Question 1

Write an expression for the nth term of the sequence $\frac { 1 } { 3 } , \frac { 2 } { 9 } , \frac { 4 } { 27 } , \frac { 8 } { 81 } , \mathrm {~K}$ .

Accepted Answer

A)  $a _ { n } = \frac { 2 ^ { n - 1 } } { 3 ^ { n } }$ 
B)  $a _ { n } = \frac { 2 ^ { n + 1 } } { 3 ^ { n } }$ 
C)  $a _ { n } = \frac { 2 ^ { n } } { 3 }$ 
D)  $a _ { n } = \frac { 6 ^ { n + 1 } } { 4 ^ { n - 1 } }$ 
E)  $a _ { n } = \frac { 6 ^ { n - 1 } } { 4 ^ { n } }$ 
A)  $a _ { n } = \frac { 2 ^ { n - 1 } } { 3 ^ { n } }$ 
B)  $a _ { n } = \frac { 2 ^ { n + 1 } } { 3 ^ { n } }$ 
C)  $a _ { n } = \frac { 2 ^ { n } } { 3 }$ 
D)  $a _ { n } = \frac { 6 ^ { n + 1 } } { 4 ^ { n - 1 } }$ 
E)  $a _ { n } = \frac { 6 ^ { n - 1 } } { 4 ^ { n } }$ A

Question 2

Find the limit of the following sequence. $a _ { n } = 1 + ( - 1 ) ^ { n }$

Accepted Answer

A)  $\infty$ 
B) 1 
C) 2 
D)  $- \infty$ 
E) The sequence diverges. 
A)  $\infty$ 
B) 1 
C) 2 
D)  $- \infty$ 
E) The sequence diverges. E

Question 3

Use the Ratio Test to determine the convergence or divergence of the series $\sum _ { n = 0 } ^ { \infty } \frac { ( - 6 ) ^ { n } 2 ^ { n } } { n ! }$ .

Accepted Answer

A) The series converges. 
B) The series diverges. 
A) The series converges. 
B) The series diverges. A

Question 4

Use a symbolic differentiation utility to find the Taylor polynomials (centred at zero) of degrees (a) 2, (b) 4, (c) 6, (d) 8. $f ( x ) = \frac { 1 } { 1 + x ^ { 2 } }$

