Question 1

Find the fourth degree Maclaurin polynomial for the function. $$f ( x ) = \frac { 1 } { x + 4 }$$

Accepted Answer

A)  $\frac { 1 } { 4 } + \frac { 1 } { 16 } x - \frac { 1 } { 64 } x ^ { 2 } + \frac { 1 } { 256 } x ^ { 3 } - \frac { 1 } { 1024 } x ^ { 4 }$ 
B)  $\frac { 1 } { 4 } + \frac { 1 } { 16 } x + \frac { 1 } { 64 } x ^ { 2 } + \frac { 1 } { 256 } x ^ { 3 } + \frac { 1 } { 1024 } x ^ { 4 }$ 
C)  $4 + 16 x + 64 x ^ { 2 } + 256 x ^ { 3 } + 1024 x ^ { 4 }$ 
D)  $4 - 16 x + 64 x ^ { 2 } - 256 x ^ { 3 } + 1024 x ^ { 4 }$ 
E)  $\frac { 1 } { 4 } - \frac { 1 } { 16 } x + \frac { 1 } { 64 } x ^ { 2 } - \frac { 1 } { 256 } x ^ { 3 } + \frac { 1 } { 1024 } x ^ { 4 }$ 
A)  $\frac { 1 } { 4 } + \frac { 1 } { 16 } x - \frac { 1 } { 64 } x ^ { 2 } + \frac { 1 } { 256 } x ^ { 3 } - \frac { 1 } { 1024 } x ^ { 4 }$ 
B)  $\frac { 1 } { 4 } + \frac { 1 } { 16 } x + \frac { 1 } { 64 } x ^ { 2 } + \frac { 1 } { 256 } x ^ { 3 } + \frac { 1 } { 1024 } x ^ { 4 }$ 
C)  $4 + 16 x + 64 x ^ { 2 } + 256 x ^ { 3 } + 1024 x ^ { 4 }$ 
D)  $4 - 16 x + 64 x ^ { 2 } - 256 x ^ { 3 } + 1024 x ^ { 4 }$ 
E)  $\frac { 1 } { 4 } - \frac { 1 } { 16 } x + \frac { 1 } { 64 } x ^ { 2 } - \frac { 1 } { 256 } x ^ { 3 } + \frac { 1 } { 1024 } x ^ { 4 }$

Question 2

Determine the convergence or divergence of the series using any appropriate test from this chapter. Identify the test used. $\sum _ { n = 1 } ^ { \infty } \frac { 6 n } { n + 9 }$

Accepted Answer

A)  both diverges; Ratio Test and diverges; Theorem $9.9 \left( n ^ { \text {th } } \right.$ Term Test for Divergence) 
B)  converges; Integral Test 
C)  diverges; Theorem $9.9$ ( $n ^ { \text {th } }$ Term Test for Divergence) 
D)  converges; $p$-series 
E)  diverges; Ratio Test 
A)  both diverges; Ratio Test and diverges; Theorem $9.9 \left( n ^ { \text {th } } \right.$ Term Test for Divergence) 
B)  converges; Integral Test 
C)  diverges; Theorem $9.9$ ( $n ^ { \text {th } }$ Term Test for Divergence) 
D)  converges; $p$-series 
E)  diverges; Ratio Test

Question 3

$\text { Write an equivalent series of the series } \sum _ { n = 0 } ^ { \infty } \frac { x ^ { n } } { n ! } \text { with the index of summation }$ beginning at $n = 5$.

