Question 1

Approximate the sum of the series by using the first six terms. $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } 4 } { n ! }$

Accepted Answer

A)  $1.457 < S < 1.477$ 
B)  $1.417 < S < 1.497$ 
C)  $1.467 < S < 1.472$ 
D)  $1.427 < S < 1.467$ 
E)  $1.461 < S < 1.473$ 
A)  $1.457 < S < 1.477$ 
B)  $1.417 < S < 1.497$ 
C)  $1.467 < S < 1.472$ 
D)  $1.427 < S < 1.467$ 
E)  $1.461 < S < 1.473$ E

Question 2

$\text { Determine the convergence or divergence of the series } \sum _ { n = 1 } ^ { \infty } \frac { n } { 7 n ^ { 2 } + 1 } \text { using any }$ appropriate test.

Accepted Answer

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

Question 3

$\text {The infinite series } \sum _ { n = 1 } ^ { \infty } \frac { n } { 13 n + 4 } \text { diverges ?}$

Accepted Answer

A) True 
 B)False  False

Question 4

Find a geometric power series for the function centered at 0, (i) by the technique shown in Examples 1 and 2 and (ii) by long division. $$f ( x ) = \frac { 10 } { 4 - x }$$

Accepted Answer

A)  
$\sum _ { n = 0 } ^ { \infty } \frac { 5 } { 2 } \left( \frac { x } { 4 } \right) ^ { n } , | x | < 4$ 
B) 
$\sum _ { n = 0 } ^ { \infty } \frac { 1 } { 4 } \left( \frac { x } { 4 } \right) ^ { n } , | x | < 4$ 
C) 
$\sum _ { n = 0 } ^ { \infty } 10 \left( - \frac { x } { 4 } \right) ^ { n } , | x | < 4$ 
D) 
$\sum _ { n = 0 } ^ { \infty } \frac { 5 } { 2 } ( - 4 x ) ^ { n } , | x | < 4$ 
E) 
$\sum _ { n = 0 } ^ { \infty } \frac { 5 } { 2 } ( - x ) ^ { n } , | x | < 1$ 
A)  
$\sum _ { n = 0 } ^ { \infty } \frac { 5 } { 2 } \left( \frac { x } { 4 } \right) ^ { n } , | x | < 4$ 
B) 
$\sum _ { n = 0 } ^ { \infty } \frac { 1 } { 4 } \left( \frac { x } { 4 } \right) ^ { n } , | x | < 4$ 
C) 
$\sum _ { n = 0 } ^ { \infty } 10 \left( - \frac { x } { 4 } \right) ^ { n } , | x | < 4$ 
D) 
$\sum _ { n = 0 } ^ { \infty } \frac { 5 } { 2 } ( - 4 x ) ^ { n } , | x | < 4$ 
E) 
$\sum _ { n = 0 } ^ { \infty } \frac { 5 } { 2 } ( - x ) ^ { n } , | x | < 1$

Question 5

$\text { Find the positive values of } p \text { for which the series } \sum _ { n = 2 } ^ { \infty } \frac { 1 } { n ( \ln n ) ^ { p } } \text { converges. }$

Accepted Answer

A)  converges for $p > 0$ 
B)  diverges only at $p = 1$ 
C)  converges for $p \geq 2$ 
D)  converges for $p > 1$ 
E)  converges for $p \geq 1$ 
A)  converges for $p > 0$ 
B)  diverges only at $p = 1$ 
C)  converges for $p \geq 2$ 
D)  converges for $p > 1$ 
E)  converges for $p \geq 1$

Question 6

Identify the interval of convergence of a power series
$\sum _ { n = 1 } ^ { \infty } ( - 3 ) ^ { n } n x ^ { n - 1 }$

