Question 1

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

Accepted Answer

A) 
The series $\sum _ { n = 1 } ^ { \infty } \frac { 7 n } { \left( n ^ { 2 } + 7 \right) ^ { 2 } }$ diverges. 
B) 
The series $\sum _ { n = 1 } ^ { \infty } \frac { 7 n } { \left( n ^ { 2 } + 7 \right) ^ { 2 } }$ converges. 
A) 
The series $\sum _ { n = 1 } ^ { \infty } \frac { 7 n } { \left( n ^ { 2 } + 7 \right) ^ { 2 } }$ diverges. 
B) 
The series $\sum _ { n = 1 } ^ { \infty } \frac { 7 n } { \left( n ^ { 2 } + 7 \right) ^ { 2 } }$ converges.

Question 2

A government program that currently costs taxpayers $3.5 billion per year is cut back by 30 percent per year. Compute the budget after the fifth year. Round your answer to the nearest integer.

Accepted Answer

A)  $\$ 1,552,968,594$ 
B)  $\$ 272,160,000$ 
C)  $\$ 8,505,000$ 
D)  $\$ 1,146,880,000$ 
E)  $\$ 588,245,000$ 
A)  $\$ 1,552,968,594$ 
B)  $\$ 272,160,000$ 
C)  $\$ 8,505,000$ 
D)  $\$ 1,146,880,000$ 
E)  $\$ 588,245,000$

Question 3

Identify the graph of the first 10 terms of the sequence of partial sum of the series $\sum _ { n = 0 } ^ { \infty } \left( \frac { x } { 3 } \right) ^ { n } \text { for } x = 2$

Accepted Answer

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

Question 4

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

Accepted Answer

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

Question 5

$\text { Determine whether the series } \sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { n ^ { 3 / 8 } } \text { converges absolutely, converges }$ conditionally, or diverges.

Accepted Answer

A)  converges conditionally 
B)  diverges 
C)  converges absolutely 
A)  converges conditionally 
B)  diverges 
C)  converges absolutely

Question 6

Write the first five terms of the sequence of partial sums. $$\sum _ { n = 1 } ^ { \infty } \frac { - 5 } { ( - 6 ) ^ { n - 1 } }$$

Accepted Answer

A)  $- 5 , - \frac { 9 } { 2 } , - \frac { 55 } { 12 } , - \frac { 200 } { 27 } , - \frac { 1855 } { 432 }$ 
B)  $- 5 , - \frac { 13 } { 3 } , - \frac { 40 } { 9 } , - \frac { 475 } { 108 } , - \frac { 695 } { 162 }$ 
C)  $- 5 , - \frac { 23 } { 6 } , - \frac { 35 } { 9 } , - \frac { 1625 } { 216 } , - \frac { 5545 } { 1296 }$ 
D)  $- 5 , - \frac { 25 } { 6 } , - \frac { 155 } { 36 } , - \frac { 925 } { 216 } , - \frac { 5555 } { 1296 }$ 
E)  $- 5 , \frac { 5 } { 6 } , - \frac { 5 } { 36 } , \frac { 5 } { 216 } , - \frac { 5 } { 1296 }$ 
A)  $- 5 , - \frac { 9 } { 2 } , - \frac { 55 } { 12 } , - \frac { 200 } { 27 } , - \frac { 1855 } { 432 }$ 
B)  $- 5 , - \frac { 13 } { 3 } , - \frac { 40 } { 9 } , - \frac { 475 } { 108 } , - \frac { 695 } { 162 }$ 
C)  $- 5 , - \frac { 23 } { 6 } , - \frac { 35 } { 9 } , - \frac { 1625 } { 216 } , - \frac { 5545 } { 1296 }$ 
D)  $- 5 , - \frac { 25 } { 6 } , - \frac { 155 } { 36 } , - \frac { 925 } { 216 } , - \frac { 5555 } { 1296 }$ 
E)  $- 5 , \frac { 5 } { 6 } , - \frac { 5 } { 36 } , \frac { 5 } { 216 } , - \frac { 5 } { 1296 }$

Question 7

Write the first five terms of the sequence of partial sums. $$- 6 + \frac { 36 } { 7 } - \frac { 216 } { 49 } + \frac { 1,296 } { 343 } - \frac { 7,776 } { 2,401 } + \ldots$$

