Question 1

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

Accepted Answer

$0.99260$

Question 2

Determine whether the geometric series converges or diverges. If it converges, find its sum. $\sum _ { n = 0 } ^ { \infty } 7 ^ { n } 8 ^ { - n + 1 }$

Accepted Answer

A)  56 
B)  7 
C)  Diverges 
D)  64 
A)  56 
B)  7 
C)  Diverges 
D)  64 D

Question 3

Write the partial sum of the converging series which represent the decimal number $0.2543$ .

Accepted Answer

$\frac { 1 } { 5 } + \frac { 3 } { 10 ^ { 2 } } + \frac { 5 } { 10 ^ { 3 } } + \frac { 4 } { 10 ^ { 4 } }$

Question 4

Express the number as a rational number. $5 . \overline { 45 } = 5.454545$

Accepted Answer

The answer of Express the number as a rational number....

Question 5

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

Accepted Answer

The answer of The terms of a series are defined...

Question 6

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

Accepted Answer

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

Question 7

Determine whether the sequence converges or diverges. If it converges, find the limit. $a _ { n } = e ^ { n / ( n + 6 ) }$

Accepted Answer

A)  $e$ 
B)  $\ln 3$ 
C)  diverges 
D)  $e ^ { 3 }$ 
E)  $\ln \left( \frac { 1 } { 3 } \right)$ 
A)  $e$ 
B)  $\ln 3$ 
C)  diverges 
D)  $e ^ { 3 }$ 
E)  $\ln \left( \frac { 1 } { 3 } \right)$

Question 8

Use the binomial series to expand the function as a power series. Find the radius of convergence. $\frac { x } { \sqrt { 9 + x ^ { 2 } } }$

Accepted Answer

A)  $| x | < 1$ 
B)  $| x | < 10$ 
C)  $| x | < 6$ 
D)  $| x | < 9$ 
E)  $| x | < 3$ 
A)  $| x | < 1$ 
B)  $| x | < 10$ 
C)  $| x | < 6$ 
D)  $| x | < 9$ 
E)  $| x | < 3$

Question 9

Determine whether the sequence defined by $a _ { n } = \frac { n ^ { 2 } - 5 } { 6 n ^ { 2 } + 1 }$ converges or diverges. If it converges, find its limit.

Accepted Answer

A)  $\frac { 1 } { 6 }$ 
B)  -5 
C)  $- \frac { 5 } { 6 }$ 
D)  Diverges 
A)  $\frac { 1 } { 6 }$ 
B)  -5 
C)  $- \frac { 5 } { 6 }$ 
D)  Diverges

Question 10

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \left( \frac { \ln \left( n ^ { 8 } \right) } { n } \right) ^ { n }$

Accepted Answer

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

Question 11

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

Question 12

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 13

Determine whether the given series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 25 n ^ { 2 } - 16 }$

Accepted Answer

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

Question 14

Determine whether the given series converges or diverges. If it converges, find its sum. $\sum _ { n = 1 } ^ { \infty } \left( 1 + \frac { 8 } { n } \right) ^ { n }$

Accepted Answer

A)  $e ^ { - 8 }$ 
B)  Diverges 
C)  1 
D)  $e ^ { 8 }$ 
A)  $e ^ { - 8 }$ 
B)  Diverges 
C)  1 
D)  $e ^ { 8 }$

Question 15

Find the Maclaurin series for $f ( x )$ using the definition of a Maclaurin serires. $f ( x ) = ( 4 + x ) ^ { - 3 }$

Accepted Answer

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

Question 16

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 $a _ { 1 } = 16 , a _ { 2 } = 8$ . Use your calculator to guess the limit of the sequence.

Accepted Answer

A)  13 
B)  $6$ 
C)  12 
D)  $17$ 
E)  $26$ 
A)  13 
B)  $6$ 
C)  12 
D)  $17$ 
E)  $26$

Question 17

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { \tan ^ { - 1 } n } { n \sqrt { n + 2 } }$

Accepted Answer

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

Question 18

Find a power series representation for the function and determine the radius of convergence. $f ( x ) = \arctan \left( \frac { x } { 4 } \right)$

Accepted Answer

The answer of Find a power series representation for the...

