Question 1

$\text { Write an expression for the } n \text {th term of the sequence } 10,22,34,46 , \ldots \text {. }$

Accepted Answer

A)  $24 n + 2$ 
B)  $12 n - 2$ 
C)  $10 n + 12$ 
D)  $12 n + 2$ 
E)  $14 n - 12$ 
A)  $24 n + 2$ 
B)  $12 n - 2$ 
C)  $10 n + 12$ 
D)  $12 n + 2$ 
E)  $14 n - 12$

Question 2

$\text { What is } P _ { 1 } \text {, a first-degree polynomial function whose value and slope agree with the }$ value and slope of $f ( x ) = 10 \tan ( x )$ at $x = \frac { \pi } { 4 }$ ?

Accepted Answer

A)  $P _ { 1 }$ is the tangent line to the curve $f ( x ) = 10 \tan ( x )$ at the point $\left( 10 , \frac { \pi } { 4 } \right)$. 
B)  $P _ { 1 }$ is the tangent line to the curve $f ( x ) = 10 \tan ( x )$ at the point $\left( \frac { \pi } { 4 } , 10 \right)$. 
C)  $P _ { 1 }$ is the tangent line to the curve $f ( x ) = 10 \tan ( x )$ at the point $\left( 13 , \frac { \pi } { 4 } \right)$. 
D)  $P _ { 1 }$ is the tangent line to the curve $f ( x ) = 10 \tan ( x )$ at the point $\left( 10 , - \frac { \pi } { 4 } \right)$. 
E)  $P _ { 1 }$ is the tangent line to the curve $f ( x ) = 10 \tan ( x )$ at the point $\left( \frac { \pi } { 4 } , - 10 \right)$. 
A)  $P _ { 1 }$ is the tangent line to the curve $f ( x ) = 10 \tan ( x )$ at the point $\left( 10 , \frac { \pi } { 4 } \right)$. 
B)  $P _ { 1 }$ is the tangent line to the curve $f ( x ) = 10 \tan ( x )$ at the point $\left( \frac { \pi } { 4 } , 10 \right)$. 
C)  $P _ { 1 }$ is the tangent line to the curve $f ( x ) = 10 \tan ( x )$ at the point $\left( 13 , \frac { \pi } { 4 } \right)$. 
D)  $P _ { 1 }$ is the tangent line to the curve $f ( x ) = 10 \tan ( x )$ at the point $\left( 10 , - \frac { \pi } { 4 } \right)$. 
E)  $P _ { 1 }$ is the tangent line to the curve $f ( x ) = 10 \tan ( x )$ at the point $\left( \frac { \pi } { 4 } , - 10 \right)$.

Question 3

Use the Root Test to determine the convergence or divergence of the series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 16 ^ { n } }$

Accepted Answer

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

Question 4

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

Accepted Answer

A) 
The series $\sum _ { n = 1 } ^ { \infty } \frac { 3 ^ { n } + 1 } { 8 ^ { n } + 1 }$ converges. 
B) 
The series $\sum _ { n = 1 } ^ { \infty } \frac { 3 ^ { n } + 1 } { 8 ^ { n } + 1 }$ diverges. 
A) 
The series $\sum _ { n = 1 } ^ { \infty } \frac { 3 ^ { n } + 1 } { 8 ^ { n } + 1 }$ converges. 
B) 
The series $\sum _ { n = 1 } ^ { \infty } \frac { 3 ^ { n } + 1 } { 8 ^ { n } + 1 }$ diverges.

Question 5

$\text {The series } \frac { \ln 2 } { 4 } + \frac { \ln 3 } { 6 } + \frac { \ln 4 } { 8 } + \frac { \ln 5 } { 10 } + \frac { \ln 6 } { 12 } + \cdots \text { converges. }$

Accepted Answer

A) True 
 B)False

Question 6

$\text { Use the power series } \frac { 1 } { 1 + x } = \sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } x ^ { n } \text { to determine a power series centered at } 0$ for the function $g ( x ) = \frac { 1 } { 11 x ^ { 2 } + 1 }$.

