Question 1

Find the fourth Maclaurin polynomial $P _ { 4 } ( x )$ for the function $f ( x ) = \sqrt { 1 + 2 x }$

Accepted Answer

The answer of Find the fourth Maclaurin polynomial \(P _...

Question 2

Find the interval of convergence of the power series: $$\sum _ { k = 1 } ^ { \infty } ( - 1 ) ^ { k } \frac { \sqrt { k } } { k ^ { 2 } } ( x - 1 ) ^ { k }$$

Accepted Answer

[0, 2]

Question 3

Find the fourth Maclaurin polynomial $P _ { 4 } ( x )$ for the function $f ( x ) = x e ^ { x }$

Accepted Answer

The answer of Find the fourth Maclaurin polynomial \(P _...

Question 4

Find the interval of convergence of the power series:  ∑k=0∞k!(x−1) k\sum _ { k = 0 } ^ { \infty } k ! ( x - 1 )  ^ { k }∑k=0∞​k!(x−1) k

Accepted Answer

A)   (-1, 1) 
B)   Converges only when x = 1
C)   (0, 1) 
D)   (-?, ?)

Question 5

Find the interval of convergence and radius of convergence of the power series:  ∑k=0∞11+3kxk\sum _ { k = 0 } ^ { \infty } \frac { 1 } { 1 + 3 ^ { k } } x ^ { k }∑k=0∞​1+3k1​xk

Accepted Answer

A)   R = 1, interval of convergence: (-1, 1) 
B)   R = 1, interval of convergence: (-3, 3) 
C)   R = 3, interval of convergence: (-1, 1) 
D)   R = 3, interval of convergence: (-3, 3)

Question 6

Find the series equal to the definite integral $\int _ { 0 } x ^ { 2 } e ^ { - x ^ { 2 } } d x$

Accepted Answer

The answer of Find the series equal to the definite...

Question 7

Find the interval of convergence of the power series:  ∑k=0∞(−1) kk22k(x−2) k\sum _ { k = 0 } ^ { \infty } ( - 1 )  ^ { k } \frac { k ^ { 2 } } { 2 ^ { k } } ( x - 2 )  ^ { k }∑k=0∞​(−1) k2kk2​(x−2) k

Accepted Answer

A)   [0, 4) 
B)   (0, 4) 
C)   (0, 4]
D)   [0, 4]

Question 8

Find the fourth Taylor polynomial $P _ { 4 } ( x )$ for the function $f ( x ) = e ^ { 2 x }$ at x = 1.

Accepted Answer

The answer of Find the fourth Taylor polynomial \(P _...

Question 9

Find the radius of convergence of the series:  ∑k=0∞k!(k+2) !xk\sum _ { k = 0 } ^ { \infty } \frac { k ! } { ( k + 2 )  ! } x ^ { k }∑k=0∞​(k+2) !k!​xk

Accepted Answer

A)   R = 0
B)   R = 1
C)   R = 2
D)   R = ?

Question 10

Find the interval of convergence of the power series:  ∑k=0∞12k!xk\sum _ { k = 0 } ^ { \infty } \frac { 1 } { 2 k ! } x ^ { k }∑k=0∞​2k!1​xk

Accepted Answer

A)   (-1, 1) 
B)   Converges only when x = 0.
C)   (0, 2) 
D)   (-?, ?)   D

Question 11

Find the Maclaurin series for the function  f(x) =xcos⁡xf ( x )  = x \cos xf(x) =xcosx

Accepted Answer

A)    ∑k=0∞(−1) kk!x2k+1\sum _ { k = 0 } ^ { \infty } \frac { ( - 1 )  ^ { k } } { k ! } x ^ { 2 k + 1 }∑k=0∞​k!(−1) k​x2k+1 
B)    ∑k=0∞(−1) k(2k) !x2k+1\sum _ { k = 0 } ^ { \infty } \frac { ( - 1 )  ^ { k } } { ( 2 k )  ! } x ^ { 2 k + 1 }∑k=0∞​(2k) !(−1) k​x2k+1 
C)    ∑k=0∞(−1) k(2k) !x2k\sum _ { k = 0 } ^ { \infty } \frac { ( - 1 )  ^ { k } } { ( 2 k )  ! } x ^ { 2 k }∑k=0∞​(2k) !(−1) k​x2k 
D)    ∑k=0∞(−1) k(2k) !xk+1\sum _ { k = 0 } ^ { \infty } \frac { ( - 1 )  ^ { k } } { ( 2 k )  ! } x ^ { k + 1 }∑k=0∞​(2k) !(−1) k​xk+1

Question 12

Use Maclaurin series to find first four nonzero terms in the Maclaurin series for the function $f ( x ) = e ^ { x } \tan ^ { - 1 } x$ Also, give the interval of convergence.

