Question 1

Use the Limit Comparison Test to determine the convergence or divergence of the series $\sum _ { n = 6 } ^ { \infty } \frac { 1 } { n ^ { 7 } - 6 }$.

Accepted Answer

A) 
The series $\sum _ { n = 6 } ^ { \infty } \frac { 1 } { n ^ { 2 } - 6 }$ diverges. 
B) 
The series $\sum _ { n = 6 } ^ { \infty } \frac { 1 } { n ^ { 7 } - 6 }$ converges. 
A) 
The series $\sum _ { n = 6 } ^ { \infty } \frac { 1 } { n ^ { 2 } - 6 }$ diverges. 
B) 
The series $\sum _ { n = 6 } ^ { \infty } \frac { 1 } { n ^ { 7 } - 6 }$ converges.

Question 2

$\text { If the rate of inflation is } 3 \frac { 1 } { 2 } \% \text { per year and the average price of a car is currently }$ $\$ 40,000$, the average price after $n$ years is $P _ { n } = \$ 40,000 ( 1.035 ) ^ { n }$. Compute the average price after 6 years. Round your answer to two decimal places.

Accepted Answer

A)  $\$ 32,301.59$ 
B)  $\$ 49,170.21$ 
C)  $\$ 248,400.00$ 
D)  $\$ 45,937.01$ 
E)  $\$ 231,600.00$ 
A)  $\$ 32,301.59$ 
B)  $\$ 49,170.21$ 
C)  $\$ 248,400.00$ 
D)  $\$ 45,937.01$ 
E)  $\$ 231,600.00$

Question 3

Find a first-degree polynomial function P1 whose value and slope agree with the value and slope of $f$ at $x = c$. What is $P _ { 1 }$ called?
$$f ( x ) = \frac { 4 } { \sqrt [ 3 ] { x } } , c = 27$$

Accepted Answer

A)  $y = \frac { 4 } { 3 } + \frac { 4 } { 243 } ( x - 27 ) ;$ tangent line to $y = f ( x )$ at $x = 27$ 
B)  $y = \frac { 4 } { 3 } - \frac { 4 } { 243 } ( x - 27 )$; secant line to $y = f ( x )$ at $x = 27$ 
C)  $y = \frac { 4 } { 3 } + \frac { 4 } { 243 } ( x - 27 ) ;$ secant line to $y = f ( x )$ at $x = 27$ 
D)  $y = \frac { 4 } { 3 } + \frac { 4 } { 243 } ( x + 27 )$; secant line to $y = f ( x )$ at $x = 27$ 
E)  $y = \frac { 4 } { 3 } - \frac { 4 } { 243 } ( x - 27 )$; tangent line to $y = f ( x )$ at $x = 27$ 
A)  $y = \frac { 4 } { 3 } + \frac { 4 } { 243 } ( x - 27 ) ;$ tangent line to $y = f ( x )$ at $x = 27$ 
B)  $y = \frac { 4 } { 3 } - \frac { 4 } { 243 } ( x - 27 )$; secant line to $y = f ( x )$ at $x = 27$ 
C)  $y = \frac { 4 } { 3 } + \frac { 4 } { 243 } ( x - 27 ) ;$ secant line to $y = f ( x )$ at $x = 27$ 
D)  $y = \frac { 4 } { 3 } + \frac { 4 } { 243 } ( x + 27 )$; secant line to $y = f ( x )$ at $x = 27$ 
E)  $y = \frac { 4 } { 3 } - \frac { 4 } { 243 } ( x - 27 )$; tangent line to $y = f ( x )$ at $x = 27$

Question 4

Find the radius of convergence of the power series. $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { n } } { 5 ^ { n } }$

Accepted Answer

A)  $\frac { 1 } { 25 }$ 
B)  $\frac { 1 } { 5 }$ 
C)  5 
D)  $\infty$ 
E)  25 
A)  $\frac { 1 } { 25 }$ 
B)  $\frac { 1 } { 5 }$ 
C)  5 
D)  $\infty$ 
E)  25

