Question 1

Find a power series representation for $\frac { 1 } { ( 1 - x ) ^ { 2 } }$ and give its radius of convergence.

Accepted Answer

@#IMG-DLM& ; radius

Question 2

Construct an example of power series that has $( 3,5 )$ as its interval of convergence.

Accepted Answer

There are

Question 3

Consider the series $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } \frac { n } { 4 ^ { n } }$ .(a) Show that the series is absolutely convergent.(b) Calculate the sum of the first 3 terms to approximate the sum of the series.(c) Is the approximation in part (b) an overestimate or an underestimate?
(d) Estimate the error involved in the approximation from part (b).

Accepted Answer

(b) About

Question 4

Find a power series representation for the function $f ( x ) = \ln \sqrt { 1 - x }$ and determine the radius of convergence.

Accepted Answer

@#IMG-DLM& , radius

Question 5

Determine whether the series $\sum _ { n = 1 } ^ { \infty } \frac { n } { ( n + 1 ) ! }$ is convergent or divergent. If it is convergent, find the sum.

Accepted Answer

The series

Question 6

Determine whether each of the following series is convergent or divergent.(a) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 2 + 3 ^ { - n } }$ (b) $\sum _ { n = 0 } ^ { \infty } \frac { \pi ^ { n } } { 3 ^ { n } }$ (c) $\sum _ { n = 1 } ^ { \infty } \frac { e ^ { n } } { 3 ^ { n } }$

Accepted Answer

(a) Diverg

Question 7

Determine whether the series $\sum _ { n = 1 } ^ { \infty } \ln \left( \frac { 2 n - 1 } { 2 n + 1 } \right)$ is convergent or divergent. If it is convergent, find the sum.

Accepted Answer

The answer of Determine whether the series \(\sum _ {...

Question 8

Find the radius of convergence of $\sum _ { n = 1 } ^ { \infty } \frac { ( x - 2 ) ^ { n } } { n 3 ^ { n } }$ .

Accepted Answer

A) 0 
B) 1 
C) 2 
D) 3 
E) 4
F)5
G)6
H) $\infty$ 
A) 0 
B) 1 
C) 2 
D) 3 
E) 4
F)5
G)6
H) $\infty$

Question 9

Let $a _ { n } = \frac { 1 + 3 ^ { n } } { 1 + 4 \cdot 3 ^ { n } }$ .
(a) Find $\lim _ { n \rightarrow \infty } a _ { n }$ .
(b) Is $\sum a _ { n }$ convergent? Justify your answer.

Accepted Answer

(a) @#IMG-DLM& 
(b)

Question 10

Find the coefficient of $x ^ { 3 }$ in the binomial series for $\sqrt { 1 + x }$ .

Accepted Answer

A)  $- \frac { 1 } { 2 }$ 
B)  $- \frac { 1 } { 4 }$ 
C)  $\frac { 1 } { 8 }$ 
D)  $\frac { 1 } { 2 }$ h. $\frac { 1 } { 4 }$ 
E)  $- \frac { 1 } { 8 }$ 
F) $- \frac { 1 } { 16 }$ 
G) $\frac { 1 } { 16 }$ 
A)  $- \frac { 1 } { 2 }$ 
B)  $- \frac { 1 } { 4 }$ 
C)  $\frac { 1 } { 8 }$ 
D)  $\frac { 1 } { 2 }$ h. $\frac { 1 } { 4 }$ 
E)  $- \frac { 1 } { 8 }$ 
F) $- \frac { 1 } { 16 }$ 
G) $\frac { 1 } { 16 }$

Question 11

Use the sum of the first 10 terms to approximate the sum of the series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ( \ln ( 2 n ) ) ^ { 4 } + 1 }$ . Estimate the error involved in this approximation.

Accepted Answer

The answer of Use the sum of the first 10...

Question 12

Find the radius of convergence of $\sum _ { n = 0 } ^ { \infty } \frac { ( - 3 ) ^ { n } ( x - 1 ) ^ { n } } { \sqrt { n + 1 } }$ .