Accepted Answer

A)  $- 6 , \frac { 6 } { 7 } , - \frac { 258 } { 49 } , - \frac { 510 } { 343 } , - \frac { 11,346 } { 2,401 }$ 
B)  $- 6 , - \frac { 6 } { 7 } , - \frac { 258 } { 49 } , - \frac { 510 } { 343 } , - \frac { 11,346 } { 2,401 }$ 
C)  $- 6 , \frac { 6 } { 7 } , \frac { 258 } { 49 } , \frac { 510 } { 343 } , \frac { 11,346 } { 2,401 }$ 
D)  $- 6 , \frac { 36 } { 7 } , - \frac { 216 } { 49 } , \frac { 1,296 } { 343 } , - \frac { 7,776 } { 2,401 }$ 
E)  $- 6 , \frac { 6 } { 7 } , - \frac { 258 } { 49 } , \frac { 510 } { 343 } , - \frac { 11,346 } { 2,401 }$ 
A)  $- 6 , \frac { 6 } { 7 } , - \frac { 258 } { 49 } , - \frac { 510 } { 343 } , - \frac { 11,346 } { 2,401 }$ 
B)  $- 6 , - \frac { 6 } { 7 } , - \frac { 258 } { 49 } , - \frac { 510 } { 343 } , - \frac { 11,346 } { 2,401 }$ 
C)  $- 6 , \frac { 6 } { 7 } , \frac { 258 } { 49 } , \frac { 510 } { 343 } , \frac { 11,346 } { 2,401 }$ 
D)  $- 6 , \frac { 36 } { 7 } , - \frac { 216 } { 49 } , \frac { 1,296 } { 343 } , - \frac { 7,776 } { 2,401 }$ 
E)  $- 6 , \frac { 6 } { 7 } , - \frac { 258 } { 49 } , \frac { 510 } { 343 } , - \frac { 11,346 } { 2,401 }$

Question 8

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

Accepted Answer

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

Question 9

$\text {The series } \sum _ { n = 1 } ^ { \infty } \frac { 6 n + 1 } { 12 n - 1 } \text { is convergent ? }$

Accepted Answer

A) True 
 B)False

Question 10

Suppose an electronic games manufacturer producing a new product estimates the annual sales to be 4,000 units. Each year $20 \%$ of the units that have been sold will become inoperative. So, 4,000 units will be in use after 1 year, $[ 4,000 + 0.80 ( 4,000 ) ]$ units will be in use after 2 years, and so on. How many units will be in use after $n$ years?

Accepted Answer

A)  $20,000 \left( 1 - 20 ^ { n } \right)$ units 
B)  $40,000 \left( 1 - 0.80 ^ { n } \right)$ units 
C)  $\left( 1 - 0.80 ^ { n } \right)$ units 
D)  $20,000 \left( 1 + 0.80 ^ { n } \right)$ units 
E)  $\left( 1 + 0.80 ^ { n } \right)$ units 
A)  $20,000 \left( 1 - 20 ^ { n } \right)$ units 
B)  $40,000 \left( 1 - 0.80 ^ { n } \right)$ units 
C)  $\left( 1 - 0.80 ^ { n } \right)$ units 
D)  $20,000 \left( 1 + 0.80 ^ { n } \right)$ units 
E)  $\left( 1 + 0.80 ^ { n } \right)$ units

Question 11

$\text { Use a power series to approximate the value of the integral } \int _ { 0.42 } ^ { 0.59 } \sqrt { 1 + x ^ { 3 } } d x \text { with an }$error less than 0.001. Round your answer to four decimal places.

Accepted Answer

A) 0.1925 
B) 0.1700 
C) 0.1813 
D) 0.1933 
E) 0.1817 
A) 0.1925 
B) 0.1700 
C) 0.1813 
D) 0.1933 
E) 0.1817

Question 12

A deposit of $225 is made at the beginning of each month into an account at an annual interest rate of $3 \%$ compounded monthly. The balance in the account after $n$ months is $A _ { n } = 225 ( 401 ) \left( 1.0025 ^ { n } - 1 \right)$. Find the balance in the account after 4 years by computing the 48 th term of the sequence. Round your answer to two decimal places.

