Question 1

Find all positive values of $u$ for which the series $\sum _ { m = 1 } ^ { \infty } 6 u ^ { m } 7 m$ converges,

Accepted Answer

A)  $u > 7$ 
B)  $6  \ln 7$ 
A)  $u > 7$ 
B)  $6  \ln 7$

Question 2

Determine whether the geometric series converges or diverges. If it converges, find its sum.
$$\sum _ { n = 0 } ^ { \infty } 5 ^ { n } 6 ^ { - n + 1 }$$

Accepted Answer

A)  30 
B)  36 
C)  5 
D)  Diverges 
A)  30 
B)  36 
C)  5 
D)  Diverges

Question 3

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 4

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 5

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 6

$$\text { Find the exact value of the limit of the sequence defined by } a _ { 1 } = \sqrt { 4 } , a _ { n + 1 } = \sqrt { 4 + a _ { n } } \text {. }$$

Accepted Answer

The answer of \[\text { Find the exact value of...

Question 7

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 8

Find the radius of convergence of the series. Select the correct answer.
$$\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } \frac { ( x + 10 ) ^ { n } } { n 6 ^ { n } }$$

Accepted Answer

A)  $( - 8,6 ]$ 
B)  $( 2,14 ]$ 
C)  $( - 14 , - 2 )$ 
D)  $[ - 16 , - 4 )$ 
E)  $[ - 1,1 ]$ 
A)  $( - 8,6 ]$ 
B)  $( 2,14 ]$ 
C)  $( - 14 , - 2 )$ 
D)  $[ - 16 , - 4 )$ 
E)  $[ - 1,1 ]$

Question 9

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 10

Test the series for convergence or divergence.
$$\sum _ { k = 1 } ^ { \infty } \frac { ( - 6 ) ^ { k + 1 } } { 7 ^ { 2 k } }$$

Accepted Answer

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

Question 11

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.
$\cos x \approx 1 - \frac { x ^ { 2 } } { 2 } + \frac { x ^ { 4 } } { 24 } \quad \mid$ error $\mid < 0.08$
Write $a$ such that $- a < x < a$.

Accepted Answer

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

Question 12

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 13

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 14

Find the radius of convergence of the series. Select the correct answer.
$$\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } \frac { ( x + 10 ) ^ { n } } { n 6 ^ { n } }$$

Accepted Answer

A)  $( - 8,6 ]$ 
B)  $( 2,14 ]$ 
C)  $( - 14 , - 2 )$ 
D)  $[ - 16 , - 4 )$ 
E)  $[ - 1,1 ]$ 
A)  $( - 8,6 ]$ 
B)  $( 2,14 ]$ 
C)  $( - 14 , - 2 )$ 
D)  $[ - 16 , - 4 )$ 
E)  $[ - 1,1 ]$

Question 15

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

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

Question 16

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 17

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

Accepted Answer

A)  13 
B)  5 
C)  Diverges 
D)  $- 3$ 
A)  13 
B)  5 
C)  Diverges 
D)  $- 3$

Question 18

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

Accepted Answer

A)  13 
B)  5 
C)  Diverges 
D)  $- 3$ 
A)  13 
B)  5 
C)  Diverges 
D)  $- 3$

Question 19

Find all values of $p$ for which the series $\sum _ { n = 1 } ^ { \infty } \frac { \ln \left( n ^ { 9 } \right) } { n ^ { p } }$
converges.

Accepted Answer

The answer of Find all values of $p$ for which...

Question 20

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

A)  $- 1.06$ 
B)  $- 0.84$ 
C)  $- 0.90$ 
D)  $- 0.98$ 
A)  $- 1.06$ 
B)  $- 0.84$ 
C)  $- 0.90$ 
D)  $- 0.98$

Find all positive values of $u$ for which the series $\sum _ { m = 1 } ^ { \infty } 6 u ^ { m } 7 m$ converges,

Determine whether the geometric series converges or diverges. If it converges, find its sum. $\sum _ { n = 0 } ^ { \infty } 5 ^ { n } 6 ^ { - n + 1 }$

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

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

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

$\text { Find the exact value of the limit of the sequence defined by } a _ { 1 } = \sqrt { 4 } , a _ { n + 1 } = \sqrt { 4 + a _ { n } } \text {. }$

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

Find the radius of convergence of the series. Select the correct answer. $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } \frac { ( x + 10 ) ^ { n } } { n 6 ^ { n } }$

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

Test the series for convergence or divergence. $\sum _ { k = 1 } ^ { \infty } \frac { ( - 6 ) ^ { k + 1 } } { 7 ^ { 2 k } }$

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

Find the radius of convergence of the series. Select the correct answer. $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } \frac { ( x + 10 ) ^ { n } } { n 6 ^ { n } }$

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

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

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

Find all values of $p$ for which the series $\sum _ { n = 1 } ^ { \infty } \frac { \ln \left( n ^ { 9 } \right) } { n ^ { p } }$ converges.

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

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

Exam 11: Infinite Sequences and Series

Find all positive values of uuu for which the series ∑m=1∞6um7m\sum _ { m = 1 } ^ { \infty } 6 u ^ { m } 7 m∑m=1∞​6um7m converges,

Determine whether the geometric series converges or diverges. If it converges, find its sum. ∑n=0∞5n6−n+1\sum _ { n = 0 } ^ { \infty } 5 ^ { n } 6 ^ { - n + 1 }∑n=0∞​5n6−n+1

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​

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​

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​

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

Find the radius of convergence of the series. Select the correct answer. ∑n=1∞(−1)n(x+10)nn6n\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } \frac { ( x + 10 ) ^ { n } } { n 6 ^ { n } }∑n=1∞​(−1)nn6n(x+10)n​

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​

Test the series for convergence or divergence. ∑k=1∞(−6)k+172k\sum _ { k = 1 } ^ { \infty } \frac { ( - 6 ) ^ { k + 1 } } { 7 ^ { 2 k } }∑k=1∞​72k(−6)k+1​

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 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)

Find the radius of convergence of the series. Select the correct answer. ∑n=1∞(−1)n(x+10)nn6n\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } \frac { ( x + 10 ) ^ { n } } { n 6 ^ { n } }∑n=1∞​(−1)nn6n(x+10)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​

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 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. Select the correct answer.

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.

Find all values of ppp for which the series ∑n=1∞ln⁡(n9)np\sum _ { n = 1 } ^ { \infty } \frac { \ln \left( n ^ { 9 } \right) } { n ^ { p } }∑n=1∞​npln(n9)​ converges.

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​

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 all positive values of $u$ for which the series $\sum _ { m = 1 } ^ { \infty } 6 u ^ { m } 7 m$ converges,

Determine whether the geometric series converges or diverges. If it converges, find its sum. $\sum _ { n = 0 } ^ { \infty } 5 ^ { n } 6 ^ { - n + 1 }$

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

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

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

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

Find the radius of convergence of the series. Select the correct answer. $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } \frac { ( x + 10 ) ^ { n } } { n 6 ^ { n } }$

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

Test the series for convergence or divergence. $\sum _ { k = 1 } ^ { \infty } \frac { ( - 6 ) ^ { k + 1 } } { 7 ^ { 2 k } }$

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

Find the radius of convergence of the series. Select the correct answer. $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } \frac { ( x + 10 ) ^ { n } } { n 6 ^ { n } }$

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

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

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

Find all values of $p$ for which the series $\sum _ { n = 1 } ^ { \infty } \frac { \ln \left( n ^ { 9 } \right) } { n ^ { p } }$ converges.

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