Question 1

Find the radius of convergence centered at $c = 0$ for the following function. $\sqrt { 1 + x }$

Accepted Answer

A) 3 
B) 2 
C) 0 
D) 1 
E)  $\infty$ 
A) 3 
B) 2 
C) 0 
D) 1 
E)  $\infty$

Question 2

Use summation notation to write the sum below. $2 + 4 + 6 + 8 + 10 + \mathrm { K } + 200$

Accepted Answer

A)  $\sum _ { n = 1 } ^ { 150 } 5 n$ 
B)  $\sum _ { n = 1 } ^ { 100 } 4 n$ 
C)  $\sum _ { n = 1 } ^ { 150 } 4 n$ 
D)  $\sum _ { n = 1 } ^ { 150 } 2 n$ 
E)  $\sum _ { n = 1 } ^ { 100 } 2 n$ 
A)  $\sum _ { n = 1 } ^ { 150 } 5 n$ 
B)  $\sum _ { n = 1 } ^ { 100 } 4 n$ 
C)  $\sum _ { n = 1 } ^ { 150 } 4 n$ 
D)  $\sum _ { n = 1 } ^ { 150 } 2 n$ 
E)  $\sum _ { n = 1 } ^ { 100 } 2 n$

Question 3

Find the next three terms of the geometric sequence. $5,15,45 , \ldots$

Accepted Answer

A)  $55,65,75 , \ldots$ 
B)  $135,405,1215 , \ldots$ 
C)  $75,105,135 , \ldots$ 
D)  $48,51,54 , \ldots$ 
E)  $45,135,405 , \ldots$ 
A)  $55,65,75 , \ldots$ 
B)  $135,405,1215 , \ldots$ 
C)  $75,105,135 , \ldots$ 
D)  $48,51,54 , \ldots$ 
E)  $45,135,405 , \ldots$

Question 4

Use a graphing utility to approximate all the real zeros of the function $$f ( x ) = x ^ { 3 } + 4.173 x ^ { 2 } + 5.761 x + 2.628$$ by Newton's Method.

Accepted Answer

A) -2.111, 2.082, 5.333 
B) -1.151, 3.236, 9.571 
C) -1.511, -1.511, -1.151 
D) -1.031, 2.970, 9.571 
E) -1.191, 3.236, 5.333 
A) -2.111, 2.082, 5.333 
B) -1.151, 3.236, 9.571 
C) -1.511, -1.511, -1.151 
D) -1.031, 2.970, 9.571 
E) -1.191, 3.236, 5.333

Question 5

Find the sum of the infinite geometric series below. $\sum _ { n = 1 } ^ { \infty } \left( \frac { 1 } { 8 } \right) ^ { n - 1 }$

Accepted Answer

A)  $\frac { 11 } { 10 }$ 
B)  $\frac { 8 } { 7 }$ 
C)  $\frac { 3 } { 2 }$ 
D)  $\frac { 10 } { 9 }$ 
E)  $\frac { 6 } { 5 }$ 
A)  $\frac { 11 } { 10 }$ 
B)  $\frac { 8 } { 7 }$ 
C)  $\frac { 3 } { 2 }$ 
D)  $\frac { 10 } { 9 }$ 
E)  $\frac { 6 } { 5 }$

Question 6

Suppose that the annual payroll $a _ { n }$ (in billions of dollars) of new car dealerships in the United States from 2000 to 2005 can be approximated by the model $a _ { n } = 44.7 + 1.51 n - 0.108 n ^ { 2 },$ $n = 0,1,2,3,4,5$ where $n$ represents the year, with $n = 0$ corresponding to 2000. Find the total payroll from 2000 to 2005 by evaluating the sum $\sum _ { n = 0 } ^ { 5 } \left( 44.7 + 1.51 n - 0.108 n ^ { 2 } \right)$ Round your answer to the nearest ten million dollars.

Accepted Answer

A) $334.78 billion 
B) $284.91 billion 
C) $377.86 billion 
D) $290.08 billion 
E) $235.36 billion 
A) $334.78 billion 
B) $284.91 billion 
C) $377.86 billion 
D) $290.08 billion 
E) $235.36 billion

Question 7

Use the fifth-degree Taylor polynomial centered at $c = 0$ for the function $f ( x ) = e ^ { - x }$ to approximate $f \left( \frac { 4 } { 5 } \right)$ . Round your answer to nearest thousandth.