Accepted Answer

A)  $\sum _ { n = 0 } ^ { \infty } ( - 11 ) ^ { n } x ^ { n }$ 
B) $\sum _ { n = 0 } ^ { \infty } ( 11 ) ^ { n } x ^ { 2 n }$ 
C)  $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } x ^ { 2 n }$ 
D) $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } x ^ { n }$ 
E) $\sum _ { n = 0 } ^ { \infty } ( - 11 ) ^ { n } x ^ { 2 n }$ 
A)  $\sum _ { n = 0 } ^ { \infty } ( - 11 ) ^ { n } x ^ { n }$ 
B) $\sum _ { n = 0 } ^ { \infty } ( 11 ) ^ { n } x ^ { 2 n }$ 
C)  $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } x ^ { 2 n }$ 
D) $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } x ^ { n }$ 
E) $\sum _ { n = 0 } ^ { \infty } ( - 11 ) ^ { n } x ^ { 2 n }$

Question 7

$\text { Write the repeating decimal } 0 . \overline { 75 } \text { as a geometric series. }$

Accepted Answer

A)  $\sum _ { n = 0 } ^ { \infty } 75 \left( \frac { 1 } { 100 } \right) ^ { n }$ 
B)  $\sum _ { n = 0 } ^ { \infty } \left( \frac { 75 } { 10 } \right) ^ { n }$ 
C) $\sum _ { n = 0 } ^ { \infty } \frac { 75 } { 100 } \left( \frac { 1 } { 100 } \right) ^ { n }$ 
D) 
$\sum _ { n = 0 } ^ { \infty } \frac { 75 } { 100 } - \left( \frac { 1 } { 100 } \right) ^ { n }$ 
E) 
$\sum _ { n = 0 } ^ { \infty } \frac { 75 } { 100 } + \left( \frac { 1 } { 100 } \right) ^ { n }$ 
A)  $\sum _ { n = 0 } ^ { \infty } 75 \left( \frac { 1 } { 100 } \right) ^ { n }$ 
B)  $\sum _ { n = 0 } ^ { \infty } \left( \frac { 75 } { 10 } \right) ^ { n }$ 
C) $\sum _ { n = 0 } ^ { \infty } \frac { 75 } { 100 } \left( \frac { 1 } { 100 } \right) ^ { n }$ 
D) 
$\sum _ { n = 0 } ^ { \infty } \frac { 75 } { 100 } - \left( \frac { 1 } { 100 } \right) ^ { n }$ 
E) 
$\sum _ { n = 0 } ^ { \infty } \frac { 75 } { 100 } + \left( \frac { 1 } { 100 } \right) ^ { n }$

Question 8

$\text { Determine the convergence or divergence of the series } \sum _ { n = 1 } ^ { \infty } \frac { \cos n } { 10 ^ { n } } \text { using any }$ appropriate test.

Accepted Answer

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

Question 9

Use the Limit Comparison Test to determine the convergence or divergence of the series $\sum _ { n = 1 } ^ { \infty } \frac { 9 } { n \sqrt { n ^ { 2 } + 6 } }$.

Accepted Answer

A)  The series $\sum _ { n = 1 } ^ { \infty } \frac { 9 } { n \sqrt { n ^ { 2 } + 6 } }$ diverges. 
B)  The series $\sum _ { n = 1 } ^ { \infty } \frac { 9 } { n \sqrt { n ^ { 2 } + 6 } }$ converges. 
A)  The series $\sum _ { n = 1 } ^ { \infty } \frac { 9 } { n \sqrt { n ^ { 2 } + 6 } }$ diverges. 
B)  The series $\sum _ { n = 1 } ^ { \infty } \frac { 9 } { n \sqrt { n ^ { 2 } + 6 } }$ converges.

Question 10

$\text { Consider the function given by } f ( x ) = \sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } ( x - 6 ) ^ { n } } { n } \text {. Find the interval of }$ convergence for $f ^ { \prime } ( x )$.