Accepted Answer

A)  $S _ { 2 } ( x ) = 1 - x ^ { 2 } , S _ { 4 } ( x ) = 1 - x ^ { 2 } + x ^ { 4 } , S _ { 6 } ( x ) = 1 - x ^ { 2 } + x ^ { 4 } - x ^ { 6 } , S _ { 8 } ( x ) = 1 - x ^ { 2 } + x ^ { 4 } - x ^ { 6 } + x ^ { 8 }$ 
B)  $S _ { 2 } ( x ) = 1 + x ^ { 2 } , S _ { 4 } ( x ) = 1 + x ^ { 2 } + x ^ { 4 } , S _ { 6 } ( x ) = 1 + x ^ { 2 } + x ^ { 4 } + x ^ { 6 } , S _ { 8 } ( x ) = 1 + x ^ { 2 } + x ^ { 4 } + x ^ { 6 } + x ^ { 8 }$ 
C)  $S _ { 2 } ( x ) = 1 - x ^ { 2 } , S _ { 4 } ( x ) = 1 - x ^ { 2 } - x ^ { 4 } , S _ { 6 } ( x ) = 1 - x ^ { 2 } - x ^ { 4 } - x ^ { 6 } , S _ { 8 } ( x ) = 1 - x ^ { 2 } - x ^ { 4 } - x ^ { 6 } - x ^ { 8 }$ 
D)  $S _ { 2 } ( x ) = 1 + x ^ { 2 } , S _ { 4 } ( x ) = 1 + x ^ { 2 } - x ^ { 4 } , S _ { 6 } ( x ) = 1 + x ^ { 2 } - x ^ { 4 } + x ^ { 6 } , S _ { 8 } ( x ) = 1 + x ^ { 2 } - x ^ { 4 } + x ^ { 6 } - x ^ { 8 }$ 
E)  $S _ { 2 } ( x ) = 1 - x ^ { 2 } , S _ { 4 } ( x ) = 1 - x ^ { 2 } + x ^ { 4 } , S _ { 6 } ( x ) = 1 - x ^ { 2 } + x ^ { 4 } + x ^ { 6 } , S _ { 8 } ( x ) = 1 - x ^ { 2 } + x ^ { 4 } + x ^ { 6 } + x ^ { 8 }$ 
A)  $S _ { 2 } ( x ) = 1 - x ^ { 2 } , S _ { 4 } ( x ) = 1 - x ^ { 2 } + x ^ { 4 } , S _ { 6 } ( x ) = 1 - x ^ { 2 } + x ^ { 4 } - x ^ { 6 } , S _ { 8 } ( x ) = 1 - x ^ { 2 } + x ^ { 4 } - x ^ { 6 } + x ^ { 8 }$ 
B)  $S _ { 2 } ( x ) = 1 + x ^ { 2 } , S _ { 4 } ( x ) = 1 + x ^ { 2 } + x ^ { 4 } , S _ { 6 } ( x ) = 1 + x ^ { 2 } + x ^ { 4 } + x ^ { 6 } , S _ { 8 } ( x ) = 1 + x ^ { 2 } + x ^ { 4 } + x ^ { 6 } + x ^ { 8 }$ 
C)  $S _ { 2 } ( x ) = 1 - x ^ { 2 } , S _ { 4 } ( x ) = 1 - x ^ { 2 } - x ^ { 4 } , S _ { 6 } ( x ) = 1 - x ^ { 2 } - x ^ { 4 } - x ^ { 6 } , S _ { 8 } ( x ) = 1 - x ^ { 2 } - x ^ { 4 } - x ^ { 6 } - x ^ { 8 }$ 
D)  $S _ { 2 } ( x ) = 1 + x ^ { 2 } , S _ { 4 } ( x ) = 1 + x ^ { 2 } - x ^ { 4 } , S _ { 6 } ( x ) = 1 + x ^ { 2 } - x ^ { 4 } + x ^ { 6 } , S _ { 8 } ( x ) = 1 + x ^ { 2 } - x ^ { 4 } + x ^ { 6 } - x ^ { 8 }$ 
E)  $S _ { 2 } ( x ) = 1 - x ^ { 2 } , S _ { 4 } ( x ) = 1 - x ^ { 2 } + x ^ { 4 } , S _ { 6 } ( x ) = 1 - x ^ { 2 } + x ^ { 4 } + x ^ { 6 } , S _ { 8 } ( x ) = 1 - x ^ { 2 } + x ^ { 4 } + x ^ { 6 } + x ^ { 8 }$

Question 5

Use Newton's Method to approximate the zero(s) of the function $f ( x ) = x - 2 \sqrt { x + 1 }$ accurate to three decimal places.

Accepted Answer

A) 4.828 
B) 4.832 
C) -4.828 
D) -4.832 
E) 4.822 
A) 4.828 
B) 4.832 
C) -4.828 
D) -4.832 
E) 4.822

Question 6

Determine whether the sequence is arithmetic. If so, find the common difference. (Assume that n begins with 1.) $\alpha _ { n } = - 2 - 7 n$

Accepted Answer

A) 7 
B) 2 
C) -2 
D) -7 
E) not arithmetic 
A) 7 
B) 2 
C) -2 
D) -7 
E) not arithmetic

Question 7

Find the sum of the finite geometric series. Round to the nearest hundredth. $\sum _ { n = 1 } ^ { 6 } \left( - \frac { 8 } { 9 } \right) ^ { n }$

Accepted Answer

A)  $- 0.24$ 
B)  $- 0.10$ 
C)  $- 0.33$ 
D)  $- 0.80$ 
E)  $- 0.63$ 
A)  $- 0.24$ 
B)  $- 0.10$ 
C)  $- 0.33$ 
D)  $- 0.80$ 
E)  $- 0.63$

Question 8

Use the Ratio Test to determine the convergence or divergence of the series. $\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 3 } } { 8 ^ { - n } }$

Accepted Answer

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

Question 9

Apply Taylor's Theorem to find the power series centered at $c = 12$ for the function $f ( x ) = e ^ { x }$ .

