergent or divergent by expressing SnS _ { n }Sn​ as a telescoping sum. If it is convergent, find its sum. ∑n=2∞5n(n2−1)\sum _ { n = 2 } ^ { \infty } \frac { 5 } { n \left( n ^ { 2 } - 1 \right) }∑n=2∞​n(n2−1)5​

Use the binomial series to expand the function as a power series. Find the radius of convergence. 1(4+x)5\frac { 1 } { ( 4 + x ) ^ { 5 } }(4+x)51​

Use series to approximate the definite integral to within the indicated accuracy. ∫00.5x2e−x2dx∣error∣<0.001\int _ { 0 } ^ { 0.5 } x ^ { 2 } e ^ { - x ^ { 2 } } d x \quad | e r r o r | < 0.001∫00.5​x2e−x2dx∣error∣<0.001

Test the series for convergence or divergence. ∑k=5∞5k(ln⁡k)7\sum _ { k = 5 } ^ { \infty } \frac { 5 } { k ( \ln k ) ^ { 7 } }∑k=5∞​k(lnk)75​

Determine whether the given series converges or diverges. If it converges, find its sum. ∑n=0∞9n2+32n2+5\sum _ { n = 0 } ^ { \infty } \frac { 9 n ^ { 2 } + 3 } { 2 n ^ { 2 } + 5 }∑n=0∞​2n2+59n2+3​

Determine whether the geometric series converges or diverges. If it converges, find its sum. ∑n=0∞3n4−n+1\sum _ { n = 0 } ^ { \infty } 3 ^ { n } 4 ^ { - n + 1 }∑n=0∞​3n4−n+1

Determine whether the series converges or diverges. ∑n=1∞(−1)nn2n\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } n } { 2 ^ { n } }∑n=1∞​2n(−1)nn​

Find an approximation of the sum of the series accurate to two decimal places. ∑n=1∞(−1)nn3\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { n ^ { 3 } }∑n=1∞​n3(−1)n​

Express the number 0.81‾ as a ratio of integers. \text { Express the number } 0 . \overline { 81 } \text { as a ratio of integers. } Express the number 0.81 as a ratio of integers.

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. Select the correct answer.

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)

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

Find the Maclaurin series for $f ( x )$ using the definition of the Maclaurin series. $f ( x ) = x \cos ( 4 x )$

Find the Maclaurin series for $f ( x )$ using the definition of the Maclaurin series. $f ( x ) = x \cos ( 4 x )$

If $\$ 600$ is invested at $4 \%$ interest, compounded annually, then after $n$ years the investment is worth $a _ { n } = 600 ( 1.04 ) ^ { n }$ dollars. Find the size of investment after 7 years.

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

Find the Maclaurin series for $f ( x )$ using the definition of a Maclaurin serires. $f ( x ) = ( 3 + x ) ^ { - 3 }$

Determine whether the series is convergent or divergent by expressing $S _ { n }$ as a telescoping sum. If it is convergent, find its sum. $\sum _ { n = 2 } ^ { \infty } \frac { 5 } { n \left( n ^ { 2 } - 1 \right) }$

Use the binomial series to expand the function as a power series. Find the radius of convergence. $\frac { 1 } { ( 4 + x ) ^ { 5 } }$

Use series to approximate the definite integral to within the indicated accuracy. $\int _ { 0 } ^ { 0.5 } x ^ { 2 } e ^ { - x ^ { 2 } } d x \quad | e r r o r | < 0.001$

Test the series for convergence or divergence. $\sum _ { k = 5 } ^ { \infty } \frac { 5 } { k ( \ln k ) ^ { 7 } }$

Determine whether the given series converges or diverges. If it converges, find its sum. $\sum _ { n = 0 } ^ { \infty } \frac { 9 n ^ { 2 } + 3 } { 2 n ^ { 2 } + 5 }$

Determine whether the geometric series converges or diverges. If it converges, find its sum. $\sum _ { n = 0 } ^ { \infty } 3 ^ { n } 4 ^ { - n + 1 }$

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

Find an approximation of the sum of the series accurate to two decimal places. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { n ^ { 3 } }$

$\text { Express the number } 0 . \overline { 81 } \text { as a ratio of integers. }$

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

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