Question 1

Find the sum of the finite geometric series. $\sum _ { n = 1 } ^ { 4 } 3 ( - 3 ) ^ { n }$

Accepted Answer

A)  $180$ 
B)  $- 63$ 
C)  $1638$ 
D)  $132,858$ 
E)  $- 549$ 
A)  $180$ 
B)  $- 63$ 
C)  $1638$ 
D)  $132,858$ 
E)  $- 549$

Question 2

Find the fifth term of the sequence that has the given nth term. $$a _ { n } = \frac { ( n + 2 ) ! } { n ! }$$

Accepted Answer

A)  $42$ 
B)  $30$ 
C)  $12$ 
D)  $20$ 
E)  $6$ 
A)  $42$ 
B)  $30$ 
C)  $12$ 
D)  $20$ 
E)  $6$

Question 3

Use Newton's Method to approximate the zero(s) of the function $f ( x ) = x ^ { 5 } + 4 x + 2$ accurate to three decimal places.

Accepted Answer

A) 0.493 
B) 0.497 
C) -0.493 
D) -0.497 
E) 0.486 
A) 0.493 
B) 0.497 
C) -0.493 
D) -0.497 
E) 0.486

Question 4

Match the geometric sequence with its graph from the choices below. $a _ { n } = 12 \left( - \frac { 4 } { 3 } \right) ^ { n - 1 }$

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 5

Write an expression for the nth term of the sequence. $9 , - \frac { 9 } { 4 } , \frac { 9 } { 9 } , - \frac { 9 } { 16 } , \mathrm {~L}$

Accepted Answer

A)  $\frac { 9 - 1 ^ { n + 1 } } { n ^ { 2 } }$ 
B)  $\frac { 9 ( - 1 ) ^ { n } } { n ^ { 2 } }$ 
C)  $\frac { 9 ( - 1 ) ^ { n - 1 } } { n ^ { 2 } }$ 
D)  $\frac { - 1 ^ { n - 1 } } { 9 n ^ { 2 } }$ 
E)  $\frac { ( - 1 ) ^ { n } } { 9 n ^ { 2 } }$ 
A)  $\frac { 9 - 1 ^ { n + 1 } } { n ^ { 2 } }$ 
B)  $\frac { 9 ( - 1 ) ^ { n } } { n ^ { 2 } }$ 
C)  $\frac { 9 ( - 1 ) ^ { n - 1 } } { n ^ { 2 } }$ 
D)  $\frac { - 1 ^ { n - 1 } } { 9 n ^ { 2 } }$ 
E)  $\frac { ( - 1 ) ^ { n } } { 9 n ^ { 2 } }$

Question 6

Find the partial sum. $\sum _ { n = 1 } ^ { 140 } ( - 4 n + 1 )$

Accepted Answer

A) -39,339 
B) -38,781 
C) -39,620 
D) -39,903 
E) -39,340 
A) -39,339 
B) -38,781 
C) -39,620 
D) -39,903 
E) -39,340

Question 7

Find the limit of the following sequence. $a _ { n } = \frac { ( n - 2 ) ! } { n ! }$

Accepted Answer

A) 3 
B) 1 
C)  $0$ 
D) -3 
E) The sequence diverges. 
A) 3 
B) 1 
C)  $0$ 
D) -3 
E) The sequence diverges.

Question 8

Determine the convergence or divergence of the series. $\sum _ { n = 1 } ^ { \infty } \frac { 4 } { n \cdot \sqrt [ 3 ] { n } }$

Accepted Answer

A) diverges 
B) converges 
C) inconclusive 
A) diverges 
B) converges 
C) inconclusive

Question 9

Find the radius of convergence of the series $\sum _ { n = 0 } ^ { \infty } \left( \frac { x } { 6 } \right) ^ { n }$ .