Accepted Answer

A)  $\sum _ { n = 5 } ^ { \infty } \frac { x ^ { n - 5 } } { ( n - 5 ) ! }$ 
B)  $\sum _ { n = 5 } ^ { \infty } \frac { x ^ { n - 5 } } { n ! }$ 
C) $\sum _ { n = 5 } ^ { \infty } \frac { x ^ { n + 5 } } { n ! }$ 
D) $\sum _ { n = 5 } ^ { \infty } \frac { x ^ { n + 5 } } { ( n + 5 ) ! }$ 
E) $\sum _ { n = 5 } ^ { \infty } \frac { x ^ { n } } { ( n - 5 ) ! }$ 
A)  $\sum _ { n = 5 } ^ { \infty } \frac { x ^ { n - 5 } } { ( n - 5 ) ! }$ 
B)  $\sum _ { n = 5 } ^ { \infty } \frac { x ^ { n - 5 } } { n ! }$ 
C) $\sum _ { n = 5 } ^ { \infty } \frac { x ^ { n + 5 } } { n ! }$ 
D) $\sum _ { n = 5 } ^ { \infty } \frac { x ^ { n + 5 } } { ( n + 5 ) ! }$ 
E) $\sum _ { n = 5 } ^ { \infty } \frac { x ^ { n } } { ( n - 5 ) ! }$

Question 4

Find the sum of the convergent series $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \frac { 1 } { 4 ^ { 2 n + 1 } ( 2 n + 1 ) } $ by using a  well-known function. Round your answer to four decimal places.

Accepted Answer

A) 0.1419 
B) 0.2450 
C) 0.8961 
D) 0.1651 
E) 0.1974 
A) 0.1419 
B) 0.2450 
C) 0.8961 
D) 0.1651 
E) 0.1974

Question 5

Use the definition to find the Taylor series centered at $c = 0 \text { for the function }$ $f ( x ) = \sin 4 x$

Accepted Answer

A)  $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } ( 4 x ) ^ { 2 n - 1 } } { ( 2 n - 1 ) ! }$ 
B)  $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( 4 x ) ^ { 2 n + 1 } } { ( 2 n + 1 ) ! }$ 
C) $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } ( 4 x ) ^ { 2 n + 1 } } { ( 2 n + 1 ) ! }$ 
D)  $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { 2 n - 1 } ( 4 x ) ^ { 2 n - 1 } } { n ! }$ 
E) $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( 4 x ) ^ { 2 n + 1 } } { ( 2 n + 1 ) ! }$ 
A)  $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } ( 4 x ) ^ { 2 n - 1 } } { ( 2 n - 1 ) ! }$ 
B)  $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( 4 x ) ^ { 2 n + 1 } } { ( 2 n + 1 ) ! }$ 
C) $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } ( 4 x ) ^ { 2 n + 1 } } { ( 2 n + 1 ) ! }$ 
D)  $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { 2 n - 1 } ( 4 x ) ^ { 2 n - 1 } } { n ! }$ 
E) $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( 4 x ) ^ { 2 n + 1 } } { ( 2 n + 1 ) ! }$

Question 6

Find the radius of convergence of the power series. $$\sum _ { n = 0 } ^ { \infty } \frac { ( 8 x ) ^ { 2 n } } { ( 2 n ) ! }$$

Accepted Answer

A)  $\infty$ 
B)  16 
C)  64 
D)  0 
E)  8 
A)  $\infty$ 
B)  16 
C)  64 
D)  0 
E)  8

Question 7

Find the values of $x$ for which the series $\sum _ { n = 0 } ^ { \infty } 5 \left( \frac { x } { 4 } \right) ^ { n }$ converges.

Accepted Answer

A)  $- 5 < x < 0$ 
B)  $0 < x < 5$ 
C)  $- 4 < x < 4$ 
D)  $- 5 < x < 5$ 
E)  $0 < x < 4$ 
A)  $- 5 < x < 0$ 
B)  $0 < x < 5$ 
C)  $- 4 < x < 4$ 
D)  $- 5 < x < 5$ 
E)  $0 < x < 4$

Question 8

$\text { Use the power series } \frac { 1 } { 1 + x } = \sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } x ^ { n } \text { to determine a power series centered at } 0$ 
for the function $f ( x ) = - \frac { 8 } { ( 8 x + 1 ) ^ { 2 } } = \frac { d } { d x } \left[ \frac { 1 } { 8 x + 1 } \right]$

