Question 1

Find the limit of the sequence $a _ { n } = \frac { e ^ { n } } { n ! }$ .

Accepted Answer

A)  $\frac { e ^ { 2 } - 1 } { e }$ 
B)  $\sqrt { e }$ 
C)  $e$ 
D)  $e ^ { 2 }$ 
E) 0
F) $\frac { e - 1 } { e }$ 
G)Divergent
H)1 
A)  $\frac { e ^ { 2 } - 1 } { e }$ 
B)  $\sqrt { e }$ 
C)  $e$ 
D)  $e ^ { 2 }$ 
E) 0
F) $\frac { e - 1 } { e }$ 
G)Divergent
H)1

Question 2

Find the limit of the sequence $a _ { n } = \cos \left( \frac { n \pi } { 2 } \right)$ .

Accepted Answer

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

Question 3

Examine the two series below for absolute convergence (A), convergence that is not absolute (C), or divergence (D).
1) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n }$ 
2) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } n ^ { - 1 }$

Accepted Answer

A) 1A, 2A 
B) 1A, 2C 
C) 1A, 2D 
D) 1C, 2A 
E) 1C, 2C
F)1C, 2D
G)1D, 2A
H)1D, 2C 
A) 1A, 2A 
B) 1A, 2C 
C) 1A, 2D 
D) 1C, 2A 
E) 1C, 2C
F)1C, 2D
G)1D, 2A
H)1D, 2C

Question 4

Find the third-degree Taylor polynomial of the function $f ( x ) = e ^ { x ^ { 2 } } - x ^ { 2 } \cos \sqrt { x } \text { at } a = 0$ .

Accepted Answer

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

Question 5

Which of the following three tests will establish that the series $\sum _ { n = 1 } ^ { \infty } \frac { n } { \sqrt { 2 n ^ { 5 } + 1 } }$ converges?
1) Comparison Test with $\sum _ { n = 1 } ^ { \infty } n ^ { - 5 / 2 }$ 
2) Comparison Test with $\sum _ { n = 1 } ^ { \infty } n ^ { - 3 / 2 }$ 
3) Comparison Test with $\sum _ { n = 1 } ^ { \infty } n ^ { - 1 / 2 }$

Accepted Answer

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

Question 6

Find the interval of convergence for $\sum _ { k = 1 } ^ { \infty } \frac { ( - 1 ) ^ { k } ( x - 3 ) ^ { k } } { 5 ^ { k } ( k + 1 ) }$ .

Accepted Answer

The answer of Find the interval of convergence for \(\sum...

Question 7

If $\sum _ { n = 2 } ^ { \infty } \left( \frac { a } { 1 + a } \right) ^ { n } = 3$ and $a > 0$ , determine the value of a.

Accepted Answer

The answer of If \(\sum _ { n = 2...

Question 8

A sequence is defined by $a _ { n } = 0.9999 ^ { n }$ .(a) Calculate $a _ { 10 ^ { 3 } }$ and $a _ { 10 ^ { 5 } }$ .(b) Determine whether $a _ { n }$ converges or diverges. If it converges, find the limit.

Accepted Answer

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

Question 9

Find a formula for the general term $a _ { n }$ of the sequence $\left\{ 1 , - \frac { 12 } { 10 } , \frac { 19 } { 15 } , - \frac { 26 } { 20 } , \frac { 33 } { 25 } , \ldots \right\}$ .

Accepted Answer

A)  $\frac { ( - 1 ) ^ { n - 1 } ( 5 + 7 n ) } { 5 ( n + 1 ) }$ 
B)  $\frac { ( - 1 ) ^ { n } ( 5 + 7 n ) } { 5 n }$ 
C)  $\frac { ( 5 + 7 n ) } { 5 ( n + 1 ) }$ 
D)  $\frac { ( - 1 ) ^ { n } ( 5 - 7 n ) } { 5 ( n + 1 ) }$ 
E)  $\frac { ( - 1 ) ^ { n } ( 5 + 7 n ) } { ( n + 1 ) }$ 
F) $\frac { ( - 1 ) ^ { n + 1 } ( 5 - 7 n ) } { 5 ( n + 1 ) }$ 
G) $\frac { ( - 1 ) ^ { n } ( 5 + 7 n ) } { 5 ( n + 1 ) }$ 
H)None of these 
A)  $\frac { ( - 1 ) ^ { n - 1 } ( 5 + 7 n ) } { 5 ( n + 1 ) }$ 
B)  $\frac { ( - 1 ) ^ { n } ( 5 + 7 n ) } { 5 n }$ 
C)  $\frac { ( 5 + 7 n ) } { 5 ( n + 1 ) }$ 
D)  $\frac { ( - 1 ) ^ { n } ( 5 - 7 n ) } { 5 ( n + 1 ) }$ 
E)  $\frac { ( - 1 ) ^ { n } ( 5 + 7 n ) } { ( n + 1 ) }$ 
F) $\frac { ( - 1 ) ^ { n + 1 } ( 5 - 7 n ) } { 5 ( n + 1 ) }$ 
G) $\frac { ( - 1 ) ^ { n } ( 5 + 7 n ) } { 5 ( n + 1 ) }$ 
H)None of these