Accepted Answer

<img  loading="lazy" src="/assets/images/blured-textbook.svg" class="d-inline m-1" alt="blured image" width="139" height="30"> interval ...

Question 13

Use Maclaurin series to find first four nonzero terms in the Maclaurin series for the function $f ( x ) = e ^ { 2 x } \cos x$ Also, give the interval of convergence.

Accepted Answer

<img  loading="lazy" src="/assets/images/blured-textbook.svg" class="d-inline m-1" alt="blured image" width="139" height="30"> interval ...

Question 14

Find the fourth Maclaurin polynomial $P _ { 4 } ( x )$ for the function $f ( x ) = e ^ { - 2 x }$

Accepted Answer

$1 - 2 x + 2 x ^ { 2 } - \frac { 4 } { 3 } x ^ { 3 } + \frac { 2 } { 3 } x ^ { 4 }$

Question 15

Find the interval of convergence of the power series:  ∑k=0∞3k+1k3(x−1) k\sum _ { k = 0 } ^ { \infty } \frac { 3 k + 1 } { k ^ { 3 } } ( x - 1 )  ^ { k }∑k=0∞​k33k+1​(x−1) k

Accepted Answer

A)   (0, 2) 
B)   [0, 2) 
C)   (0, 2]
D)   [0, 2]

Question 16

Find the series equal to the definite integral $\int _ { 0 } ^ { 2 } \sin \left( x ^ { 2 } \right) d x$

Accepted Answer

The answer of Find the series equal to the definite...

Question 17

Find the interval of convergence and radius of convergence of the power series:  ∑k=0∞(3k) kx2k\sum _ { k = 0 } ^ { \infty } \left( \frac { 3 } { k } \right)  ^ { k } x ^ { 2 k }∑k=0∞​(k3​) kx2k

Accepted Answer

A)   R = 1, interval of convergence: (-1, 1) 
B)   R = ?, interval of convergence: (-?, ?) 
C)   R = 1, interval of convergence: (0, 2) 
D)   R = 3, interval of convergence: (-3, 3)

Question 18

Find the series equal to the definite integral $\int _ { 0 } x \cos \left( x ^ { 2 } \right) d x$

Accepted Answer

The answer of Find the series equal to the definite...

Question 19

Find the Maclaurin series for the function  f(x) =cos⁡(2x2) f ( x )  = \cos \left( 2 x ^ { 2 } \right) f(x) =cos(2x2)   , and give the interval of convergence of the series.

Accepted Answer

A)    ∑k=0∞22k(2k) !x2k\sum _ { k = 0 } ^ { \infty } \frac { 2 ^ { 2 k } } { ( 2 k )  ! } x ^ { 2 k }∑k=0∞​(2k) !22k​x2k  interval of convergence: (-?, ?) 
B)    ∑k=0∞22k(2k) !x2k\sum _ { k = 0 } ^ { \infty } \frac { 2 ^ { 2 k } } { ( 2 k )  ! } x ^ { 2 k }∑k=0∞​(2k) !22k​x2k  interval of convergence: (0, 1) 
C)    ∑k=0∞(−1) k22k(2k) !x2k\sum _ { k = 0 } ^ { \infty } \frac { ( - 1 )  ^ { k } 2 ^ { 2 k } } { ( 2 k )  ! } x ^ { 2 k }∑k=0∞​(2k) !(−1) k22k​x2k  interval of convergence: (-?, ?) 
D)    ∑k=0∞(−1) k22k(2k) !x4k\sum _ { k = 0 } ^ { \infty } \frac { ( - 1 )  ^ { k } 2 ^ { 2 k } } { ( 2 k )  ! } x ^ { 4 k }∑k=0∞​(2k) !(−1) k22k​x4k  interval of convergence: (-?, ?)

Question 20

Find the Maclaurin series for the function $f ( x ) = x \sin \left( x ^ { 2 } \right)$ , and give the interval of convergence of the series.

Accepted Answer

<img  loading="lazy" src="/assets/images/blured-textbook.svg" class="d-inline m-1" alt="blured image" width="139" height="30"> interval ...