Accepted Answer

A)  $\$ 11,488.17$ 
B)  $\$ 905.64$ 
C)  $\$ 9,172.05$ 
D)  $\$ 24,404.88$ 
E)  $\$ 1,818.08$ 
A)  $\$ 11,488.17$ 
B)  $\$ 905.64$ 
C)  $\$ 9,172.05$ 
D)  $\$ 24,404.88$ 
E)  $\$ 1,818.08$

Question 13

Use the binomial series to find the Maclaurin series for the function $f ( x ) = \frac { 1 } { \sqrt { 81 + x ^ { 2 } } }$

Accepted Answer

A)  $\frac { 1 } { \sqrt { 81 + x ^ { 2 } } } = \frac { 1 } { 9 } + \sum _ { n = 0 } ^ { \infty } \frac { 1 \cdot 3 \cdot 5 \cdots ( 2 n - 1 ) x ^ { n } } { 9 ^ { n } n ! }$ 
B) $\frac { 1 } { \sqrt { 81 + x ^ { 2 } } } = \frac { 1 } { 9 } + \sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } 1 \cdot 3 \cdot 5 \cdots ( 2 n - 1 ) x ^ { 2 n } } { 2 ^ { n } ( 2 n - 1 ) ! 9 ^ { 2 n + 1 } }$ 
C) $\frac { 1 } { \sqrt { 81 + x ^ { 2 } } } = \frac { 1 } { 9 } + \sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } 1 \cdot 3 \cdot 5 \cdots ( 2 n - 1 ) x ^ { 2 n } } { 2 ^ { n } n ! 9 ^ { 2 n + 1 } }$ 
D) $\frac { 1 } { \sqrt { 81 + x ^ { 2 } } } = \frac { 1 } { 9 } + \sum _ { n = 0 } ^ { \infty } \frac { 1 \cdot 3 \cdot 5 \cdots ( 2 n - 1 ) x ^ { n } } { n ! }$ 
E) $\frac { 1 } { \sqrt { 81 + x ^ { 2 } } } = \frac { 1 } { 9 } + \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } 1 \cdot 3 \cdot 5 \cdots ( 2 n - 1 ) x ^ { 2 n } } { 2 ^ { n } 9 ^ { 2 n + 1 } }$ 
A)  $\frac { 1 } { \sqrt { 81 + x ^ { 2 } } } = \frac { 1 } { 9 } + \sum _ { n = 0 } ^ { \infty } \frac { 1 \cdot 3 \cdot 5 \cdots ( 2 n - 1 ) x ^ { n } } { 9 ^ { n } n ! }$ 
B) $\frac { 1 } { \sqrt { 81 + x ^ { 2 } } } = \frac { 1 } { 9 } + \sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } 1 \cdot 3 \cdot 5 \cdots ( 2 n - 1 ) x ^ { 2 n } } { 2 ^ { n } ( 2 n - 1 ) ! 9 ^ { 2 n + 1 } }$ 
C) $\frac { 1 } { \sqrt { 81 + x ^ { 2 } } } = \frac { 1 } { 9 } + \sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } 1 \cdot 3 \cdot 5 \cdots ( 2 n - 1 ) x ^ { 2 n } } { 2 ^ { n } n ! 9 ^ { 2 n + 1 } }$ 
D) $\frac { 1 } { \sqrt { 81 + x ^ { 2 } } } = \frac { 1 } { 9 } + \sum _ { n = 0 } ^ { \infty } \frac { 1 \cdot 3 \cdot 5 \cdots ( 2 n - 1 ) x ^ { n } } { n ! }$ 
E) $\frac { 1 } { \sqrt { 81 + x ^ { 2 } } } = \frac { 1 } { 9 } + \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } 1 \cdot 3 \cdot 5 \cdots ( 2 n - 1 ) x ^ { 2 n } } { 2 ^ { n } 9 ^ { 2 n + 1 } }$

Question 14

Use the Limit Comparison Test (if possible) to determine whether the series $\sum _ { n = 1 } ^ { \infty } \frac { 4 ^ { n + 1 } } { 5 ^ { n } - 6 }$ converges or diverges.

Accepted Answer

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

Question 15

Determine the degree of the Maclaurin polynomial required for the error in the approximation of the function $\sin ( 0.6 )$ to be less than $0.001$.