Accepted Answer

A)  $[ - 6,6 ]$ 
B)  $[ 5,7 ]$ 
C)  $( 0,7 )$ 
D)  $( - 6,6 )$ 
E)  $( 5,7 )$ 
A)  $[ - 6,6 ]$ 
B)  $[ 5,7 ]$ 
C)  $( 0,7 )$ 
D)  $( - 6,6 )$ 
E)  $( 5,7 )$

Question 11

Determine the convergence or divergence of the series using any appropriate test from this chapter. Identify the test used. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } 9 } { 7 n }$

Accepted Answer

A)  converges; Integral Test 
B)  converges; Ratio Test 
C)  converges; Alternating Series Test 
D)  diverges; Ratio Test 
E)  diverges; Integral Test 
A)  converges; Integral Test 
B)  converges; Ratio Test 
C)  converges; Alternating Series Test 
D)  diverges; Ratio Test 
E)  diverges; Integral Test

Question 12

$\text { The terms of a series } \sum _ { n = 1 } ^ { \infty } a _ { n } \text { are defined recursively. Determine the convergence or }$ divergence of the series. Explain your reasoning. 
$a _ { 1 } = 6 , a _ { n + 1 } = \frac { 4 n - 1 } { - 3 n - 2 } a _ { n }$

Accepted Answer

A)  converges; Ratio Test 
B)  diverges; Ratio Test 
C)  diverges; Integral Test 
D)  converges; Alternating Series Test 
E)  converges; Root Test 
A)  converges; Ratio Test 
B)  diverges; Ratio Test 
C)  diverges; Integral Test 
D)  converges; Alternating Series Test 
E)  converges; Root Test

Question 13

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

Accepted Answer

A) True 
 B)False

Question 14

$\text { Find a power series for the function } \frac { 1 } { 12 - x } \text { centered at } 1 \text {. }$

Accepted Answer

A)  $\sum _ { n = 0 } ^ { \infty } \frac { ( x - 1 ) ^ { n } } { 11 ^ { n } }$ 
B)  $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( x - 1 ) ^ { n } } { 11 ^ { n + 1 } }$ 
C) $\sum _ { n = 0 } ^ { \infty } \frac { ( x - 1 ) ^ { n } } { 11 ^ { n + 1 } }$ 
D) $\sum _ { n = 0 } ^ { \infty } \frac { ( x - 1 ) ^ { n } } { 11 ^ { n } + 1 }$ 
E) $\sum _ { n = 0 } ^ { \infty } \frac { ( x - 1 ) ^ { n } } { 11 ^ { n } } + 1$ 
A)  $\sum _ { n = 0 } ^ { \infty } \frac { ( x - 1 ) ^ { n } } { 11 ^ { n } }$ 
B)  $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( x - 1 ) ^ { n } } { 11 ^ { n + 1 } }$ 
C) $\sum _ { n = 0 } ^ { \infty } \frac { ( x - 1 ) ^ { n } } { 11 ^ { n + 1 } }$ 
D) $\sum _ { n = 0 } ^ { \infty } \frac { ( x - 1 ) ^ { n } } { 11 ^ { n } + 1 }$ 
E) $\sum _ { n = 0 } ^ { \infty } \frac { ( x - 1 ) ^ { n } } { 11 ^ { n } } + 1$

Question 15

Use the Ratio Test to determine the convergence or divergence of the series $$\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( 7 ) ^ { 8 n } } { ( 7 n + 1 ) ! }$$

Accepted Answer

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

Question 16

Determine the minimal number of terms required to approximate the sum of the series with an error of less than 0.005. $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { ( 2 n ) ! }$

Accepted Answer

A)  6 
B)  4 
C)  5 
D)  8 
E)  3 
A)  6 
B)  4 
C)  5 
D)  8 
E)  3

Question 17

Use the Limit Comparison Test (if possible) to determine whether the series $\sum _ { n = 1 } ^ { \infty } \frac { 2 n ^ { 2 } - 7 } { 7 n ^ { 7 } + 3 n + 2 }$ converges or diverges.