Accepted Answer

A) 0.452 
B) 0.449 
C) 0.574 
D) 0.560 
E) 0.461 
A) 0.452 
B) 0.449 
C) 0.574 
D) 0.560 
E) 0.461

Question 8

Find the radius of convergence of the series $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { 3 n + 7 } } { 3 n + 7 }$ .

Accepted Answer

A) 1 
B) 3 
C) 10 
D) 4 
E) 6 
A) 1 
B) 3 
C) 10 
D) 4 
E) 6

Question 9

A deposit of $\$$ 200 is made each month in an account that earns 8.4% interest, compounded monthly. The balance in the account after n months is given by $A _ { n } = 200 ( 201 ) \left[ ( 1.007 ) ^ { n } - 1 \right]$ . Find the balance after 22 years by computing the 264th term of the sequence. Round your answer to two decimal places.

Accepted Answer

A) $213,316.53 
B) $293,716.53 
C) $6,667.78 
D) $86,634.61 
E) $281.40 
A) $213,316.53 
B) $293,716.53 
C) $6,667.78 
D) $86,634.61 
E) $281.40

Question 10

Find the nth term of the geometric sequence. $3 , - 12,48 , \mathrm {~K}$

Accepted Answer

A)  $- 4 ( 3 ) ^ { n - 1 }$ 
B)  $4 ( 3 ) ^ { n - 1 }$ 
C)  $- 3 ( 4 ) ^ { n - 1 }$ 
D)  $4 ( - 3 ) ^ { n - 1 }$ 
E)  $3 ( - 4 ) ^ { n - 1 }$ 
A)  $- 4 ( 3 ) ^ { n - 1 }$ 
B)  $4 ( 3 ) ^ { n - 1 }$ 
C)  $- 3 ( 4 ) ^ { n - 1 }$ 
D)  $4 ( - 3 ) ^ { n - 1 }$ 
E)  $3 ( - 4 ) ^ { n - 1 }$

Question 11

Find the sum of the convergent series. $\sum _ { n = 0 } ^ { \infty } 9 \left( - \frac { 7 } { 8 } \right) ^ { n }$

Accepted Answer

A)  $\frac { 27 } { 5 }$ 
B)  $\frac { 21 } { 5 }$ 
C)  $\frac { 72 } { 13 }$ 
D)  $\frac { 24 } { 5 }$ 
E)  $\frac { 81 } { 13 }$ 
A)  $\frac { 27 } { 5 }$ 
B)  $\frac { 21 } { 5 }$ 
C)  $\frac { 72 } { 13 }$ 
D)  $\frac { 24 } { 5 }$ 
E)  $\frac { 81 } { 13 }$

Question 12

Find the rational number representation of the repeating decimal. $0 . \overline { 437 }$

Accepted Answer

A)  $\frac { 437 } { 9999 }$ 
B)  $\frac { 43.7 } { 999 }$ 
C)  $\frac { 437 } { 9 }$ 
D)  $\frac { 437 } { 999 }$ 
E)  $\frac { 437 } { 99 }$ 
A)  $\frac { 437 } { 9999 }$ 
B)  $\frac { 43.7 } { 999 }$ 
C)  $\frac { 437 } { 9 }$ 
D)  $\frac { 437 } { 999 }$ 
E)  $\frac { 437 } { 99 }$

Question 13

Find the power series for the function $f ( x ) = \frac { 8 x } { x + 1 }$ using the power series for $\frac { 1 } { x + 1 }$ .

Accepted Answer

A)  $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n + 1 } 8 x ^ { n + 1 }$ 
B)  $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } 8 x ^ { n - 1 }$ 
C)  $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } 8 x ^ { n - 1 }$ 
D)  $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } 8 x ^ { n + 1 }$ 
E)  $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } 8 x ^ { n + 1 }$ 
A)  $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n + 1 } 8 x ^ { n + 1 }$ 
B)  $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } 8 x ^ { n - 1 }$ 
C)  $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } 8 x ^ { n - 1 }$ 
D)  $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } 8 x ^ { n + 1 }$ 
E)  $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } 8 x ^ { n + 1 }$

Question 14

Use Newton´s Method to approximate the zero(s) of the function $f ( x ) = x ^ { 3 } + 3 x + 4$ accurate to three decimal places.

