Question 1

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 2

Which of the given series are absolutely convergent? Select the correct answer.

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 3

Test the series for convergence or divergence.
$$\sum _ { m = 1 } ^ { \infty } ( - 4 ) ^ { m } \frac { \ln m } { \sqrt { m } }$$

Accepted Answer

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

Question 4

Determine whether the series is absolutely convergent, conditionally convergent, or divergent.
$\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } \arctan n } { n ^ { 4 } }$

Accepted Answer

A)  conditionally convergent 
B)  absolutely convergent 
C)  divergent 
A)  conditionally convergent 
B)  absolutely convergent 
C)  divergent

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

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 6

Determine whether the given series converges or diverges. If it converges, find its sum.
$$\sum _ { n = 0 } ^ { \infty } \frac { 9 ^ { n } + 8 ^ { n } } { 12 ^ { n } }$$

Accepted Answer

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

Question 7

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 8

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 9

Use the Alternating Series Estimation Theorem or Taylor's Inequality to estimate the range of values of $x$ for which the given approximation is accurate to within the stated error.
$\cos x \approx 1 - \frac { x ^ { 2 } } { 2 } + \frac { x ^ { 4 } } { 24 } \quad$ error $\mid < 0.08$
Write $a$ such that $- a < x < a$.

Accepted Answer

The answer of Use the Alternating Series Estimation Theorem or...

Question 10

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

Accepted Answer

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

Question 11

Use series to evaluate the limit correct to three decimal places.
$\lim _ { x \rightarrow 0 } \frac { 7 x - \tan ^ { - 1 } 7 x } { x ^ { 3 } }$
Select the correct answer.

Accepted Answer

A)  $118.933$ 
B)  $114.133$ 
C)  $34.3233$ 
D)  $114.333$ 
E)  $115.933$ 
A)  $118.933$ 
B)  $114.133$ 
C)  $34.3233$ 
D)  $114.333$ 
E)  $115.933$

Question 12

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

Accepted Answer

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

Question 13

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

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

Question 14

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

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 15

Find a formula for the general term $a _ { n }$ of the sequence, assuming that the pattern of the first few terms continues.
$$\left\{ - \frac { 1 } { 2 } , \frac { 16 } { 3 } , - \frac { 81 } { 4 } , \frac { 256 } { 5 } , - \frac { 625 } { 6 } , \ldots \right\}$$

Accepted Answer

The answer of Find a formula for the general term...

Question 16

Test the series for convergence or divergence.
$$\sum _ { m = 1 } ^ { \infty } \frac { 4 ^ { m } m ^ { 3 } } { m ! }$$

Accepted Answer

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

Question 17

Determine whether the series is convergent or divergent.
$$\sum _ { n = 1 } ^ { \infty } \frac { ( n ! ) ^ { 4 } } { ( 7 n ) ! }$$

Accepted Answer

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

Question 18

Find the radius of convergence of the series.
$$\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 3 } x ^ { n } } { 2 ^ { n } }$$

Accepted Answer

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

Question 19

A rubber ball is dropped from a height of $8 \mathrm {~m}$ onto a flat surface. Each time the ball hits the surface, it rebounds to $50 \%$ of its previous height. Find the total distance the ball travels.

Accepted Answer

A)  16 
B)  24 
C)  8 
D)  32 
A)  16 
B)  24 
C)  8 
D)  32

Question 20

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)  $R = \infty , I = ( - \infty , \infty )$

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

Which of the given series are absolutely convergent? Select the correct answer.

Test the series for convergence or divergence. $\sum _ { m = 1 } ^ { \infty } ( - 4 ) ^ { m } \frac { \ln m } { \sqrt { m } }$

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } \arctan n } { n ^ { 4 } }$

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 given series converges or diverges. If it converges, find its sum. $\sum _ { n = 0 } ^ { \infty } \frac { 9 ^ { n } + 8 ^ { n } } { 12 ^ { n } }$

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.

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

Use series to evaluate the limit correct to three decimal places. $\lim _ { x \rightarrow 0 } \frac { 7 x - \tan ^ { - 1 } 7 x } { x ^ { 3 } }$ Select the correct answer.

