Question 1

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 2

Find all positive values of $u$ for which the series $\sum _ { m - 1 } ^ { \infty } 6 u ^ { \mathrm { m } 7 m }$ converges. Select the correct answer.

Accepted Answer

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

Question 3

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

Accepted Answer

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

Question 4

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 5

Find the sum of the series.
$$\frac { 2 } { 1 \cdot 3 } - \frac { 2 ^ { 2 } } { 2 \cdot 3 ^ { 2 } } + \frac { 2 ^ { 3 } } { 3 \cdot 3 ^ { 3 } } - \frac { 2 ^ { 4 } } { 4 \cdot 3 ^ { 4 } } + \ldots$$

Accepted Answer

The answer of Find the sum of the series.
\[\frac {...

Question 6

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 7

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

Accepted Answer

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

Question 8

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

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

Question 9

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 10

$$\text { Find the partial sum } S _ { \gamma } \text { of the series } \sum _ { m - 1 } ^ { \infty } \frac { 6 } { 10 + 8 ^ { m } } \text {. Give your answer to five decimal places. }$$

Accepted Answer

The answer of \[\text { Find the partial sum }...

Question 11

Find the value of the limit of the sequence defined by
$$a _ { 1 } = 1 , a _ { n + 1 } = 6 - \frac { 1 } { a _ { n } }$$
Select the correct answer.

Accepted Answer

A)  $\frac { 6 + \sqrt { 5 } } { 2 }$ 
B)  $\frac { 6 - \sqrt { 10 } } { 2 }$ 
C)  $\frac { 6 + \sqrt { 10 } } { 2 }$ 
D)  $\frac { 6 - \sqrt { 5 } } { 2 }$ 
E)  $6 + \sqrt { 10 }$ 
A)  $\frac { 6 + \sqrt { 5 } } { 2 }$ 
B)  $\frac { 6 - \sqrt { 10 } } { 2 }$ 
C)  $\frac { 6 + \sqrt { 10 } } { 2 }$ 
D)  $\frac { 6 - \sqrt { 5 } } { 2 }$ 
E)  $6 + \sqrt { 10 }$

Question 12

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

Accepted Answer

A)  $p  9$ 
D)  $p > 1$ 
A)  $p  9$ 
D)  $p > 1$

Question 13

Determine which one of the  p -series below is convergent.

Accepted Answer

The answer of Determine which one of the  p...

Question 14

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

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

Question 15

Test the series for convergence or divergence.
$$\sum _ { n = 2 } ^ { \infty } ( - 1 ) ^ { n } \frac { n } { 5 \ln n }$$

Accepted Answer

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

Question 16

Find a power series representation for
$$f ( t ) = \ln ( 14 - t )$$

Accepted Answer

The answer of Find a power series representation for
\[f (...

Question 17

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

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

Question 18

A sequenceis $\left\{ a _ { n } \right\}$ defined recursively by the equation $a _ { n } = 0.5 \left( a _ { n - 1 } + a _ { n - 2 } \right)$ for $n \geq 3$ where $a _ { 1 } = 14 , a _ { 2 } = 14$.
Use your calculator to guess the limit of the sequence. Select the correct answer.

Accepted Answer

A)  6 
B)  14 
C)  26 
D)  17 
E)  15 
A)  6 
B)  14 
C)  26 
D)  17 
E)  15

Question 19

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 20

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

Accepted Answer

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

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

Find all positive values of $u$ for which the series $\sum _ { m - 1 } ^ { \infty } 6 u ^ { \mathrm { m } 7 m }$ converges. Select the correct answer.

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

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

Find the sum of the series. $\frac { 2 } { 1 \cdot 3 } - \frac { 2 ^ { 2 } } { 2 \cdot 3 ^ { 2 } } + \frac { 2 ^ { 3 } } { 3 \cdot 3 ^ { 3 } } - \frac { 2 ^ { 4 } } { 4 \cdot 3 ^ { 4 } } + \ldots$

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

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

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

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

$\text { Find the partial sum } S _ { \gamma } \text { of the series } \sum _ { m - 1 } ^ { \infty } \frac { 6 } { 10 + 8 ^ { m } } \text {. Give your answer to five decimal places. }$

Find the value of the limit of the sequence defined by $a _ { 1 } = 1 , a _ { n + 1 } = 6 - \frac { 1 } { a _ { n } }$ Select the correct answer.

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

Determine which one of the p -series below is convergent.

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

Test the series for convergence or divergence. $\sum _ { n = 2 } ^ { \infty } ( - 1 ) ^ { n } \frac { n } { 5 \ln n }$

Find a power series representation for $f ( t ) = \ln ( 14 - t )$

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

A sequenceis $\left\{ a _ { n } \right\}$ defined recursively by the equation $a _ { n } = 0.5 \left( a _ { n - 1 } + a _ { n - 2 } \right)$ for $n \geq 3$ where $a _ { 1 } = 14 , a _ { 2 } = 14$ . Use your calculator to guess the limit of the sequence. Select the correct answer.

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

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

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

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​

Find all positive values of uuu for which the series ∑m−1∞6um7m\sum _ { m - 1 } ^ { \infty } 6 u ^ { \mathrm { m } 7 m }∑m−1∞​6um7m converges. Select the correct answer.

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

Find the sum of the series. 21⋅3−222⋅32+233⋅33−244⋅34+…\frac { 2 } { 1 \cdot 3 } - \frac { 2 ^ { 2 } } { 2 \cdot 3 ^ { 2 } } + \frac { 2 ^ { 3 } } { 3 \cdot 3 ^ { 3 } } - \frac { 2 ^ { 4 } } { 4 \cdot 3 ^ { 4 } } + \ldots1⋅32​−2⋅3222​+3⋅3323​−4⋅3424​+…

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

Find the value of the limit for the sequence given. Select the correct answer. {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)​}

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 value of the limit of the sequence defined by a1=1,an+1=6−1ana _ { 1 } = 1 , a _ { n + 1 } = 6 - \frac { 1 } { a _ { n } }a1​=1,an+1​=6−an​1​ Select the correct answer.

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.

Determine which one of the p -series below is convergent.

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

Test the series for convergence or divergence. ∑n=2∞(−1)nn5ln⁡n\sum _ { n = 2 } ^ { \infty } ( - 1 ) ^ { n } \frac { n } { 5 \ln n }∑n=2∞​(−1)n5lnnn​

Find a power series representation for f(t)=ln⁡(14−t)f ( t ) = \ln ( 14 - t )f(t)=ln(14−t)

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

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​

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

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

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

Find all positive values of $u$ for which the series $\sum _ { m - 1 } ^ { \infty } 6 u ^ { \mathrm { m } 7 m }$ converges. Select the correct answer.

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

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

Find the sum of the series. $\frac { 2 } { 1 \cdot 3 } - \frac { 2 ^ { 2 } } { 2 \cdot 3 ^ { 2 } } + \frac { 2 ^ { 3 } } { 3 \cdot 3 ^ { 3 } } - \frac { 2 ^ { 4 } } { 4 \cdot 3 ^ { 4 } } + \ldots$

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

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

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 value of the limit of the sequence defined by $a _ { 1 } = 1 , a _ { n + 1 } = 6 - \frac { 1 } { a _ { n } }$ Select the correct answer.

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

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

Test the series for convergence or divergence. $\sum _ { n = 2 } ^ { \infty } ( - 1 ) ^ { n } \frac { n } { 5 \ln n }$

Find a power series representation for $f ( t ) = \ln ( 14 - t )$

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

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

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