Accepted Answer

A)  $- 3 < x < 3$ 
B)  $- 1 < x < 1$ 
C)  $- \frac { 1 } { 3 } < x < \frac { 1 } { 3 }$ 
D)  $0 < x < 3$ 
E)  $- \frac { 1 } { 6 } < x < \frac { 1 } { 6 }$ 
A)  $- 3 < x < 3$ 
B)  $- 1 < x < 1$ 
C)  $- \frac { 1 } { 3 } < x < \frac { 1 } { 3 }$ 
D)  $0 < x < 3$ 
E)  $- \frac { 1 } { 6 } < x < \frac { 1 } { 6 }$

Question 7

$\text { Use the Ratio Test to determine the convergence or divergence of the series } \sum _ { n = 0 } ^ { \infty } \frac { n ! } { 8 ^ { n } } \text {. }$

Accepted Answer

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

Question 8

$\text { Sketch the graph of the sequence of partial sum of the series } \sum _ { n = 1 } ^ { \infty } \frac { 2 } { \sqrt [ 4 ] { n ^ { 3 } } } \text {. }$

Accepted Answer

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

Question 9

Use the Integral Test to determine the convergence or divergence of the series. $$\sum _ { n = 2 } ^ { \infty } \frac { \ln n } { n ^ { 10 } }$$

Accepted Answer

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

Question 10

Determine the minimal number of terms required to approximate the sum of the series with an error of less than 0.008. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } } { 2 n ^ { 3 } - 1 }$

Accepted Answer

A)  3 
B)  1 
C)  4 
D)  6 
E)  2 
A)  3 
B)  1 
C)  4 
D)  6 
E)  2

Question 11

Find all values of x for which the series converges. For these values of x, write the sum of the series as a function of x. $$\sum _ { n = 0 } ^ { \infty } 10 \left( \frac { x - 8 } { 10 } \right) ^ { n }$$

Accepted Answer

A)  $- \frac { 100 } { 18 - x } , - 2 < x < 18$ 
B)  $\frac { 100 } { 18 - x } , - 2 < x < 18$ 
C)  $\frac { 100 } { 28 + x } , - 12 < x < 28$ 
D)  $\frac { 100 } { 28 - x } , - 12 < x < 28$ 
E)  The series diverges for all $x$. 
A)  $- \frac { 100 } { 18 - x } , - 2 < x < 18$ 
B)  $\frac { 100 } { 18 - x } , - 2 < x < 18$ 
C)  $\frac { 100 } { 28 + x } , - 12 < x < 28$ 
D)  $\frac { 100 } { 28 - x } , - 12 < x < 28$ 
E)  The series diverges for all $x$.

Question 12

Determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. $a _ { n } = \frac { 3 ^ { n } } { 4 ^ { n } }$

Accepted Answer

A)  The sequence diverges. 
B)  The sequence converges to 2 . 
C)  The sequence converges to $- 1$. 
D)  The sequence converges to 1 . 
E)  The sequence converges to 0 . 
A)  The sequence diverges. 
B)  The sequence converges to 2 . 
C)  The sequence converges to $- 1$. 
D)  The sequence converges to 1 . 
E)  The sequence converges to 0 .

Question 13

Find the values of $x$ for which the series
$\sum _ { n = 0 } ^ { \infty } 9 ( x - 1 ) ^ { n }$ converges.

Accepted Answer

A)  $8 < x < 10$ 
B)  $- 1 < x < 1$ 
C)  $- 9 < x < 0$ 
D)  $0 < x < 2$ 
E)  $- 2 < x < 0$ 
A)  $8 < x < 10$ 
B)  $- 1 < x < 1$ 
C)  $- 9 < x < 0$ 
D)  $0 < x < 2$ 
E)  $- 2 < x < 0$

Question 14

Use the Direct Comparison Test to determine the convergence or divergence of the series $\sum _ { n = 0 } ^ { \infty } \frac { 8 ^ { n } } { 9 ^ { n } + 7 }$.