Accepted Answer

A)  4 
B)  2 
C)  5 
D)  1 
E)  3 
A)  4 
B)  2 
C)  5 
D)  1 
E)  3

Question 16

$\text { Find the interval of convergence of the power series } \sum _ { n = 0 } ^ { \infty } \frac { x ^ { 3 n + 1 } } { ( 3 n + 1 ) ! } \text {. (Be sure to }$ include a check for convergence at the endpoints of the interval.)

Accepted Answer

A)  $( - \infty , 4 )$ 
B)  $[ - 4,4 ]$ 
C)  $[ - 3,3 ]$ 
D)  $( 4 , \infty )$ 
E)  $( - \infty , \infty )$ 
A)  $( - \infty , 4 )$ 
B)  $[ - 4,4 ]$ 
C)  $[ - 3,3 ]$ 
D)  $( 4 , \infty )$ 
E)  $( - \infty , \infty )$

Question 17

Find the Maclaurin series for the function $$f ( x ) = \cos \left( x ^ { 17 / 2 } \right)$$

Accepted Answer

A)  $\cos \left( x ^ { 17 / 2 } \right) = 1 - \frac { x ^ { 17 } } { 2 ! } + \frac { x ^ { 34 } } { 4 ! } - \cdots$ 
B)  $\cos \left( x ^ { 17 / 2 } \right) = x - \frac { x ^ { 2 } } { 2 ! } + \frac { x ^ { 4 } } { 4 ! } - \cdots$ 
C)  $\cos \left( x ^ { 17 / 2 } \right) = x + \frac { x ^ { 17 } } { 17 ! } + \frac { x ^ { 34 } } { 34 ! } + \cdots$ 
D)  $\cos \left( x ^ { 17 / 2 } \right) = 1 + \frac { x ^ { 17 } } { 2 ! } + \frac { x ^ { 34 } } { 4 ! } + \cdots$ 
E)  $\cos \left( x ^ { 17 / 2 } \right) = x - \frac { x ^ { 17 } } { 17 ! } + \frac { x ^ { 34 } } { 34 ! } - \cdots$ 
A)  $\cos \left( x ^ { 17 / 2 } \right) = 1 - \frac { x ^ { 17 } } { 2 ! } + \frac { x ^ { 34 } } { 4 ! } - \cdots$ 
B)  $\cos \left( x ^ { 17 / 2 } \right) = x - \frac { x ^ { 2 } } { 2 ! } + \frac { x ^ { 4 } } { 4 ! } - \cdots$ 
C)  $\cos \left( x ^ { 17 / 2 } \right) = x + \frac { x ^ { 17 } } { 17 ! } + \frac { x ^ { 34 } } { 34 ! } + \cdots$ 
D)  $\cos \left( x ^ { 17 / 2 } \right) = 1 + \frac { x ^ { 17 } } { 2 ! } + \frac { x ^ { 34 } } { 4 ! } + \cdots$ 
E)  $\cos \left( x ^ { 17 / 2 } \right) = x - \frac { x ^ { 17 } } { 17 ! } + \frac { x ^ { 34 } } { 34 ! } - \cdots$

Question 18

$\text { Find a power series for the function } \frac { 1 } { 1 - 9 x } \text { centered at } 0 \text {. }$

Accepted Answer

A)  $\sum _ { n = 0 } ^ { \infty } \left( \frac { x } { 9 } \right) ^ { n }$ 
B)  $\sum _ { n = 0 } ^ { \infty } ( - 9 x ) ^ { n }$ 
C)  $\sum _ { n = 0 } ^ { \infty } \left( \frac { 9 } { x } \right) ^ { n }$ 
D) $\sum _ { n = 0 } ^ { \infty } ( 9 x ) ^ { n }$ 
E)  $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } ( 9 x ) ^ { n }$ 
A)  $\sum _ { n = 0 } ^ { \infty } \left( \frac { x } { 9 } \right) ^ { n }$ 
B)  $\sum _ { n = 0 } ^ { \infty } ( - 9 x ) ^ { n }$ 
C)  $\sum _ { n = 0 } ^ { \infty } \left( \frac { 9 } { x } \right) ^ { n }$ 
D) $\sum _ { n = 0 } ^ { \infty } ( 9 x ) ^ { n }$ 
E)  $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } ( 9 x ) ^ { n }$