Accepted Answer

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

Question 18

Use the binomial series to find the Maclaurin series for the function 
$f ( x ) = \frac { 8 } { ( 1 + x ) ^ { 2 } }$

Accepted Answer

A) $\frac { 8 } { ( x + 1 ) ^ { 2 } } = 8 + 8 \sum _ { n = 0 } ^ { \infty } \frac { 1 \cdot 3 \cdot 5 \cdots ( 2 n - 1 ) x ^ { n } } { 2 ^ { n } n ! }$ 
B) $\frac { 8 } { ( x + 1 ) ^ { 2 } } = 8 + 8 \sum _ { n = 1 } ^ { \infty } \frac { 1 \cdot 3 \cdot 5 \cdots ( 2 n - 1 ) x ^ { n } } { 2 ^ { n } n ! }$ 
C)  $\frac { 8 } { ( x + 1 ) ^ { 2 } } = 8 + 8 \sum _ { n = 0 } ^ { \infty } \frac { 1 \cdot 3 \cdot 5 \cdots ( 2 n - 1 ) x ^ { n } } { n ! }$ 
D)  $\frac { 8 } { ( x + 1 ) ^ { 2 } } = 8 \sum _ { n = 0 } ^ { \infty } \frac { 1 \cdot 3 \cdot 5 \cdots ( 2 n - 1 ) x ^ { n } } { 2 ^ { n } n ! }$ 
E) $\frac { 8 } { ( x + 1 ) ^ { 2 } } = 8 \sum _ { n = 1 } ^ { \infty } \frac { ( 2 n - 1 ) x ^ { n } } { 2 ^ { n } }$ 
A) $\frac { 8 } { ( x + 1 ) ^ { 2 } } = 8 + 8 \sum _ { n = 0 } ^ { \infty } \frac { 1 \cdot 3 \cdot 5 \cdots ( 2 n - 1 ) x ^ { n } } { 2 ^ { n } n ! }$ 
B) $\frac { 8 } { ( x + 1 ) ^ { 2 } } = 8 + 8 \sum _ { n = 1 } ^ { \infty } \frac { 1 \cdot 3 \cdot 5 \cdots ( 2 n - 1 ) x ^ { n } } { 2 ^ { n } n ! }$ 
C)  $\frac { 8 } { ( x + 1 ) ^ { 2 } } = 8 + 8 \sum _ { n = 0 } ^ { \infty } \frac { 1 \cdot 3 \cdot 5 \cdots ( 2 n - 1 ) x ^ { n } } { n ! }$ 
D)  $\frac { 8 } { ( x + 1 ) ^ { 2 } } = 8 \sum _ { n = 0 } ^ { \infty } \frac { 1 \cdot 3 \cdot 5 \cdots ( 2 n - 1 ) x ^ { n } } { 2 ^ { n } n ! }$ 
E) $\frac { 8 } { ( x + 1 ) ^ { 2 } } = 8 \sum _ { n = 1 } ^ { \infty } \frac { ( 2 n - 1 ) x ^ { n } } { 2 ^ { n } }$

Question 19

Determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. $a _ { n } = \frac { \ln \left( n ^ { 3 } \right) } { 7 n }$

Accepted Answer

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

Question 20

Find the Maclaurin polynomial of degree 4 for the function. $$f ( x ) = e ^ { 9 x }$$

Accepted Answer

A)  $1 + 9 x + \frac { 81 } { 2 } x ^ { 2 } + \frac { 243 } { 2 } x ^ { 3 } + \frac { 2187 } { 8 } x ^ { 4 }$ 
B)  $1 - 9 x + \frac { 27 } { 2 } x ^ { 2 } + \frac { 243 } { 2 } x ^ { 3 } + \frac { 2187 } { 8 } x ^ { 4 }$ 
C)  $1 - 9 x + \frac { 27 } { 2 } x ^ { 2 } - \frac { 243 } { 4 } x ^ { 3 } + \frac { 2187 } { 16 } x ^ { 4 }$ 
D)  $1 + 9 x + \frac { 27 } { 2 } x ^ { 2 } + \frac { 243 } { 4 } x ^ { 3 } + \frac { 2187 } { 16 } x ^ { 4 }$ 
E)  $1 + 9 x + \frac { 81 } { 2 } x ^ { 2 } + \frac { 729 } { 4 } x ^ { 3 } + \frac { 2187 } { 2 } x ^ { 4 }$ 
A)  $1 + 9 x + \frac { 81 } { 2 } x ^ { 2 } + \frac { 243 } { 2 } x ^ { 3 } + \frac { 2187 } { 8 } x ^ { 4 }$ 
B)  $1 - 9 x + \frac { 27 } { 2 } x ^ { 2 } + \frac { 243 } { 2 } x ^ { 3 } + \frac { 2187 } { 8 } x ^ { 4 }$ 
C)  $1 - 9 x + \frac { 27 } { 2 } x ^ { 2 } - \frac { 243 } { 4 } x ^ { 3 } + \frac { 2187 } { 16 } x ^ { 4 }$ 
D)  $1 + 9 x + \frac { 27 } { 2 } x ^ { 2 } + \frac { 243 } { 4 } x ^ { 3 } + \frac { 2187 } { 16 } x ^ { 4 }$ 
E)  $1 + 9 x + \frac { 81 } { 2 } x ^ { 2 } + \frac { 729 } { 4 } x ^ { 3 } + \frac { 2187 } { 2 } x ^ { 4 }$