Accepted Answer

A) -1 
B) 1.007 
C) 1 
D) -1.008 
E) 0.995 
A) -1 
B) 1.007 
C) 1 
D) -1.008 
E) 0.995

Question 15

Test the series $\sum _ { n = 1 } ^ { \infty } \frac { n 4 ^ { n } } { n ! }$ for convergence or divergence using any appropriate test.

Accepted Answer

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

Question 16

Find the Taylor polynomials (centred at zero) of degree (a) 1, (b) 2, (c) 3, and (d) 4. $f ( x ) = \sqrt { x + 1 }$

Accepted Answer

A)  $S _ { 1 } ( x ) = 1 + \frac { x } { 2 } , S _ { 2 } ( x ) = 1 + \frac { x } { 2 } - \frac { x ^ { 2 } } { 8 } , S _ { 3 } ( x ) = 1 + \frac { x } { 2 } - \frac { x ^ { 2 } } { 8 } + \frac { x ^ { 3 } } { 16 } , S _ { 4 } ( x ) = 1 + \frac { x } { 2 } - \frac { x ^ { 2 } } { 8 } + \frac { x ^ { 3 } } { 16 } - \frac { x ^ { 4 } } { 128 }$ 
B)  $S _ { 1 } ( x ) = 1 - \frac { x } { 2 } , S _ { 2 } ( x ) = 1 - \frac { x } { 2 } + \frac { x ^ { 2 } } { 8 } , S _ { 3 } ( x ) = 1 - \frac { x } { 2 } + \frac { x ^ { 2 } } { 8 } - \frac { x ^ { 3 } } { 16 } , S _ { 4 } ( x ) = 1 - \frac { x } { 2 } + \frac { x ^ { 2 } } { 8 } - \frac { x ^ { 3 } } { 16 } + \frac { x ^ { 4 } } { 128 }$ 
C)  $S _ { 1 } ( x ) = 1 + \frac { x } { 2 } , S _ { 2 } ( x ) = 1 + \frac { x } { 2 } + \frac { x ^ { 2 } } { 8 } , S _ { 3 } ( x ) = 1 + \frac { x } { 2 } + \frac { x ^ { 2 } } { 8 } + \frac { x ^ { 3 } } { 16 } , S _ { 4 } ( x ) = 1 + \frac { x } { 2 } + \frac { x ^ { 2 } } { 8 } + \frac { x ^ { 3 } } { 16 } + \frac { x ^ { 4 } } { 128 }$ 
D)  $S _ { 1 } ( x ) = 1 - \frac { x } { 2 } , S _ { 2 } ( x ) = 1 - \frac { x } { 2 } - \frac { x ^ { 2 } } { 8 } , S _ { 3 } ( x ) = 1 - \frac { x } { 2 } - \frac { x ^ { 2 } } { 8 } - \frac { x ^ { 3 } } { 16 } , S _ { 4 } ( x ) = 1 - \frac { x } { 2 } - \frac { x ^ { 2 } } { 8 } - \frac { x ^ { 3 } } { 16 } - \frac { x ^ { 4 } } { 128 }$ 
E)  $S _ { 1 } ( x ) = 1 - \frac { x } { 2 } , S _ { 2 } ( x ) = 1 + \frac { x } { 2 } - \frac { x ^ { 2 } } { 8 } , S _ { 3 } ( x ) = 1 + \frac { x } { 2 } + \frac { x ^ { 2 } } { 8 } + \frac { x ^ { 3 } } { 16 } , S _ { 4 } ( x ) = 1 + \frac { x } { 2 } + \frac { x ^ { 2 } } { 8 } + \frac { x ^ { 3 } } { 16 } - \frac { x ^ { 4 } } { 128 }$ 
A)  $S _ { 1 } ( x ) = 1 + \frac { x } { 2 } , S _ { 2 } ( x ) = 1 + \frac { x } { 2 } - \frac { x ^ { 2 } } { 8 } , S _ { 3 } ( x ) = 1 + \frac { x } { 2 } - \frac { x ^ { 2 } } { 8 } + \frac { x ^ { 3 } } { 16 } , S _ { 4 } ( x ) = 1 + \frac { x } { 2 } - \frac { x ^ { 2 } } { 8 } + \frac { x ^ { 3 } } { 16 } - \frac { x ^ { 4 } } { 128 }$ 
B)  $S _ { 1 } ( x ) = 1 - \frac { x } { 2 } , S _ { 2 } ( x ) = 1 - \frac { x } { 2 } + \frac { x ^ { 2 } } { 8 } , S _ { 3 } ( x ) = 1 - \frac { x } { 2 } + \frac { x ^ { 2 } } { 8 } - \frac { x ^ { 3 } } { 16 } , S _ { 4 } ( x ) = 1 - \frac { x } { 2 } + \frac { x ^ { 2 } } { 8 } - \frac { x ^ { 3 } } { 16 } + \frac { x ^ { 4 } } { 128 }$ 
C)  $S _ { 1 } ( x ) = 1 + \frac { x } { 2 } , S _ { 2 } ( x ) = 1 + \frac { x } { 2 } + \frac { x ^ { 2 } } { 8 } , S _ { 3 } ( x ) = 1 + \frac { x } { 2 } + \frac { x ^ { 2 } } { 8 } + \frac { x ^ { 3 } } { 16 } , S _ { 4 } ( x ) = 1 + \frac { x } { 2 } + \frac { x ^ { 2 } } { 8 } + \frac { x ^ { 3 } } { 16 } + \frac { x ^ { 4 } } { 128 }$ 
D)  $S _ { 1 } ( x ) = 1 - \frac { x } { 2 } , S _ { 2 } ( x ) = 1 - \frac { x } { 2 } - \frac { x ^ { 2 } } { 8 } , S _ { 3 } ( x ) = 1 - \frac { x } { 2 } - \frac { x ^ { 2 } } { 8 } - \frac { x ^ { 3 } } { 16 } , S _ { 4 } ( x ) = 1 - \frac { x } { 2 } - \frac { x ^ { 2 } } { 8 } - \frac { x ^ { 3 } } { 16 } - \frac { x ^ { 4 } } { 128 }$ 
E)  $S _ { 1 } ( x ) = 1 - \frac { x } { 2 } , S _ { 2 } ( x ) = 1 + \frac { x } { 2 } - \frac { x ^ { 2 } } { 8 } , S _ { 3 } ( x ) = 1 + \frac { x } { 2 } + \frac { x ^ { 2 } } { 8 } + \frac { x ^ { 3 } } { 16 } , S _ { 4 } ( x ) = 1 + \frac { x } { 2 } + \frac { x ^ { 2 } } { 8 } + \frac { x ^ { 3 } } { 16 } - \frac { x ^ { 4 } } { 128 }$