Accepted Answer

A) The series $\sum _ { n = 0 } ^ { \infty } \frac { 8 ^ { n } } { 9 ^ { n } + 7 }$ converges. 
B) The series $\sum _ { n = 0 } ^ { \infty } \frac { 8 ^ { n } } { 9 ^ { n } + 7 }$ diverges. 
A) The series $\sum _ { n = 0 } ^ { \infty } \frac { 8 ^ { n } } { 9 ^ { n } + 7 }$ converges. 
B) The series $\sum _ { n = 0 } ^ { \infty } \frac { 8 ^ { n } } { 9 ^ { n } + 7 }$ diverges.

Question 15

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?
$$f ( x ) = \tan x , c = - \frac { \pi } { 6 }$$

Accepted Answer

A) 
$3 y + \sqrt { 3 } = - 4 \left( x + \frac { \pi } { 6 } \right) ;$ tangent line to $y = f ( x )$ at $x = - \frac { \pi } { 6 }$ 
B) 
$3 y + \sqrt { 3 } = 4 \left( x + \frac { \pi } { 6 } \right) ;$ secant line to $y = f ( x )$ at $x = - \frac { \pi } { 6 }$ 
C) 
$3 y + \sqrt { 2 } = 4 \left( x + \frac { \pi } { 6 } \right) ;$ tangent line to $y = f ( x )$ at $x = - \frac { \pi } { 6 }$ 
D)  
$3 x + \sqrt { 3 } = 4 \left( y + \frac { \pi } { 6 } \right) ;$ differential of $y = f ( x )$ at $x = - \frac { \pi } { 6 }$ 
E) 
$3 y + \sqrt { 3 } = 4 \left( x + \frac { \pi } { 6 } \right) ;$ tangent line to $y = f ( x )$ at $x = - \frac { \pi } { 6 }$ 
A) 
$3 y + \sqrt { 3 } = - 4 \left( x + \frac { \pi } { 6 } \right) ;$ tangent line to $y = f ( x )$ at $x = - \frac { \pi } { 6 }$ 
B) 
$3 y + \sqrt { 3 } = 4 \left( x + \frac { \pi } { 6 } \right) ;$ secant line to $y = f ( x )$ at $x = - \frac { \pi } { 6 }$ 
C) 
$3 y + \sqrt { 2 } = 4 \left( x + \frac { \pi } { 6 } \right) ;$ tangent line to $y = f ( x )$ at $x = - \frac { \pi } { 6 }$ 
D)  
$3 x + \sqrt { 3 } = 4 \left( y + \frac { \pi } { 6 } \right) ;$ differential of $y = f ( x )$ at $x = - \frac { \pi } { 6 }$ 
E) 
$3 y + \sqrt { 3 } = 4 \left( x + \frac { \pi } { 6 } \right) ;$ tangent line to $y = f ( x )$ at $x = - \frac { \pi } { 6 }$

Question 16

Find the interval of convergence of the power series $\sum _ { n = 0 } ^ { \infty } \frac { ( x - 12 ) ^ { n - 1 } } { 12 ^ { n - 1 } }$.
(Be sure to include a check for convergence at the endpoints of the interval.)

Accepted Answer

A)  $( - \infty , 0 ) \cup ( 24 , \infty )$ 
B)  $( - \infty , 0 ] \cup [ 12 , \infty )$ 
C)  $[ 0,24 ]$ 
D)  $( 0,24 )$ 
E)  $( 0,12 )$ 
A)  $( - \infty , 0 ) \cup ( 24 , \infty )$ 
B)  $( - \infty , 0 ] \cup [ 12 , \infty )$ 
C)  $[ 0,24 ]$ 
D)  $( 0,24 )$ 
E)  $( 0,12 )$

Question 17

Find all values of $x$ for which the series
$\sum _ { n = 0 } ^ { \infty } 9 \left( \frac { x - 8 } { 9 } \right) ^ { n }$ converges.