Question 10

Find the Maclaurin series expansion with $n = 5$ for $f ( x ) = 2 ^ { x }$ . Use this expansion to approximate $2 ^ { 0.1 }$ .

Accepted Answer

The answer of Find the Maclaurin series expansion with \(n...

Question 11

Consider the power series $\sum _ { k = 1 } ^ { \infty } \frac { ( - 1 ) ^ { k } } { k } x ^ { k }$ .(a) Find the radius of convergence.(b) Determine what happens at the end points (absolute or conditional convergence, or divergence).

Accepted Answer

(a) @#IMG-DLM& (b) for @#IMG-DLM& , conv

Question 12

Find the limit of the sequence $a _ { n } = 1 + \left( - \frac { 4 } { 5 } \right) ^ { n }$ .

Accepted Answer

A) 0 
B)  $- \frac { 4 } { 5 }$ 
C)  $\frac { 4 } { 5 }$ 
D)  $- \frac { 1 } { 5 }$ 
E) 1

F) $\frac { 1 } { 5 }$ 
G)-1
H)Divergent 
A) 0 
B)  $- \frac { 4 } { 5 }$ 
C)  $\frac { 4 } { 5 }$ 
D)  $- \frac { 1 } { 5 }$ 
E) 1

F) $\frac { 1 } { 5 }$ 
G)-1
H)Divergent

Question 13

Find an approximation for $\sin ( 0.1 )$ accurate to 6 decimal places.(Note: sin's argument is measured in radians.)

Accepted Answer

The answer of Find an approximation for \(\sin ( 0.1...

Question 14

Which one of the following series is divergent?

Accepted Answer

A)  $\sum _ { n = 1 } ^ { \infty } \left( \frac { 2 } { 3 } \right) ^ { n }$ 
B)  $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 } + 1 }$ 
C)  $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n 5 ^ { n } }$ 
D)  $\sum _ { n = 2 } ^ { \infty } \frac { n } { n ^ { 2 } - 1 }$ 
E)  $\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 3 } } { n ^ { 5 } + 2 }$ 
F) $\sum _ { n = 1 } ^ { \infty } \frac { 2 ^ { n } } { n ! }$ 
G) $\sum _ { n = 1 } ^ { \infty } \frac { \pi } { n ^ { 2 } }$ 
H) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } \frac { \pi } { n }$ 
A)  $\sum _ { n = 1 } ^ { \infty } \left( \frac { 2 } { 3 } \right) ^ { n }$ 
B)  $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 } + 1 }$ 
C)  $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n 5 ^ { n } }$ 
D)  $\sum _ { n = 2 } ^ { \infty } \frac { n } { n ^ { 2 } - 1 }$ 
E)  $\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 3 } } { n ^ { 5 } + 2 }$ 
F) $\sum _ { n = 1 } ^ { \infty } \frac { 2 ^ { n } } { n ! }$ 
G) $\sum _ { n = 1 } ^ { \infty } \frac { \pi } { n ^ { 2 } }$ 
H) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } \frac { \pi } { n }$

Question 15

Find the terms of the Maclaurin series for $\frac { 1 } { \sqrt { 1 + 2 x } }$ , as far as the term in $x ^ { 3 }$ .

Accepted Answer

The answer of Find the terms of the Maclaurin series...

Question 16

Find the radius of convergence of $\sum _ { n = 1 } ^ { \infty } \frac { 2 \cdot 5 \cdot 8 \cdots ( 3 n - 1 ) } { 3 \cdot 7 \cdot 11 \cdots ( 4 n - 1 ) } ( x + 1 ) ^ { n }$ .

Accepted Answer

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

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

G) $\frac { 2 } { 3 }$ 
H) $\infty$

Question 17

Find the Taylor polynomial $T _ { 3 } ( x )$ for the function $f ( x ) = \frac { 5 x } { 2 + 4 x }$ at the point $x _ { 0 } = 0$ .

Accepted Answer

The answer of Find the Taylor polynomial \(T _ {...

Question 18

Find the Maclaurin series expansion for $f ( x ) = \ln ( 1 - x )$ and determine the interval of convergence.