Accepted Answer

A)  $e ^ { 12 } \sum _ { n = 0 } ^ { \infty } \frac { ( x ) ^ { n } } { n ! }$ 
B)  $\sum _ { n = 1 } ^ { \infty } \frac { ( x - 12 ) ^ { n } } { ( n + 1 ) ! }$ 
C)  $e ^ { 12 } \sum _ { n = 0 } ^ { \infty } \frac { ( x - 12 ) ^ { n } } { n ! }$ 
D)  $e ^ { 12 } \sum _ { n = 0 } ^ { \infty } \frac { ( x ) ^ { n + 1 } } { ( n + 1 ) ! }$ 
E)  $\sum _ { n = 0 } ^ { \infty } \frac { ( x - 12 ) ^ { 2 n } } { 2 n ! }$ 
A)  $e ^ { 12 } \sum _ { n = 0 } ^ { \infty } \frac { ( x ) ^ { n } } { n ! }$ 
B)  $\sum _ { n = 1 } ^ { \infty } \frac { ( x - 12 ) ^ { n } } { ( n + 1 ) ! }$ 
C)  $e ^ { 12 } \sum _ { n = 0 } ^ { \infty } \frac { ( x - 12 ) ^ { n } } { n ! }$ 
D)  $e ^ { 12 } \sum _ { n = 0 } ^ { \infty } \frac { ( x ) ^ { n + 1 } } { ( n + 1 ) ! }$ 
E)  $\sum _ { n = 0 } ^ { \infty } \frac { ( x - 12 ) ^ { 2 n } } { 2 n ! }$

Question 10

What are the next three terms in the arithmetic sequence $7,3 , - 1 , \ldots$ ?

Accepted Answer

A)  $- 5 , - 9 , - 13$   
B)  $11,15,19$ 
C)  $20 , - 80,320$ 
D)  $28,35,42$   
E)  $- 1 , - 5 , - 9$ 
A)  $- 5 , - 9 , - 13$   
B)  $11,15,19$ 
C)  $20 , - 80,320$ 
D)  $28,35,42$   
E)  $- 1 , - 5 , - 9$

Question 11

Find the indicated nth term of the geometric sequence. 7th term: $a _ { 5 } = \frac { 4 } { 81 } , a _ { 10 } = \frac { 4 } { 19,683 }$

Accepted Answer

A)  $\frac { 4 } { 2187 }$ 
B)  $\frac { 3 } { 4096 }$ 
C)  $\frac { 4 } { 6561 }$ 
D)  $\frac { 4 } { 729 }$ 
E)  $\frac { 4 } { 243 }$ 
A)  $\frac { 4 } { 2187 }$ 
B)  $\frac { 3 } { 4096 }$ 
C)  $\frac { 4 } { 6561 }$ 
D)  $\frac { 4 } { 729 }$ 
E)  $\frac { 4 } { 243 }$

Question 12

Determine the convergence or divergence of the following series. Use a symbolic algebra utility to verify your result. $\sum _ { n = 0 } ^ { \infty } \frac { 4 } { 2 ^ { n } }$

Accepted Answer

A) The series diverges. 
B) The series converges. 
A) The series diverges. 
B) The series converges.

Question 13

Evaluate the series. $\sum _ { i = 1 } ^ { 4 } ( 4 i + 3 ) ( 3 i - 4 )$

Accepted Answer

A)  $15$ 
B)  $242$ 
C)  $873$ 
D)  $495$ 
E)  $2100$ 
A)  $15$ 
B)  $242$ 
C)  $873$ 
D)  $495$ 
E)  $2100$

Question 14

The repeating decimal $0 . \overline { 2 }$ is expressed as a geometric series $0.2 + 0.02 + 0.002 + 0.0002 + \ldots$ . Write the decimal $0 . \overline { 2 }$ as the ratio of two integers.

