Question 1

Find the coefficient of $( x - 2 ) ^ { 2 }$ in the Taylor polynomial $T _ { 2 } ( x )$ for the function $x ^ { 3 }$ at the number 2.

Accepted Answer

A) 3 
B) 0 
C) 1 
D) 6 
E) 2
F)5
G)8
H)4 
A) 3 
B) 0 
C) 1 
D) 6 
E) 2
F)5
G)8
H)4

Question 2

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

Accepted Answer

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

Question 3

Estimate the range of values of x for which the approximation $e ^ { x } \cos x = 1 + x - \frac { 1 } { 3 } x ^ { 3 }$ is accurate to within 0.001.

Accepted Answer

The answer of Estimate the range of values of x...

Question 4

Determine whether the series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { ( 3 n - 1 ) ( 3 n + 2 ) }$ is convergent or divergent. If it is convergent, find the sum.

Accepted Answer

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

Question 5

Which of the three series below converges?
1) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n }$ 
2) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 1.1 } }$ 
3) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 09 } }$

Accepted Answer

A) 1, 2 
B) 1 
C) None 
D) 3 
E) 2
F)2, 3
G)1, 2, 3
H)1, 3 
A) 1, 2 
B) 1 
C) None 
D) 3 
E) 2
F)2, 3
G)1, 2, 3
H)1, 3

Question 6

Determine whether the given series is convergent (but not absolutely convergent), absolutely convergent, or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } n } { n ^ { 2 } + 1 }$

Accepted Answer

Convergent

Question 7

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

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 8

Consider the series $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } \frac { 1 } { n ^ { 4 } + 1 }$ .(a) Show that the series is absolutely convergent.(b) How many terms of the series do we need to add in order to find the sum to within 0.01?
(c) What is the approximation sum in part (b)?

Accepted Answer

The answer of Consider the series \(\sum _ { n...

Question 9

(a) Express $\int _ { 0 } ^ { x } \frac { \sin \left( t ^ { 2 } \right) } { t ^ { 2 } } d t$ as a Maclaurin series.(b) Evaluate $\int _ { 0 } ^ { 03 } \frac { \sin \left( x ^ { 2 } \right) } { x ^ { 2 } } d x$ as a series.

Accepted Answer

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

Question 10

Use the binomial series to expand $\sqrt [ 3 ] { 1 + x ^ { 2 } }$ as a power series. State the radius of convergence.

Accepted Answer

The answer of Use the binomial series to expand \(\sqrt...

Question 11

Which of the following series is convergent, but not absolutely convergent?
(a) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n }$ (b) $\sum _ { n = 1 } ^ { \infty } \frac { \sin n } { n ^ { 2 } }$ (c) $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { \sqrt { n } }$ (d) $\sum _ { n = 1 } ^ { \infty } \frac { 3 ^ { n } } { 2 ^ { n } + \sqrt { n } }$ (e) $\sum _ { n = 1 } ^ { \infty } \frac { 1 - 2 n } { n + 1 }$

Accepted Answer

(c) is convergent by the Alter

Question 12

Find the interval of convergence of $\sum _ { n = 0 } ^ { \infty } \frac { n } { 4 ^ { n } } ( x + 3 ) ^ { n }$ .

Accepted Answer

A)  $( - 3,3 )$ 
B)  $[ - 3,3 )$ 
C)  $[ - 3,3 ]$ 
D)  $( - 3,3 ]$ 
E)  $( - 7,1 )$ 
F) $[ - 7,1 )$ 
G) $[ - 7,1 ]$ 
H) $( - 7,1 ]$ 
A)  $( - 3,3 )$ 
B)  $[ - 3,3 )$ 
C)  $[ - 3,3 ]$ 
D)  $( - 3,3 ]$ 
E)  $( - 7,1 )$ 
F) $[ - 7,1 )$ 
G) $[ - 7,1 ]$ 
H) $( - 7,1 ]$

Question 13

Find the values of x for which the series $\sum _ { n = 1 } ^ { \infty } ( 2 x ) ^ { n }$ converges.

Accepted Answer

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

Question 14

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

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 15

Consider the series $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { ( 2 n ) ! }$ .(a) Show that the series is absolutely convergent.(b) How many terms of the series do we need to add in order to find the sum to within 0.0001?
(c) What is the approximation sum in part (b)?

Accepted Answer

The answer of Consider the series \(\sum _ { n...

Question 16

Find the value of $\sum _ { n = 2 } ^ { \infty } \frac { 3 ^ { n } + 5 ^ { n } } { 15 ^ { n } }$ .

Accepted Answer

The answer of Find the value of \(\sum _ {...

Question 17

(a) Express $\int _ { 0 } ^ { x } \sin \left( t ^ { 2 } \right) d t$ as a Maclaurin series.(b) Evaluate $\int _ { 0 } ^ { 03 } \sin \left( x ^ { 2 } \right) d x$ as a series.