$\text { Write an expression for the } n \text {th term of the sequence } 10,22,34,46 , \ldots \text {. }$

$\text { What is } P _ { 1 } \text {, a first-degree polynomial function whose value and slope agree with the }$ value and slope of $f ( x ) = 10 \tan ( x )$ at $x = \frac { \pi } { 4 }$ ?

Use the Root Test to determine the convergence or divergence of the series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 16 ^ { n } }$

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

$\text {The series } \frac { \ln 2 } { 4 } + \frac { \ln 3 } { 6 } + \frac { \ln 4 } { 8 } + \frac { \ln 5 } { 10 } + \frac { \ln 6 } { 12 } + \cdots \text { converges. }$

$\text { Use the power series } \frac { 1 } { 1 + x } = \sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } x ^ { n } \text { to determine a power series centered at } 0$ for the function $g ( x ) = \frac { 1 } { 11 x ^ { 2 } + 1 }$ .

$\text { Write the repeating decimal } 0 . \overline { 75 } \text { as a geometric series. }$

$\text { Determine the convergence or divergence of the series } \sum _ { n = 1 } ^ { \infty } \frac { \cos n } { 10 ^ { n } } \text { using any }$ appropriate test.

Use the Limit Comparison Test to determine the convergence or divergence of the series $\sum _ { n = 1 } ^ { \infty } \frac { 9 } { n \sqrt { n ^ { 2 } + 6 } }$ .

$\text { Consider the function given by } f ( x ) = \sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } ( x - 6 ) ^ { n } } { n } \text {. Find the interval of }$ convergence for $f ^ { \prime } ( x )$ .

Determine the convergence or divergence of the series using any appropriate test from this chapter. Identify the test used. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } 9 } { 7 n }$

$\text { The terms of a series } \sum _ { n = 1 } ^ { \infty } a _ { n } \text { are defined recursively. Determine the convergence or }$ divergence of the series. Explain your reasoning. $a _ { 1 } = 6 , a _ { n + 1 } = \frac { 4 n - 1 } { - 3 n - 2 } a _ { n }$

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

$\text { Find a power series for the function } \frac { 1 } { 12 - x } \text { centered at } 1 \text {. }$

Use the Ratio Test to determine the convergence or divergence of the series $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( 7 ) ^ { 8 n } } { ( 7 n + 1 ) ! }$

Determine the minimal number of terms required to approximate the sum of the series with an error of less than 0.005. $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { ( 2 n ) ! }$

Use the Limit Comparison Test (if possible) to determine whether the series $\sum _ { n = 1 } ^ { \infty } \frac { 2 n ^ { 2 } - 7 } { 7 n ^ { 7 } + 3 n + 2 }$ converges or diverges.

Use the binomial series to find the Maclaurin series for the function $f ( x ) = \frac { 8 } { ( 1 + x ) ^ { 2 } }$

Determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. $a _ { n } = \frac { \ln \left( n ^ { 3 } \right) } { 7 n }$

Find the Maclaurin polynomial of degree 4 for the function. $f ( x ) = e ^ { 9 x }$

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

Write an expression for the nth term of the sequence 10,22,34,46,…. \text { Write an expression for the } n \text {th term of the sequence } 10,22,34,46 , \ldots \text {. } Write an expression for the nth term of the sequence 10,22,34,46,….

Use the Root Test to determine the convergence or divergence of the series ∑n=1∞116n\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 16 ^ { n } }∑n=1∞​16n1​

Use the Limit Comparison Test to determine the convergence or divergence of the series ∑n=1∞3n+18n+1\sum _ { n = 1 } ^ { \infty } \frac { 3 ^ { n } + 1 } { 8 ^ { n } + 1 }∑n=1∞​8n+13n+1​ .