Question 17

Find the third degree Taylor polynomial centered at c = 4 for the function. $f ( x ) = \sqrt { x }$

Accepted Answer

A)  $2 + \frac { 1 } { 4 } ( x - 4 ) - \frac { 1 } { 64 } ( x - 4 ) ^ { 2 } + \frac { 1 } { 512 } ( x - 4 ) ^ { 3 }$ 
B)  $2 - \frac { 1 } { 4 } ( x - 4 ) + \frac { 1 } { 64 } ( x - 4 ) ^ { 2 } - \frac { 1 } { 512 } ( x - 4 ) ^ { 3 }$ 
C)  $2 - \frac { 1 } { 4 } ( x - 4 ) - \frac { 1 } { 64 } ( x - 4 ) ^ { 2 } - \frac { 1 } { 512 } ( x - 4 ) ^ { 3 }$ 
D)  $2 + \frac { 1 } { 4 } ( x + 4 ) - \frac { 1 } { 64 } ( x + 4 ) ^ { 2 } + \frac { 1 } { 512 } ( x + 4 ) ^ { 3 }$ 
E)  $2 - \frac { 1 } { 4 } ( x + 4 ) + \frac { 1 } { 64 } ( x + 4 ) ^ { 2 } - \frac { 1 } { 512 } ( x + 4 ) ^ { 3 }$ 
A)  $2 + \frac { 1 } { 4 } ( x - 4 ) - \frac { 1 } { 64 } ( x - 4 ) ^ { 2 } + \frac { 1 } { 512 } ( x - 4 ) ^ { 3 }$ 
B)  $2 - \frac { 1 } { 4 } ( x - 4 ) + \frac { 1 } { 64 } ( x - 4 ) ^ { 2 } - \frac { 1 } { 512 } ( x - 4 ) ^ { 3 }$ 
C)  $2 - \frac { 1 } { 4 } ( x - 4 ) - \frac { 1 } { 64 } ( x - 4 ) ^ { 2 } - \frac { 1 } { 512 } ( x - 4 ) ^ { 3 }$ 
D)  $2 + \frac { 1 } { 4 } ( x + 4 ) - \frac { 1 } { 64 } ( x + 4 ) ^ { 2 } + \frac { 1 } { 512 } ( x + 4 ) ^ { 3 }$ 
E)  $2 - \frac { 1 } { 4 } ( x + 4 ) + \frac { 1 } { 64 } ( x + 4 ) ^ { 2 } - \frac { 1 } { 512 } ( x + 4 ) ^ { 3 }$

