Question 1

Match the sequence with its graph. $a _ { n } = \frac { 1 ^ { n } } { n ! }$

Accepted Answer

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

Question 2

$\text { Write an expression for the } n \text {th term of the sequence } \frac { 10 } { 11 } , \frac { 19 } { 20 } , \frac { 28 } { 29 } , \frac { 37 } { 38 } , \ldots$

Accepted Answer

A)  $\frac { 9 n } { 9 n + 2 }$ 
B)  $\frac { 9 n } { 9 n + 1 }$ 
C)  $\frac { 9 n - 1 } { 9 n + 1 }$ 
D)  $\frac { 9 n + 1 } { 9 n + 2 }$ 
E)  $\frac { 9 n - 1 } { 9 n + 2 }$ 
A)  $\frac { 9 n } { 9 n + 2 }$ 
B)  $\frac { 9 n } { 9 n + 1 }$ 
C)  $\frac { 9 n - 1 } { 9 n + 1 }$ 
D)  $\frac { 9 n + 1 } { 9 n + 2 }$ 
E)  $\frac { 9 n - 1 } { 9 n + 2 }$

Question 3

$\text { Write the next two apparent terms of the sequence } 1 , - \frac { 1 } { 3 } , \frac { 1 } { 9 } , - \frac { 1 } { 27 } , \ldots$

Accepted Answer

A)  $\frac { 1 } { 243 } ; - \frac { 1 } { 729 }$ 
B)  $\frac { 1 } { 81 } ; - \frac { 1 } { 243 }$ 
C)  $\frac { 1 } { 81 } ; \frac { 1 } { 2,187 }$ 
D)  $- \frac { 1 } { 81 } ; \frac { 1 } { 729 }$ 
E)  $- \frac { 1 } { 27 } ; \frac { 1 } { 729 }$ 
A)  $\frac { 1 } { 243 } ; - \frac { 1 } { 729 }$ 
B)  $\frac { 1 } { 81 } ; - \frac { 1 } { 243 }$ 
C)  $\frac { 1 } { 81 } ; \frac { 1 } { 2,187 }$ 
D)  $- \frac { 1 } { 81 } ; \frac { 1 } { 729 }$ 
E)  $- \frac { 1 } { 27 } ; \frac { 1 } { 729 }$

Question 4

$\text { Determine whether the series } \sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { \ln 2 n } \text { converges conditionally or }$ 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 5

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

Accepted Answer

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

Question 6

Use the definition to find the Taylor series centered at 
$c = 0 \text { for the function }$ $f ( x ) = 3 \ln \left( x ^ { 2 } + 1 \right)$

Accepted Answer

A) $3 \ln \left( x ^ { 2 } + 1 \right) = \sum _ { n = 0 } ^ { \infty } \frac { ( 3 x ) ^ { 2 n + 2 } } { n + 1 }$ 
B) $3 \ln \left( x ^ { 2 } + 1 \right) = 3 \sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \frac { x ^ { n + 1 } } { n + 1 }$ 
C) $3 \ln \left( x ^ { 2 } + 1 \right) = 3 \sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \frac { x ^ { 2 n + 2 } } { n + 1 }$ 
D) $3 \ln \left( x ^ { 2 } + 1 \right) = 3 \sum _ { n = 0 } ^ { \infty } \frac { x ^ { 2 n + 2 } } { n }$ 
E) $3 \ln \left( x ^ { 2 } + 1 \right) = \sum _ { n = 0 } ^ { \infty } \frac { ( 3 x ) ^ { n + 1 } } { n + 1 }$ 
A) $3 \ln \left( x ^ { 2 } + 1 \right) = \sum _ { n = 0 } ^ { \infty } \frac { ( 3 x ) ^ { 2 n + 2 } } { n + 1 }$ 
B) $3 \ln \left( x ^ { 2 } + 1 \right) = 3 \sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \frac { x ^ { n + 1 } } { n + 1 }$ 
C) $3 \ln \left( x ^ { 2 } + 1 \right) = 3 \sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \frac { x ^ { 2 n + 2 } } { n + 1 }$ 
D) $3 \ln \left( x ^ { 2 } + 1 \right) = 3 \sum _ { n = 0 } ^ { \infty } \frac { x ^ { 2 n + 2 } } { n }$ 
E) $3 \ln \left( x ^ { 2 } + 1 \right) = \sum _ { n = 0 } ^ { \infty } \frac { ( 3 x ) ^ { n + 1 } } { n + 1 }$

Question 7

Identify the most appropriate test to be used to determine whether the series $\sum _ { n = 1 } ^ { \infty } \frac { \cos n } { 6 ^ { n } }$ converges or diverges.