Accepted Answer

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

Question 9

Match the sequence with its graph. $a _ { n } = \frac { 4 ( - 1 ) ^ { n } } { n }$

Accepted Answer

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

Question 10

Find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) $$\sum _ { n = 1 } ^ { \infty } \frac { ( x - 4 ) ^ { n - 1 } } { 4 ^ { n - 1 } }$$

Accepted Answer

A)  $( 0,4 )$ 
B)  $( - 4,4 )$ 
C)  $( 4,8 )$ 
D)  $( - 8,8 )$ 
E)  $( 0,8 )$ 
A)  $( 0,4 )$ 
B)  $( - 4,4 )$ 
C)  $( 4,8 )$ 
D)  $( - 8,8 )$ 
E)  $( 0,8 )$

Question 11

Use the Direct Comparison Test to determine the convergence or divergence of the series $\sum _ { n = 0 } ^ { \infty } e ^ { - n ^ { 6 } }$.

Accepted Answer

A) The series $\sum _ { n = 0 } ^ { \infty } e ^ { - n ^ { 6 } }$ diverges. 
B)  The series $\sum _ { n = 0 } ^ { \infty } e ^ { - n ^ { 6 } }$ converges. 
A) The series $\sum _ { n = 0 } ^ { \infty } e ^ { - n ^ { 6 } }$ diverges. 
B)  The series $\sum _ { n = 0 } ^ { \infty } e ^ { - n ^ { 6 } }$ converges.

Question 12

Find the Maclaurin series for the function $$f ( x ) = \sin 7 x$$

Accepted Answer

A)   $$\sin 7 x = \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( 7 x ) ^ { 2 n + 1 } } { ( 2 n + 1 ) ! }$$ 
B) 
$$\sin 7 x = \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( 7 x ) ^ { 2 n } } { ( 2 n ) ! }$$ 
C) 
$$\sin 7 x = \sum _ { n = 0 } ^ { \infty } \frac { ( 7 x ) ^ { 2 n } } { ( 2 n ) ! }$$ 
D) 
 $$\sin 7 x = \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( 7 x ) ^ { 2 n + 1 } } { ( n + 1 ) ! }$$ 
E) 
$$\sin 7 x = \sum _ { n = 0 } ^ { \infty } \frac { ( 7 x ) ^ { 2 n + 1 } } { ( 2 n + 1 ) ! }$$ 
A)   $$\sin 7 x = \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( 7 x ) ^ { 2 n + 1 } } { ( 2 n + 1 ) ! }$$ 
B) 
$$\sin 7 x = \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( 7 x ) ^ { 2 n } } { ( 2 n ) ! }$$ 
C) 
$$\sin 7 x = \sum _ { n = 0 } ^ { \infty } \frac { ( 7 x ) ^ { 2 n } } { ( 2 n ) ! }$$ 
D) 
 $$\sin 7 x = \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( 7 x ) ^ { 2 n + 1 } } { ( n + 1 ) ! }$$ 
E) 
$$\sin 7 x = \sum _ { n = 0 } ^ { \infty } \frac { ( 7 x ) ^ { 2 n + 1 } } { ( 2 n + 1 ) ! }$$

Question 13

Find the third degree Taylor polynomial centered at $c = 1$ for the function. $$f ( x ) = \sqrt { x }$$