Question 19

Use the Integral Test to determine the convergence or divergence of the series. $$\sum _ { n = 1 } ^ { \infty } n e ^ { - \frac { n } { 2 } }$$

Accepted Answer

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

Question 20

$\text { The series } \sum _ { n = 0 } ^ { \infty } \frac { 20 } { 10 ^ { n } } \text { is divergent? }$

Accepted Answer

A) True 
 B)False

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

A government program that currently costs taxpayers $3.5 billion per year is cut back by 30 percent per year. Compute the budget after the fifth year. Round your answer to the nearest integer.

Identify the graph of the first 10 terms of the sequence of partial sum of the series $\sum _ { n = 0 } ^ { \infty } \left( \frac { x } { 3 } \right) ^ { n } \text { for } x = 2$

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

$\text { Determine whether the series } \sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { n ^ { 3 / 8 } } \text { converges absolutely, converges }$ conditionally, or diverges.

Write the first five terms of the sequence of partial sums. $\sum _ { n = 1 } ^ { \infty } \frac { - 5 } { ( - 6 ) ^ { n - 1 } }$

Write the first five terms of the sequence of partial sums. $- 6 + \frac { 36 } { 7 } - \frac { 216 } { 49 } + \frac { 1,296 } { 343 } - \frac { 7,776 } { 2,401 } + \ldots$

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

$\text {The series } \sum _ { n = 1 } ^ { \infty } \frac { 6 n + 1 } { 12 n - 1 } \text { is convergent ? }$

$\text { Use a power series to approximate the value of the integral } \int _ { 0.42 } ^ { 0.59 } \sqrt { 1 + x ^ { 3 } } d x \text { with an }$ error less than 0.001. Round your answer to four decimal places.

Use the binomial series to find the Maclaurin series for the function $f ( x ) = \frac { 1 } { \sqrt { 81 + x ^ { 2 } } }$

Use the Limit Comparison Test (if possible) to determine whether the series $\sum _ { n = 1 } ^ { \infty } \frac { 4 ^ { n + 1 } } { 5 ^ { n } - 6 }$ converges or diverges.

Determine the degree of the Maclaurin polynomial required for the error in the approximation of the function $\sin ( 0.6 )$ to be less than $0.001$ .

$\text { Find the interval of convergence of the power series } \sum _ { n = 0 } ^ { \infty } \frac { x ^ { 3 n + 1 } } { ( 3 n + 1 ) ! } \text {. (Be sure to }$ include a check for convergence at the endpoints of the interval.)

Find the Maclaurin series for the function $f ( x ) = \cos \left( x ^ { 17 / 2 } \right)$

$\text { Find a power series for the function } \frac { 1 } { 1 - 9 x } \text { centered at } 0 \text {. }$

Use the Integral Test to determine the convergence or divergence of the series. $\sum _ { n = 1 } ^ { \infty } n e ^ { - \frac { n } { 2 } }$

$\text { The series } \sum _ { n = 0 } ^ { \infty } \frac { 20 } { 10 ^ { n } } \text { is divergent? }$

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 Direct Comparison Test to determine the convergence or divergence of the series ∑n=1∞7n(n2+7)2\sum _ { n = 1 } ^ { \infty } \frac { 7 n } { \left( n ^ { 2 } + 7 \right) ^ { 2 } }∑n=1∞​(n2+7)27n​ .

A government program that currently costs taxpayers $3.5 billion per year is cut back by 30 percent per year. Compute the budget after the fifth year. Round your answer to the nearest integer.