Question 5

$\text {The series } \sum _ { n = 1 } ^ { \infty } \frac { 1 } { ( 6 n + 7 ) ^ { 3 } } \text { converges. }$

Accepted Answer

A) True 
 B)False

Question 6

$\text { Find the sum of the convergent series } 8 - 1 + \frac { 1 } { 8 } - \frac { 1 } { 64 } + \cdots \cdots$

Accepted Answer

A)  $\frac { 64 } { 9 }$ 
B)  $\frac { 64 } { 65 }$ 
C)  $\frac { 512 } { 9 }$ 
D)  $\frac { 65 } { 9 }$ 
E)  $\frac { 9 } { 64 }$ 
A)  $\frac { 64 } { 9 }$ 
B)  $\frac { 64 } { 65 }$ 
C)  $\frac { 512 } { 9 }$ 
D)  $\frac { 65 } { 9 }$ 
E)  $\frac { 9 } { 64 }$

Question 7

Use the Direct Comparison Test (if possible) to determine whether the series $\sum _ { n = 1 } ^ { \infty } \frac { 2 ^ { n } } { 8 ^ { n } + 1 }$ converges or diverges.

Accepted Answer

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

Question 8

Use Theorem 9.11 to determine the convergence or divergence of the series. $$1 + \frac { 1 } { \sqrt [ 3 ] { 2 ^ { 2 } } } + \frac { 1 } { \sqrt [ 3 ] { 3 ^ { 2 } } } + \frac { 1 } { \sqrt [ 3 ] { 4 ^ { 2 } } } + \frac { 1 } { \sqrt [ 3 ] { 5 ^ { 2 } } } + \cdots$$

Accepted Answer

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

Question 9

$\text { Use the polynomial test to determine whether the series } \frac { 1 } { 20 } + \frac { 2 } { 23 } + \frac { 3 } { 28 } + \frac { 4 } { 35 } + \frac { 5 } { 44 } \ldots$ converges or diverges.

Accepted Answer

A)  The series $\frac { 1 } { 20 } + \frac { 2 } { 23 } + \frac { 3 } { 28 } + \frac { 4 } { 35 } + \frac { 5 } { 44 } \ldots$ diverges. 
B)  The series $\frac { 1 } { 20 } + \frac { 2 } { 23 } + \frac { 3 } { 28 } + \frac { 4 } { 35 } + \frac { 5 } { 44 } \ldots$ converges. 
A)  The series $\frac { 1 } { 20 } + \frac { 2 } { 23 } + \frac { 3 } { 28 } + \frac { 4 } { 35 } + \frac { 5 } { 44 } \ldots$ diverges. 
B)  The series $\frac { 1 } { 20 } + \frac { 2 } { 23 } + \frac { 3 } { 28 } + \frac { 4 } { 35 } + \frac { 5 } { 44 } \ldots$ converges.

Question 10

Use the Direct Comparison Test (if possible) to determine whether the series $\sum _ { n = 5 } ^ { \infty } \frac { 1 } { 7 n ^ { 2 } + 4 }$ converges or diverges.

Accepted Answer

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

Question 11

Use a power series to approximate the value of the integral $\int e ^ { - x ^ { 4 } } d x \text { with an error of }$ less than 0.01. Round your answer to two decimal places.

Accepted Answer

A) 0.81 
B) 0.74 
C) 0.89 
D) 0.88 
E) 0.84 
A) 0.81 
B) 0.74 
C) 0.89 
D) 0.88 
E) 0.84

Question 12

Identify the most appropriate test to be used to determine whether the series $\sum _ { n = 1 } ^ { \infty } \frac { 15 ( - 1 ) ^ { n + 1 } } { n }$ converges or diverges.