Accepted Answer

A)  $1 + \frac { 1 } { 2 } ( x + 1 ) - \frac { 1 } { 8 } ( x - 1 ) ^ { 2 } + \frac { 1 } { 16 } ( x + 1 ) ^ { 3 }$ 
B)  $1 + \frac { 1 } { 2 } ( x - 1 ) + \frac { 1 } { 8 } ( x - 1 ) ^ { 2 } - \frac { 1 } { 16 } ( x - 1 ) ^ { 3 }$ 
C)  $1 + \frac { 1 } { 2 } ( x - 1 ) - \frac { 1 } { 8 } ( x - 1 ) ^ { 2 } + \frac { 1 } { 16 } ( x - 1 ) ^ { 3 }$ 
D)  $1 + \frac { 1 } { 2 } ( x - 1 ) - \frac { 1 } { 8 } ( x - 1 ) ^ { 2 } - \frac { 1 } { 16 } ( x - 1 ) ^ { 3 }$ 
E)  $1 + \frac { 1 } { 2 } ( x + 1 ) + \frac { 1 } { 8 } ( x - 1 ) ^ { 2 } - \frac { 1 } { 16 } ( x + 1 ) ^ { 3 }$ 
A)  $1 + \frac { 1 } { 2 } ( x + 1 ) - \frac { 1 } { 8 } ( x - 1 ) ^ { 2 } + \frac { 1 } { 16 } ( x + 1 ) ^ { 3 }$ 
B)  $1 + \frac { 1 } { 2 } ( x - 1 ) + \frac { 1 } { 8 } ( x - 1 ) ^ { 2 } - \frac { 1 } { 16 } ( x - 1 ) ^ { 3 }$ 
C)  $1 + \frac { 1 } { 2 } ( x - 1 ) - \frac { 1 } { 8 } ( x - 1 ) ^ { 2 } + \frac { 1 } { 16 } ( x - 1 ) ^ { 3 }$ 
D)  $1 + \frac { 1 } { 2 } ( x - 1 ) - \frac { 1 } { 8 } ( x - 1 ) ^ { 2 } - \frac { 1 } { 16 } ( x - 1 ) ^ { 3 }$ 
E)  $1 + \frac { 1 } { 2 } ( x + 1 ) + \frac { 1 } { 8 } ( x - 1 ) ^ { 2 } - \frac { 1 } { 16 } ( x + 1 ) ^ { 3 }$

Question 14

Determine the convergence or divergence of the series using any appropriate test from this chapter. Identify the test used. $\sum _ { n = 1 } ^ { \infty } \frac { 10 } { n ^ { - } }$

Accepted Answer

A)  converges; Ratio Test 
B)  both civerges; $p$-series and civerges; Integral Test 
C)  civerges; $p$-series 
D)  civerges; Integral Test 
E)  converges; $p$-series 
A)  converges; Ratio Test 
B)  both civerges; $p$-series and civerges; Integral Test 
C)  civerges; $p$-series 
D)  civerges; Integral Test 
E)  converges; $p$-series

Question 15

$\text { Identify the interval of convergence of a power series } \sum _ { n = 1 } ^ { \infty } ( - 9 ) ^ { n } x ^ { 2 n } \text {. }$

Accepted Answer

A)  $- \frac { 1 } { 3 } < x < \frac { 1 } { 3 }$ 
B)  $0 < x < \frac { 1 } { 3 }$ 
C)  $- \frac { 1 } { 9 } < x < \frac { 1 } { 9 }$ 
D)  $- \frac { 1 } { 9 } < x < 0$ 
E)  $0 < x < \frac { 1 } { 9 }$ 
A)  $- \frac { 1 } { 3 } < x < \frac { 1 } { 3 }$ 
B)  $0 < x < \frac { 1 } { 3 }$ 
C)  $- \frac { 1 } { 9 } < x < \frac { 1 } { 9 }$ 
D)  $- \frac { 1 } { 9 } < x < 0$ 
E)  $0 < x < \frac { 1 } { 9 }$

Question 16

Approximate the sum of the series by using the first six terms. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } 2 } { n ^ { 3 } }$

Accepted Answer

A)  $1.799 < S < 1.805$ 
B)  $1.794 < S < 1.806$ 
C)  $1.794 < S < 1.805$ 
D)  $1.792 < S < 1.792$ 
E) $1.698 < S < 1.902$ 
A)  $1.799 < S < 1.805$ 
B)  $1.794 < S < 1.806$ 
C)  $1.794 < S < 1.805$ 
D)  $1.792 < S < 1.792$ 
E) $1.698 < S < 1.902$

Question 17

$\text { Determine whether the series } \sum _ { n = 0 } ^ { \infty } \frac { \cos ( n \pi ) } { n + 3 } \text { converges conditionally or }$ absolutely, or diverges.