Accepted Answer

A) 0 
B)  $\frac { 1 } { 3 }$ 
C)  $\frac { 1 } { 2 }$ 
D) 1 
E) 2
F)3
G)4
H) $\infty$ 
A) 0 
B)  $\frac { 1 } { 3 }$ 
C)  $\frac { 1 } { 2 }$ 
D) 1 
E) 2
F)3
G)4
H) $\infty$

Question 13

Consider a sequence of rectangles, $R _ { 1 }$ , $R _ { 2 }$ , $R _ { 3 }$ , . . . illustrated in the figure below:   
(a) The height $L _ { n }$ of $R _ { n }$ is given by $L _ { n } = f ( n )$ where $f ( x ) = \frac { 1 } { x }$ . Write down the first five terms of $\left\{ L _ { n } \right\}$ and determine the limit of $\left\{ L _ { n } \right\}$ .
(b) Let $b _ { n } = \sum _ { k = 1 } ^ { n } L _ { k }$ . Compare $b _ { n }$ to $\int _ { n } ^ { n + 1 } ( 1 / x ) d x$ .
(c) Determine whether $\left\{ b _ { n } \right\}$ converges or diverges. Justify your answer.

Accepted Answer

(a) @#IMG-DLM& (b) For any n, the region

Question 14

Find the sum of the series $0.9 + 0.09 + 0.009 + 0.0009 + \cdots$ .

Accepted Answer

A) 9 
B) 0 
C) 0.999 
D) 0.9999 
E) 9.9
F)2
G)1
H)Divergent 
A) 9 
B) 0 
C) 0.999 
D) 0.9999 
E) 9.9
F)2
G)1
H)Divergent

Question 15

Find the radius of convergence of $\sum _ { n = 0 } ^ { \infty } ( 3 x ) ^ { n }$ .

Accepted Answer

A) 3 
B) 0 
C) 2 
D)  $\frac { 1 } { 6 }$ 
E) 6
F) $\frac { 1 } { 3 }$ 
G)1

H) $\frac { 1 } { 2 }$ 
A) 3 
B) 0 
C) 2 
D)  $\frac { 1 } { 6 }$ 
E) 6
F) $\frac { 1 } { 3 }$ 
G)1

H) $\frac { 1 } { 2 }$

Question 16

Find the radius of convergence of $\sum _ { n = 0 } ^ { \infty } \frac { ( - 3 ) ^ { n } ( x - 1 ) ^ { n } } { \sqrt { n + 1 } }$ .

Accepted Answer

A)  $\left( \frac { 2 } { 3 } , \frac { 4 } { 3 } \right]$ 
B)  $\left[ \frac { 2 } { 3 } , \frac { 4 } { 3 } \right)$ 
C)  $\left( \frac { 1 } { 2 } , \frac { 3 } { 2 } \right)$ 
D)  $\left[ \frac { 1 } { 2 } , \frac { 3 } { 2 } \right]$ 
E)  $( 0,2 ]$ 
F) $[ 0,2 )$ 
G) $( - 1,3 )$ 
H) $[ - 2,4 ]$ 
A)  $\left( \frac { 2 } { 3 } , \frac { 4 } { 3 } \right]$ 
B)  $\left[ \frac { 2 } { 3 } , \frac { 4 } { 3 } \right)$ 
C)  $\left( \frac { 1 } { 2 } , \frac { 3 } { 2 } \right)$ 
D)  $\left[ \frac { 1 } { 2 } , \frac { 3 } { 2 } \right]$ 
E)  $( 0,2 ]$ 
F) $[ 0,2 )$ 
G) $( - 1,3 )$ 
H) $[ - 2,4 ]$

Question 17

Find the second-degree Taylor polynomial of the function $f ( x ) = x \ln x \text { at } a = \frac { 1 } { e }$ .

Accepted Answer

The answer of Find the second-degree Taylor polynomial of the...

Question 18

Construct an example of power series that has $( - 3,5 ]$ as its interval of convergence.

Accepted Answer

There are

Question 19

Which one of the following series converges?

Accepted Answer

A)  $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 1.0001 } }$ 
B)  $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n }$ 
C)  $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 099 } }$ 
D)  $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { - 4 } }$ 
E) None of these 
A)  $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 1.0001 } }$ 
B)  $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n }$ 
C)  $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 099 } }$ 
D)  $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { - 4 } }$ 
E) None of these

Question 20

Determine if the series $\sum _ { n = 1 } ^ { \infty } \frac { 6 ^ { n } } { n ! }$ converges or diverges by the Ratio Test or Root Test.

Accepted Answer

The answer of Determine if the series \(\sum _ {...

Find a power series representation for $\frac { 1 } { ( 1 - x ) ^ { 2 } }$ and give its radius of convergence.

Construct an example of power series that has $( 3,5 )$ as its interval of convergence.