Accepted Answer

@#IMG-DLM& ; converg

Question 19

If $f ( x ) = e ^ { x ^ { 2 } }$ , compute $f ^ { ( 11 ) } ( 0 )$ .

Accepted Answer

The answer of If \(f ( x ) = e...

Question 20

Evaluate $\int \frac { 1 } { 1 + x ^ { 5 } } d x$ as a power series.

Accepted Answer

The answer of Evaluate \(\int \frac { 1 } {...

Find the limit of the sequence $a _ { n } = \frac { e ^ { n } } { n ! }$ .

Find the limit of the sequence $a _ { n } = \cos \left( \frac { n \pi } { 2 } \right)$ .

Examine the two series below for absolute convergence (A), convergence that is not absolute (C), or divergence (D). 1) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n }$ 2) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } n ^ { - 1 }$

Find the third-degree Taylor polynomial of the function $f ( x ) = e ^ { x ^ { 2 } } - x ^ { 2 } \cos \sqrt { x } \text { at } a = 0$ .

Find the interval of convergence for $\sum _ { k = 1 } ^ { \infty } \frac { ( - 1 ) ^ { k } ( x - 3 ) ^ { k } } { 5 ^ { k } ( k + 1 ) }$ .

If $\sum _ { n = 2 } ^ { \infty } \left( \frac { a } { 1 + a } \right) ^ { n } = 3$ and $a > 0$ , determine the value of a.

A sequence is defined by $a _ { n } = 0.9999 ^ { n }$ .(a) Calculate $a _ { 10 ^ { 3 } }$ and $a _ { 10 ^ { 5 } }$ .(b) Determine whether $a _ { n }$ converges or diverges. If it converges, find the limit.

Find a formula for the general term $a _ { n }$ of the sequence $\left\{ 1 , - \frac { 12 } { 10 } , \frac { 19 } { 15 } , - \frac { 26 } { 20 } , \frac { 33 } { 25 } , \ldots \right\}$ .

Find the Maclaurin series expansion with $n = 5$ for $f ( x ) = 2 ^ { x }$ . Use this expansion to approximate $2 ^ { 0.1 }$ .

Consider the power series $\sum _ { k = 1 } ^ { \infty } \frac { ( - 1 ) ^ { k } } { k } x ^ { k }$ .(a) Find the radius of convergence.(b) Determine what happens at the end points (absolute or conditional convergence, or divergence).

Find the limit of the sequence $a _ { n } = 1 + \left( - \frac { 4 } { 5 } \right) ^ { n }$ .

Find an approximation for $\sin ( 0.1 )$ accurate to 6 decimal places.(Note: sin's argument is measured in radians.)

Which one of the following series is divergent?

Find the terms of the Maclaurin series for $\frac { 1 } { \sqrt { 1 + 2 x } }$ , as far as the term in $x ^ { 3 }$ .

Find the radius of convergence of $\sum _ { n = 1 } ^ { \infty } \frac { 2 \cdot 5 \cdot 8 \cdots ( 3 n - 1 ) } { 3 \cdot 7 \cdot 11 \cdots ( 4 n - 1 ) } ( x + 1 ) ^ { n }$ .

Find the Taylor polynomial $T _ { 3 } ( x )$ for the function $f ( x ) = \frac { 5 x } { 2 + 4 x }$ at the point $x _ { 0 } = 0$ .

Find the Maclaurin series expansion for $f ( x ) = \ln ( 1 - x )$ and determine the interval of convergence.

If $f ( x ) = e ^ { x ^ { 2 } }$ , compute $f ^ { ( 11 ) } ( 0 )$ .

Evaluate $\int \frac { 1 } { 1 + x ^ { 5 } } d x$ as a power series.

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Exam 8: Infinite Sequences and Series

Find the limit of the sequence an=enn!a _ { n } = \frac { e ^ { n } } { n ! }an​=n!en​ .

Find the limit of the sequence an=cos⁡(nπ2)a _ { n } = \cos \left( \frac { n \pi } { 2 } \right)an​=cos(2nπ​) .

Find the third-degree Taylor polynomial of the function f(x)=ex2−x2cos⁡x at a=0f ( x ) = e ^ { x ^ { 2 } } - x ^ { 2 } \cos \sqrt { x } \text { at } a = 0f(x)=ex2−x2cosx​ at a=0 .

Find the interval of convergence for ∑k=1∞(−1)k(x−3)k5k(k+1)\sum _ { k = 1 } ^ { \infty } \frac { ( - 1 ) ^ { k } ( x - 3 ) ^ { k } } { 5 ^ { k } ( k + 1 ) }∑k=1∞​5k(k+1)(−1)k(x−3)k​ .