Accepted Answer

A) The series converges absolutely. 
B) The series diverges. 
C) The series converges absolutely but does not converge conditionally. 
D) The series converges conditionally but does not converge absolutely. 
A) The series converges absolutely. 
B) The series diverges. 
C) The series converges absolutely but does not converge conditionally. 
D) The series converges conditionally but does not converge absolutely.

Question 18

Determine the convergence or divergence of the series. $$\sum _ { n = 0 } ^ { \infty } \left( \frac { 1 } { 4 } \right) ^ { n }$$

Accepted Answer

A)  converges 
B)  diverges 
C)  cannot be determined 
A)  converges 
B)  diverges 
C)  cannot be determined

Question 19

$\text { Write an equivalent series of the series } \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { 6 n + 1 } } { 6 n + 1 } \text { with the index of }$ summation beginning at $n = 5$.

Accepted Answer

A)  $\sum _ { n = 5 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { 6 n + 29 } } { 6 n + 29 }$ 
B)  $\sum _ { n = 5 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { 6 n - 31 } } { 6 n - 31 }$ 
C)  $\sum _ { n = 5 } ^ { \infty } \frac { ( - 1 ) ^ { n - 5 } x ^ { 6 n - 29 } } { 6 n - 29 }$ 
D) $\sum _ { n = 5 } ^ { \infty } \frac { ( - 1 ) ^ { n - 5 } x ^ { 6 n + 29 } } { 6 n + 29 }$ 
E) $\sum _ { n = 5 } ^ { \infty } \frac { ( - 1 ) ^ { n + 5 } x ^ { 7 n - 29 } } { 7 n - 29 }$ 
A)  $\sum _ { n = 5 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { 6 n + 29 } } { 6 n + 29 }$ 
B)  $\sum _ { n = 5 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { 6 n - 31 } } { 6 n - 31 }$ 
C)  $\sum _ { n = 5 } ^ { \infty } \frac { ( - 1 ) ^ { n - 5 } x ^ { 6 n - 29 } } { 6 n - 29 }$ 
D) $\sum _ { n = 5 } ^ { \infty } \frac { ( - 1 ) ^ { n - 5 } x ^ { 6 n + 29 } } { 6 n + 29 }$ 
E) $\sum _ { n = 5 } ^ { \infty } \frac { ( - 1 ) ^ { n + 5 } x ^ { 7 n - 29 } } { 7 n - 29 }$

Question 20

UseTheorem 9.11 to determine the convergence or divergence of the series. $$\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 0.86 } }$$

Accepted Answer

A)  converges 
B)  diverges 
C)  Theorem $9.11$ inconclusive 
A)  converges 
B)  diverges 
C)  Theorem $9.11$ inconclusive

Find the fourth degree Maclaurin polynomial for the function. $f ( x ) = \frac { 1 } { x + 4 }$

Determine the convergence or divergence of the series using any appropriate test from this chapter. Identify the test used. $\sum _ { n = 1 } ^ { \infty } \frac { 6 n } { n + 9 }$

$\text { Write an equivalent series of the series } \sum _ { n = 0 } ^ { \infty } \frac { x ^ { n } } { n ! } \text { with the index of summation }$ beginning at $n = 5$ .

Find the sum of the convergent series $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \frac { 1 } { 4 ^ { 2 n + 1 } ( 2 n + 1 ) }$ by using a well-known function. Round your answer to four decimal places.

Use the definition to find the Taylor series centered at $c = 0 \text { for the function }$ $f ( x ) = \sin 4 x$

Find the radius of convergence of the power series. $\sum _ { n = 0 } ^ { \infty } \frac { ( 8 x ) ^ { 2 n } } { ( 2 n ) ! }$

Find the values of $x$ for which the series $\sum _ { n = 0 } ^ { \infty } 5 \left( \frac { x } { 4 } \right) ^ { n }$ converges.