Question 19

Find the $n ^ { \text {th } }$ partial sum $S _ { n }$ of the telescoping series, and use it to determine whether the series converges or diverges. If it converges, find its sum. $\sum _ { n = 0 } ^ { \infty } \left( \frac { 1 } { n + 10 } - \frac { 1 } { n + 8 } \right)$

Accepted Answer

The answer of Find the \(n ^ { \text {th...

Question 20

Determine whether the series converges or diverges. $\sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n } n } { \ln \left( n ^ { 7 } \right) }$

Accepted Answer

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

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

Determine whether the geometric series converges or diverges. If it converges, find its sum. $\sum _ { n = 0 } ^ { \infty } 7 ^ { n } 8 ^ { - n + 1 }$

Write the partial sum of the converging series which represent the decimal number $0.2543$ .

Express the number as a rational number. $5 . \overline { 45 } = 5.454545$

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

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

Determine whether the sequence converges or diverges. If it converges, find the limit. $a _ { n } = e ^ { n / ( n + 6 ) }$

Use the binomial series to expand the function as a power series. Find the radius of convergence. $\frac { x } { \sqrt { 9 + x ^ { 2 } } }$

Determine whether the sequence defined by $a _ { n } = \frac { n ^ { 2 } - 5 } { 6 n ^ { 2 } + 1 }$ converges or diverges. If it converges, find its limit.

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \left( \frac { \ln \left( n ^ { 8 } \right) } { n } \right) ^ { n }$

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

Determine whether the given series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 25 n ^ { 2 } - 16 }$

Determine whether the given series converges or diverges. If it converges, find its sum. $\sum _ { n = 1 } ^ { \infty } \left( 1 + \frac { 8 } { n } \right) ^ { n }$

Find the Maclaurin series for $f ( x )$ using the definition of a Maclaurin serires. $f ( x ) = ( 4 + x ) ^ { - 3 }$

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 $a _ { 1 } = 16 , a _ { 2 } = 8$ . Use your calculator to guess the limit of the sequence.

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { \tan ^ { - 1 } n } { n \sqrt { n + 2 } }$

Find a power series representation for the function and determine the radius of convergence. $f ( x ) = \arctan \left( \frac { x } { 4 } \right)$

Find the $n ^ { \text {th } }$ partial sum $S _ { n }$ of the telescoping series, and use it to determine whether the series converges or diverges. If it converges, find its sum. $\sum _ { n = 0 } ^ { \infty } \left( \frac { 1 } { n + 10 } - \frac { 1 } { n + 8 } \right)$

Determine whether the series converges or diverges. $\sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n } n } { \ln \left( n ^ { 7 } \right) }$

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

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

Final Exam

Filters

Exam 11: Infinite Sequences and Series

Approximate the sum to the indicated accuracy. ∑n=1∞(−1)n−1n7 (five decimal places) \sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } } { n ^ { 7 } } \text { (five decimal places) }∑n=1∞​n7(−1)n−1​ (five decimal places)

Determine whether the geometric series converges or diverges. If it converges, find its sum. ∑n=0∞7n8−n+1\sum _ { n = 0 } ^ { \infty } 7 ^ { n } 8 ^ { - n + 1 }∑n=0∞​7n8−n+1

Write the partial sum of the converging series which represent the decimal number 0.25430.25430.2543 .

Express the number as a rational number. 5.45‾=5.4545455 . \overline { 45 } = 5.4545455.45=5.454545

The terms of a series are defined recursively by the equations a1=6,an+1=7n+16n+3ana _ { 1 } = 6 , a _ { n + 1 } = \frac { 7 n + 1 } { 6 n + 3 } a _ { n }a1​=6,an+1​=6n+37n+1​an​ . Determine whether ∑an\sum a _ { n }∑an​ converges or diverges.

