Question 1

Use the Root Test to determine the convergence or divergence of the series
$\sum _ { n = 1 } ^ { \infty } e ^ { 3 n }$

Accepted Answer

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

Question 2

Use the definition to find the Taylor series (centered at c) for the function. 
$f ( x ) = \ln \left( x ^ { 2 } \right) , c = 1$

Accepted Answer

A) $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } ( 2 ) } { n } ( x - 1 ) ^ { n }$ 
B)  $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( 2 ) } { n } ( x - 1 ) ^ { n }$ 
C) $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } ( 2 ) } { n } ( x - 1 ) ^ { n - 1 }$ 
D) $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } ( 2 ) } { n - 1 } ( x - 1 ) ^ { n - 1 }$ 
E) $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } ( 2 ) } { n - 1 } ( x - 1 ) ^ { n }$ 
A) $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } ( 2 ) } { n } ( x - 1 ) ^ { n }$ 
B)  $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( 2 ) } { n } ( x - 1 ) ^ { n }$ 
C) $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } ( 2 ) } { n } ( x - 1 ) ^ { n - 1 }$ 
D) $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } ( 2 ) } { n - 1 } ( x - 1 ) ^ { n - 1 }$ 
E) $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } ( 2 ) } { n - 1 } ( x - 1 ) ^ { n }$

Question 3

Suppose the annual spending by tourists in a resort city is $100 million. Approximately 90% of that revenue is again spent in the resort city, and of that amount approximately 90% is again spent in the same city, and so on. Write the expression that gives the total amount of Spending generated by the $100 million after n years.

Accepted Answer

A)  $1,000 \left( 1 + 0.90 ^ { n } \right)$ million dollars 
B)  $1,001 \left( 1 + 0.90 ^ { n } \right)$ million dollars 
C)  $2,000 \left( 1 - 0.90 ^ { n } \right)$ million dollars 
D)  $999 \left( 1 - 0.90 ^ { n } \right)$ million dollars 
E)  $1,000 \left( 1 - 0.90 ^ { n } \right)$ million dollars 
A)  $1,000 \left( 1 + 0.90 ^ { n } \right)$ million dollars 
B)  $1,001 \left( 1 + 0.90 ^ { n } \right)$ million dollars 
C)  $2,000 \left( 1 - 0.90 ^ { n } \right)$ million dollars 
D)  $999 \left( 1 - 0.90 ^ { n } \right)$ million dollars 
E)  $1,000 \left( 1 - 0.90 ^ { n } \right)$ million dollars

Question 4

$\text { Find the sum of the convergent series } \sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } \frac { 1 } { 11 ^ { n } n } \text { by using a well-known }$ function. Round your answer to four decimal places.

Accepted Answer

A) 0.1671 
B) 0.7802 
C) 2.4061 
D) 0.0870 
E) 2.3979 
A) 0.1671 
B) 0.7802 
C) 2.4061 
D) 0.0870 
E) 2.3979

Question 5

Write the first five terms of the sequence. $a _ { n } = \left( - \frac { 4 } { 5 } \right) ^ { n }$

Accepted Answer

A)  $\frac { 4 } { 5 } , \frac { 16 } { 25 } , \frac { 64 } { 125 } , \frac { 256 } { 625 } , \frac { 1024 } { 3125 }$ 
B)  $\frac { 4 } { 5 } , - \frac { 16 } { 25 } , \frac { 64 } { 125 } , - \frac { 256 } { 625 } , \frac { 1024 } { 3125 }$ 
C)  $- \frac { 4 } { 5 } , \frac { 16 } { 5 } , - \frac { 64 } { 5 } , \frac { 256 } { 5 } , - \frac { 1024 } { 5 }$ 
D)  $- \frac { 4 } { 5 } , \frac { 16 } { 25 } , - \frac { 64 } { 125 } , \frac { 256 } { 625 } , - \frac { 1024 } { 3125 }$ 
E)  $- \frac { 4 } { 5 } , - \frac { 16 } { 25 } , - \frac { 64 } { 125 } , - \frac { 256 } { 625 } , - \frac { 1024 } { 3125 }$ 
A)  $\frac { 4 } { 5 } , \frac { 16 } { 25 } , \frac { 64 } { 125 } , \frac { 256 } { 625 } , \frac { 1024 } { 3125 }$ 
B)  $\frac { 4 } { 5 } , - \frac { 16 } { 25 } , \frac { 64 } { 125 } , - \frac { 256 } { 625 } , \frac { 1024 } { 3125 }$ 
C)  $- \frac { 4 } { 5 } , \frac { 16 } { 5 } , - \frac { 64 } { 5 } , \frac { 256 } { 5 } , - \frac { 1024 } { 5 }$ 
D)  $- \frac { 4 } { 5 } , \frac { 16 } { 25 } , - \frac { 64 } { 125 } , \frac { 256 } { 625 } , - \frac { 1024 } { 3125 }$ 
E)  $- \frac { 4 } { 5 } , - \frac { 16 } { 25 } , - \frac { 64 } { 125 } , - \frac { 256 } { 625 } , - \frac { 1024 } { 3125 }$