Accepted Answer

A)  $- 1 < x < 17$ 
B)  $- 10 < x < 18$ 
C)  $- 1 < x < 18$ 
D)  $- 18 < x < 19$ 
E)  $- 9 < x < 19$ 
A)  $- 1 < x < 17$ 
B)  $- 10 < x < 18$ 
C)  $- 1 < x < 18$ 
D)  $- 18 < x < 19$ 
E)  $- 9 < x < 19$

Question 18

$\text { Find a power series for the function } \frac { 12 } { 12 x + 13 } \text { centered at } 0 \text {. }$

Accepted Answer

A)  $\sum _ { n = 0 } ^ { \infty } - \left( \frac { 12 } { 13 } \right) ^ { n + 1 } x ^ { n }$ 
B) 
$\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \left( \frac { 12 } { 13 } \right) ^ { n + 1 } x ^ { n }$ 
C) 
$\sum _ { n = 0 } ^ { \infty } \left( \frac { 12 } { 13 } \right) ^ { n + 1 } x ^ { n }$ 
D) 
$\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \left( \frac { 12 x } { 13 } \right) ^ { n + 1 }$ 
E) 
$\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \left( \frac { 12 x } { 13 } \right) ^ { n } + 1$ 
A)  $\sum _ { n = 0 } ^ { \infty } - \left( \frac { 12 } { 13 } \right) ^ { n + 1 } x ^ { n }$ 
B) 
$\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \left( \frac { 12 } { 13 } \right) ^ { n + 1 } x ^ { n }$ 
C) 
$\sum _ { n = 0 } ^ { \infty } \left( \frac { 12 } { 13 } \right) ^ { n + 1 } x ^ { n }$ 
D) 
$\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \left( \frac { 12 x } { 13 } \right) ^ { n + 1 }$ 
E) 
$\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \left( \frac { 12 x } { 13 } \right) ^ { n } + 1$

Question 19

$\text { The series } \sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { 9 ^ { n } } \text { diverges. }$

Accepted Answer

A) True 
 B)False

Question 20

Use the Direct Comparison Test to determine the convergence or divergence of the series $$\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 5 ^ { 4 } \sqrt { n } - 1 }$$

Accepted Answer

A)  The series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 5 ^ { 4 } \sqrt { n } - 1 }$ converges. 
B)  The series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 5 ^ { 4 } \sqrt { n } - 1 }$ diverges. 
A)  The series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 5 ^ { 4 } \sqrt { n } - 1 }$ converges. 
B)  The series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 5 ^ { 4 } \sqrt { n } - 1 }$ diverges.

Approximate the sum of the series by using the first six terms. $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } 4 } { n ! }$

$\text { Determine the convergence or divergence of the series } \sum _ { n = 1 } ^ { \infty } \frac { n } { 7 n ^ { 2 } + 1 } \text { using any }$ appropriate test.

$\text {The infinite series } \sum _ { n = 1 } ^ { \infty } \frac { n } { 13 n + 4 } \text { diverges ?}$

Find a geometric power series for the function centered at 0, (i) by the technique shown in Examples 1 and 2 and (ii) by long division. $f ( x ) = \frac { 10 } { 4 - x }$

$\text { Find the positive values of } p \text { for which the series } \sum _ { n = 2 } ^ { \infty } \frac { 1 } { n ( \ln n ) ^ { p } } \text { converges. }$

Identify the interval of convergence of a power series $\sum _ { n = 1 } ^ { \infty } ( - 3 ) ^ { n } n x ^ { n - 1 }$

$\text { Use the Ratio Test to determine the convergence or divergence of the series } \sum _ { n = 0 } ^ { \infty } \frac { n ! } { 8 ^ { n } } \text {. }$

$\text { Sketch the graph of the sequence of partial sum of the series } \sum _ { n = 1 } ^ { \infty } \frac { 2 } { \sqrt [ 4 ] { n ^ { 3 } } } \text {. }$