Accepted Answer

A)  Ratio Test 
B)  $\rho$-Series Test 
C)  Alternating Series Test 
D)  Telescoping Series Test 
E)  Root Test 
A)  Ratio Test 
B)  $\rho$-Series Test 
C)  Alternating Series Test 
D)  Telescoping Series Test 
E)  Root Test

Question 13

$\text { Consider the function } f ( x ) = \frac { 20 } { \sqrt { x } } \text {, and its second-degree polynomial }$ 
$P _ { 2 } ( x ) = 20 - 10 ( x - 1 ) + \frac { 15 } { 2 } ( x - 1 ) ^ { 2 }$ at $x = 0.8$. Compute the value of $f ( 0.8 )$ and $P _ { 2 } ( 0.8 )$. Round your answer to four decimal places.

Accepted Answer

A)  $f ( 0.8 ) \approx 22.3607 , P _ { 2 } ( 0.8 ) \approx 44.6000$ 
B)  $f ( 0.8 ) \approx 22.3607 , P _ { 2 } ( 0.8 ) \approx 22.3000$ 
C)  $f ( 0.8 ) \approx 45.7214 ; P _ { 2 } ( 0.8 ) \approx 45.6000$ 
D)  $f ( 0.8 ) \approx 21.3607 , P _ { 2 } ( 0.8 ) \approx 21.3000$ 
E)  $f ( 0.8 ) \approx 23.3607 , P _ { 2 } ( 0.8 ) \approx 22.3000$ 
A)  $f ( 0.8 ) \approx 22.3607 , P _ { 2 } ( 0.8 ) \approx 44.6000$ 
B)  $f ( 0.8 ) \approx 22.3607 , P _ { 2 } ( 0.8 ) \approx 22.3000$ 
C)  $f ( 0.8 ) \approx 45.7214 ; P _ { 2 } ( 0.8 ) \approx 45.6000$ 
D)  $f ( 0.8 ) \approx 21.3607 , P _ { 2 } ( 0.8 ) \approx 21.3000$ 
E)  $f ( 0.8 ) \approx 23.3607 , P _ { 2 } ( 0.8 ) \approx 22.3000$

Question 14

Use the definition to find the Taylor series (centered at c) for the function. $$f ( x ) = e ^ { 2 x } , c = 0$$

Accepted Answer

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

Question 15

Use Theorem 9.11 to determine the convergence or divergence of the series. $\sum _ { n = 1 } ^ { \infty } \frac { 2 } { \frac { 5 } { 9 } }$

Accepted Answer

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

Question 16

$\text { State where the power series } \sum _ { n = 1 } ^ { \infty } \frac { ( x - 2 ) ^ { n } } { n ^ { 3 } } \text { is centered. }$

Accepted Answer

A)  0 
B)  $- 3$ 
C)  2 
D)  3 
E)  $- 2$ 
A)  0 
B)  $- 3$ 
C)  2 
D)  3 
E)  $- 2$

Question 17

Find the Maclaurin polynomial of degree 3 for the function. 
$f ( x ) = e ^ { - 9 x }$

Accepted Answer

A)  $- 1 + 9 x - \frac { 81 } { 2 } x ^ { 2 } + \frac { 243 } { 2 } x ^ { 3 }$ 
B)  $1 - 9 x + \frac { 81 } { 2 } x ^ { 2 } - \frac { 243 } { 2 } x ^ { 3 }$ 
C)  $1 + 9 x + \frac { 81 } { 2 } x ^ { 2 } + \frac { 243 } { 2 } x ^ { 3 }$ 
D)  $1 - 9 x - \frac { 81 } { 2 } x ^ { 2 } - \frac { 243 } { 2 } x ^ { 3 }$ 
E)  $1 - 9 x + \frac { 81 } { 2 } x ^ { 2 } + \frac { 243 } { 2 } x ^ { 3 }$ 
A)  $- 1 + 9 x - \frac { 81 } { 2 } x ^ { 2 } + \frac { 243 } { 2 } x ^ { 3 }$ 
B)  $1 - 9 x + \frac { 81 } { 2 } x ^ { 2 } - \frac { 243 } { 2 } x ^ { 3 }$ 
C)  $1 + 9 x + \frac { 81 } { 2 } x ^ { 2 } + \frac { 243 } { 2 } x ^ { 3 }$ 
D)  $1 - 9 x - \frac { 81 } { 2 } x ^ { 2 } - \frac { 243 } { 2 } x ^ { 3 }$ 
E)  $1 - 9 x + \frac { 81 } { 2 } x ^ { 2 } + \frac { 243 } { 2 } x ^ { 3 }$