Question 6

Use the Ratio Test to determine the convergence or divergence of the series. $$\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 6 } } { 10 ^ { n } }$$

Accepted Answer

A)  converges 
B)  diverges 
C)  Ratio Test inconclusive 
A)  converges 
B)  diverges 
C)  Ratio Test inconclusive

Question 7

$\text { Find a geometric power series for the function } \frac { 1 } { 6 + x } \text { centered at } 0 \text {. }$

Accepted Answer

A)  
$\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { n } } { 6 ^ { n + 1 } }$ 
B) 
$\sum _ { n = 0 } ^ { \infty } \frac { - x ^ { n } } { 6 ^ { n + 1 } }$ 
C)  
$\sum _ { n = 0 } ^ { \infty } \frac { x ^ { n } } { 6 ^ { n + 1 } }$ 
D) 
$\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { n } } { 6 ^ { n } } + 1$ 
E)  
$\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { n } } { 6 ^ { n } + 1 }$ 
A)  
$\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { n } } { 6 ^ { n + 1 } }$ 
B) 
$\sum _ { n = 0 } ^ { \infty } \frac { - x ^ { n } } { 6 ^ { n + 1 } }$ 
C)  
$\sum _ { n = 0 } ^ { \infty } \frac { x ^ { n } } { 6 ^ { n + 1 } }$ 
D) 
$\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { n } } { 6 ^ { n } } + 1$ 
E)  
$\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { n } } { 6 ^ { n } + 1 }$

Question 8

$\text { Find a power series for the function } \frac { 24 x - 35 } { 6 x ^ { 2 } + 29 x - 5 } \text { centered at } 0 \text {. }$

Accepted Answer

A)  
$\sum _ { n = 0 } ^ { \infty } \left( - \frac { 1 } { 5 ^ { n } } + 6 ^ { n + 1 } \right) x ^ { n }$ 
B) 
$\sum _ { n = 0 } ^ { \infty } \left( \frac { x } { 5 ^ { n } } + 6 ^ { n + 1 } \right) x ^ { n }$ 
C) 
$\sum _ { n = 0 } ^ { \infty } \left( \frac { ( - 1 ) ^ { n } } { 5 ^ { n } } + 6 ^ { n + 1 } \right) x ^ { n }$ 
D) 
$\sum _ { n = 0 } ^ { \infty } \left( - \frac { x } { 6 ^ { n } } + 5 ^ { n + 1 } \right) x ^ { n }$ 
E) 
$\sum _ { n = 0 } ^ { \infty } \left( \frac { ( - 1 ) ^ { n } } { 6 ^ { n } } + 5 ^ { n + 1 } \right) x ^ { n }$ 
A)  
$\sum _ { n = 0 } ^ { \infty } \left( - \frac { 1 } { 5 ^ { n } } + 6 ^ { n + 1 } \right) x ^ { n }$ 
B) 
$\sum _ { n = 0 } ^ { \infty } \left( \frac { x } { 5 ^ { n } } + 6 ^ { n + 1 } \right) x ^ { n }$ 
C) 
$\sum _ { n = 0 } ^ { \infty } \left( \frac { ( - 1 ) ^ { n } } { 5 ^ { n } } + 6 ^ { n + 1 } \right) x ^ { n }$ 
D) 
$\sum _ { n = 0 } ^ { \infty } \left( - \frac { x } { 6 ^ { n } } + 5 ^ { n + 1 } \right) x ^ { n }$ 
E) 
$\sum _ { n = 0 } ^ { \infty } \left( \frac { ( - 1 ) ^ { n } } { 6 ^ { n } } + 5 ^ { n + 1 } \right) x ^ { n }$

Question 9

$\text { Consider the series } \sum _ { n = 1 } ^ { \infty } \frac { 1 } { ( 6 n - 1 ) ^ { 2 } } \text {. The sum of the series is } \pi ^ { 2 } / 6 \text { Find the sum of }$ the series $\sum _ { n = 5 } ^ { \infty } \frac { 1 } { ( 6 n - 1 ) ^ { 2 } }$.