Use the Integral Test to determine the convergence or divergence of the series. $\sum _ { n = 2 } ^ { \infty } \frac { \ln n } { n ^ { 10 } }$

Determine the minimal number of terms required to approximate the sum of the series with an error of less than 0.008. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } } { 2 n ^ { 3 } - 1 }$

Find all values of x for which the series converges. For these values of x, write the sum of the series as a function of x. $\sum _ { n = 0 } ^ { \infty } 10 \left( \frac { x - 8 } { 10 } \right) ^ { n }$

Determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. $a _ { n } = \frac { 3 ^ { n } } { 4 ^ { n } }$

Find the values of $x$ for which the series $\sum _ { n = 0 } ^ { \infty } 9 ( x - 1 ) ^ { n }$ converges.

Use the Direct Comparison Test to determine the convergence or divergence of the series $\sum _ { n = 0 } ^ { \infty } \frac { 8 ^ { n } } { 9 ^ { n } + 7 }$ .

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? $f ( x ) = \tan x , c = - \frac { \pi } { 6 }$

Find the interval of convergence of the power series $\sum _ { n = 0 } ^ { \infty } \frac { ( x - 12 ) ^ { n - 1 } } { 12 ^ { n - 1 } }$ . (Be sure to include a check for convergence at the endpoints of the interval.)

Find all values of $x$ for which the series $\sum _ { n = 0 } ^ { \infty } 9 \left( \frac { x - 8 } { 9 } \right) ^ { n }$ converges.

$\text { Find a power series for the function } \frac { 12 } { 12 x + 13 } \text { centered at } 0 \text {. }$

$\text { The series } \sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { 9 ^ { n } } \text { diverges. }$

Use the Direct Comparison Test to determine the convergence or divergence of the series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 5 ^ { 4 } \sqrt { n } - 1 }$

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

Approximate the sum of the series by using the first six terms. ∑n=0∞(−1)n4n!\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } 4 } { n ! }∑n=0∞​n!(−1)n4​

The infinite series ∑n=1∞n13n+4 diverges ?\text {The infinite series } \sum _ { n = 1 } ^ { \infty } \frac { n } { 13 n + 4 } \text { diverges ?}The infinite series ∑n=1∞​13n+4n​ diverges ?

Find a geometric power series for the function centered at 0, (i) by the technique shown in Examples 1 and 2 and (ii) by long division. f(x)=104−xf ( x ) = \frac { 10 } { 4 - x }f(x)=4−x10​

Identify the interval of convergence of a power series ∑n=1∞(−3)nnxn−1\sum _ { n = 1 } ^ { \infty } ( - 3 ) ^ { n } n x ^ { n - 1 }∑n=1∞​(−3)nnxn−1

Use the Integral Test to determine the convergence or divergence of the series. ∑n=2∞ln⁡nn10\sum _ { n = 2 } ^ { \infty } \frac { \ln n } { n ^ { 10 } }∑n=2∞​n10lnn​

Determine the minimal number of terms required to approximate the sum of the series with an error of less than 0.008. ∑n=1∞(−1)n+12n3−1\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } } { 2 n ^ { 3 } - 1 }∑n=1∞​2n3−1(−1)n+1​

Find all values of x for which the series converges. For these values of x, write the sum of the series as a function of x. ∑n=0∞10(x−810)n\sum _ { n = 0 } ^ { \infty } 10 \left( \frac { x - 8 } { 10 } \right) ^ { n }∑n=0∞​10(10x−8​)n

Determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. an=3n4na _ { n } = \frac { 3 ^ { n } } { 4 ^ { n } }an​=4n3n​

Find the values of xxx for which the series ∑n=0∞9(x−1)n\sum _ { n = 0 } ^ { \infty } 9 ( x - 1 ) ^ { n }∑n=0∞​9(x−1)n converges.