Question 18

Integrate the series for $\frac { 1 } { x + 6 }$ to find the power series for the function $f ( x ) = \ln ( x + 6 )$ .

Accepted Answer

A)  $\frac { 1 } { 7 } \sum _ { n = 0 } ^ { \infty } \left( \frac { - 1 } { 7 } \right) ^ { n } \frac { ( x ) ^ { n } } { n + 1 }$ 
B)  $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( 6 x - 1 ) ^ { n - 1 } } { n - 1 }$ 
C)  $\frac { 1 } { 6 } \sum _ { n = 0 } ^ { \infty } \frac { ( x ) ^ { n + 1 } } { n + 1 }$ 
D)  $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( 6 x - 1 ) ^ { n } } { n - 1 }$ 
E)  $\frac { 1 } { 6 } \sum _ { n = 0 } ^ { \infty } \left( \frac { - 1 } { 6 } \right) ^ { n } \frac { ( x ) ^ { n + 1 } } { n + 1 }$ 
A)  $\frac { 1 } { 7 } \sum _ { n = 0 } ^ { \infty } \left( \frac { - 1 } { 7 } \right) ^ { n } \frac { ( x ) ^ { n } } { n + 1 }$ 
B)  $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( 6 x - 1 ) ^ { n - 1 } } { n - 1 }$ 
C)  $\frac { 1 } { 6 } \sum _ { n = 0 } ^ { \infty } \frac { ( x ) ^ { n + 1 } } { n + 1 }$ 
D)  $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( 6 x - 1 ) ^ { n } } { n - 1 }$ 
E)  $\frac { 1 } { 6 } \sum _ { n = 0 } ^ { \infty } \left( \frac { - 1 } { 6 } \right) ^ { n } \frac { ( x ) ^ { n + 1 } } { n + 1 }$

Question 19

A company produces a new product for which it estimates the annual sales to be 5000 units. Suppose that in any given year $10$ % of the units (regardless of age) will become inoperative. How many units will be in use after n years?

Accepted Answer

A)  $50,000 \left( 1 - 0.9 ^ { n } \right)$ 
B)  $50,000 \left( 0.1 ^ { n } \right)$ 
C)  $5000 \left( 1 - 0.9 ^ { n } \right)$ 
D)  $5000 \left( 0.9 ^ { n } \right)$ 
E)  $50,000 \left( 1 - 0.1 ^ { n } \right)$ 
A)  $50,000 \left( 1 - 0.9 ^ { n } \right)$ 
B)  $50,000 \left( 0.1 ^ { n } \right)$ 
C)  $5000 \left( 1 - 0.9 ^ { n } \right)$ 
D)  $5000 \left( 0.9 ^ { n } \right)$ 
E)  $50,000 \left( 1 - 0.1 ^ { n } \right)$

Question 20

Determine the convergence or divergence of the following p-series. $1 + \frac { 1 } { 2 } + \frac { 1 } { 3 } + \frac { 1 } { 4 } + \frac { 1 } { 5 } + \cdots$

Accepted Answer

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

Find the radius of convergence centered at $c = 0$ for the following function. $\sqrt { 1 + x }$

Use summation notation to write the sum below. $2 + 4 + 6 + 8 + 10 + \mathrm { K } + 200$

Find the next three terms of the geometric sequence. $5,15,45 , \ldots$

Use a graphing utility to approximate all the real zeros of the function $f ( x ) = x ^ { 3 } + 4.173 x ^ { 2 } + 5.761 x + 2.628$ by Newton's Method.