Accepted Answer

A)  $1.5932$ 
B)  $1.5632$ 
C)  $1.6132$ 
D)  $1.6232$ 
E)  $1.5732$ 
A)  $1.5932$ 
B)  $1.5632$ 
C)  $1.6132$ 
D)  $1.6232$ 
E)  $1.5732$

Question 10

Write the first five terms of the sequence of partial sums. $$5 + \frac { 5 } { 4 } + \frac { 5 } { 9 } + \frac { 5 } { 16 } + \frac { 5 } { 25 } + \cdots$$

Accepted Answer

A)  $5 , \frac { 5 } { 4 } , \frac { 5 } { 9 } , \frac { 5 } { 16 } , \frac { 1 } { 5 }$ 
B)  $5 , \frac { 5 } { 2 } , \frac { 225 } { 32 } , \frac { 250 } { 33 } , \frac { 525 } { 64 }$ 
C)  $5 , \frac { 25 } { 6 } , \frac { 245 } { 32 } , \frac { 1025 } { 132 } , \frac { 5269 } { 640 }$ 
D)  $5 , \frac { 35 } { 4 } , \frac { 275 } { 36 } , \frac { 1075 } { 144 } , \frac { 265 } { 36 }$ 
E)  $5 , \frac { 25 } { 4 } , \frac { 245 } { 36 } , \frac { 1025 } { 144 } , \frac { 5269 } { 720 }$ 
A)  $5 , \frac { 5 } { 4 } , \frac { 5 } { 9 } , \frac { 5 } { 16 } , \frac { 1 } { 5 }$ 
B)  $5 , \frac { 5 } { 2 } , \frac { 225 } { 32 } , \frac { 250 } { 33 } , \frac { 525 } { 64 }$ 
C)  $5 , \frac { 25 } { 6 } , \frac { 245 } { 32 } , \frac { 1025 } { 132 } , \frac { 5269 } { 640 }$ 
D)  $5 , \frac { 35 } { 4 } , \frac { 275 } { 36 } , \frac { 1075 } { 144 } , \frac { 265 } { 36 }$ 
E)  $5 , \frac { 25 } { 4 } , \frac { 245 } { 36 } , \frac { 1025 } { 144 } , \frac { 5269 } { 720 }$

Question 11

Use the binomial series to find the Maclaurin series for the function. $f ( x ) = \sqrt { 1 + x ^ { 2 } }$