Accepted Answer

A)  $\frac { 2 } { 99 }$ 
B)  $\frac { 7 } { 33 }$ 
C)  $\frac { 2 } { 9 }$ 
D)  $\frac { 5 } { 11 }$ 
E)  $\frac { 9 } { 2 }$ 
A)  $\frac { 2 } { 99 }$ 
B)  $\frac { 7 } { 33 }$ 
C)  $\frac { 2 } { 9 }$ 
D)  $\frac { 5 } { 11 }$ 
E)  $\frac { 9 } { 2 }$

Question 15

Find the limit of the following sequence. $a _ { n } = \frac { n ^ { 2 } - 25 } { n + 5 }$

Accepted Answer

A)  $\infty$ 
B)  $0$ 
C)  $0.5$ 
D)  $- \infty $ 
E) The sequence diverges. 
A)  $\infty$ 
B)  $0$ 
C)  $0.5$ 
D)  $- \infty $ 
E) The sequence diverges.

Question 16

Write the first five terms of the sequence of partial sums. $2 + \frac { 2 } { 4 } + \frac { 2 } { 9 } + \frac { 2 } { 16 } + \frac { 2 } { 25 } + L$

Accepted Answer

A)  $2 , \frac { 5 } { 3 } , \frac { 49 } { 16 } , \frac { 205 } { 66 } , \frac { 5269 } { 1600 }$ 
B)  $2 , \frac { 1 } { 2 } , \frac { 2 } { 9 } , \frac { 1 } { 8 } , \frac { 2 } { 25 }$ 
C)  $2 , \frac { 5 } { 2 } , \frac { 49 } { 18 } , \frac { 205 } { 72 } , \frac { 5269 } { 1800 }$ 
D)  $2 , \frac { 7 } { 2 } , \frac { 55 } { 18 } , \frac { 215 } { 72 } , \frac { 53 } { 18 }$ 
E)  $2 , \frac { 1 } { 1 } , \frac { 45 } { 16 } , \frac { 100 } { 33 } , \frac { 105 } { 32 }$ 
A)  $2 , \frac { 5 } { 3 } , \frac { 49 } { 16 } , \frac { 205 } { 66 } , \frac { 5269 } { 1600 }$ 
B)  $2 , \frac { 1 } { 2 } , \frac { 2 } { 9 } , \frac { 1 } { 8 } , \frac { 2 } { 25 }$ 
C)  $2 , \frac { 5 } { 2 } , \frac { 49 } { 18 } , \frac { 205 } { 72 } , \frac { 5269 } { 1800 }$ 
D)  $2 , \frac { 7 } { 2 } , \frac { 55 } { 18 } , \frac { 215 } { 72 } , \frac { 53 } { 18 }$ 
E)  $2 , \frac { 1 } { 1 } , \frac { 45 } { 16 } , \frac { 100 } { 33 } , \frac { 105 } { 32 }$

Question 17

Match the sequence with the graph of its first 10 terms. $a _ { n } = \frac { 3 n } { n + 1 }$

Accepted Answer

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

Question 18

Find the limit of the sequence $a _ { n } = \frac { 1 } { n ^ { 9 / 2 } }$ .

Accepted Answer

A)  $\infty$ 
B)  $1$ 
C)  $0$ 
D)  $\frac { 1 } { 2 }$ 
E) The sequence diverges. 
A)  $\infty$ 
B)  $1$ 
C)  $0$ 
D)  $\frac { 1 } { 2 }$ 
E) The sequence diverges.

Question 19

Approximate, to three decimal places, the x-value of the point of intersection of the graphs of f(x) and g(x). Round your answer to three decimal places. $$f ( x ) = - x$$ $$g ( x ) = \ln ( x )$$

Accepted Answer

A) 0.572 
B) 0.562 
C) 0.567 
D) 0.517 
E) 0.617 
A) 0.572 
B) 0.562 
C) 0.567 
D) 0.517 
E) 0.617

Question 20

Determine the maximum error guaranteed by Taylor's Theorem with Remainder when the fifth-degree Taylor polynomial is used to approximate $e ^ { - x }$ in the interval $[ 0,1 ]$ centered at 0. Round your answer to five decimal places.