Write the repeating decimal 0.75‾ as a geometric series. \text { Write the repeating decimal } 0 . \overline { 75 } \text { as a geometric series. } Write the repeating decimal 0.75 as a geometric series.

Use the Limit Comparison Test to determine the convergence or divergence of the series ∑n=1∞9nn2+6\sum _ { n = 1 } ^ { \infty } \frac { 9 } { n \sqrt { n ^ { 2 } + 6 } }∑n=1∞​nn2+6​9​ .

Determine the convergence or divergence of the series using any appropriate test from this chapter. Identify the test used. ∑n=1∞(−1)n97n\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } 9 } { 7 n }∑n=1∞​7n(−1)n9​

The series ∑n=1∞(−1)n(6n+3)!\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { ( 6 n + 3 ) ! }∑n=1∞​(6n+3)!(−1)n​ converges

Find a power series for the function 112−x centered at 1. \text { Find a power series for the function } \frac { 1 } { 12 - x } \text { centered at } 1 \text {. } Find a power series for the function 12−x1​ centered at 1.

Use the Ratio Test to determine the convergence or divergence of the series ∑n=0∞(−1)n(7)8n(7n+1)!\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( 7 ) ^ { 8 n } } { ( 7 n + 1 ) ! }∑n=0∞​(7n+1)!(−1)n(7)8n​

Determine the minimal number of terms required to approximate the sum of the series with an error of less than 0.005. ∑n=0∞(−1)n(2n)!\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { ( 2 n ) ! }∑n=0∞​(2n)!(−1)n​

Use the Limit Comparison Test (if possible) to determine whether the series ∑n=1∞2n2−77n7+3n+2\sum _ { n = 1 } ^ { \infty } \frac { 2 n ^ { 2 } - 7 } { 7 n ^ { 7 } + 3 n + 2 }∑n=1∞​7n7+3n+22n2−7​ converges or diverges.

Use the binomial series to find the Maclaurin series for the function f(x)=8(1+x)2f ( x ) = \frac { 8 } { ( 1 + x ) ^ { 2 } }f(x)=(1+x)28​

Determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. an=ln⁡(n3)7na _ { n } = \frac { \ln \left( n ^ { 3 } \right) } { 7 n }an​=7nln(n3)​

Find the Maclaurin polynomial of degree 4 for the function. f(x)=e9xf ( x ) = e ^ { 9 x }f(x)=e9x

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

$\text { Write an expression for the } n \text {th term of the sequence } 10,22,34,46 , \ldots \text {. }$

Use the Root Test to determine the convergence or divergence of the series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 16 ^ { n } }$

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

$\text { Write the repeating decimal } 0 . \overline { 75 } \text { as a geometric series. }$

Use the Limit Comparison Test to determine the convergence or divergence of the series $\sum _ { n = 1 } ^ { \infty } \frac { 9 } { n \sqrt { n ^ { 2 } + 6 } }$ .

Determine the convergence or divergence of the series using any appropriate test from this chapter. Identify the test used. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } 9 } { 7 n }$

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

$\text { Find a power series for the function } \frac { 1 } { 12 - x } \text { centered at } 1 \text {. }$

Use the Ratio Test to determine the convergence or divergence of the series $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( 7 ) ^ { 8 n } } { ( 7 n + 1 ) ! }$

Determine the minimal number of terms required to approximate the sum of the series with an error of less than 0.005. $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { ( 2 n ) ! }$

Use the Limit Comparison Test (if possible) to determine whether the series $\sum _ { n = 1 } ^ { \infty } \frac { 2 n ^ { 2 } - 7 } { 7 n ^ { 7 } + 3 n + 2 }$ converges or diverges.

Use the binomial series to find the Maclaurin series for the function $f ( x ) = \frac { 8 } { ( 1 + x ) ^ { 2 } }$

Determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. $a _ { n } = \frac { \ln \left( n ^ { 3 } \right) } { 7 n }$

Find the Maclaurin polynomial of degree 4 for the function. $f ( x ) = e ^ { 9 x }$