Accepted Answer

A)  $1 - \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 2 ! 2 ^ { 2 } } x ^ { 4 } - \frac { 3 } { 3 ! 2 ^ { 3 } } x ^ { 6 } + \frac { 3 ( 5 ) } { 4 ! 2 ^ { 4 } } x ^ { 8 } - \frac { 3 ( 5 ) ( 7 ) } { 5 ! 2 ^ { 5 } } x ^ { 10 } + \cdots \cdots$
$$= 1 - \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 8 } x ^ { 4 } - \frac { 1 } { 16 } x ^ { 6 } + \frac { 5 } { 128 } x ^ { 8 } - \frac { 7 } { 256 } x ^ { 10 } + \cdots \cdots$$ 
B) 
$$\begin{array} { l } 
1 - \frac { 1 } { 2 } x ^ { 2 } - \frac { 1 } { 2 ! 2 ^ { 2 } } x ^ { 4 } - \frac { 3 } { 3 ! 2 ^ { 3 } } x ^ { 6 } - \frac { 3 ( 5 ) } { 4 ! 2 ^ { 4 } } x ^ { 8 } - \frac { 3 ( 5 ) ( 7 ) } { 5 ! 2 ^ { 5 } } x ^ { 10 } - \cdots \cdots \
= 1 - \frac { 1 } { 2 } x ^ { 2 } - \frac { 1 } { 8 } x ^ { 4 } - \frac { 1 } { 16 } x ^ { 6 } - \frac { 5 } { 128 } x ^ { 8 } - \frac { 7 } { 256 } x ^ { 10 } - \cdots \cdots
\end{array}$$ 
C) 
$$\begin{array} { l } 
1 + \frac { 1 } { 2 } x ^ { 2 } - \frac { 1 } { 2 ! 2 ^ { 2 } } x ^ { 4 } + \frac { 3 } { 3 ! 2 ^ { 3 } } x ^ { 6 } - \frac { 3 ( 5 ) } { 4 ! 2 ^ { 4 } } x ^ { 8 } + \frac { 3 ( 5 ) ( 7 ) } { 5 ! 2 ^ { 5 } } x ^ { 10 } - \cdots \cdots \
= 1 + \frac { 1 } { 2 } x ^ { 2 } - \frac { 1 } { 8 } x ^ { 4 } + \frac { 1 } { 16 } x ^ { 6 } - \frac { 5 } { 128 } x ^ { 8 } + \frac { 7 } { 256 } x ^ { 10 } - \cdots \cdots
\end{array}$$ 
D) 
$$\begin{array} { l } 
1 + \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 2 ! 2 ^ { 2 } } x ^ { 4 } + \frac { 3 } { 3 ! 2 ^ { 3 } } x ^ { 6 } + \frac { 3 ( 5 ) } { 4 ! 2 ^ { 4 } } x ^ { 8 } + \frac { 3 ( 5 ) ( 7 ) } { 5 ! 2 ^ { 5 } } x ^ { 10 } + \cdots \cdots \
= 1 + \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 8 } x ^ { 4 } + \frac { 1 } { 16 } x ^ { 6 } + \frac { 5 } { 128 } x ^ { 8 } + \frac { 7 } { 256 } x ^ { 10 } + \cdots \cdots
\end{array}$$ 
E) 
$$\begin{array} { l } 
- 1 + \frac { 1 } { 2 } x ^ { 2 } - \frac { 1 } { 2 ! 2 ^ { 2 } } x ^ { 4 } + \frac { 3 } { 3 ! 2 ^ { 3 } } x ^ { 6 } - \frac { 3 ( 5 ) } { 4 ! 2 ^ { 4 } } x ^ { 8 } + \frac { 3 ( 5 ) ( 7 ) } { 5 ! 2 ^ { 5 } } x ^ { 10 } - \cdots \cdots \
= - 1 + \frac { 1 } { 2 } x ^ { 2 } - \frac { 1 } { 8 } x ^ { 4 } + \frac { 1 } { 16 } x ^ { 6 } - \frac { 5 } { 128 } x ^ { 8 } + \frac { 7 } { 256 } x ^ { 10 } - \cdots \cdots
\end{array}$$ 
A)  $1 - \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 2 ! 2 ^ { 2 } } x ^ { 4 } - \frac { 3 } { 3 ! 2 ^ { 3 } } x ^ { 6 } + \frac { 3 ( 5 ) } { 4 ! 2 ^ { 4 } } x ^ { 8 } - \frac { 3 ( 5 ) ( 7 ) } { 5 ! 2 ^ { 5 } } x ^ { 10 } + \cdots \cdots$
$$= 1 - \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 8 } x ^ { 4 } - \frac { 1 } { 16 } x ^ { 6 } + \frac { 5 } { 128 } x ^ { 8 } - \frac { 7 } { 256 } x ^ { 10 } + \cdots \cdots$$ 
B) 
$$\begin{array} { l } 
1 - \frac { 1 } { 2 } x ^ { 2 } - \frac { 1 } { 2 ! 2 ^ { 2 } } x ^ { 4 } - \frac { 3 } { 3 ! 2 ^ { 3 } } x ^ { 6 } - \frac { 3 ( 5 ) } { 4 ! 2 ^ { 4 } } x ^ { 8 } - \frac { 3 ( 5 ) ( 7 ) } { 5 ! 2 ^ { 5 } } x ^ { 10 } - \cdots \cdots \
= 1 - \frac { 1 } { 2 } x ^ { 2 } - \frac { 1 } { 8 } x ^ { 4 } - \frac { 1 } { 16 } x ^ { 6 } - \frac { 5 } { 128 } x ^ { 8 } - \frac { 7 } { 256 } x ^ { 10 } - \cdots \cdots
\end{array}$$ 
C) 
$$\begin{array} { l } 
1 + \frac { 1 } { 2 } x ^ { 2 } - \frac { 1 } { 2 ! 2 ^ { 2 } } x ^ { 4 } + \frac { 3 } { 3 ! 2 ^ { 3 } } x ^ { 6 } - \frac { 3 ( 5 ) } { 4 ! 2 ^ { 4 } } x ^ { 8 } + \frac { 3 ( 5 ) ( 7 ) } { 5 ! 2 ^ { 5 } } x ^ { 10 } - \cdots \cdots \
= 1 + \frac { 1 } { 2 } x ^ { 2 } - \frac { 1 } { 8 } x ^ { 4 } + \frac { 1 } { 16 } x ^ { 6 } - \frac { 5 } { 128 } x ^ { 8 } + \frac { 7 } { 256 } x ^ { 10 } - \cdots \cdots
\end{array}$$ 
D) 
$$\begin{array} { l } 
1 + \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 2 ! 2 ^ { 2 } } x ^ { 4 } + \frac { 3 } { 3 ! 2 ^ { 3 } } x ^ { 6 } + \frac { 3 ( 5 ) } { 4 ! 2 ^ { 4 } } x ^ { 8 } + \frac { 3 ( 5 ) ( 7 ) } { 5 ! 2 ^ { 5 } } x ^ { 10 } + \cdots \cdots \
= 1 + \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 8 } x ^ { 4 } + \frac { 1 } { 16 } x ^ { 6 } + \frac { 5 } { 128 } x ^ { 8 } + \frac { 7 } { 256 } x ^ { 10 } + \cdots \cdots
\end{array}$$ 
E) 
$$\begin{array} { l } 
- 1 + \frac { 1 } { 2 } x ^ { 2 } - \frac { 1 } { 2 ! 2 ^ { 2 } } x ^ { 4 } + \frac { 3 } { 3 ! 2 ^ { 3 } } x ^ { 6 } - \frac { 3 ( 5 ) } { 4 ! 2 ^ { 4 } } x ^ { 8 } + \frac { 3 ( 5 ) ( 7 ) } { 5 ! 2 ^ { 5 } } x ^ { 10 } - \cdots \cdots \
= - 1 + \frac { 1 } { 2 } x ^ { 2 } - \frac { 1 } { 8 } x ^ { 4 } + \frac { 1 } { 16 } x ^ { 6 } - \frac { 5 } { 128 } x ^ { 8 } + \frac { 7 } { 256 } x ^ { 10 } - \cdots \cdots
\end{array}$$