Exam 8: Power Series

Find the fourth Maclaurin polynomial P4(x)P _ { 4 } ( x )P4​(x) for the function f(x)=1+2xf ( x ) = \sqrt { 1 + 2 x }f(x)=1+2x​

Correct AnswerverifiedShow Answer

Find the interval of convergence of the power series: ∑k=1∞(−1)kkk2(x−1)k\sum _ { k = 1 } ^ { \infty } ( - 1 ) ^ { k } \frac { \sqrt { k } } { k ^ { 2 } } ( x - 1 ) ^ { k }∑k=1∞​(−1)kk2k​​(x−1)k

Correct Answerverified[0, 2]

Find the fourth Maclaurin polynomial P4(x)P _ { 4 } ( x )P4​(x) for the function f(x)=xexf ( x ) = x e ^ { x }f(x)=xex

Correct AnswerverifiedShow Answer

Find the interval of convergence of the power series: ∑k=0∞k!(x−1) k\sum _ { k = 0 } ^ { \infty } k ! ( x - 1 ) ^ { k }∑k=0∞​k!(x−1) k

Correct AnswerverifiedShow Answer

Find the interval of convergence and radius of convergence of the power series: ∑k=0∞11+3kxk\sum _ { k = 0 } ^ { \infty } \frac { 1 } { 1 + 3 ^ { k } } x ^ { k }∑k=0∞​1+3k1​xk

Correct AnswerverifiedShow Answer

Find the series equal to the definite integral ∫0x2e−x2dx\int _ { 0 } x ^ { 2 } e ^ { - x ^ { 2 } } d x∫0​x2e−x2dx

Correct AnswerverifiedShow Answer

Find the interval of convergence of the power series: ∑k=0∞(−1) kk22k(x−2) k\sum _ { k = 0 } ^ { \infty } ( - 1 ) ^ { k } \frac { k ^ { 2 } } { 2 ^ { k } } ( x - 2 ) ^ { k }∑k=0∞​(−1) k2kk2​(x−2) k

Correct AnswerverifiedShow Answer

Find the fourth Taylor polynomial P4(x)P _ { 4 } ( x )P4​(x) for the function f(x)=e2xf ( x ) = e ^ { 2 x }f(x)=e2x at x = 1.

Correct AnswerverifiedShow Answer

Find the radius of convergence of the series: ∑k=0∞k!(k+2) !xk\sum _ { k = 0 } ^ { \infty } \frac { k ! } { ( k + 2 ) ! } x ^ { k }∑k=0∞​(k+2) !k!​xk

Correct AnswerverifiedShow Answer

Find the interval of convergence of the power series: ∑k=0∞12k!xk\sum _ { k = 0 } ^ { \infty } \frac { 1 } { 2 k ! } x ^ { k }∑k=0∞​2k!1​xk

Correct AnswerverifiedD

Find the Maclaurin series for the function f(x) =xcos⁡xf ( x ) = x \cos xf(x) =xcosx

Correct AnswerverifiedShow Answer

Use Maclaurin series to find first four nonzero terms in the Maclaurin series for the function f(x)=extan⁡−1xf ( x ) = e ^ { x } \tan ^ { - 1 } xf(x)=extan−1x Also, give the interval of convergence.

Correct Answerverified interval ...View AnswerShow Answer

Use Maclaurin series to find first four nonzero terms in the Maclaurin series for the function f(x)=e2xcos⁡xf ( x ) = e ^ { 2 x } \cos xf(x)=e2xcosx Also, give the interval of convergence.

Correct Answerverified interval ...View AnswerShow Answer

Find the fourth Maclaurin polynomial P4(x)P _ { 4 } ( x )P4​(x) for the function f(x)=e−2xf ( x ) = e ^ { - 2 x }f(x)=e−2x

Correct Answerverified\(1 - 2 x + 2 x ^ { 2 } - \frac { 4 } { 3 } x ^ { 3 } + \frac { 2 } { 3 } x ^ { 4 }\)

Find the interval of convergence of the power series: ∑k=0∞3k+1k3(x−1) k\sum _ { k = 0 } ^ { \infty } \frac { 3 k + 1 } { k ^ { 3 } } ( x - 1 ) ^ { k }∑k=0∞​k33k+1​(x−1) k

Correct AnswerverifiedShow Answer

Find the series equal to the definite integral ∫02sin⁡(x2)dx\int _ { 0 } ^ { 2 } \sin \left( x ^ { 2 } \right) d x∫02​sin(x2)dx

Correct AnswerverifiedShow Answer

Find the interval of convergence and radius of convergence of the power series: ∑k=0∞(3k) kx2k\sum _ { k = 0 } ^ { \infty } \left( \frac { 3 } { k } \right) ^ { k } x ^ { 2 k }∑k=0∞​(k3​) kx2k

Correct AnswerverifiedShow Answer

Find the series equal to the definite integral ∫0xcos⁡(x2)dx\int _ { 0 } x \cos \left( x ^ { 2 } \right) d x∫0​xcos(x2)dx

Correct AnswerverifiedShow Answer

Find the Maclaurin series for the function f(x) =cos⁡(2x2) f ( x ) = \cos \left( 2 x ^ { 2 } \right) f(x) =cos(2x2) , and give the interval of convergence of the series.