Accepted Answer

A) 12 
B) 14 
C) 1 
D) 7 
E) 6 
A) 12 
B) 14 
C) 1 
D) 7 
E) 6

Question 10

Find a formula for $a _ { n }$ for the arithmetic sequence below. $- 2 , - 1,0,1,2 , \mathbf { K }$

Accepted Answer

A)  $a _ { n } = 3 n + 1$ 
B)  $a _ { n } = - 3 n - 1$ 
C)  $a _ { n } = - n - 3$ 
D)  $a _ { n } = n - 3$ 
E)  $a _ { n } = n + 3$ 
A)  $a _ { n } = 3 n + 1$ 
B)  $a _ { n } = - 3 n - 1$ 
C)  $a _ { n } = - n - 3$ 
D)  $a _ { n } = n - 3$ 
E)  $a _ { n } = n + 3$

Question 11

The seating section in a theater has 29 seats in the first row, 34 seats in the second row, and so on, increasing by 5 seats each row for a total of 15 rows. How many seats are in the thirteenth row?

Accepted Answer

A)  $79$ 
B)  $89$ 
C)  $84$ 
D)  $59$ 
E)  $54$ 
A)  $79$ 
B)  $89$ 
C)  $84$ 
D)  $59$ 
E)  $54$

Question 12

Use a symbolic differentiation utility to find the fourth-degree Taylor polynomials (centred at zero) . $f ( x ) = \frac { 1 } { \sqrt [ 3 ] { x + 1 } }$

Accepted Answer

A)  $S _ { 4 } ( x ) = 1 - \frac { x } { 3 } + \frac { 2 x ^ { 2 } } { 9 } - \frac { 14 x ^ { 3 } } { 81 } + \frac { 35 x ^ { 4 } } { 243 }$ 
B)  $S _ { 4 } ( x ) = 1 + \frac { x } { 3 } + \frac { 2 x ^ { 2 } } { 9 } + \frac { 14 x ^ { 3 } } { 81 } + \frac { 35 x ^ { 4 } } { 243 }$ 
C)  $S _ { 4 } ( x ) = 1 - \frac { x } { 3 } - \frac { 2 x ^ { 2 } } { 9 } - \frac { 14 x ^ { 3 } } { 81 } - \frac { 35 x ^ { 4 } } { 243 }$ 
D)  $S _ { 4 } ( x ) = 1 + \frac { x } { 3 } - \frac { 2 x ^ { 2 } } { 9 } + \frac { 14 x ^ { 3 } } { 81 } - \frac { 35 x ^ { 4 } } { 243 }$ 
E)  $S _ { 4 } ( x ) = 1 - \frac { x } { 3 } + \frac { 2 x ^ { 2 } } { 9 } + \frac { 14 x ^ { 3 } } { 81 } - \frac { 35 x ^ { 4 } } { 243 }$ 
A)  $S _ { 4 } ( x ) = 1 - \frac { x } { 3 } + \frac { 2 x ^ { 2 } } { 9 } - \frac { 14 x ^ { 3 } } { 81 } + \frac { 35 x ^ { 4 } } { 243 }$ 
B)  $S _ { 4 } ( x ) = 1 + \frac { x } { 3 } + \frac { 2 x ^ { 2 } } { 9 } + \frac { 14 x ^ { 3 } } { 81 } + \frac { 35 x ^ { 4 } } { 243 }$ 
C)  $S _ { 4 } ( x ) = 1 - \frac { x } { 3 } - \frac { 2 x ^ { 2 } } { 9 } - \frac { 14 x ^ { 3 } } { 81 } - \frac { 35 x ^ { 4 } } { 243 }$ 
D)  $S _ { 4 } ( x ) = 1 + \frac { x } { 3 } - \frac { 2 x ^ { 2 } } { 9 } + \frac { 14 x ^ { 3 } } { 81 } - \frac { 35 x ^ { 4 } } { 243 }$ 
E)  $S _ { 4 } ( x ) = 1 - \frac { x } { 3 } + \frac { 2 x ^ { 2 } } { 9 } + \frac { 14 x ^ { 3 } } { 81 } - \frac { 35 x ^ { 4 } } { 243 }$

Question 13

Use the sixth-degree Taylor polynomial centered at zero for the function $f ( x ) = \frac { 1 } { \sqrt { 1 + x ^ { 2 } } }$ to approximate the integral $\int _ { 0 } ^ { 2 / 3 } \frac { 1 } { \sqrt { 1 + x ^ { 2 } } }$ . Round your answer to nearest ten thousandth.