Question 12

Determine the convergence or divergence of the series $\sum _ { n = 1 } ^ { \infty } \frac { 7 ( - 1 ) ^ { n + 1 } } { n }$ using any appropriate test.

Accepted Answer

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

Question 13

Consider the function given by $f ( x ) = \sum _ { n = 0 } ^ { \infty } \left( \frac { x } { 14 } \right) ^ { n }$.
Find the interval of convergence for $f ^ { t } ( x )$

Accepted Answer

A)  $[ - 14,14 )$ 
B)  $( - 14,0 ]$ 
C)  $( 0,14 )$ 
D)  $[ - 14,14 ]$ 
E)  $( - 14,14 )$ 
A)  $[ - 14,14 )$ 
B)  $( - 14,0 ]$ 
C)  $( 0,14 )$ 
D)  $[ - 14,14 ]$ 
E)  $( - 14,14 )$

Question 14

Write the first three terms of the sequence. $a _ { n } = 1 - \frac { 3 } { n } + \frac { 5 } { n ^ { 2 } }$

Accepted Answer

A)  $3 , \frac { 3 } { 4 } , - \frac { 13 } { 9 }$ 
B)  $3 , \frac { 3 } { 4 } , \frac { 5 } { 9 }$ 
C)  $3 , - \frac { 3 } { 4 } , \frac { 5 } { 9 }$ 
D)  $3 , \frac { 1 } { 2 } , - \frac { 1 } { 3 }$ 
E)  $3 , - \frac { 3 } { 4 } , - \frac { 13 } { 9 }$ 
A)  $3 , \frac { 3 } { 4 } , - \frac { 13 } { 9 }$ 
B)  $3 , \frac { 3 } { 4 } , \frac { 5 } { 9 }$ 
C)  $3 , - \frac { 3 } { 4 } , \frac { 5 } { 9 }$ 
D)  $3 , \frac { 1 } { 2 } , - \frac { 1 } { 3 }$ 
E)  $3 , - \frac { 3 } { 4 } , - \frac { 13 } { 9 }$

Question 15

Determine the minimal number of terms required to approximate the sum of the series with an error of less than 0.004. $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { n ! }$

Accepted Answer

A)  10 
B)  5 
C)  8 
D)  6 
E)  4 
A)  10 
B)  5 
C)  8 
D)  6 
E)  4

Question 16

Determine the convergence or divergence of the series. $\sum _ { n = 0 } ^ { \infty } \frac { 5 } { 3 ^ { - n } }$

Accepted Answer

A)  cannot be determined from the methods in the chapter 
B)  Diverges 
C)  Converges 
A)  cannot be determined from the methods in the chapter 
B)  Diverges 
C)  Converges

