Question 1

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 2

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

Accepted Answer

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

Question 3

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 4

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

Accepted Answer

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

Question 5

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

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) 
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 6

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

Accepted Answer

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

Question 7

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

Accepted Answer

A)  $\sum _ { k = 0 } ^ { \infty } \frac { ( - 1 ) ^ { k } } { k ! } x ^ { 2 k + 1 }$ 
B)  $\sum _ { k = 0 } ^ { \infty } \frac { ( - 1 ) ^ { k } } { ( 2 k ) ! } x ^ { 2 k + 1 }$ 
C)  $\sum _ { k = 0 } ^ { \infty } \frac { ( - 1 ) ^ { k } } { ( 2 k ) ! } x ^ { 2 k }$ 
D)  $\sum _ { k = 0 } ^ { \infty } \frac { ( - 1 ) ^ { k } } { ( 2 k ) ! } x ^ { k + 1 }$ 
A)  $\sum _ { k = 0 } ^ { \infty } \frac { ( - 1 ) ^ { k } } { k ! } x ^ { 2 k + 1 }$ 
B)  $\sum _ { k = 0 } ^ { \infty } \frac { ( - 1 ) ^ { k } } { ( 2 k ) ! } x ^ { 2 k + 1 }$ 
C)  $\sum _ { k = 0 } ^ { \infty } \frac { ( - 1 ) ^ { k } } { ( 2 k ) ! } x ^ { 2 k }$ 
D)  $\sum _ { k = 0 } ^ { \infty } \frac { ( - 1 ) ^ { k } } { ( 2 k ) ! } x ^ { k + 1 }$

Question 8

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 9

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 10

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 11

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 12

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

Accepted Answer

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

Question 13

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

Accepted Answer

A)  $\sum _ { k = 0 } ^ { \infty } \frac { 2 ^ { 2 k } } { ( 2 k ) ! } x ^ { 2 k }$ interval of convergence: (-?, ?) 
B)  $\sum _ { k = 0 } ^ { \infty } \frac { 2 ^ { 2 k } } { ( 2 k ) ! } x ^ { 2 k }$ interval of convergence: (0, 1) 
C)  $\sum _ { k = 0 } ^ { \infty } \frac { ( - 1 ) ^ { k } 2 ^ { 2 k } } { ( 2 k ) ! } x ^ { 2 k }$ interval of convergence: (-?, ?) 
D)  $\sum _ { k = 0 } ^ { \infty } \frac { ( - 1 ) ^ { k } 2 ^ { 2 k } } { ( 2 k ) ! } x ^ { 4 k }$ interval of convergence: (-?, ?) 
A)  $\sum _ { k = 0 } ^ { \infty } \frac { 2 ^ { 2 k } } { ( 2 k ) ! } x ^ { 2 k }$ interval of convergence: (-?, ?) 
B)  $\sum _ { k = 0 } ^ { \infty } \frac { 2 ^ { 2 k } } { ( 2 k ) ! } x ^ { 2 k }$ interval of convergence: (0, 1) 
C)  $\sum _ { k = 0 } ^ { \infty } \frac { ( - 1 ) ^ { k } 2 ^ { 2 k } } { ( 2 k ) ! } x ^ { 2 k }$ interval of convergence: (-?, ?) 
D)  $\sum _ { k = 0 } ^ { \infty } \frac { ( - 1 ) ^ { k } 2 ^ { 2 k } } { ( 2 k ) ! } x ^ { 4 k }$ interval of convergence: (-?, ?)

Question 14

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 15

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 16

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

Accepted Answer

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

Question 17

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

@#LAT-DLM& interval

Question 18

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

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) 
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 19

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

@#LAT-DLM& interval

Question 20

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

@#LAT-DLM& interval

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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.

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Exam 8: Power Series

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

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

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

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

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

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

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

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

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.

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​

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

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

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.

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

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

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

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.

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

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.

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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.