Accepted Answer

A) 0.6222 
B) 0.6322 
C) 0.8772 
D) 0.6272 
E) 0.3772 
A) 0.6222 
B) 0.6322 
C) 0.8772 
D) 0.6272 
E) 0.3772

Question 14

Use summation notation to write the sum. $4 - 8 + 16 - K + 64$

Accepted Answer

A)  $\sum _ { n = 0 } ^ { 4 } 4 ( - 2 ) ^ { n - 1 }$ 
B)  $\sum _ { n = 1 } ^ { 3 } 4 ( - 2 ) ^ { n }$ 
C)  $\sum _ { n = 1 } ^ { 5 } 4 ( - 2 ) ^ { n - 1 }$ 
D)  $\sum _ { n = 1 } ^ { 3 } 4 ( - 2 ) ^ { n - 1 }$ 
E)  $\sum _ { n = 1 } ^ { 4 } 4 ( - 2 ) ^ { n + 1 }$ 
A)  $\sum _ { n = 0 } ^ { 4 } 4 ( - 2 ) ^ { n - 1 }$ 
B)  $\sum _ { n = 1 } ^ { 3 } 4 ( - 2 ) ^ { n }$ 
C)  $\sum _ { n = 1 } ^ { 5 } 4 ( - 2 ) ^ { n - 1 }$ 
D)  $\sum _ { n = 1 } ^ { 3 } 4 ( - 2 ) ^ { n - 1 }$ 
E)  $\sum _ { n = 1 } ^ { 4 } 4 ( - 2 ) ^ { n + 1 }$

Question 15

Write the first five terms of the sequence. (Assume that n begins with 1.) $$a _ { n } = 3 n + 7$$

Accepted Answer

A) -4, -1, 2, 5, 8 
B) 7, 10, 13, 16, 19 
C) 10, 6, 9, 12, 15 
D) 10, 13, 16, 19, 22 
E) 10, 17, 24, 31, 38 
A) -4, -1, 2, 5, 8 
B) 7, 10, 13, 16, 19 
C) 10, 6, 9, 12, 15 
D) 10, 13, 16, 19, 22 
E) 10, 17, 24, 31, 38

Question 16

Give an example of a sequence that converges to $\frac { 1 } { 4 }$ .

Accepted Answer

A)  $a _ { n } = \frac { 8 n ^ { 4 } - 5 } { 2 n ^ { 4 } - 6 }$ 
B)  $a _ { n } = \frac { 2 + 5 n ^ { 4 } } { 8 + 6 n ^ { 4 } }$ 
C)  $a _ { n } = \frac { 3 n ^ { 4 } - 5 } { 9 n ^ { 4 } - 6 }$ 
D)  $a _ { n } = \frac { 2 n ^ { 4 } - 5 } { 8 n ^ { 4 } - 6 }$ 
E)  $a _ { n } = \frac { 3 + 5 n ^ { 4 } } { 9 + 6 n ^ { 4 } }$ 
A)  $a _ { n } = \frac { 8 n ^ { 4 } - 5 } { 2 n ^ { 4 } - 6 }$ 
B)  $a _ { n } = \frac { 2 + 5 n ^ { 4 } } { 8 + 6 n ^ { 4 } }$ 
C)  $a _ { n } = \frac { 3 n ^ { 4 } - 5 } { 9 n ^ { 4 } - 6 }$ 
D)  $a _ { n } = \frac { 2 n ^ { 4 } - 5 } { 8 n ^ { 4 } - 6 }$ 
E)  $a _ { n } = \frac { 3 + 5 n ^ { 4 } } { 9 + 6 n ^ { 4 } }$