Question 17

$\text { Find the differential equation having the solution } \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( 4 x ) ^ { 2 n + 1 } } { ( 2 n + 1 ) ! } \text {. }$

Accepted Answer

A)  $y ^ {tt } - 16 y = 0$ 
B)  $y ^ {tt } + 16 y ^ { \prime } = 0$ 
C)  $y ^ { t } - 16 y = 0$ 
D)  $y ^ { tt } + 16 y = 0$ 
E)  $y ^ { t } + 16 y = 0$ 
A)  $y ^ {tt } - 16 y = 0$ 
B)  $y ^ {tt } + 16 y ^ { \prime } = 0$ 
C)  $y ^ { t } - 16 y = 0$ 
D)  $y ^ { tt } + 16 y = 0$ 
E)  $y ^ { t } + 16 y = 0$

Question 18

Explain how to use the geometric series $g ( x ) = \frac { 1 } { 1 - x } = \sum _ { n = 0 } ^ { \infty } x ^ { n } , | x | < 1 \text { to find the }$ series for the function $\frac { 9 } { 1 + x }$.

Accepted Answer

A)  replace $x$ with $\frac { 9 } { ( - x ) }$ 
B)  replace $x$ with $( - x )$ and multiply the series by 9 
C)  replace $x$ with $\frac { 1 } { x }$ and divide the series by 9 
D)  replace $x$ with $( - x )$ and divide the series by 9 
E)  replace $x$ with $\frac { 9 } { x }$ 
A)  replace $x$ with $\frac { 9 } { ( - x ) }$ 
B)  replace $x$ with $( - x )$ and multiply the series by 9 
C)  replace $x$ with $\frac { 1 } { x }$ and divide the series by 9 
D)  replace $x$ with $( - x )$ and divide the series by 9 
E)  replace $x$ with $\frac { 9 } { x }$

Question 19

$\text { The series } \sum _ { n = 1 } ^ { \infty } \sec ( n \pi ) \text { converges. }$

Accepted Answer

A) True 
 B)False

Question 20

Find the Maclaurin series for the function $f ( x ) = e ^ { \frac { x ^ { 7 } } { 9 } }$

Accepted Answer

A)  $e ^ { \frac { x ^ { 7 } } { 9 } } = \sum _ { n = 1 } ^ { \infty } \frac { x ^ { 7 n } } { 9 ^ { n } n ! }$ 
B)  $e ^ { \frac { x ^ { 7 } } { 9 } } = \frac { 1 } { 9 } \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { 7 n } } { n ! }$ 
C)  $e ^ { \frac { x ^ { 7 } } { 9 } } = \sum _ { n = 0 } ^ { \infty } \frac { x ^ { 7 n } } { 9 ^ { n } n ! }$ 
D) $e ^ { \frac { x ^ { 7 } } { 9 } } = \frac { 1 } { 9 } \sum _ { n = 0 } ^ { \infty } \frac { x ^ { 7 n } } { n ! }$ 
E) $e ^ { \frac { x ^ { 7 } } { 9 } } = \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { 7 n } } { 9 ^ { n } n ! }$ 
A)  $e ^ { \frac { x ^ { 7 } } { 9 } } = \sum _ { n = 1 } ^ { \infty } \frac { x ^ { 7 n } } { 9 ^ { n } n ! }$ 
B)  $e ^ { \frac { x ^ { 7 } } { 9 } } = \frac { 1 } { 9 } \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { 7 n } } { n ! }$ 
C)  $e ^ { \frac { x ^ { 7 } } { 9 } } = \sum _ { n = 0 } ^ { \infty } \frac { x ^ { 7 n } } { 9 ^ { n } n ! }$ 
D) $e ^ { \frac { x ^ { 7 } } { 9 } } = \frac { 1 } { 9 } \sum _ { n = 0 } ^ { \infty } \frac { x ^ { 7 n } } { n ! }$ 
E) $e ^ { \frac { x ^ { 7 } } { 9 } } = \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { 7 n } } { 9 ^ { n } n ! }$

Use the Root Test to determine the convergence or divergence of the series $\sum _ { n = 1 } ^ { \infty } e ^ { 3 n }$

Use the definition to find the Taylor series (centered at c) for the function. $f ( x ) = \ln \left( x ^ { 2 } \right) , c = 1$

$\text { Find the sum of the convergent series } \sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } \frac { 1 } { 11 ^ { n } n } \text { by using a well-known }$ function. Round your answer to four decimal places.