$\text { Use the power series } \frac { 1 } { 1 + x } = \sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } x ^ { n } \text { to determine a power series centered at } 0$ for the function $f ( x ) = - \frac { 8 } { ( 8 x + 1 ) ^ { 2 } } = \frac { d } { d x } \left[ \frac { 1 } { 8 x + 1 } \right]$

Match the sequence with its graph. $a _ { n } = \frac { 4 ( - 1 ) ^ { n } } { n }$

Find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) $\sum _ { n = 1 } ^ { \infty } \frac { ( x - 4 ) ^ { n - 1 } } { 4 ^ { n - 1 } }$

Use the Direct Comparison Test to determine the convergence or divergence of the series $\sum _ { n = 0 } ^ { \infty } e ^ { - n ^ { 6 } }$ .

Find the Maclaurin series for the function $f ( x ) = \sin 7 x$

Find the third degree Taylor polynomial centered at $c = 1$ for the function. $f ( x ) = \sqrt { x }$

Determine the convergence or divergence of the series using any appropriate test from this chapter. Identify the test used. $\sum _ { n = 1 } ^ { \infty } \frac { 10 } { n ^ { - } }$

$\text { Identify the interval of convergence of a power series } \sum _ { n = 1 } ^ { \infty } ( - 9 ) ^ { n } x ^ { 2 n } \text {. }$

Approximate the sum of the series by using the first six terms. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } 2 } { n ^ { 3 } }$

$\text { Determine whether the series } \sum _ { n = 0 } ^ { \infty } \frac { \cos ( n \pi ) } { n + 3 } \text { converges conditionally or }$ absolutely, or diverges.

Determine the convergence or divergence of the series. $\sum _ { n = 0 } ^ { \infty } \left( \frac { 1 } { 4 } \right) ^ { n }$

$\text { Write an equivalent series of the series } \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { 6 n + 1 } } { 6 n + 1 } \text { with the index of }$ summation beginning at $n = 5$ .

UseTheorem 9.11 to determine the convergence or divergence of the series. $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 0.86 } }$

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

Find the fourth degree Maclaurin polynomial for the function. f(x)=1x+4f ( x ) = \frac { 1 } { x + 4 }f(x)=x+41​

Determine the convergence or divergence of the series using any appropriate test from this chapter. Identify the test used. ∑n=1∞6nn+9\sum _ { n = 1 } ^ { \infty } \frac { 6 n } { n + 9 }∑n=1∞​n+96n​

Find the sum of the convergent series ∑n=0∞(−1)n142n+1(2n+1)\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \frac { 1 } { 4 ^ { 2 n + 1 } ( 2 n + 1 ) } ∑n=0∞​(−1)n42n+1(2n+1)1​ by using a well-known function. Round your answer to four decimal places.

Use the definition to find the Taylor series centered at c=0 for the function c = 0 \text { for the function }c=0 for the function f(x)=sin⁡4xf ( x ) = \sin 4 xf(x)=sin4x

Find the radius of convergence of the power series. ∑n=0∞(8x)2n(2n)!\sum _ { n = 0 } ^ { \infty } \frac { ( 8 x ) ^ { 2 n } } { ( 2 n ) ! }∑n=0∞​(2n)!(8x)2n​

Find the values of xxx for which the series ∑n=0∞5(x4)n\sum _ { n = 0 } ^ { \infty } 5 \left( \frac { x } { 4 } \right) ^ { n }∑n=0∞​5(4x​)n converges.

Match the sequence with its graph. an=4(−1)nna _ { n } = \frac { 4 ( - 1 ) ^ { n } } { n }an​=n4(−1)n​

Find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) ∑n=1∞(x−4)n−14n−1\sum _ { n = 1 } ^ { \infty } \frac { ( x - 4 ) ^ { n - 1 } } { 4 ^ { n - 1 } }∑n=1∞​4n−1(x−4)n−1​

Use the Direct Comparison Test to determine the convergence or divergence of the series ∑n=0∞e−n6\sum _ { n = 0 } ^ { \infty } e ^ { - n ^ { 6 } }∑n=0∞​e−n6 .