Use the Direct Comparison Test to determine the convergence or divergence of the series ∑n=0∞8n9n+7\sum _ { n = 0 } ^ { \infty } \frac { 8 ^ { n } } { 9 ^ { n } + 7 }∑n=0∞​9n+78n​ .

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)=tan⁡x,c=−π6f ( x ) = \tan x , c = - \frac { \pi } { 6 }f(x)=tanx,c=−6π​

Find the interval of convergence of the power series ∑n=0∞(x−12)n−112n−1\sum _ { n = 0 } ^ { \infty } \frac { ( x - 12 ) ^ { n - 1 } } { 12 ^ { n - 1 } }∑n=0∞​12n−1(x−12)n−1​ . (Be sure to include a check for convergence at the endpoints of the interval.)

Find all values of xxx for which the series ∑n=0∞9(x−89)n\sum _ { n = 0 } ^ { \infty } 9 \left( \frac { x - 8 } { 9 } \right) ^ { n }∑n=0∞​9(9x−8​)n converges.

Find a power series for the function 1212x+13 centered at 0. \text { Find a power series for the function } \frac { 12 } { 12 x + 13 } \text { centered at } 0 \text {. } Find a power series for the function 12x+1312​ centered at 0.

The series ∑n=1∞(−1)n9n diverges. \text { The series } \sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { 9 ^ { n } } \text { diverges. } The series ∑n=1∞​9n(−1)n​ diverges.

Use the Direct Comparison Test to determine the convergence or divergence of the series ∑n=1∞154n−1\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 5 ^ { 4 } \sqrt { n } - 1 }∑n=1∞​54n​−11​

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

Approximate the sum of the series by using the first six terms. $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } 4 } { n ! }$

$\text {The infinite series } \sum _ { n = 1 } ^ { \infty } \frac { n } { 13 n + 4 } \text { diverges ?}$

Find a geometric power series for the function centered at 0, (i) by the technique shown in Examples 1 and 2 and (ii) by long division. $f ( x ) = \frac { 10 } { 4 - x }$

Identify the interval of convergence of a power series $\sum _ { n = 1 } ^ { \infty } ( - 3 ) ^ { n } n x ^ { n - 1 }$

Use the Integral Test to determine the convergence or divergence of the series. $\sum _ { n = 2 } ^ { \infty } \frac { \ln n } { n ^ { 10 } }$

Determine the minimal number of terms required to approximate the sum of the series with an error of less than 0.008. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } } { 2 n ^ { 3 } - 1 }$

Find all values of x for which the series converges. For these values of x, write the sum of the series as a function of x. $\sum _ { n = 0 } ^ { \infty } 10 \left( \frac { x - 8 } { 10 } \right) ^ { n }$

Determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. $a _ { n } = \frac { 3 ^ { n } } { 4 ^ { n } }$

Find the values of $x$ for which the series $\sum _ { n = 0 } ^ { \infty } 9 ( x - 1 ) ^ { n }$ converges.

Use the Direct Comparison Test to determine the convergence or divergence of the series $\sum _ { n = 0 } ^ { \infty } \frac { 8 ^ { n } } { 9 ^ { n } + 7 }$ .

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? $f ( x ) = \tan x , c = - \frac { \pi } { 6 }$

Find the interval of convergence of the power series $\sum _ { n = 0 } ^ { \infty } \frac { ( x - 12 ) ^ { n - 1 } } { 12 ^ { n - 1 } }$ . (Be sure to include a check for convergence at the endpoints of the interval.)

Find all values of $x$ for which the series $\sum _ { n = 0 } ^ { \infty } 9 \left( \frac { x - 8 } { 9 } \right) ^ { n }$ converges.

$\text { Find a power series for the function } \frac { 12 } { 12 x + 13 } \text { centered at } 0 \text {. }$

$\text { The series } \sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { 9 ^ { n } } \text { diverges. }$

Use the Direct Comparison Test to determine the convergence or divergence of the series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 5 ^ { 4 } \sqrt { n } - 1 }$