Identify the graph of the first 10 terms of the sequence of partial sum of the series ∑n=0∞(x3)n for x=2\sum _ { n = 0 } ^ { \infty } \left( \frac { x } { 3 } \right) ^ { n } \text { for } x = 2∑n=0∞​(3x​)n for x=2

Write the first five terms of the sequence of partial sums. ∑n=1∞−5(−6)n−1\sum _ { n = 1 } ^ { \infty } \frac { - 5 } { ( - 6 ) ^ { n - 1 } }∑n=1∞​(−6)n−1−5​

Write the first five terms of the sequence of partial sums. −6+367−21649+1,296343−7,7762,401+…- 6 + \frac { 36 } { 7 } - \frac { 216 } { 49 } + \frac { 1,296 } { 343 } - \frac { 7,776 } { 2,401 } + \ldots−6+736​−49216​+3431,296​−2,4017,776​+…

Use the Limit Comparison Test to determine the convergence or divergence of the series ∑n=1∞9n9n2+2\sum _ { n = 1 } ^ { \infty } \frac { 9 n } { 9 n ^ { 2 } + 2 }∑n=1∞​9n2+29n​ .

The series ∑n=1∞6n+112n−1 is convergent ? \text {The series } \sum _ { n = 1 } ^ { \infty } \frac { 6 n + 1 } { 12 n - 1 } \text { is convergent ? }The series ∑n=1∞​12n−16n+1​ is convergent ?

Use the binomial series to find the Maclaurin series for the function f(x)=181+x2f ( x ) = \frac { 1 } { \sqrt { 81 + x ^ { 2 } } }f(x)=81+x2​1​

Use the Limit Comparison Test (if possible) to determine whether the series ∑n=1∞4n+15n−6\sum _ { n = 1 } ^ { \infty } \frac { 4 ^ { n + 1 } } { 5 ^ { n } - 6 }∑n=1∞​5n−64n+1​ converges or diverges.

Determine the degree of the Maclaurin polynomial required for the error in the approximation of the function sin⁡(0.6)\sin ( 0.6 )sin(0.6) to be less than 0.0010.0010.001 .

Find the Maclaurin series for the function f(x)=cos⁡(x17/2)f ( x ) = \cos \left( x ^ { 17 / 2 } \right)f(x)=cos(x17/2)

Find a power series for the function 11−9x centered at 0. \text { Find a power series for the function } \frac { 1 } { 1 - 9 x } \text { centered at } 0 \text {. } Find a power series for the function 1−9x1​ centered at 0.

Use the Integral Test to determine the convergence or divergence of the series. ∑n=1∞ne−n2\sum _ { n = 1 } ^ { \infty } n e ^ { - \frac { n } { 2 } }∑n=1∞​ne−2n​

The series ∑n=0∞2010n is divergent? \text { The series } \sum _ { n = 0 } ^ { \infty } \frac { 20 } { 10 ^ { n } } \text { is divergent? } The series ∑n=0∞​10n20​ is divergent?

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

Identify the graph of the first 10 terms of the sequence of partial sum of the series $\sum _ { n = 0 } ^ { \infty } \left( \frac { x } { 3 } \right) ^ { n } \text { for } x = 2$

Write the first five terms of the sequence of partial sums. $\sum _ { n = 1 } ^ { \infty } \frac { - 5 } { ( - 6 ) ^ { n - 1 } }$

Write the first five terms of the sequence of partial sums. $- 6 + \frac { 36 } { 7 } - \frac { 216 } { 49 } + \frac { 1,296 } { 343 } - \frac { 7,776 } { 2,401 } + \ldots$

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

$\text {The series } \sum _ { n = 1 } ^ { \infty } \frac { 6 n + 1 } { 12 n - 1 } \text { is convergent ? }$

Use the binomial series to find the Maclaurin series for the function $f ( x ) = \frac { 1 } { \sqrt { 81 + x ^ { 2 } } }$

Use the Limit Comparison Test (if possible) to determine whether the series $\sum _ { n = 1 } ^ { \infty } \frac { 4 ^ { n + 1 } } { 5 ^ { n } - 6 }$ converges or diverges.

Determine the degree of the Maclaurin polynomial required for the error in the approximation of the function $\sin ( 0.6 )$ to be less than $0.001$ .

Find the Maclaurin series for the function $f ( x ) = \cos \left( x ^ { 17 / 2 } \right)$

$\text { Find a power series for the function } \frac { 1 } { 1 - 9 x } \text { centered at } 0 \text {. }$

Use the Integral Test to determine the convergence or divergence of the series. $\sum _ { n = 1 } ^ { \infty } n e ^ { - \frac { n } { 2 } }$

$\text { The series } \sum _ { n = 0 } ^ { \infty } \frac { 20 } { 10 ^ { n } } \text { is divergent? }$