Accepted Answer

A) 0.02139 
B) 0.00339 
C) 0.00833 
D) 0.00139 
E) 0.05139 
A) 0.02139 
B) 0.00339 
C) 0.00833 
D) 0.00139 
E) 0.05139

Write an expression for the nth term of the sequence $\frac { 1 } { 3 } , \frac { 2 } { 9 } , \frac { 4 } { 27 } , \frac { 8 } { 81 } , \mathrm {~K}$ .

Find the limit of the following sequence. $a _ { n } = 1 + ( - 1 ) ^ { n }$

Use the Ratio Test to determine the convergence or divergence of the series $\sum _ { n = 0 } ^ { \infty } \frac { ( - 6 ) ^ { n } 2 ^ { n } } { n ! }$ .

Use a symbolic differentiation utility to find the Taylor polynomials (centred at zero) of degrees (a) 2, (b) 4, (c) 6, (d) 8. $f ( x ) = \frac { 1 } { 1 + x ^ { 2 } }$

Use Newton's Method to approximate the zero(s) of the function $f ( x ) = x - 2 \sqrt { x + 1 }$ accurate to three decimal places.

Determine whether the sequence is arithmetic. If so, find the common difference. (Assume that n begins with 1.) $\alpha _ { n } = - 2 - 7 n$

Find the sum of the finite geometric series. Round to the nearest hundredth. $\sum _ { n = 1 } ^ { 6 } \left( - \frac { 8 } { 9 } \right) ^ { n }$

Use the Ratio Test to determine the convergence or divergence of the series. $\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 3 } } { 8 ^ { - n } }$

Apply Taylor's Theorem to find the power series centered at $c = 12$ for the function $f ( x ) = e ^ { x }$ .

What are the next three terms in the arithmetic sequence $7,3 , - 1 , \ldots$ ?

Find the indicated nth term of the geometric sequence. 7th term: $a _ { 5 } = \frac { 4 } { 81 } , a _ { 10 } = \frac { 4 } { 19,683 }$

Determine the convergence or divergence of the following series. Use a symbolic algebra utility to verify your result. $\sum _ { n = 0 } ^ { \infty } \frac { 4 } { 2 ^ { n } }$

Evaluate the series. $\sum _ { i = 1 } ^ { 4 } ( 4 i + 3 ) ( 3 i - 4 )$

The repeating decimal $0 . \overline { 2 }$ is expressed as a geometric series $0.2 + 0.02 + 0.002 + 0.0002 + \ldots$ . Write the decimal $0 . \overline { 2 }$ as the ratio of two integers.

Find the limit of the following sequence. $a _ { n } = \frac { n ^ { 2 } - 25 } { n + 5 }$

Write the first five terms of the sequence of partial sums. $2 + \frac { 2 } { 4 } + \frac { 2 } { 9 } + \frac { 2 } { 16 } + \frac { 2 } { 25 } + L$

Match the sequence with the graph of its first 10 terms. $a _ { n } = \frac { 3 n } { n + 1 }$

Find the limit of the sequence $a _ { n } = \frac { 1 } { n ^ { 9 / 2 } }$ .

Determine the maximum error guaranteed by Taylor's Theorem with Remainder when the fifth-degree Taylor polynomial is used to approximate $e ^ { - x }$ in the interval $[ 0,1 ]$ centered at 0. Round your answer to five decimal places.