Correct AnswerverifiedShow Answer

Find the Maclaurin series for the function f(x)=xsin⁡(x2)f ( x ) = x \sin \left( x ^ { 2 } \right)f(x)=xsin(x2) , and give the interval of convergence of the series.

Correct Answerverified interval ...View AnswerShow Answer

Find the fourth Maclaurin polynomial $P _ { 4 } ( x )$ for the function $f ( x ) = \sqrt { 1 + 2 x }$

Correct Answer
verified

Find the interval of convergence of the power series: $\sum _ { k = 1 } ^ { \infty } ( - 1 ) ^ { k } \frac { \sqrt { k } } { k ^ { 2 } } ( x - 1 ) ^ { k }$

Correct Answer
verified
[0, 2]

Find the fourth Maclaurin polynomial $P _ { 4 } ( x )$ for the function $f ( x ) = x e ^ { x }$

Correct Answer
verified

Find the interval of convergence of the power series: $\sum _ { k = 0 } ^ { \infty } k ! ( x - 1 ) ^ { k }$

Correct Answer
verified

Find the interval of convergence and radius of convergence of the power series: $\sum _ { k = 0 } ^ { \infty } \frac { 1 } { 1 + 3 ^ { k } } x ^ { k }$

Correct Answer
verified

Find the series equal to the definite integral $\int _ { 0 } x ^ { 2 } e ^ { - x ^ { 2 } } d x$

Correct Answer
verified

Find the interval of convergence of the power series: $\sum _ { k = 0 } ^ { \infty } ( - 1 ) ^ { k } \frac { k ^ { 2 } } { 2 ^ { k } } ( x - 2 ) ^ { k }$

Correct Answer
verified

Find the fourth Taylor polynomial $P _ { 4 } ( x )$ for the function $f ( x ) = e ^ { 2 x }$ at x = 1.

Correct Answer
verified

Find the radius of convergence of the series: $\sum _ { k = 0 } ^ { \infty } \frac { k ! } { ( k + 2 ) ! } x ^ { k }$

Correct Answer
verified

Find the interval of convergence of the power series: $\sum _ { k = 0 } ^ { \infty } \frac { 1 } { 2 k ! } x ^ { k }$

Correct Answer
verified
D

Find the Maclaurin series for the function $f ( x ) = x \cos x$

Correct Answer
verified

Use Maclaurin series to find first four nonzero terms in the Maclaurin series for the function $f ( x ) = e ^ { x } \tan ^ { - 1 } x$ Also, give the interval of convergence.

Correct Answer
verified
interval ...
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Use Maclaurin series to find first four nonzero terms in the Maclaurin series for the function $f ( x ) = e ^ { 2 x } \cos x$ Also, give the interval of convergence.

Correct Answer
verified
interval ...
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Find the fourth Maclaurin polynomial $P _ { 4 } ( x )$ for the function $f ( x ) = e ^ { - 2 x }$

Correct Answer
verified
\(1 - 2 x + 2 x ^ { 2 } - \frac { 4 } { 3 } x ^ { 3 } + \frac { 2 } { 3 } x ^ { 4 }\)

Find the interval of convergence of the power series: $\sum _ { k = 0 } ^ { \infty } \frac { 3 k + 1 } { k ^ { 3 } } ( x - 1 ) ^ { k }$

Correct Answer
verified

Find the series equal to the definite integral $\int _ { 0 } ^ { 2 } \sin \left( x ^ { 2 } \right) d x$

Correct Answer
verified

Find the interval of convergence and radius of convergence of the power series: $\sum _ { k = 0 } ^ { \infty } \left( \frac { 3 } { k } \right) ^ { k } x ^ { 2 k }$

Correct Answer
verified

Find the series equal to the definite integral $\int _ { 0 } x \cos \left( x ^ { 2 } \right) d x$

Correct Answer
verified

Find the Maclaurin series for the function $f ( x ) = \cos \left( 2 x ^ { 2 } \right)$ , and give the interval of convergence of the series.

Correct Answer
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Find the Maclaurin series for the function $f ( x ) = x \sin \left( x ^ { 2 } \right)$ , and give the interval of convergence of the series.

Correct Answer
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interval ...
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