Question 17

Find the indicated nth partial sum of the arithmetic sequence. 3.4, 6.2, 9, 11.8, ..., n = 10

Accepted Answer

A) 181 
B) 188 
C) 160 
D) 159.4 
E) 160.6 
A) 181 
B) 188 
C) 160 
D) 159.4 
E) 160.6

Question 18

A Taylor polynomial approximation of $f ( x ) = e ^ { - \frac { x ^ { 2 } } { 2 } }$ is given below. Use a graphing utility to graph both functions. $y = - \frac { 1 } { 2 } x ^ { 2 } + 1$

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 19

Find the radius of convergence of $f ^ { \prime } ( x )$ where $f ( x ) = \sum _ { n = 0 } ^ { \infty } \left( \frac { x } { 6 } \right) ^ { n }$ .

Accepted Answer

A) 1 
B)  $\infty$ 
C)  $\frac { 1 } { 6 }$ 
D) 6 
E) 12 
A) 1 
B)  $\infty$ 
C)  $\frac { 1 } { 6 }$ 
D) 6 
E) 12

Question 20

Approximate the sum of the convergent series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 } }$ using four terms. Estimate the maximum error of your approximation. Round your answers to four decimal places.

Accepted Answer

A) The approximate value is 2.4470.The maximum error of your approximation is 0.2532. 
B) The approximate value is 1.4236.The maximum error of your approximation is 0.2500. 
C) The approximate value is 2.4292.The maximum error of your approximation is 0.2665. 
D) The approximate value is 3.4207.The maximum error of your approximation is 1.2471. 
E) The approximate value is 1.4260.The maximum error of your approximation is 0.2517. 
A) The approximate value is 2.4470.The maximum error of your approximation is 0.2532. 
B) The approximate value is 1.4236.The maximum error of your approximation is 0.2500. 
C) The approximate value is 2.4292.The maximum error of your approximation is 0.2665. 
D) The approximate value is 3.4207.The maximum error of your approximation is 1.2471. 
E) The approximate value is 1.4260.The maximum error of your approximation is 0.2517.

Find the sum of the finite geometric series. $\sum _ { n = 1 } ^ { 4 } 3 ( - 3 ) ^ { n }$

Find the fifth term of the sequence that has the given nth term. $a _ { n } = \frac { ( n + 2 ) ! } { n ! }$

Use Newton's Method to approximate the zero(s) of the function $f ( x ) = x ^ { 5 } + 4 x + 2$ accurate to three decimal places.

Match the geometric sequence with its graph from the choices below. $a _ { n } = 12 \left( - \frac { 4 } { 3 } \right) ^ { n - 1 }$

Write an expression for the nth term of the sequence. $9 , - \frac { 9 } { 4 } , \frac { 9 } { 9 } , - \frac { 9 } { 16 } , \mathrm {~L}$

Find the partial sum. $\sum _ { n = 1 } ^ { 140 } ( - 4 n + 1 )$

Find the limit of the following sequence. $a _ { n } = \frac { ( n - 2 ) ! } { n ! }$

Determine the convergence or divergence of the series. $\sum _ { n = 1 } ^ { \infty } \frac { 4 } { n \cdot \sqrt [ 3 ] { n } }$

Find the radius of convergence of the series $\sum _ { n = 0 } ^ { \infty } \left( \frac { x } { 6 } \right) ^ { n }$ .

Find a formula for $a _ { n }$ for the arithmetic sequence below. $- 2 , - 1,0,1,2 , \mathbf { K }$

The seating section in a theater has 29 seats in the first row, 34 seats in the second row, and so on, increasing by 5 seats each row for a total of 15 rows. How many seats are in the thirteenth row?