Fundamental Concepts of Algebra

Equations and Inequalities

Functions and Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Systems of Equations and Inequalities

Matrices and Determinants

Limits and Derivatives

Applications of the Derivative

Further Applications of the Derivative

Derivatives of Exponential and Logarithmic Functions

Integration and Its Applications

Techniques of Integration

Functions of Several Variables

Trigonometric Functions Web

Probability Web

Filters

Exam 16: Series and Taylor Polynomials Web

Write an expression for the nth term of the sequence 13,29,427,881, K\frac { 1 } { 3 } , \frac { 2 } { 9 } , \frac { 4 } { 27 } , \frac { 8 } { 81 } , \mathrm {~K}31​,92​,274​,818​, K .

Find the limit of the following sequence. an=1+(−1)na _ { n } = 1 + ( - 1 ) ^ { n }an​=1+(−1)n

Use the Ratio Test to determine the convergence or divergence of the series ∑n=0∞(−6)n2nn!\sum _ { n = 0 } ^ { \infty } \frac { ( - 6 ) ^ { n } 2 ^ { n } } { n ! }∑n=0∞​n!(−6)n2n​ .

Use a symbolic differentiation utility to find the Taylor polynomials (centred at zero) of degrees (a) 2, (b) 4, (c) 6, (d) 8. f(x)=11+x2f ( x ) = \frac { 1 } { 1 + x ^ { 2 } }f(x)=1+x21​

Use Newton's Method to approximate the zero(s) of the function f(x)=x−2x+1f ( x ) = x - 2 \sqrt { x + 1 }f(x)=x−2x+1​ accurate to three decimal places.

Determine whether the sequence is arithmetic. If so, find the common difference. (Assume that n begins with 1.) αn=−2−7n\alpha _ { n } = - 2 - 7 nαn​=−2−7n

Find the sum of the finite geometric series. Round to the nearest hundredth. ∑n=16(−89)n\sum _ { n = 1 } ^ { 6 } \left( - \frac { 8 } { 9 } \right) ^ { n }∑n=16​(−98​)n

Use the Ratio Test to determine the convergence or divergence of the series. ∑n=1∞n38−n\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 3 } } { 8 ^ { - n } }∑n=1∞​8−nn3​

Apply Taylor's Theorem to find the power series centered at c=12c = 12c=12 for the function f(x)=exf ( x ) = e ^ { x }f(x)=ex .

What are the next three terms in the arithmetic sequence 7,3,−1,…7,3 , - 1 , \ldots7,3,−1,… ?

Find the indicated nth term of the geometric sequence. 7th term: a5=481,a10=419,683a _ { 5 } = \frac { 4 } { 81 } , a _ { 10 } = \frac { 4 } { 19,683 }a5​=814​,a10​=19,6834​

Determine the convergence or divergence of the following series. Use a symbolic algebra utility to verify your result. ∑n=0∞42n\sum _ { n = 0 } ^ { \infty } \frac { 4 } { 2 ^ { n } }∑n=0∞​2n4​

Evaluate the series. ∑i=14(4i+3)(3i−4)\sum _ { i = 1 } ^ { 4 } ( 4 i + 3 ) ( 3 i - 4 )∑i=14​(4i+3)(3i−4)

The repeating decimal 0.2‾0 . \overline { 2 }0.2 is expressed as a geometric series 0.2+0.02+0.002+0.0002+…0.2 + 0.02 + 0.002 + 0.0002 + \ldots0.2+0.02+0.002+0.0002+… . Write the decimal 0.2‾0 . \overline { 2 }0.2 as the ratio of two integers.

Find the limit of the following sequence. an=n2−25n+5a _ { n } = \frac { n ^ { 2 } - 25 } { n + 5 }an​=n+5n2−25​

Write the first five terms of the sequence of partial sums. 2+24+29+216+225+L2 + \frac { 2 } { 4 } + \frac { 2 } { 9 } + \frac { 2 } { 16 } + \frac { 2 } { 25 } + L2+42​+92​+162​+252​+L

Match the sequence with the graph of its first 10 terms. an=3nn+1a _ { n } = \frac { 3 n } { n + 1 }an​=n+13n​

Find the limit of the sequence an=1n9/2a _ { n } = \frac { 1 } { n ^ { 9 / 2 } }an​=n9/21​ .