Find a power series representation for the function $f ( x ) = \ln \sqrt { 1 - x }$ and determine the radius of convergence.

Determine whether the series $\sum _ { n = 1 } ^ { \infty } \frac { n } { ( n + 1 ) ! }$ is convergent or divergent. If it is convergent, find the sum.

Determine whether each of the following series is convergent or divergent.(a) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 2 + 3 ^ { - n } }$ (b) $\sum _ { n = 0 } ^ { \infty } \frac { \pi ^ { n } } { 3 ^ { n } }$ (c) $\sum _ { n = 1 } ^ { \infty } \frac { e ^ { n } } { 3 ^ { n } }$

Determine whether the series $\sum _ { n = 1 } ^ { \infty } \ln \left( \frac { 2 n - 1 } { 2 n + 1 } \right)$ is convergent or divergent. If it is convergent, find the sum.

Find the radius of convergence of $\sum _ { n = 1 } ^ { \infty } \frac { ( x - 2 ) ^ { n } } { n 3 ^ { n } }$ .

Let $a _ { n } = \frac { 1 + 3 ^ { n } } { 1 + 4 \cdot 3 ^ { n } }$ . (a) Find $\lim _ { n \rightarrow \infty } a _ { n }$ . (b) Is $\sum a _ { n }$ convergent? Justify your answer.

Find the coefficient of $x ^ { 3 }$ in the binomial series for $\sqrt { 1 + x }$ .

Use the sum of the first 10 terms to approximate the sum of the series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ( \ln ( 2 n ) ) ^ { 4 } + 1 }$ . Estimate the error involved in this approximation.

Find the radius of convergence of $\sum _ { n = 0 } ^ { \infty } \frac { ( - 3 ) ^ { n } ( x - 1 ) ^ { n } } { \sqrt { n + 1 } }$ .

Find the sum of the series $0.9 + 0.09 + 0.009 + 0.0009 + \cdots$ .

Find the radius of convergence of $\sum _ { n = 0 } ^ { \infty } ( 3 x ) ^ { n }$ .

Find the radius of convergence of $\sum _ { n = 0 } ^ { \infty } \frac { ( - 3 ) ^ { n } ( x - 1 ) ^ { n } } { \sqrt { n + 1 } }$ .

Find the second-degree Taylor polynomial of the function $f ( x ) = x \ln x \text { at } a = \frac { 1 } { e }$ .

Construct an example of power series that has $( - 3,5 ]$ as its interval of convergence.

Which one of the following series converges?

Determine if the series $\sum _ { n = 1 } ^ { \infty } \frac { 6 ^ { n } } { n ! }$ converges or diverges by the Ratio Test or Root Test.

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Exam 8: Infinite Sequences and Series

Find a power series representation for 1(1−x)2\frac { 1 } { ( 1 - x ) ^ { 2 } }(1−x)21​ and give its radius of convergence.

Construct an example of power series that has (3,5)( 3,5 )(3,5) as its interval of convergence.

Find a power series representation for the function f(x)=ln⁡1−xf ( x ) = \ln \sqrt { 1 - x }f(x)=ln1−x​ and determine the radius of convergence.

Determine whether the series ∑n=1∞n(n+1)!\sum _ { n = 1 } ^ { \infty } \frac { n } { ( n + 1 ) ! }∑n=1∞​(n+1)!n​ is convergent or divergent. If it is convergent, find the sum.

Determine whether the series ∑n=1∞ln⁡(2n−12n+1)\sum _ { n = 1 } ^ { \infty } \ln \left( \frac { 2 n - 1 } { 2 n + 1 } \right)∑n=1∞​ln(2n+12n−1​) is convergent or divergent. If it is convergent, find the sum.

Find the radius of convergence of ∑n=1∞(x−2)nn3n\sum _ { n = 1 } ^ { \infty } \frac { ( x - 2 ) ^ { n } } { n 3 ^ { n } }∑n=1∞​n3n(x−2)n​ .

Let an=1+3n1+4⋅3na _ { n } = \frac { 1 + 3 ^ { n } } { 1 + 4 \cdot 3 ^ { n } }an​=1+4⋅3n1+3n​ . (a) Find lim⁡n→∞an\lim _ { n \rightarrow \infty } a _ { n }limn→∞​an​ . (b) Is ∑an\sum a _ { n }∑an​ convergent? Justify your answer.

Find the coefficient of x3x ^ { 3 }x3 in the binomial series for 1+x\sqrt { 1 + x }1+x​ .

