Question 1

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

Accepted Answer

A)  1 
B)  $\frac { 5 } { 8 }$ 
C)  0 
D)  Diverges 
A)  1 
B)  $\frac { 5 } { 8 }$ 
C)  0 
D)  Diverges

Question 2

Find an expression for the $n ^ { \text {th } }$ term of the sequence. (Assume that the pattern continues.)
$$\left\{ \frac { 2 } { 25 } , \frac { 4 } { 36 } , \frac { 6 } { 49 } , \frac { 8 } { 64 } , \frac { 10 } { 81 } , \cdots \right\}$$

Accepted Answer

The answer of Find an expression for the \(n ^...

Question 3

Determine which series is convergent.

Accepted Answer

A) 
$- \frac { 2 } { 7 } + \frac { 3 } { 8 } - \frac { 4 } { 9 } + \frac { 5 } { 10 } - \frac { 6 } { 11 } - \ldots$ 
B) 
$\frac { 4 } { 3 } - \frac { 4 } { 4 } + \frac { 4 } { 5 } - \frac { 4 } { 6 } + \frac { 4 } { 7 } - \ldots$ 
A) 
$- \frac { 2 } { 7 } + \frac { 3 } { 8 } - \frac { 4 } { 9 } + \frac { 5 } { 10 } - \frac { 6 } { 11 } - \ldots$ 
B) 
$\frac { 4 } { 3 } - \frac { 4 } { 4 } + \frac { 4 } { 5 } - \frac { 4 } { 6 } + \frac { 4 } { 7 } - \ldots$

Question 4

Determine whether the given series is convergent or divergent. $$\sum _ { n = 2 } ^ { \infty } \frac { 1 } { n ( \ln n ) ^ { 2 } }$$

Accepted Answer

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

Question 5

Use multiplication or division of power series to find the first three nonzero terms in the Maclaurin series for the function. $$f ( x ) = 5 e ^ { - x ^ { 2 } } \cos 4 x$$

Accepted Answer

A)  $5 \left( 1 - 17 x ^ { 2 } + \frac { 115 } { 6 } x ^ { 4 } \right)$ 
B)  $5 \left( 1 - 9 x ^ { 2 } + \frac { 115 } { 6 } x ^ { 4 } \right)$ 
C)  $5 \left( 1 - 9 x + \frac { 115 } { 6 } x ^ { 4 } \right)$ 
D)  $5 \left( 1 - 9 x ^ { 2 } + \frac { 97 } { 6 } x ^ { 4 } \right)$ 
E)  $5 \left( 1 - 17 x ^ { 2 } + \frac { 67 } { 6 } x ^ { 4 } \right)$ 
A)  $5 \left( 1 - 17 x ^ { 2 } + \frac { 115 } { 6 } x ^ { 4 } \right)$ 
B)  $5 \left( 1 - 9 x ^ { 2 } + \frac { 115 } { 6 } x ^ { 4 } \right)$ 
C)  $5 \left( 1 - 9 x + \frac { 115 } { 6 } x ^ { 4 } \right)$ 
D)  $5 \left( 1 - 9 x ^ { 2 } + \frac { 97 } { 6 } x ^ { 4 } \right)$ 
E)  $5 \left( 1 - 17 x ^ { 2 } + \frac { 67 } { 6 } x ^ { 4 } \right)$

Question 6

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

Question 7

Which of the given series are absolutely convergent?

Accepted Answer

A) 
$\sum _ { n = 1 } ^ { \infty } \frac { \sin 2 n } { n }$ 
B) 
$\sum _ { n = 1 } ^ { \infty } \frac { \cos \frac { \pi n } { 7 } } { n \sqrt { n } }$ 
A) 
$\sum _ { n = 1 } ^ { \infty } \frac { \sin 2 n } { n }$ 
B) 
$\sum _ { n = 1 } ^ { \infty } \frac { \cos \frac { \pi n } { 7 } } { n \sqrt { n } }$

Question 8

Determine whether the sequence convergent or divergent. $$\sum _ { n = 1 } ^ { \infty } \frac { 3 } { n ^ { 2 } + 3 }$$

Accepted Answer

A)  converges 
B)  diverges 
A)  converges 
B)  diverges

Question 9

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 10

Find the value of the limit for the sequence given. $$\left\{ \frac { 1 \cdot 9 \cdot 17 \cdots ( 7 n + 1 ) } { ( 7 n ) ^ { 2 } } \right\}$$

Accepted Answer

A)  0 
B)  -1 
C)  \pi 
D)  3 
E)  1 
A)  0 
B)  -1 
C)  \pi 
D)  3 
E)  1

Question 11

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 12

Test the series for convergence or divergence. $$\sum _ { m - 1 } ^ { \infty } \frac { ( - 6 ) ^ { m + 1 } } { 4 ^ { 8 m } }$$

Accepted Answer

A)  The series is convergent. 
B)  The series is divergent. 
A)  The series is convergent. 
B)  The series is divergent.

Question 13

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

A)  $\$ 430.21$ 
B)  $\$ 1,860.81$ 
C)  $\$ 1,230.81$ 
D)  $\$ 789.56$ 
E)  $\$ 1,321.06$ 
A)  $\$ 430.21$ 
B)  $\$ 1,860.81$ 
C)  $\$ 1,230.81$ 
D)  $\$ 789.56$ 
E)  $\$ 1,321.06$

Question 14

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 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....