Question 18

Find the Maclaurin polynomial of degree 5 for the function. $f ( x ) = \sin ( 2 x )$

Accepted Answer

A)  $2 x - \frac { 4 } { 3 } x ^ { 3 } + \frac { 4 } { 15 } x ^ { 5 }$ 
B)  $2 + \frac { 4 } { 3 } x ^ { 2 } + \frac { 2 } { 3 } x ^ { 2 }$ 
C)  $2 x - \frac { 8 } { 3 } x ^ { 3 } + \frac { 32 } { 5 } x ^ { 5 }$ 
D)  $2 x - \frac { 4 } { 3 } x ^ { 3 } + \frac { 2 } { 3 } x ^ { 5 }$ 
E)  $2 x + \frac { 4 } { 3 } x ^ { 3 } + \frac { 4 } { 15 } x ^ { 5 }$ 
A)  $2 x - \frac { 4 } { 3 } x ^ { 3 } + \frac { 4 } { 15 } x ^ { 5 }$ 
B)  $2 + \frac { 4 } { 3 } x ^ { 2 } + \frac { 2 } { 3 } x ^ { 2 }$ 
C)  $2 x - \frac { 8 } { 3 } x ^ { 3 } + \frac { 32 } { 5 } x ^ { 5 }$ 
D)  $2 x - \frac { 4 } { 3 } x ^ { 3 } + \frac { 2 } { 3 } x ^ { 5 }$ 
E)  $2 x + \frac { 4 } { 3 } x ^ { 3 } + \frac { 4 } { 15 } x ^ { 5 }$

Question 19

$\text {The series } \sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } n } { 6 n + 4 } \text { converges. }$

Accepted Answer

A) True 
 B)False

Question 20

Determine the convergence or divergence of the series. $\sum _ { n = 1 } ^ { \infty } \frac { 4 } { n \cdot \sqrt [ 8 ] { n } }$

Accepted Answer

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

Use the Limit Comparison Test to determine the convergence or divergence of the series $\sum _ { n = 6 } ^ { \infty } \frac { 1 } { n ^ { 7 } - 6 }$ .

Find a first-degree polynomial function P1 whose value and slope agree with the value and slope of $f$ at $x = c$ . What is $P _ { 1 }$ called? $f ( x ) = \frac { 4 } { \sqrt [ 3 ] { x } } , c = 27$

Find the radius of convergence of the power series. $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { n } } { 5 ^ { n } }$

$\text {The series } \sum _ { n = 1 } ^ { \infty } \frac { 1 } { ( 6 n + 7 ) ^ { 3 } } \text { converges. }$

$\text { Find the sum of the convergent series } 8 - 1 + \frac { 1 } { 8 } - \frac { 1 } { 64 } + \cdots \cdots$

Use the Direct Comparison Test (if possible) to determine whether the series $\sum _ { n = 1 } ^ { \infty } \frac { 2 ^ { n } } { 8 ^ { n } + 1 }$ converges or diverges.

Use Theorem 9.11 to determine the convergence or divergence of the series. $1 + \frac { 1 } { \sqrt [ 3 ] { 2 ^ { 2 } } } + \frac { 1 } { \sqrt [ 3 ] { 3 ^ { 2 } } } + \frac { 1 } { \sqrt [ 3 ] { 4 ^ { 2 } } } + \frac { 1 } { \sqrt [ 3 ] { 5 ^ { 2 } } } + \cdots$

$\text { Use the polynomial test to determine whether the series } \frac { 1 } { 20 } + \frac { 2 } { 23 } + \frac { 3 } { 28 } + \frac { 4 } { 35 } + \frac { 5 } { 44 } \ldots$ converges or diverges.