Find the Maclaurin series for the function f(x)=sin⁡7xf ( x ) = \sin 7 xf(x)=sin7x

Find the third degree Taylor polynomial centered at c=1c = 1c=1 for the function. f(x)=xf ( x ) = \sqrt { x }f(x)=x​

Determine the convergence or divergence of the series using any appropriate test from this chapter. Identify the test used. ∑n=1∞10n−\sum _ { n = 1 } ^ { \infty } \frac { 10 } { n ^ { - } }∑n=1∞​n−10​

Identify the interval of convergence of a power series ∑n=1∞(−9)nx2n. \text { Identify the interval of convergence of a power series } \sum _ { n = 1 } ^ { \infty } ( - 9 ) ^ { n } x ^ { 2 n } \text {. } Identify the interval of convergence of a power series ∑n=1∞​(−9)nx2n.

Approximate the sum of the series by using the first six terms. ∑n=1∞(−1)n+12n3\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } 2 } { n ^ { 3 } }∑n=1∞​n3(−1)n+12​

Determine the convergence or divergence of the series. ∑n=0∞(14)n\sum _ { n = 0 } ^ { \infty } \left( \frac { 1 } { 4 } \right) ^ { n }∑n=0∞​(41​)n

UseTheorem 9.11 to determine the convergence or divergence of the series. ∑n=1∞1n0.86\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 0.86 } }∑n=1∞​n0.861​

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

Find the fourth degree Maclaurin polynomial for the function. $f ( x ) = \frac { 1 } { x + 4 }$

Determine the convergence or divergence of the series using any appropriate test from this chapter. Identify the test used. $\sum _ { n = 1 } ^ { \infty } \frac { 6 n } { n + 9 }$

Find the sum of the convergent series $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \frac { 1 } { 4 ^ { 2 n + 1 } ( 2 n + 1 ) }$ by using a well-known function. Round your answer to four decimal places.

Use the definition to find the Taylor series centered at $c = 0 \text { for the function }$ $f ( x ) = \sin 4 x$

Find the radius of convergence of the power series. $\sum _ { n = 0 } ^ { \infty } \frac { ( 8 x ) ^ { 2 n } } { ( 2 n ) ! }$

Find the values of $x$ for which the series $\sum _ { n = 0 } ^ { \infty } 5 \left( \frac { x } { 4 } \right) ^ { n }$ converges.

Match the sequence with its graph. $a _ { n } = \frac { 4 ( - 1 ) ^ { n } } { n }$

Find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) $\sum _ { n = 1 } ^ { \infty } \frac { ( x - 4 ) ^ { n - 1 } } { 4 ^ { n - 1 } }$

Use the Direct Comparison Test to determine the convergence or divergence of the series $\sum _ { n = 0 } ^ { \infty } e ^ { - n ^ { 6 } }$ .

Find the Maclaurin series for the function $f ( x ) = \sin 7 x$

Find the third degree Taylor polynomial centered at $c = 1$ for the function. $f ( x ) = \sqrt { x }$

Determine the convergence or divergence of the series using any appropriate test from this chapter. Identify the test used. $\sum _ { n = 1 } ^ { \infty } \frac { 10 } { n ^ { - } }$

$\text { Identify the interval of convergence of a power series } \sum _ { n = 1 } ^ { \infty } ( - 9 ) ^ { n } x ^ { 2 n } \text {. }$

Approximate the sum of the series by using the first six terms. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } 2 } { n ^ { 3 } }$

Determine the convergence or divergence of the series. $\sum _ { n = 0 } ^ { \infty } \left( \frac { 1 } { 4 } \right) ^ { n }$

UseTheorem 9.11 to determine the convergence or divergence of the series. $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 0.86 } }$