Accepted Answer

A)  Limit Comparison Test with $b _ { n } = \frac { 1 } { 6 ^ { n } }$ 
B)  Direct Comparison Test with $b _ { \mathrm { n } } = \frac { 1 } { 6 ^ { n } }$ 
C)  Alternating Series Test 
D)  Root Test 
E)  Ratio Test 
A)  Limit Comparison Test with $b _ { n } = \frac { 1 } { 6 ^ { n } }$ 
B)  Direct Comparison Test with $b _ { \mathrm { n } } = \frac { 1 } { 6 ^ { n } }$ 
C)  Alternating Series Test 
D)  Root Test 
E)  Ratio Test

Question 8

Find the sum of the convergent series. $$\sum _ { n = 0 } ^ { \infty } 9 \left( \frac { 4 } { 5 } \right) ^ { n }$$

Accepted Answer

A)  4 
B)  36 
C)  9 
D)  27 
E)  45 
A)  4 
B)  36 
C)  9 
D)  27 
E)  45

Question 9

$\text { Find the third Taylor polynomial for } f ( x ) = \frac { 7 } { x } \text {, expanded about } c = 1 \text {. }$

Accepted Answer

A)  $P _ { 3 } ( x ) = 7 - 7 ( x - 1 ) - 14 ( x - 1 ) ^ { 2 } - 7 ( x - 1 ) ^ { 3 }$ 
B)  $P _ { 3 } ( x ) = 7 - 7 ( x - 1 ) + 7 ( x - 1 ) ^ { 2 } - 7 ( x - 1 ) ^ { 3 }$ 
C)  $P _ { 3 } ( x ) = 7 - 7 ( x - 1 ) + 14 ( x - 1 ) ^ { 2 } - 14 ( x - 1 ) ^ { 3 }$ 
D)  $P _ { 3 } ( x ) = 7 - 7 ( x + 1 ) + 7 ( x + 1 ) ^ { 2 } - 7 ( x + 1 ) ^ { 3 }$ 
E)  $P _ { 3 } ( x ) = 7 - 7 ( x - 1 ) - 42 ( x - 1 ) ^ { 2 } - 7 ( x - 1 ) ^ { 3 }$ 
A)  $P _ { 3 } ( x ) = 7 - 7 ( x - 1 ) - 14 ( x - 1 ) ^ { 2 } - 7 ( x - 1 ) ^ { 3 }$ 
B)  $P _ { 3 } ( x ) = 7 - 7 ( x - 1 ) + 7 ( x - 1 ) ^ { 2 } - 7 ( x - 1 ) ^ { 3 }$ 
C)  $P _ { 3 } ( x ) = 7 - 7 ( x - 1 ) + 14 ( x - 1 ) ^ { 2 } - 14 ( x - 1 ) ^ { 3 }$ 
D)  $P _ { 3 } ( x ) = 7 - 7 ( x + 1 ) + 7 ( x + 1 ) ^ { 2 } - 7 ( x + 1 ) ^ { 3 }$ 
E)  $P _ { 3 } ( x ) = 7 - 7 ( x - 1 ) - 42 ( x - 1 ) ^ { 2 } - 7 ( x - 1 ) ^ { 3 }$

Question 10

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

Accepted Answer

A)  $- \frac { 1 } { 2 } < x < 0$ 
B)  $- \frac { 1 } { 2 } < x < \frac { 1 } { 2 }$ 
C)  $0 < x < \frac { 1 } { 2 }$ 
D)  $- \frac { 1 } { 4 } < x < \frac { 1 } { 4 }$ 
E)  $0 < x < \frac { 1 } { 4 }$ 
A)  $- \frac { 1 } { 2 } < x < 0$ 
B)  $- \frac { 1 } { 2 } < x < \frac { 1 } { 2 }$ 
C)  $0 < x < \frac { 1 } { 2 }$ 
D)  $- \frac { 1 } { 4 } < x < \frac { 1 } { 4 }$ 
E)  $0 < x < \frac { 1 } { 4 }$

Question 11

Use the Ratio Test to determine the convergence or divergence of the series. $$\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } \left( \frac { 7 } { 2 } \right) ^ { n } } { n ^ { 2 } }$$

Accepted Answer

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

Question 12

$\text { Find the positive values of } p \text { for which the series } \sum _ { n = 1 } ^ { \infty } 3 n \left( 8 + n ^ { 2 } \right) ^ { p } \text { converges. }$

Accepted Answer

A)  The series converges for $p > 4$. 
B)  The series converges for $p \geq 11$. 
C)  The series diverges for all positive values of $p$. 
D)  The series converges for $p > 1$. 
E)  The series converges for $p \geq 8$. 
A)  The series converges for $p > 4$. 
B)  The series converges for $p \geq 11$. 
C)  The series diverges for all positive values of $p$. 
D)  The series converges for $p > 1$. 
E)  The series converges for $p \geq 8$.