Use the sum of the first 10 terms to approximate the sum of the series ∑n=1∞1n(ln⁡(2n))4+1\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ( \ln ( 2 n ) ) ^ { 4 } + 1 }∑n=1∞​n(ln(2n))4+11​ . Estimate the error involved in this approximation.

Find the radius of convergence of ∑n=0∞(−3)n(x−1)nn+1\sum _ { n = 0 } ^ { \infty } \frac { ( - 3 ) ^ { n } ( x - 1 ) ^ { n } } { \sqrt { n + 1 } }∑n=0∞​n+1​(−3)n(x−1)n​ .

Find the sum of the series 0.9+0.09+0.009+0.0009+⋯0.9 + 0.09 + 0.009 + 0.0009 + \cdots0.9+0.09+0.009+0.0009+⋯ .

Find the radius of convergence of ∑n=0∞(3x)n\sum _ { n = 0 } ^ { \infty } ( 3 x ) ^ { n }∑n=0∞​(3x)n .

Find the radius of convergence of ∑n=0∞(−3)n(x−1)nn+1\sum _ { n = 0 } ^ { \infty } \frac { ( - 3 ) ^ { n } ( x - 1 ) ^ { n } } { \sqrt { n + 1 } }∑n=0∞​n+1​(−3)n(x−1)n​ .

Find the second-degree Taylor polynomial of the function f(x)=xln⁡x at a=1ef ( x ) = x \ln x \text { at } a = \frac { 1 } { e }f(x)=xlnx at a=e1​ .

Construct an example of power series that has (−3,5]( - 3,5 ](−3,5] as its interval of convergence.

Which one of the following series converges?

Determine if the series ∑n=1∞6nn!\sum _ { n = 1 } ^ { \infty } \frac { 6 ^ { n } } { n ! }∑n=1∞​n!6n​ converges or diverges by the Ratio Test or Root Test.

Functions and Models

Limits and Derivatives

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Integrals

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Vectors and the Geometry of Space

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Filters

Find a power series representation for $\frac { 1 } { ( 1 - x ) ^ { 2 } }$ and give its radius of convergence.

Construct an example of power series that has $( 3,5 )$ as its interval of convergence.

Find a power series representation for the function $f ( x ) = \ln \sqrt { 1 - x }$ and determine the radius of convergence.

Determine whether the series $\sum _ { n = 1 } ^ { \infty } \frac { n } { ( n + 1 ) ! }$ is convergent or divergent. If it is convergent, find the sum.

Determine whether the series $\sum _ { n = 1 } ^ { \infty } \ln \left( \frac { 2 n - 1 } { 2 n + 1 } \right)$ is convergent or divergent. If it is convergent, find the sum.

Find the radius of convergence of $\sum _ { n = 1 } ^ { \infty } \frac { ( x - 2 ) ^ { n } } { n 3 ^ { n } }$ .

Let $a _ { n } = \frac { 1 + 3 ^ { n } } { 1 + 4 \cdot 3 ^ { n } }$ . (a) Find $\lim _ { n \rightarrow \infty } a _ { n }$ . (b) Is $\sum a _ { n }$ convergent? Justify your answer.

Find the coefficient of $x ^ { 3 }$ in the binomial series for $\sqrt { 1 + x }$ .

Use the sum of the first 10 terms to approximate the sum of the series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ( \ln ( 2 n ) ) ^ { 4 } + 1 }$ . Estimate the error involved in this approximation.

Find the radius of convergence of $\sum _ { n = 0 } ^ { \infty } \frac { ( - 3 ) ^ { n } ( x - 1 ) ^ { n } } { \sqrt { n + 1 } }$ .

Find the sum of the series $0.9 + 0.09 + 0.009 + 0.0009 + \cdots$ .

Find the radius of convergence of $\sum _ { n = 0 } ^ { \infty } ( 3 x ) ^ { n }$ .

Find the radius of convergence of $\sum _ { n = 0 } ^ { \infty } \frac { ( - 3 ) ^ { n } ( x - 1 ) ^ { n } } { \sqrt { n + 1 } }$ .

Find the second-degree Taylor polynomial of the function $f ( x ) = x \ln x \text { at } a = \frac { 1 } { e }$ .

Construct an example of power series that has $( - 3,5 ]$ as its interval of convergence.

Determine if the series $\sum _ { n = 1 } ^ { \infty } \frac { 6 ^ { n } } { n ! }$ converges or diverges by the Ratio Test or Root Test.