Find the sum of the infinite geometric series below. $\sum _ { n = 1 } ^ { \infty } \left( \frac { 1 } { 8 } \right) ^ { n - 1 }$

Use the fifth-degree Taylor polynomial centered at $c = 0$ for the function $f ( x ) = e ^ { - x }$ to approximate $f \left( \frac { 4 } { 5 } \right)$ . Round your answer to nearest thousandth.

Find the radius of convergence of the series $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { 3 n + 7 } } { 3 n + 7 }$ .

Find the nth term of the geometric sequence. $3 , - 12,48 , \mathrm {~K}$

Find the sum of the convergent series. $\sum _ { n = 0 } ^ { \infty } 9 \left( - \frac { 7 } { 8 } \right) ^ { n }$

Find the rational number representation of the repeating decimal. $0 . \overline { 437 }$

Find the power series for the function $f ( x ) = \frac { 8 x } { x + 1 }$ using the power series for $\frac { 1 } { x + 1 }$ .

Use Newton´s Method to approximate the zero(s) of the function $f ( x ) = x ^ { 3 } + 3 x + 4$ accurate to three decimal places.

Test the series $\sum _ { n = 1 } ^ { \infty } \frac { n 4 ^ { n } } { n ! }$ for convergence or divergence using any appropriate test.

Find the Taylor polynomials (centred at zero) of degree (a) 1, (b) 2, (c) 3, and (d) 4. $f ( x ) = \sqrt { x + 1 }$

Find the third degree Taylor polynomial centered at c = 4 for the function. $f ( x ) = \sqrt { x }$

Integrate the series for $\frac { 1 } { x + 6 }$ to find the power series for the function $f ( x ) = \ln ( x + 6 )$ .

A company produces a new product for which it estimates the annual sales to be 5000 units. Suppose that in any given year $10$ % of the units (regardless of age) will become inoperative. How many units will be in use after n years?

Determine the convergence or divergence of the following p-series. $1 + \frac { 1 } { 2 } + \frac { 1 } { 3 } + \frac { 1 } { 4 } + \frac { 1 } { 5 } + \cdots$

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Exam 16: Series and Taylor Polynomials Web

Find the radius of convergence centered at c=0c = 0c=0 for the following function. 1+x\sqrt { 1 + x }1+x​

Use summation notation to write the sum below. 2+4+6+8+10+K+2002 + 4 + 6 + 8 + 10 + \mathrm { K } + 2002+4+6+8+10+K+200

Find the next three terms of the geometric sequence. 5,15,45,…5,15,45 , \ldots5,15,45,…

Use a graphing utility to approximate all the real zeros of the function f(x)=x3+4.173x2+5.761x+2.628f ( x ) = x ^ { 3 } + 4.173 x ^ { 2 } + 5.761 x + 2.628f(x)=x3+4.173x2+5.761x+2.628 by Newton's Method.

Find the sum of the infinite geometric series below. ∑n=1∞(18)n−1\sum _ { n = 1 } ^ { \infty } \left( \frac { 1 } { 8 } \right) ^ { n - 1 }∑n=1∞​(81​)n−1

Use the fifth-degree Taylor polynomial centered at c=0c = 0c=0 for the function f(x)=e−xf ( x ) = e ^ { - x }f(x)=e−x to approximate f(45)f \left( \frac { 4 } { 5 } \right)f(54​) . Round your answer to nearest thousandth.

Find the radius of convergence of the series ∑n=1∞(−1)nx3n+73n+7\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { 3 n + 7 } } { 3 n + 7 }∑n=1∞​3n+7(−1)nx3n+7​ .

Find the nth term of the geometric sequence. 3,−12,48, K3 , - 12,48 , \mathrm {~K}3,−12,48, K

Find the sum of the convergent series. ∑n=0∞9(−78)n\sum _ { n = 0 } ^ { \infty } 9 \left( - \frac { 7 } { 8 } \right) ^ { n }∑n=0∞​9(−87​)n

Find the rational number representation of the repeating decimal. 0.437‾0 . \overline { 437 }0.437

Find the power series for the function f(x)=8xx+1f ( x ) = \frac { 8 x } { x + 1 }f(x)=x+18x​ using the power series for 1x+1\frac { 1 } { x + 1 }x+11​ .