Determine whether the sequence converges or diverges. If it converges, find the limit. an=en/(n+6)a _ { n } = e ^ { n / ( n + 6 ) }an​=en/(n+6)

Use the binomial series to expand the function as a power series. Find the radius of convergence. x9+x2\frac { x } { \sqrt { 9 + x ^ { 2 } } }9+x2​x​

Determine whether the sequence defined by an=n2−56n2+1a _ { n } = \frac { n ^ { 2 } - 5 } { 6 n ^ { 2 } + 1 }an​=6n2+1n2−5​ converges or diverges. If it converges, find its limit.

Determine whether the series is convergent or divergent. ∑n=1∞(ln⁡(n8)n)n\sum _ { n = 1 } ^ { \infty } \left( \frac { \ln \left( n ^ { 8 } \right) } { n } \right) ^ { n }∑n=1∞​(nln(n8)​)n

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 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.

Determine whether the given series is convergent or divergent. ∑n=1∞125n2−16\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 25 n ^ { 2 } - 16 }∑n=1∞​25n2−161​

Determine whether the given series converges or diverges. If it converges, find its sum. ∑n=1∞(1+8n)n\sum _ { n = 1 } ^ { \infty } \left( 1 + \frac { 8 } { n } \right) ^ { n }∑n=1∞​(1+n8​)n

Find the Maclaurin series for f(x)f ( x )f(x) using the definition of a Maclaurin serires. f(x)=(4+x)−3f ( x ) = ( 4 + x ) ^ { - 3 }f(x)=(4+x)−3

Determine whether the series is convergent or divergent. ∑n=1∞tan⁡−1nnn+2\sum _ { n = 1 } ^ { \infty } \frac { \tan ^ { - 1 } n } { n \sqrt { n + 2 } }∑n=1∞​nn+2​tan−1n​

Find a power series representation for the function and determine the radius of convergence. f(x)=arctan⁡(x4)f ( x ) = \arctan \left( \frac { x } { 4 } \right)f(x)=arctan(4x​)

Determine whether the series converges or diverges. ∑n=2∞(−1)nnln⁡(n7)\sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n } n } { \ln \left( n ^ { 7 } \right) }∑n=2∞​ln(n7)(−1)nn​

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

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

Final Exam

Filters

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

Determine whether the geometric series converges or diverges. If it converges, find its sum. $\sum _ { n = 0 } ^ { \infty } 7 ^ { n } 8 ^ { - n + 1 }$

Write the partial sum of the converging series which represent the decimal number $0.2543$ .

Express the number as a rational number. $5 . \overline { 45 } = 5.454545$

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

Determine whether the sequence converges or diverges. If it converges, find the limit. $a _ { n } = e ^ { n / ( n + 6 ) }$

Use the binomial series to expand the function as a power series. Find the radius of convergence. $\frac { x } { \sqrt { 9 + x ^ { 2 } } }$

Determine whether the sequence defined by $a _ { n } = \frac { n ^ { 2 } - 5 } { 6 n ^ { 2 } + 1 }$ converges or diverges. If it converges, find its limit.

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \left( \frac { \ln \left( n ^ { 8 } \right) } { n } \right) ^ { n }$

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

Determine whether the given series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 25 n ^ { 2 } - 16 }$

Determine whether the given series converges or diverges. If it converges, find its sum. $\sum _ { n = 1 } ^ { \infty } \left( 1 + \frac { 8 } { n } \right) ^ { n }$

Find the Maclaurin series for $f ( x )$ using the definition of a Maclaurin serires. $f ( x ) = ( 4 + x ) ^ { - 3 }$

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { \tan ^ { - 1 } n } { n \sqrt { n + 2 } }$

Find a power series representation for the function and determine the radius of convergence. $f ( x ) = \arctan \left( \frac { x } { 4 } \right)$

Determine whether the series converges or diverges. $\sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n } n } { \ln \left( n ^ { 7 } \right) }$