Accepted Answer

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

Question 18

Find the limit of the sequence $a _ { n } = \left( n + e ^ { n } \right) ^ { 1 / n }$ .

Accepted Answer

A) 1 
B) 2 
C)  $e$ 
D)  $1 + e$ 
E) -1
F)3
G)Divergent 
A) 1 
B) 2 
C)  $e$ 
D)  $1 + e$ 
E) -1
F)3
G)Divergent

Question 19

Determine whether the series $1 - \frac { 1 } { 2 } + \frac { 1 } { 4 } - \frac { 1 } { 8 } + \cdots$ is convergent or divergent. If it is convergent, find its sum.

Accepted Answer

The answer of Determine whether the series \(1 - \frac...

Question 20

Find the interval of convergence of $\sum _ { n = 0 } ^ { \infty } \frac { x ^ { n } } { 3 n + 1 }$ .

Accepted Answer

A)  $[ - 3,3 ]$ 
B)  $( - 1,1 )$ 
C)  $( - 3,3 )$ 
D)  $[ - 1,1 ]$ 
E)  $( - 1,1 ]$ 
F) $( - 3,3 ]$ 
G) $[ - 3,3 )$ 
H) $[ - 1,1 )$ 
A)  $[ - 3,3 ]$ 
B)  $( - 1,1 )$ 
C)  $( - 3,3 )$ 
D)  $[ - 1,1 ]$ 
E)  $( - 1,1 ]$ 
F) $( - 3,3 ]$ 
G) $[ - 3,3 )$ 
H) $[ - 1,1 )$

Find the coefficient of $( x - 2 ) ^ { 2 }$ in the Taylor polynomial $T _ { 2 } ( x )$ for the function $x ^ { 3 }$ at the number 2.

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

Estimate the range of values of x for which the approximation $e ^ { x } \cos x = 1 + x - \frac { 1 } { 3 } x ^ { 3 }$ is accurate to within 0.001.

Determine whether the series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { ( 3 n - 1 ) ( 3 n + 2 ) }$ is convergent or divergent. If it is convergent, find the sum.

Which of the three series below converges? 1) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n }$ 2) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 1.1 } }$ 3) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 09 } }$

Determine whether the given series is convergent (but not absolutely convergent), absolutely convergent, or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } n } { n ^ { 2 } + 1 }$

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

Consider the series $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } \frac { 1 } { n ^ { 4 } + 1 }$ .(a) Show that the series is absolutely convergent.(b) How many terms of the series do we need to add in order to find the sum to within 0.01? (c) What is the approximation sum in part (b)?

(a) Express $\int _ { 0 } ^ { x } \frac { \sin \left( t ^ { 2 } \right) } { t ^ { 2 } } d t$ as a Maclaurin series.(b) Evaluate $\int _ { 0 } ^ { 03 } \frac { \sin \left( x ^ { 2 } \right) } { x ^ { 2 } } d x$ as a series.

Use the binomial series to expand $\sqrt [ 3 ] { 1 + x ^ { 2 } }$ as a power series. State the radius of convergence.

Find the interval of convergence of $\sum _ { n = 0 } ^ { \infty } \frac { n } { 4 ^ { n } } ( x + 3 ) ^ { n }$ .

Find the values of x for which the series $\sum _ { n = 1 } ^ { \infty } ( 2 x ) ^ { n }$ converges.

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

Consider the series $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { ( 2 n ) ! }$ .(a) Show that the series is absolutely convergent.(b) How many terms of the series do we need to add in order to find the sum to within 0.0001? (c) What is the approximation sum in part (b)?

Find the value of $\sum _ { n = 2 } ^ { \infty } \frac { 3 ^ { n } + 5 ^ { n } } { 15 ^ { n } }$ .

(a) Express $\int _ { 0 } ^ { x } \sin \left( t ^ { 2 } \right) d t$ as a Maclaurin series.(b) Evaluate $\int _ { 0 } ^ { 03 } \sin \left( x ^ { 2 } \right) d x$ as a series.

Find the limit of the sequence $a _ { n } = \left( n + e ^ { n } \right) ^ { 1 / n }$ .

Determine whether the series $1 - \frac { 1 } { 2 } + \frac { 1 } { 4 } - \frac { 1 } { 8 } + \cdots$ is convergent or divergent. If it is convergent, find its sum.

Find the interval of convergence of $\sum _ { n = 0 } ^ { \infty } \frac { x ^ { n } } { 3 n + 1 }$ .

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Exam 8: Infinite Sequences and Series

Find the coefficient of (x−2)2( x - 2 ) ^ { 2 }(x−2)2 in the Taylor polynomial T2(x)T _ { 2 } ( x )T2​(x) for the function x3x ^ { 3 }x3 at the number 2.

Find the radius of convergence of the series ∑n=1∞3n(x+1)nnn\sum _ { n = 1 } ^ { \infty } \frac { 3 ^ { n } ( x + 1 ) ^ { n } } { n ^ { n } }∑n=1∞​nn3n(x+1)n​ .