Use a symbolic differentiation utility to find the fourth-degree Taylor polynomials (centred at zero) . $f ( x ) = \frac { 1 } { \sqrt [ 3 ] { x + 1 } }$

Use the sixth-degree Taylor polynomial centered at zero for the function $f ( x ) = \frac { 1 } { \sqrt { 1 + x ^ { 2 } } }$ to approximate the integral $\int _ { 0 } ^ { 2 / 3 } \frac { 1 } { \sqrt { 1 + x ^ { 2 } } }$ . Round your answer to nearest ten thousandth.

Use summation notation to write the sum. $4 - 8 + 16 - K + 64$

Write the first five terms of the sequence. (Assume that n begins with 1.) $a _ { n } = 3 n + 7$

Give an example of a sequence that converges to $\frac { 1 } { 4 }$ .

Find the indicated nth partial sum of the arithmetic sequence. 3.4, 6.2, 9, 11.8, ..., n = 10

A Taylor polynomial approximation of $f ( x ) = e ^ { - \frac { x ^ { 2 } } { 2 } }$ is given below. Use a graphing utility to graph both functions. $y = - \frac { 1 } { 2 } x ^ { 2 } + 1$

Find the radius of convergence of $f ^ { \prime } ( x )$ where $f ( x ) = \sum _ { n = 0 } ^ { \infty } \left( \frac { x } { 6 } \right) ^ { n }$ .

Approximate the sum of the convergent series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 } }$ using four terms. Estimate the maximum error of your approximation. Round your answers to four decimal places.

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Exam 16: Series and Taylor Polynomials Web

Find the sum of the finite geometric series. ∑n=143(−3)n\sum _ { n = 1 } ^ { 4 } 3 ( - 3 ) ^ { n }∑n=14​3(−3)n

Find the fifth term of the sequence that has the given nth term. an=(n+2)!n!a _ { n } = \frac { ( n + 2 ) ! } { n ! }an​=n!(n+2)!​

Use Newton's Method to approximate the zero(s) of the function f(x)=x5+4x+2f ( x ) = x ^ { 5 } + 4 x + 2f(x)=x5+4x+2 accurate to three decimal places.

Match the geometric sequence with its graph from the choices below. an=12(−43)n−1a _ { n } = 12 \left( - \frac { 4 } { 3 } \right) ^ { n - 1 }an​=12(−34​)n−1

Write an expression for the nth term of the sequence. 9,−94,99,−916, L9 , - \frac { 9 } { 4 } , \frac { 9 } { 9 } , - \frac { 9 } { 16 } , \mathrm {~L}9,−49​,99​,−169​, L

Find the partial sum. ∑n=1140(−4n+1)\sum _ { n = 1 } ^ { 140 } ( - 4 n + 1 )∑n=1140​(−4n+1)

Find the limit of the following sequence. an=(n−2)!n!a _ { n } = \frac { ( n - 2 ) ! } { n ! }an​=n!(n−2)!​

Determine the convergence or divergence of the series. ∑n=1∞4n⋅n3\sum _ { n = 1 } ^ { \infty } \frac { 4 } { n \cdot \sqrt [ 3 ] { n } }∑n=1∞​n⋅3n​4​

Find the radius of convergence of the series ∑n=0∞(x6)n\sum _ { n = 0 } ^ { \infty } \left( \frac { x } { 6 } \right) ^ { n }∑n=0∞​(6x​)n .

Find a formula for ana _ { n }an​ for the arithmetic sequence below. −2,−1,0,1,2,K- 2 , - 1,0,1,2 , \mathbf { K }−2,−1,0,1,2,K

The seating section in a theater has 29 seats in the first row, 34 seats in the second row, and so on, increasing by 5 seats each row for a total of 15 rows. How many seats are in the thirteenth row?