Question 13

$\text { Evaluate } \left( \begin{array} { l } 
7 \
5
\end{array} \right) \text { using the formula } \left( \begin{array} { l } 
k \
n
\end{array} \right) = \frac { k ( k - 1 ) ( k - 2 ) ( k - 3 ) \cdots \cdots ( k - n + 1 ) } { n ! } \text { where }$
 $k$ is a real number, $n$ is a positive integer, and $\left( \begin{array} { l } k \ 0 \end{array} \right) = 1$

Accepted Answer

A)  31 
B)  45 
C)  56 
D)  21 
E)  35 
A)  31 
B)  45 
C)  56 
D)  21 
E)  35

Question 14

Match the sequence with its graph. $a _ { n } = \frac { 6 } { n + 1 }$

Accepted Answer

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

Question 15

Find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) $$\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } n ! ( x - 10 ) ^ { n } } { ( 9 ) ^ { n } }$$

Accepted Answer

A)  $\{ 10 \}$ 
B)  $( - 10,10 )$ 
C)  $\{ 0 \}$ 
D)  $[ - 10,10 )$ 
E)  $( - \infty , \infty )$ 
A)  $\{ 10 \}$ 
B)  $( - 10,10 )$ 
C)  $\{ 0 \}$ 
D)  $[ - 10,10 )$ 
E)  $( - \infty , \infty )$

Question 16

Use the Limit Comparison Test (if possible) to determine whether the series $$\sum _ { n = 1 } ^ { \infty } \frac { 2 } { \sqrt [ 9 ] { n ^ { 2 } + 9 } }$$

Accepted Answer

A)  diverges 
B)  converges 
C)  Limit Comparison Test does not apply 
A)  diverges 
B)  converges 
C)  Limit Comparison Test does not apply

Question 17

Use the Root Test to determine the convergence or divergence of the series. $$\sum _ { n = 1 } ^ { \infty } \left( \frac { 7 n + 1 } { 4 n - 1 } \right) ^ { n }$$

Accepted Answer

A)  converges 
B)  diverges 
C)  Root Test inconclusive 
A)  converges 
B)  diverges 
C)  Root Test inconclusive

Question 18

Use the Root Test to determine the convergence or divergence of the series. $$\sum _ { n = 1 } ^ { \infty } \left( \frac { 4 n } { 3 n + 1 } \right) ^ { n }$$

Accepted Answer

A)  converges 
B)  diverges 
C)  Root Test inconclusive 
A)  converges 
B)  diverges 
C)  Root Test inconclusive

Question 19

Identify the most appropriate test to be used to determine whether the series $\sum _ { n = 1 } ^ { \infty } \frac { n } { 19 n ^ { 2 } + 1 }$ converges or diverges.

Accepted Answer

A)  Limit Comparison Test with $b _ { \mathrm { n } } = \frac { 1 } { n }$ 
B)  Ratio Test 
C)  Direct Comparison Test with $b _ { \mathrm { n } } = \frac { 1 } { n }$ 
D)  Root Test 
E)  Alternating Series Test 
A)  Limit Comparison Test with $b _ { \mathrm { n } } = \frac { 1 } { n }$ 
B)  Ratio Test 
C)  Direct Comparison Test with $b _ { \mathrm { n } } = \frac { 1 } { n }$ 
D)  Root Test 
E)  Alternating Series Test

Question 20

Find the Maclaurin polynomial of degree two for the function $f ( x ) = \sec ( 11 x )$