Approximate, to three decimal places, the x-value of the point of intersection of the graphs of f(x) and g(x). Round your answer to three decimal places. f(x)=−xf ( x ) = - xf(x)=−x g(x)=ln⁡(x)g ( x ) = \ln ( x )g(x)=ln(x)

Determine the maximum error guaranteed by Taylor's Theorem with Remainder when the fifth-degree Taylor polynomial is used to approximate e−xe ^ { - x }e−x in the interval [0,1][ 0,1 ][0,1] centered at 0. Round your answer to five decimal places.

Fundamental Concepts of Algebra

Equations and Inequalities

Functions and Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Systems of Equations and Inequalities

Matrices and Determinants

Limits and Derivatives

Applications of the Derivative

Further Applications of the Derivative

Derivatives of Exponential and Logarithmic Functions

Integration and Its Applications

Techniques of Integration

Functions of Several Variables

Trigonometric Functions Web

Probability Web

Filters

Write an expression for the nth term of the sequence $\frac { 1 } { 3 } , \frac { 2 } { 9 } , \frac { 4 } { 27 } , \frac { 8 } { 81 } , \mathrm {~K}$ .

Find the limit of the following sequence. $a _ { n } = 1 + ( - 1 ) ^ { n }$

Use the Ratio Test to determine the convergence or divergence of the series $\sum _ { n = 0 } ^ { \infty } \frac { ( - 6 ) ^ { n } 2 ^ { n } } { n ! }$ .

Use a symbolic differentiation utility to find the Taylor polynomials (centred at zero) of degrees (a) 2, (b) 4, (c) 6, (d) 8. $f ( x ) = \frac { 1 } { 1 + x ^ { 2 } }$

Use Newton's Method to approximate the zero(s) of the function $f ( x ) = x - 2 \sqrt { x + 1 }$ accurate to three decimal places.

Determine whether the sequence is arithmetic. If so, find the common difference. (Assume that n begins with 1.) $\alpha _ { n } = - 2 - 7 n$

Find the sum of the finite geometric series. Round to the nearest hundredth. $\sum _ { n = 1 } ^ { 6 } \left( - \frac { 8 } { 9 } \right) ^ { n }$

Use the Ratio Test to determine the convergence or divergence of the series. $\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 3 } } { 8 ^ { - n } }$

Apply Taylor's Theorem to find the power series centered at $c = 12$ for the function $f ( x ) = e ^ { x }$ .

What are the next three terms in the arithmetic sequence $7,3 , - 1 , \ldots$ ?

Find the indicated nth term of the geometric sequence. 7th term: $a _ { 5 } = \frac { 4 } { 81 } , a _ { 10 } = \frac { 4 } { 19,683 }$

Determine the convergence or divergence of the following series. Use a symbolic algebra utility to verify your result. $\sum _ { n = 0 } ^ { \infty } \frac { 4 } { 2 ^ { n } }$

Evaluate the series. $\sum _ { i = 1 } ^ { 4 } ( 4 i + 3 ) ( 3 i - 4 )$

The repeating decimal $0 . \overline { 2 }$ is expressed as a geometric series $0.2 + 0.02 + 0.002 + 0.0002 + \ldots$ . Write the decimal $0 . \overline { 2 }$ as the ratio of two integers.

Find the limit of the following sequence. $a _ { n } = \frac { n ^ { 2 } - 25 } { n + 5 }$

Write the first five terms of the sequence of partial sums. $2 + \frac { 2 } { 4 } + \frac { 2 } { 9 } + \frac { 2 } { 16 } + \frac { 2 } { 25 } + L$

Match the sequence with the graph of its first 10 terms. $a _ { n } = \frac { 3 n } { n + 1 }$

Find the limit of the sequence $a _ { n } = \frac { 1 } { n ^ { 9 / 2 } }$ .

Determine the maximum error guaranteed by Taylor's Theorem with Remainder when the fifth-degree Taylor polynomial is used to approximate $e ^ { - x }$ in the interval $[ 0,1 ]$ centered at 0. Round your answer to five decimal places.