Use Newton´s Method to approximate the zero(s) of the function f(x)=x3+3x+4f ( x ) = x ^ { 3 } + 3 x + 4f(x)=x3+3x+4 accurate to three decimal places.

Test the series ∑n=1∞n4nn!\sum _ { n = 1 } ^ { \infty } \frac { n 4 ^ { n } } { n ! }∑n=1∞​n!n4n​ for convergence or divergence using any appropriate test.

Find the Taylor polynomials (centred at zero) of degree (a) 1, (b) 2, (c) 3, and (d) 4. f(x)=x+1f ( x ) = \sqrt { x + 1 }f(x)=x+1​

Find the third degree Taylor polynomial centered at c = 4 for the function. f(x)=xf ( x ) = \sqrt { x }f(x)=x​

Integrate the series for 1x+6\frac { 1 } { x + 6 }x+61​ to find the power series for the function f(x)=ln⁡(x+6)f ( x ) = \ln ( x + 6 )f(x)=ln(x+6) .

A company produces a new product for which it estimates the annual sales to be 5000 units. Suppose that in any given year 101010 % of the units (regardless of age) will become inoperative. How many units will be in use after n years?

Determine the convergence or divergence of the following p-series. 1+12+13+14+15+⋯1 + \frac { 1 } { 2 } + \frac { 1 } { 3 } + \frac { 1 } { 4 } + \frac { 1 } { 5 } + \cdots1+21​+31​+41​+51​+⋯

Fundamental Concepts of Algebra

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Find the radius of convergence centered at $c = 0$ for the following function. $\sqrt { 1 + x }$

Use summation notation to write the sum below. $2 + 4 + 6 + 8 + 10 + \mathrm { K } + 200$

Find the next three terms of the geometric sequence. $5,15,45 , \ldots$

Use a graphing utility to approximate all the real zeros of the function $f ( x ) = x ^ { 3 } + 4.173 x ^ { 2 } + 5.761 x + 2.628$ by Newton's Method.

Find the sum of the infinite geometric series below. $\sum _ { n = 1 } ^ { \infty } \left( \frac { 1 } { 8 } \right) ^ { n - 1 }$

Use the fifth-degree Taylor polynomial centered at $c = 0$ for the function $f ( x ) = e ^ { - x }$ to approximate $f \left( \frac { 4 } { 5 } \right)$ . Round your answer to nearest thousandth.

Find the radius of convergence of the series $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { 3 n + 7 } } { 3 n + 7 }$ .

Find the nth term of the geometric sequence. $3 , - 12,48 , \mathrm {~K}$

Find the sum of the convergent series. $\sum _ { n = 0 } ^ { \infty } 9 \left( - \frac { 7 } { 8 } \right) ^ { n }$

Find the rational number representation of the repeating decimal. $0 . \overline { 437 }$

Find the power series for the function $f ( x ) = \frac { 8 x } { x + 1 }$ using the power series for $\frac { 1 } { x + 1 }$ .

Use Newton´s Method to approximate the zero(s) of the function $f ( x ) = x ^ { 3 } + 3 x + 4$ accurate to three decimal places.

Test the series $\sum _ { n = 1 } ^ { \infty } \frac { n 4 ^ { n } } { n ! }$ for convergence or divergence using any appropriate test.

Find the Taylor polynomials (centred at zero) of degree (a) 1, (b) 2, (c) 3, and (d) 4. $f ( x ) = \sqrt { x + 1 }$

Find the third degree Taylor polynomial centered at c = 4 for the function. $f ( x ) = \sqrt { x }$

Integrate the series for $\frac { 1 } { x + 6 }$ to find the power series for the function $f ( x ) = \ln ( x + 6 )$ .

A company produces a new product for which it estimates the annual sales to be 5000 units. Suppose that in any given year $10$ % of the units (regardless of age) will become inoperative. How many units will be in use after n years?

Determine the convergence or divergence of the following p-series. $1 + \frac { 1 } { 2 } + \frac { 1 } { 3 } + \frac { 1 } { 4 } + \frac { 1 } { 5 } + \cdots$