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

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

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

Find a formula for the general term $a _ { n }$ of the sequence, assuming that the pattern of the first few terms continues. $\left\{ - \frac { 1 } { 2 } , \frac { 16 } { 3 } , - \frac { 81 } { 4 } , \frac { 256 } { 5 } , - \frac { 625 } { 6 } , \ldots \right\}$

Test the series for convergence or divergence. $\sum _ { m = 1 } ^ { \infty } \frac { 4 ^ { m } m ^ { 3 } } { m ! }$

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { ( n ! ) ^ { 4 } } { ( 7 n ) ! }$

Find the radius of convergence of the series. $\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 3 } x ^ { n } } { 2 ^ { n } }$

A rubber ball is dropped from a height of $8 \mathrm {~m}$ onto a flat surface. Each time the ball hits the surface, it rebounds to $50 \%$ of its previous height. Find the total distance the ball travels.

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

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

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Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Filters

Exam 11: Infinite Sequences and Series

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​

Which of the given series are absolutely convergent? Select the correct answer.

Test the series for convergence or divergence. ∑m=1∞(−4)mln⁡mm\sum _ { m = 1 } ^ { \infty } ( - 4 ) ^ { m } \frac { \ln m } { \sqrt { m } }∑m=1∞​(−4)mm​lnm​

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. ∑n=1∞(−1)narctan⁡nn4\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } \arctan n } { n ^ { 4 } }∑n=1∞​n4(−1)narctann​

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 given series converges or diverges. If it converges, find its sum. ∑n=0∞9n+8n12n\sum _ { n = 0 } ^ { \infty } \frac { 9 ^ { n } + 8 ^ { n } } { 12 ^ { n } }∑n=0∞​12n9n+8n​

Determine which series is convergent.

Use series to evaluate the limit correct to three decimal places. lim⁡x→07x−tan⁡−17xx3\lim _ { x \rightarrow 0 } \frac { 7 x - \tan ^ { - 1 } 7 x } { x ^ { 3 } }limx→0​x37x−tan−17x​ Select the correct answer.

Determine whether the series is convergent or divergent. ∑n=1∞tan⁡−1nnn+7\sum _ { n = 1 } ^ { \infty } \frac { \tan ^ { - 1 } n } { n \sqrt { n + 7 } }∑n=1∞​nn+7​tan−1n​

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​

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

Test the series for convergence or divergence. ∑m=1∞4mm3m!\sum _ { m = 1 } ^ { \infty } \frac { 4 ^ { m } m ^ { 3 } } { m ! }∑m=1∞​m!4mm3​

Determine whether the series is convergent or divergent. ∑n=1∞(n!)4(7n)!\sum _ { n = 1 } ^ { \infty } \frac { ( n ! ) ^ { 4 } } { ( 7 n ) ! }∑n=1∞​(7n)!(n!)4​

Find the radius of convergence of the series. ∑n=1∞n3xn2n\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 3 } x ^ { n } } { 2 ^ { n } }∑n=1∞​2nn3xn​

A rubber ball is dropped from a height of 8 m8 \mathrm {~m}8 m onto a flat surface. Each time the ball hits the surface, it rebounds to 50%50 \%50% of its previous height. Find the total distance the ball travels.

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​

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

Test the series for convergence or divergence. $\sum _ { m = 1 } ^ { \infty } ( - 4 ) ^ { m } \frac { \ln m } { \sqrt { m } }$

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } \arctan n } { n ^ { 4 } }$

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 given series converges or diverges. If it converges, find its sum. $\sum _ { n = 0 } ^ { \infty } \frac { 9 ^ { n } + 8 ^ { n } } { 12 ^ { n } }$

Use series to evaluate the limit correct to three decimal places. $\lim _ { x \rightarrow 0 } \frac { 7 x - \tan ^ { - 1 } 7 x } { x ^ { 3 } }$ Select the correct answer.

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

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

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

Test the series for convergence or divergence. $\sum _ { m = 1 } ^ { \infty } \frac { 4 ^ { m } m ^ { 3 } } { m ! }$

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { ( n ! ) ^ { 4 } } { ( 7 n ) ! }$

Find the radius of convergence of the series. $\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 3 } x ^ { n } } { 2 ^ { n } }$

A rubber ball is dropped from a height of $8 \mathrm {~m}$ onto a flat surface. Each time the ball hits the surface, it rebounds to $50 \%$ of its previous height. Find the total distance the ball travels.

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