Estimate the range of values of x for which the approximation excos⁡x=1+x−13x3e ^ { x } \cos x = 1 + x - \frac { 1 } { 3 } x ^ { 3 }excosx=1+x−31​x3 is accurate to within 0.001.

Determine whether the series ∑n=1∞1(3n−1)(3n+2)\sum _ { n = 1 } ^ { \infty } \frac { 1 } { ( 3 n - 1 ) ( 3 n + 2 ) }∑n=1∞​(3n−1)(3n+2)1​ is convergent or divergent. If it is convergent, find the sum.

Determine whether the given series is convergent (but not absolutely convergent), absolutely convergent, or divergent. ∑n=1∞(−1)n+1nn2+1\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } n } { n ^ { 2 } + 1 }∑n=1∞​n2+1(−1)n+1n​

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

Use the binomial series to expand 1+x23\sqrt [ 3 ] { 1 + x ^ { 2 } }31+x2​ as a power series. State the radius of convergence.

Find the interval of convergence of ∑n=0∞n4n(x+3)n\sum _ { n = 0 } ^ { \infty } \frac { n } { 4 ^ { n } } ( x + 3 ) ^ { n }∑n=0∞​4nn​(x+3)n .

Find the values of x for which the series ∑n=1∞(2x)n\sum _ { n = 1 } ^ { \infty } ( 2 x ) ^ { n }∑n=1∞​(2x)n converges.

Find the radius of convergence of ∑n=1∞(x+2)nn23n\sum _ { n = 1 } ^ { \infty } \frac { ( x + 2 ) ^ { n } } { n ^ { 2 } 3 ^ { n } }∑n=1∞​n23n(x+2)n​ .

Find the value of ∑n=2∞3n+5n15n\sum _ { n = 2 } ^ { \infty } \frac { 3 ^ { n } + 5 ^ { n } } { 15 ^ { n } }∑n=2∞​15n3n+5n​ .

(a) Express ∫0xsin⁡(t2)dt\int _ { 0 } ^ { x } \sin \left( t ^ { 2 } \right) d t∫0x​sin(t2)dt as a Maclaurin series.(b) Evaluate ∫003sin⁡(x2)dx\int _ { 0 } ^ { 03 } \sin \left( x ^ { 2 } \right) d x∫003​sin(x2)dx as a series.

Find the limit of the sequence an=(n+en)1/na _ { n } = \left( n + e ^ { n } \right) ^ { 1 / n }an​=(n+en)1/n .

Determine whether the series 1−12+14−18+⋯1 - \frac { 1 } { 2 } + \frac { 1 } { 4 } - \frac { 1 } { 8 } + \cdots1−21​+41​−81​+⋯ is convergent or divergent. If it is convergent, find its sum.

Find the interval of convergence of ∑n=0∞xn3n+1\sum _ { n = 0 } ^ { \infty } \frac { x ^ { n } } { 3 n + 1 }∑n=0∞​3n+1xn​ .

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Find the coefficient of $( x - 2 ) ^ { 2 }$ in the Taylor polynomial $T _ { 2 } ( x )$ for the function $x ^ { 3 }$ at the number 2.

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

Estimate the range of values of x for which the approximation $e ^ { x } \cos x = 1 + x - \frac { 1 } { 3 } x ^ { 3 }$ is accurate to within 0.001.

Determine whether the series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { ( 3 n - 1 ) ( 3 n + 2 ) }$ is convergent or divergent. If it is convergent, find the sum.

Determine whether the given series is convergent (but not absolutely convergent), absolutely convergent, or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } n } { n ^ { 2 } + 1 }$

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

Use the binomial series to expand $\sqrt [ 3 ] { 1 + x ^ { 2 } }$ as a power series. State the radius of convergence.

Find the interval of convergence of $\sum _ { n = 0 } ^ { \infty } \frac { n } { 4 ^ { n } } ( x + 3 ) ^ { n }$ .

Find the values of x for which the series $\sum _ { n = 1 } ^ { \infty } ( 2 x ) ^ { n }$ converges.

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

Find the value of $\sum _ { n = 2 } ^ { \infty } \frac { 3 ^ { n } + 5 ^ { n } } { 15 ^ { n } }$ .

(a) Express $\int _ { 0 } ^ { x } \sin \left( t ^ { 2 } \right) d t$ as a Maclaurin series.(b) Evaluate $\int _ { 0 } ^ { 03 } \sin \left( x ^ { 2 } \right) d x$ as a series.

Find the limit of the sequence $a _ { n } = \left( n + e ^ { n } \right) ^ { 1 / n }$ .

Determine whether the series $1 - \frac { 1 } { 2 } + \frac { 1 } { 4 } - \frac { 1 } { 8 } + \cdots$ is convergent or divergent. If it is convergent, find its sum.

Find the interval of convergence of $\sum _ { n = 0 } ^ { \infty } \frac { x ^ { n } } { 3 n + 1 }$ .