Question 16

Find the interval of convergence of the series. $$\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { n } } { n + 3 }$$

Accepted Answer

A)  $[ - 1,1 ]$ 
B)  $( - 1,1 )$ 
C)  $( - 1,1 ]$ 
D)  diverges everywhere 
E)  $[ - 1,1 )$ 
A)  $[ - 1,1 ]$ 
B)  $( - 1,1 )$ 
C)  $( - 1,1 ]$ 
D)  diverges everywhere 
E)  $[ - 1,1 )$

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

Question 18

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

Question 19

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

Accepted Answer

A)  $6.97036$ 
B)  $4.97036$ 
C)  $7.97036$ 
D)  $3.97036$ 
E)  $5.97036$ 
A)  $6.97036$ 
B)  $4.97036$ 
C)  $7.97036$ 
D)  $3.97036$ 
E)  $5.97036$

Question 20

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)  $1.7200$

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

Find an expression for the $n ^ { \text {th } }$ term of the sequence. (Assume that the pattern continues.) $\left\{ \frac { 2 } { 25 } , \frac { 4 } { 36 } , \frac { 6 } { 49 } , \frac { 8 } { 64 } , \frac { 10 } { 81 } , \cdots \right\}$

Determine which series is convergent.

Determine whether the given series is convergent or divergent. $\sum _ { n = 2 } ^ { \infty } \frac { 1 } { n ( \ln n ) ^ { 2 } }$

Use multiplication or division of power series to find the first three nonzero terms in the Maclaurin series for the function. $f ( x ) = 5 e ^ { - x ^ { 2 } } \cos 4 x$

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

Which of the given series are absolutely convergent?

Determine whether the sequence convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { 3 } { n ^ { 2 } + 3 }$

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

Find the value of the limit for the sequence given. $\left\{ \frac { 1 \cdot 9 \cdot 17 \cdots ( 7 n + 1 ) } { ( 7 n ) ^ { 2 } } \right\}$

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

Test the series for convergence or divergence. $\sum _ { m - 1 } ^ { \infty } \frac { ( - 6 ) ^ { m + 1 } } { 4 ^ { 8 m } }$

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.

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

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

Find the interval of convergence of the series. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { n } } { n + 3 }$

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

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

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.

Functions and Models

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Differentiation Rules

Applications of Differentiation

Integrals

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Techniques of Integration

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Exam 11: Infinite Sequences and Series

Determine whether the sequence defined by an=5n8n+1a _ { n } = \frac { 5 ^ { n } } { 8 ^ { n } + 1 }an​=8n+15n​ converges or diverges. If it converges, find its limit.

Determine which series is convergent.

Determine whether the given series is convergent or divergent. ∑n=2∞1n(ln⁡n)2\sum _ { n = 2 } ^ { \infty } \frac { 1 } { n ( \ln n ) ^ { 2 } }∑n=2∞​n(lnn)21​

Use multiplication or division of power series to find the first three nonzero terms in the Maclaurin series for the function. f(x)=5e−x2cos⁡4xf ( x ) = 5 e ^ { - x ^ { 2 } } \cos 4 xf(x)=5e−x2cos4x

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​

Which of the given series are absolutely convergent?

Determine whether the sequence convergent or divergent. ∑n=1∞3n2+3\sum _ { n = 1 } ^ { \infty } \frac { 3 } { n ^ { 2 } + 3 }∑n=1∞​n2+33​

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​

Find the value of the limit for the sequence given. {1⋅9⋅17⋯(7n+1)(7n)2}\left\{ \frac { 1 \cdot 9 \cdot 17 \cdots ( 7 n + 1 ) } { ( 7 n ) ^ { 2 } } \right\}{(7n)21⋅9⋅17⋯(7n+1)​}

Test the series for convergence or divergence. ∑m−1∞(−6)m+148m\sum _ { m - 1 } ^ { \infty } \frac { ( - 6 ) ^ { m + 1 } } { 4 ^ { 8 m } }∑m−1∞​48m(−6)m+1​

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.

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​

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 the interval of convergence of the series. ∑n=1∞(−1)nxnn+3\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { n } } { n + 3 }∑n=1∞​n+3(−1)nxn​

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​

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​

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.

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 $a _ { n } = \frac { 5 ^ { n } } { 8 ^ { n } + 1 }$ converges or diverges. If it converges, find its limit.

Determine whether the given series is convergent or divergent. $\sum _ { n = 2 } ^ { \infty } \frac { 1 } { n ( \ln n ) ^ { 2 } }$

Use multiplication or division of power series to find the first three nonzero terms in the Maclaurin series for the function. $f ( x ) = 5 e ^ { - x ^ { 2 } } \cos 4 x$

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

Determine whether the sequence convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { 3 } { n ^ { 2 } + 3 }$

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

Find the value of the limit for the sequence given. $\left\{ \frac { 1 \cdot 9 \cdot 17 \cdots ( 7 n + 1 ) } { ( 7 n ) ^ { 2 } } \right\}$

Test the series for convergence or divergence. $\sum _ { m - 1 } ^ { \infty } \frac { ( - 6 ) ^ { m + 1 } } { 4 ^ { 8 m } }$

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.

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

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

Find the interval of convergence of the series. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { n } } { n + 3 }$

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

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

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.