Accepted Answer

A)  $P _ { 2 } ( x ) = 1 + \frac { 121 } { 4 } x ^ { 2 }$ 
B)  $P _ { 2 } ( x ) = 1 - \frac { 121 } { 2 } x ^ { 2 }$ 
C)  $P _ { 2 } ( x ) = 1 + \frac { 121 } { 2 } x ^ { 2 }$ 
D)  $P _ { 2 } ( x ) = x + \frac { 121 } { 2 } x ^ { 2 }$ 
E)  $P _ { 2 } ( x ) = x - \frac { 121 } { 2 } x ^ { 2 }$ 
A)  $P _ { 2 } ( x ) = 1 + \frac { 121 } { 4 } x ^ { 2 }$ 
B)  $P _ { 2 } ( x ) = 1 - \frac { 121 } { 2 } x ^ { 2 }$ 
C)  $P _ { 2 } ( x ) = 1 + \frac { 121 } { 2 } x ^ { 2 }$ 
D)  $P _ { 2 } ( x ) = x + \frac { 121 } { 2 } x ^ { 2 }$ 
E)  $P _ { 2 } ( x ) = x - \frac { 121 } { 2 } x ^ { 2 }$

Match the sequence with its graph. $a _ { n } = \frac { 1 ^ { n } } { n ! }$

$\text { Write an expression for the } n \text {th term of the sequence } \frac { 10 } { 11 } , \frac { 19 } { 20 } , \frac { 28 } { 29 } , \frac { 37 } { 38 } , \ldots$

$\text { Write the next two apparent terms of the sequence } 1 , - \frac { 1 } { 3 } , \frac { 1 } { 9 } , - \frac { 1 } { 27 } , \ldots$

$\text { Determine whether the series } \sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { \ln 2 n } \text { converges conditionally or }$ absolutely, or diverges.

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

Use the definition to find the Taylor series centered at $c = 0 \text { for the function }$ $f ( x ) = 3 \ln \left( x ^ { 2 } + 1 \right)$

Identify the most appropriate test to be used to determine whether the series $\sum _ { n = 1 } ^ { \infty } \frac { \cos n } { 6 ^ { n } }$ converges or diverges.

Find the sum of the convergent series. $\sum _ { n = 0 } ^ { \infty } 9 \left( \frac { 4 } { 5 } \right) ^ { n }$

$\text { Find the third Taylor polynomial for } f ( x ) = \frac { 7 } { x } \text {, expanded about } c = 1 \text {. }$

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

Use the Ratio Test to determine the convergence or divergence of the series. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } \left( \frac { 7 } { 2 } \right) ^ { n } } { n ^ { 2 } }$

$\text { Find the positive values of } p \text { for which the series } \sum _ { n = 1 } ^ { \infty } 3 n \left( 8 + n ^ { 2 } \right) ^ { p } \text { converges. }$

Match the sequence with its graph. $a _ { n } = \frac { 6 } { n + 1 }$

Find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } n ! ( x - 10 ) ^ { n } } { ( 9 ) ^ { n } }$

Use the Limit Comparison Test (if possible) to determine whether the series $\sum _ { n = 1 } ^ { \infty } \frac { 2 } { \sqrt [ 9 ] { n ^ { 2 } + 9 } }$

Use the Root Test to determine the convergence or divergence of the series. $\sum _ { n = 1 } ^ { \infty } \left( \frac { 7 n + 1 } { 4 n - 1 } \right) ^ { n }$

Use the Root Test to determine the convergence or divergence of the series. $\sum _ { n = 1 } ^ { \infty } \left( \frac { 4 n } { 3 n + 1 } \right) ^ { n }$

Identify the most appropriate test to be used to determine whether the series $\sum _ { n = 1 } ^ { \infty } \frac { n } { 19 n ^ { 2 } + 1 }$ converges or diverges.

Find the Maclaurin polynomial of degree two for the function $f ( x ) = \sec ( 11 x )$

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

Match the sequence with its graph. an=1nn!a _ { n } = \frac { 1 ^ { n } } { n ! }an​=n!1n​

Write the next two apparent terms of the sequence 1,−13,19,−127,…\text { Write the next two apparent terms of the sequence } 1 , - \frac { 1 } { 3 } , \frac { 1 } { 9 } , - \frac { 1 } { 27 } , \ldots Write the next two apparent terms of the sequence 1,−31​,91​,−271​,…

Use the definition to find the Taylor series centered at c=π4 for the function c = \frac { \pi } { 4 } \text { for the function }c=4π​ for the function f(x)=cos⁡xf ( x ) = \cos xf(x)=cosx

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)=3ln⁡(x2+1)f ( x ) = 3 \ln \left( x ^ { 2 } + 1 \right)f(x)=3ln(x2+1)

Identify the most appropriate test to be used to determine whether the series ∑n=1∞cos⁡n6n\sum _ { n = 1 } ^ { \infty } \frac { \cos n } { 6 ^ { n } }∑n=1∞​6ncosn​ converges or diverges.