Use the Direct Comparison Test (if possible) to determine whether the series $\sum _ { n = 5 } ^ { \infty } \frac { 1 } { 7 n ^ { 2 } + 4 }$ converges or diverges.

Use a power series to approximate the value of the integral $\int e ^ { - x ^ { 4 } } d x \text { with an error of }$ less than 0.01. Round your answer to two decimal places.

Identify the most appropriate test to be used to determine whether the series $\sum _ { n = 1 } ^ { \infty } \frac { 15 ( - 1 ) ^ { n + 1 } } { n }$ converges or diverges.

Use the definition to find the Taylor series (centered at c) for the function. $f ( x ) = e ^ { 2 x } , c = 0$

Use Theorem 9.11 to determine the convergence or divergence of the series. $\sum _ { n = 1 } ^ { \infty } \frac { 2 } { \frac { 5 } { 9 } }$

$\text { State where the power series } \sum _ { n = 1 } ^ { \infty } \frac { ( x - 2 ) ^ { n } } { n ^ { 3 } } \text { is centered. }$

Find the Maclaurin polynomial of degree 3 for the function. $f ( x ) = e ^ { - 9 x }$

Find the Maclaurin polynomial of degree 5 for the function. $f ( x ) = \sin ( 2 x )$

$\text {The series } \sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } n } { 6 n + 4 } \text { converges. }$

Determine the convergence or divergence of the series. $\sum _ { n = 1 } ^ { \infty } \frac { 4 } { n \cdot \sqrt [ 8 ] { n } }$

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 Limit Comparison Test to determine the convergence or divergence of the series ∑n=6∞1n7−6\sum _ { n = 6 } ^ { \infty } \frac { 1 } { n ^ { 7 } - 6 }∑n=6∞​n7−61​ .

Find a first-degree polynomial function P1 whose value and slope agree with the value and slope of fff at x=cx = cx=c . What is P1P _ { 1 }P1​ called? f(x)=4x3,c=27f ( x ) = \frac { 4 } { \sqrt [ 3 ] { x } } , c = 27f(x)=3x​4​,c=27

Find the radius of convergence of the power series. ∑n=0∞(−1)nxn5n\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { n } } { 5 ^ { n } }∑n=0∞​5n(−1)nxn​

The series ∑n=1∞1(6n+7)3 converges. \text {The series } \sum _ { n = 1 } ^ { \infty } \frac { 1 } { ( 6 n + 7 ) ^ { 3 } } \text { converges. }The series ∑n=1∞​(6n+7)31​ converges.

Find the sum of the convergent series 8−1+18−164+⋯⋯\text { Find the sum of the convergent series } 8 - 1 + \frac { 1 } { 8 } - \frac { 1 } { 64 } + \cdots \cdots Find the sum of the convergent series 8−1+81​−641​+⋯⋯

Use the Direct Comparison Test (if possible) to determine whether the series ∑n=1∞2n8n+1\sum _ { n = 1 } ^ { \infty } \frac { 2 ^ { n } } { 8 ^ { n } + 1 }∑n=1∞​8n+12n​ converges or diverges.

Use the Direct Comparison Test (if possible) to determine whether the series ∑n=5∞17n2+4\sum _ { n = 5 } ^ { \infty } \frac { 1 } { 7 n ^ { 2 } + 4 }∑n=5∞​7n2+41​ converges or diverges.

Use a power series to approximate the value of the integral ∫e−x4dx with an error of \int e ^ { - x ^ { 4 } } d x \text { with an error of }∫e−x4dx with an error of less than 0.01. Round your answer to two decimal places.

Identify the most appropriate test to be used to determine whether the series ∑n=1∞15(−1)n+1n\sum _ { n = 1 } ^ { \infty } \frac { 15 ( - 1 ) ^ { n + 1 } } { n }∑n=1∞​n15(−1)n+1​ converges or diverges.