Use a symbolic differentiation utility to find the fourth-degree Taylor polynomials (centred at zero) . f(x)=1x+13f ( x ) = \frac { 1 } { \sqrt [ 3 ] { x + 1 } }f(x)=3x+1​1​

Use summation notation to write the sum. 4−8+16−K+644 - 8 + 16 - K + 644−8+16−K+64

Write the first five terms of the sequence. (Assume that n begins with 1.) an=3n+7a _ { n } = 3 n + 7an​=3n+7

Give an example of a sequence that converges to 14\frac { 1 } { 4 }41​ .

Find the indicated nth partial sum of the arithmetic sequence. 3.4, 6.2, 9, 11.8, ..., n = 10

A Taylor polynomial approximation of f(x)=e−x22f ( x ) = e ^ { - \frac { x ^ { 2 } } { 2 } }f(x)=e−2x2​ is given below. Use a graphing utility to graph both functions. y=−12x2+1y = - \frac { 1 } { 2 } x ^ { 2 } + 1y=−21​x2+1

Find the radius of convergence of f′(x)f ^ { \prime } ( x )f′(x) where f(x)=∑n=0∞(x6)nf ( x ) = \sum _ { n = 0 } ^ { \infty } \left( \frac { x } { 6 } \right) ^ { n }f(x)=∑n=0∞​(6x​)n .

Approximate the sum of the convergent series ∑n=1∞1n2\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 } }∑n=1∞​n21​ using four terms. Estimate the maximum error of your approximation. Round your answers to four decimal places.

Fundamental Concepts of Algebra

Equations and Inequalities

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Systems of Equations and Inequalities

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Find the sum of the finite geometric series. $\sum _ { n = 1 } ^ { 4 } 3 ( - 3 ) ^ { n }$

Find the fifth term of the sequence that has the given nth term. $a _ { n } = \frac { ( n + 2 ) ! } { n ! }$

Use Newton's Method to approximate the zero(s) of the function $f ( x ) = x ^ { 5 } + 4 x + 2$ accurate to three decimal places.

Match the geometric sequence with its graph from the choices below. $a _ { n } = 12 \left( - \frac { 4 } { 3 } \right) ^ { n - 1 }$

Write an expression for the nth term of the sequence. $9 , - \frac { 9 } { 4 } , \frac { 9 } { 9 } , - \frac { 9 } { 16 } , \mathrm {~L}$

Find the partial sum. $\sum _ { n = 1 } ^ { 140 } ( - 4 n + 1 )$

Find the limit of the following sequence. $a _ { n } = \frac { ( n - 2 ) ! } { n ! }$

Determine the convergence or divergence of the series. $\sum _ { n = 1 } ^ { \infty } \frac { 4 } { n \cdot \sqrt [ 3 ] { n } }$

Find the radius of convergence of the series $\sum _ { n = 0 } ^ { \infty } \left( \frac { x } { 6 } \right) ^ { n }$ .

Find a formula for $a _ { n }$ for the arithmetic sequence below. $- 2 , - 1,0,1,2 , \mathbf { K }$

Use a symbolic differentiation utility to find the fourth-degree Taylor polynomials (centred at zero) . $f ( x ) = \frac { 1 } { \sqrt [ 3 ] { x + 1 } }$

Use summation notation to write the sum. $4 - 8 + 16 - K + 64$

Write the first five terms of the sequence. (Assume that n begins with 1.) $a _ { n } = 3 n + 7$

Give an example of a sequence that converges to $\frac { 1 } { 4 }$ .

A Taylor polynomial approximation of $f ( x ) = e ^ { - \frac { x ^ { 2 } } { 2 } }$ is given below. Use a graphing utility to graph both functions. $y = - \frac { 1 } { 2 } x ^ { 2 } + 1$

Find the radius of convergence of $f ^ { \prime } ( x )$ where $f ( x ) = \sum _ { n = 0 } ^ { \infty } \left( \frac { x } { 6 } \right) ^ { n }$ .

Approximate the sum of the convergent series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 } }$ using four terms. Estimate the maximum error of your approximation. Round your answers to four decimal places.