Write the first five terms of the sequence. $a _ { n } = \left( - \frac { 4 } { 5 } \right) ^ { n }$

Use the Ratio Test to determine the convergence or divergence of the series. $\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 6 } } { 10 ^ { n } }$

$\text { Find a geometric power series for the function } \frac { 1 } { 6 + x } \text { centered at } 0 \text {. }$

$\text { Find a power series for the function } \frac { 24 x - 35 } { 6 x ^ { 2 } + 29 x - 5 } \text { centered at } 0 \text {. }$

$\text { Consider the series } \sum _ { n = 1 } ^ { \infty } \frac { 1 } { ( 6 n - 1 ) ^ { 2 } } \text {. The sum of the series is } \pi ^ { 2 } / 6 \text { Find the sum of }$ the series $\sum _ { n = 5 } ^ { \infty } \frac { 1 } { ( 6 n - 1 ) ^ { 2 } }$ .

Write the first five terms of the sequence of partial sums. $5 + \frac { 5 } { 4 } + \frac { 5 } { 9 } + \frac { 5 } { 16 } + \frac { 5 } { 25 } + \cdots$

Use the binomial series to find the Maclaurin series for the function. $f ( x ) = \sqrt { 1 + x ^ { 2 } }$

Determine the convergence or divergence of the series $\sum _ { n = 1 } ^ { \infty } \frac { 7 ( - 1 ) ^ { n + 1 } } { n }$ using any appropriate test.

Consider the function given by $f ( x ) = \sum _ { n = 0 } ^ { \infty } \left( \frac { x } { 14 } \right) ^ { n }$ . Find the interval of convergence for $f ^ { t } ( x )$

Write the first three terms of the sequence. $a _ { n } = 1 - \frac { 3 } { n } + \frac { 5 } { n ^ { 2 } }$

Determine the minimal number of terms required to approximate the sum of the series with an error of less than 0.004. $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { n ! }$

Determine the convergence or divergence of the series. $\sum _ { n = 0 } ^ { \infty } \frac { 5 } { 3 ^ { - n } }$

$\text { Find the differential equation having the solution } \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( 4 x ) ^ { 2 n + 1 } } { ( 2 n + 1 ) ! } \text {. }$

Explain how to use the geometric series $g ( x ) = \frac { 1 } { 1 - x } = \sum _ { n = 0 } ^ { \infty } x ^ { n } , | x | < 1 \text { to find the }$ series for the function $\frac { 9 } { 1 + x }$ .

$\text { The series } \sum _ { n = 1 } ^ { \infty } \sec ( n \pi ) \text { converges. }$

Find the Maclaurin series for the function $f ( x ) = e ^ { \frac { x ^ { 7 } } { 9 } }$

Graphs and Models

A Preview of Calculus

The Derivative and the Tangent Line Problem

Extrema on an Interval

Antiderivatives and Indefinite Integration

Slope Fields and Eulers Method

Area of a Region Between Two Curves

Basic Integration Rules

Conics and Calculus

Vectors in the Plane

Vector-Valued Functions

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Vector Fields

Exact First-Order Equations

Filters

Exam 9: Sequences

Use the Root Test to determine the convergence or divergence of the series ∑n=1∞e3n\sum _ { n = 1 } ^ { \infty } e ^ { 3 n }∑n=1∞​e3n

Use the definition to find the Taylor series (centered at c) for the function. f(x)=ln⁡(x2),c=1f ( x ) = \ln \left( x ^ { 2 } \right) , c = 1f(x)=ln(x2),c=1

Write the first five terms of the sequence. an=(−45)na _ { n } = \left( - \frac { 4 } { 5 } \right) ^ { n }an​=(−54​)n

Use the Ratio Test to determine the convergence or divergence of the series. ∑n=1∞n610n\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 6 } } { 10 ^ { n } }∑n=1∞​10nn6​

Find a geometric power series for the function 16+x centered at 0. \text { Find a geometric power series for the function } \frac { 1 } { 6 + x } \text { centered at } 0 \text {. } Find a geometric power series for the function 6+x1​ centered at 0.

Find a power series for the function 24x−356x2+29x−5 centered at 0. \text { Find a power series for the function } \frac { 24 x - 35 } { 6 x ^ { 2 } + 29 x - 5 } \text { centered at } 0 \text {. } Find a power series for the function 6x2+29x−524x−35​ centered at 0.