Find the sum of the convergent series. ∑n=0∞9(45)n\sum _ { n = 0 } ^ { \infty } 9 \left( \frac { 4 } { 5 } \right) ^ { n }∑n=0∞​9(54​)n

Find the third Taylor polynomial for f(x)=7x, expanded about c=1. \text { Find the third Taylor polynomial for } f ( x ) = \frac { 7 } { x } \text {, expanded about } c = 1 \text {. } Find the third Taylor polynomial for f(x)=x7​, expanded about c=1.

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

Use the Ratio Test to determine the convergence or divergence of the series. ∑n=1∞(−1)n−1(72)nn2\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } \left( \frac { 7 } { 2 } \right) ^ { n } } { n ^ { 2 } }∑n=1∞​n2(−1)n−1(27​)n​

Match the sequence with its graph. an=6n+1a _ { n } = \frac { 6 } { n + 1 }an​=n+16​

Find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) ∑n=0∞(−1)nn!(x−10)n(9)n\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } n ! ( x - 10 ) ^ { n } } { ( 9 ) ^ { n } }∑n=0∞​(9)n(−1)nn!(x−10)n​

Use the Limit Comparison Test (if possible) to determine whether the series ∑n=1∞2n2+99\sum _ { n = 1 } ^ { \infty } \frac { 2 } { \sqrt [ 9 ] { n ^ { 2 } + 9 } }∑n=1∞​9n2+9​2​

Use the Root Test to determine the convergence or divergence of the series. ∑n=1∞(7n+14n−1)n\sum _ { n = 1 } ^ { \infty } \left( \frac { 7 n + 1 } { 4 n - 1 } \right) ^ { n }∑n=1∞​(4n−17n+1​)n

Use the Root Test to determine the convergence or divergence of the series. ∑n=1∞(4n3n+1)n\sum _ { n = 1 } ^ { \infty } \left( \frac { 4 n } { 3 n + 1 } \right) ^ { n }∑n=1∞​(3n+14n​)n

Identify the most appropriate test to be used to determine whether the series ∑n=1∞n19n2+1\sum _ { n = 1 } ^ { \infty } \frac { n } { 19 n ^ { 2 } + 1 }∑n=1∞​19n2+1n​ converges or diverges.

Find the Maclaurin polynomial of degree two for the function f(x)=sec⁡(11x)f ( x ) = \sec ( 11 x )f(x)=sec(11x)

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

Match the sequence with its graph. $a _ { n } = \frac { 1 ^ { n } } { n ! }$

$\text { Write the next two apparent terms of the sequence } 1 , - \frac { 1 } { 3 } , \frac { 1 } { 9 } , - \frac { 1 } { 27 } , \ldots$

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

Use the definition to find the Taylor series centered at $c = 0 \text { for the function }$ $f ( x ) = 3 \ln \left( x ^ { 2 } + 1 \right)$

Identify the most appropriate test to be used to determine whether the series $\sum _ { n = 1 } ^ { \infty } \frac { \cos n } { 6 ^ { n } }$ converges or diverges.

Find the sum of the convergent series. $\sum _ { n = 0 } ^ { \infty } 9 \left( \frac { 4 } { 5 } \right) ^ { n }$

$\text { Find the third Taylor polynomial for } f ( x ) = \frac { 7 } { x } \text {, expanded about } c = 1 \text {. }$

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

Use the Ratio Test to determine the convergence or divergence of the series. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } \left( \frac { 7 } { 2 } \right) ^ { n } } { n ^ { 2 } }$

Match the sequence with its graph. $a _ { n } = \frac { 6 } { n + 1 }$

Find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } n ! ( x - 10 ) ^ { n } } { ( 9 ) ^ { n } }$

Use the Limit Comparison Test (if possible) to determine whether the series $\sum _ { n = 1 } ^ { \infty } \frac { 2 } { \sqrt [ 9 ] { n ^ { 2 } + 9 } }$

Use the Root Test to determine the convergence or divergence of the series. $\sum _ { n = 1 } ^ { \infty } \left( \frac { 7 n + 1 } { 4 n - 1 } \right) ^ { n }$

Use the Root Test to determine the convergence or divergence of the series. $\sum _ { n = 1 } ^ { \infty } \left( \frac { 4 n } { 3 n + 1 } \right) ^ { n }$

Identify the most appropriate test to be used to determine whether the series $\sum _ { n = 1 } ^ { \infty } \frac { n } { 19 n ^ { 2 } + 1 }$ converges or diverges.

Find the Maclaurin polynomial of degree two for the function $f ( x ) = \sec ( 11 x )$