Use the definition to find the Taylor series (centered at c) for the function. f(x)=e2x,c=0f ( x ) = e ^ { 2 x } , c = 0f(x)=e2x,c=0

Use Theorem 9.11 to determine the convergence or divergence of the series. ∑n=1∞259\sum _ { n = 1 } ^ { \infty } \frac { 2 } { \frac { 5 } { 9 } }∑n=1∞​95​2​

State where the power series ∑n=1∞(x−2)nn3 is centered. \text { State where the power series } \sum _ { n = 1 } ^ { \infty } \frac { ( x - 2 ) ^ { n } } { n ^ { 3 } } \text { is centered. } State where the power series ∑n=1∞​n3(x−2)n​ is centered.

Find the Maclaurin polynomial of degree 3 for the function. f(x)=e−9xf ( x ) = e ^ { - 9 x }f(x)=e−9x

Find the Maclaurin polynomial of degree 5 for the function. f(x)=sin⁡(2x)f ( x ) = \sin ( 2 x )f(x)=sin(2x)

The series ∑n=1∞(−1)n+1n6n+4 converges. \text {The series } \sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } n } { 6 n + 4 } \text { converges. }The series ∑n=1∞​6n+4(−1)n+1n​ converges.

Determine the convergence or divergence of the series. ∑n=1∞4n⋅n8\sum _ { n = 1 } ^ { \infty } \frac { 4 } { n \cdot \sqrt [ 8 ] { n } }∑n=1∞​n⋅8n​4​

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 Limit Comparison Test to determine the convergence or divergence of the series $\sum _ { n = 6 } ^ { \infty } \frac { 1 } { n ^ { 7 } - 6 }$ .

Find a first-degree polynomial function P1 whose value and slope agree with the value and slope of $f$ at $x = c$ . What is $P _ { 1 }$ called? $f ( x ) = \frac { 4 } { \sqrt [ 3 ] { x } } , c = 27$

Find the radius of convergence of the power series. $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { n } } { 5 ^ { n } }$

$\text {The series } \sum _ { n = 1 } ^ { \infty } \frac { 1 } { ( 6 n + 7 ) ^ { 3 } } \text { converges. }$

$\text { Find the sum of the convergent series } 8 - 1 + \frac { 1 } { 8 } - \frac { 1 } { 64 } + \cdots \cdots$

Use the Direct Comparison Test (if possible) to determine whether the series $\sum _ { n = 1 } ^ { \infty } \frac { 2 ^ { n } } { 8 ^ { n } + 1 }$ converges or diverges.

Use the Direct Comparison Test (if possible) to determine whether the series $\sum _ { n = 5 } ^ { \infty } \frac { 1 } { 7 n ^ { 2 } + 4 }$ converges or diverges.

Use a power series to approximate the value of the integral $\int e ^ { - x ^ { 4 } } d x \text { with an error of }$ less than 0.01. Round your answer to two decimal places.

Identify the most appropriate test to be used to determine whether the series $\sum _ { n = 1 } ^ { \infty } \frac { 15 ( - 1 ) ^ { n + 1 } } { n }$ converges or diverges.

Use the definition to find the Taylor series (centered at c) for the function. $f ( x ) = e ^ { 2 x } , c = 0$

Use Theorem 9.11 to determine the convergence or divergence of the series. $\sum _ { n = 1 } ^ { \infty } \frac { 2 } { \frac { 5 } { 9 } }$

$\text { State where the power series } \sum _ { n = 1 } ^ { \infty } \frac { ( x - 2 ) ^ { n } } { n ^ { 3 } } \text { is centered. }$

Find the Maclaurin polynomial of degree 3 for the function. $f ( x ) = e ^ { - 9 x }$

Find the Maclaurin polynomial of degree 5 for the function. $f ( x ) = \sin ( 2 x )$

$\text {The series } \sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } n } { 6 n + 4 } \text { converges. }$

Determine the convergence or divergence of the series. $\sum _ { n = 1 } ^ { \infty } \frac { 4 } { n \cdot \sqrt [ 8 ] { n } }$