Write the first five terms of the sequence of partial sums. 5+54+59+516+525+⋯5 + \frac { 5 } { 4 } + \frac { 5 } { 9 } + \frac { 5 } { 16 } + \frac { 5 } { 25 } + \cdots5+45​+95​+165​+255​+⋯

Use the binomial series to find the Maclaurin series for the function. f(x)=1+x2f ( x ) = \sqrt { 1 + x ^ { 2 } }f(x)=1+x2​

Determine the convergence or divergence of the series ∑n=1∞7(−1)n+1n\sum _ { n = 1 } ^ { \infty } \frac { 7 ( - 1 ) ^ { n + 1 } } { n }∑n=1∞​n7(−1)n+1​ using any appropriate test.

Consider the function given by f(x)=∑n=0∞(x14)nf ( x ) = \sum _ { n = 0 } ^ { \infty } \left( \frac { x } { 14 } \right) ^ { n }f(x)=∑n=0∞​(14x​)n . Find the interval of convergence for ft(x)f ^ { t } ( x )ft(x)

Write the first three terms of the sequence. an=1−3n+5n2a _ { n } = 1 - \frac { 3 } { n } + \frac { 5 } { n ^ { 2 } }an​=1−n3​+n25​

Determine the minimal number of terms required to approximate the sum of the series with an error of less than 0.004. ∑n=0∞(−1)nn!\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { n ! }∑n=0∞​n!(−1)n​

Determine the convergence or divergence of the series. ∑n=0∞53−n\sum _ { n = 0 } ^ { \infty } \frac { 5 } { 3 ^ { - n } }∑n=0∞​3−n5​

The series ∑n=1∞sec⁡(nπ) converges. \text { The series } \sum _ { n = 1 } ^ { \infty } \sec ( n \pi ) \text { converges. } The series ∑n=1∞​sec(nπ) converges.

Find the Maclaurin series for the function f(x)=ex79f ( x ) = e ^ { \frac { x ^ { 7 } } { 9 } }f(x)=e9x7​

Graphs and Models

A Preview of Calculus

The Derivative and the Tangent Line Problem

Extrema on an Interval

Antiderivatives and Indefinite Integration

Slope Fields and Eulers Method

Area of a Region Between Two Curves

Basic Integration Rules

Conics and Calculus

Vectors in the Plane

Vector-Valued Functions

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Vector Fields

Exact First-Order Equations

Filters

Use the Root Test to determine the convergence or divergence of the series $\sum _ { n = 1 } ^ { \infty } e ^ { 3 n }$

Use the definition to find the Taylor series (centered at c) for the function. $f ( x ) = \ln \left( x ^ { 2 } \right) , c = 1$

Write the first five terms of the sequence. $a _ { n } = \left( - \frac { 4 } { 5 } \right) ^ { n }$

Use the Ratio Test to determine the convergence or divergence of the series. $\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 6 } } { 10 ^ { n } }$

$\text { Find a geometric power series for the function } \frac { 1 } { 6 + x } \text { centered at } 0 \text {. }$

$\text { Find a power series for the function } \frac { 24 x - 35 } { 6 x ^ { 2 } + 29 x - 5 } \text { centered at } 0 \text {. }$

Write the first five terms of the sequence of partial sums. $5 + \frac { 5 } { 4 } + \frac { 5 } { 9 } + \frac { 5 } { 16 } + \frac { 5 } { 25 } + \cdots$

Use the binomial series to find the Maclaurin series for the function. $f ( x ) = \sqrt { 1 + x ^ { 2 } }$

Determine the convergence or divergence of the series $\sum _ { n = 1 } ^ { \infty } \frac { 7 ( - 1 ) ^ { n + 1 } } { n }$ using any appropriate test.

Consider the function given by $f ( x ) = \sum _ { n = 0 } ^ { \infty } \left( \frac { x } { 14 } \right) ^ { n }$ . Find the interval of convergence for $f ^ { t } ( x )$

Write the first three terms of the sequence. $a _ { n } = 1 - \frac { 3 } { n } + \frac { 5 } { n ^ { 2 } }$

Determine the minimal number of terms required to approximate the sum of the series with an error of less than 0.004. $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { n ! }$

Determine the convergence or divergence of the series. $\sum _ { n = 0 } ^ { \infty } \frac { 5 } { 3 ^ { - n } }$

$\text { The series } \sum _ { n = 1 } ^ { \infty } \sec ( n \pi ) \text { converges. }$

Find the Maclaurin series for the function $f ( x ) = e ^ { \frac { x ^ { 7 } } { 9 } }$