If ∑n=2∞(a1+a)n=3\sum _ { n = 2 } ^ { \infty } \left( \frac { a } { 1 + a } \right) ^ { n } = 3∑n=2∞​(1+aa​)n=3 and a>0a > 0a>0 , determine the value of a.

A sequence is defined by an=0.9999na _ { n } = 0.9999 ^ { n }an​=0.9999n .(a) Calculate a103a _ { 10 ^ { 3 } }a103​ and a105a _ { 10 ^ { 5 } }a105​ .(b) Determine whether ana _ { n }an​ converges or diverges. If it converges, find the limit.

Find a formula for the general term ana _ { n }an​ of the sequence {1,−1210,1915,−2620,3325,…}\left\{ 1 , - \frac { 12 } { 10 } , \frac { 19 } { 15 } , - \frac { 26 } { 20 } , \frac { 33 } { 25 } , \ldots \right\}{1,−1012​,1519​,−2026​,2533​,…} .

Find the Maclaurin series expansion with n=5n = 5n=5 for f(x)=2xf ( x ) = 2 ^ { x }f(x)=2x . Use this expansion to approximate 20.12 ^ { 0.1 }20.1 .

Consider the power series ∑k=1∞(−1)kkxk\sum _ { k = 1 } ^ { \infty } \frac { ( - 1 ) ^ { k } } { k } x ^ { k }∑k=1∞​k(−1)k​xk .(a) Find the radius of convergence.(b) Determine what happens at the end points (absolute or conditional convergence, or divergence).

Find the limit of the sequence an=1+(−45)na _ { n } = 1 + \left( - \frac { 4 } { 5 } \right) ^ { n }an​=1+(−54​)n .

Find an approximation for sin⁡(0.1)\sin ( 0.1 )sin(0.1) accurate to 6 decimal places.(Note: sin's argument is measured in radians.)

Which one of the following series is divergent?

Find the terms of the Maclaurin series for 11+2x\frac { 1 } { \sqrt { 1 + 2 x } }1+2x​1​ , as far as the term in x3x ^ { 3 }x3 .

Find the radius of convergence of ∑n=1∞2⋅5⋅8⋯(3n−1)3⋅7⋅11⋯(4n−1)(x+1)n\sum _ { n = 1 } ^ { \infty } \frac { 2 \cdot 5 \cdot 8 \cdots ( 3 n - 1 ) } { 3 \cdot 7 \cdot 11 \cdots ( 4 n - 1 ) } ( x + 1 ) ^ { n }∑n=1∞​3⋅7⋅11⋯(4n−1)2⋅5⋅8⋯(3n−1)​(x+1)n .

Find the Taylor polynomial T3(x)T _ { 3 } ( x )T3​(x) for the function f(x)=5x2+4xf ( x ) = \frac { 5 x } { 2 + 4 x }f(x)=2+4x5x​ at the point x0=0x _ { 0 } = 0x0​=0 .

Find the Maclaurin series expansion for f(x)=ln⁡(1−x)f ( x ) = \ln ( 1 - x )f(x)=ln(1−x) and determine the interval of convergence.

If f(x)=ex2f ( x ) = e ^ { x ^ { 2 } }f(x)=ex2 , compute f(11)(0)f ^ { ( 11 ) } ( 0 )f(11)(0) .

Evaluate ∫11+x5dx\int \frac { 1 } { 1 + x ^ { 5 } } d x∫1+x51​dx as a power series.

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Find the limit of the sequence $a _ { n } = \frac { e ^ { n } } { n ! }$ .

Find the limit of the sequence $a _ { n } = \cos \left( \frac { n \pi } { 2 } \right)$ .

Find the third-degree Taylor polynomial of the function $f ( x ) = e ^ { x ^ { 2 } } - x ^ { 2 } \cos \sqrt { x } \text { at } a = 0$ .

Find the interval of convergence for $\sum _ { k = 1 } ^ { \infty } \frac { ( - 1 ) ^ { k } ( x - 3 ) ^ { k } } { 5 ^ { k } ( k + 1 ) }$ .

If $\sum _ { n = 2 } ^ { \infty } \left( \frac { a } { 1 + a } \right) ^ { n } = 3$ and $a > 0$ , determine the value of a.

A sequence is defined by $a _ { n } = 0.9999 ^ { n }$ .(a) Calculate $a _ { 10 ^ { 3 } }$ and $a _ { 10 ^ { 5 } }$ .(b) Determine whether $a _ { n }$ converges or diverges. If it converges, find the limit.

Find a formula for the general term $a _ { n }$ of the sequence $\left\{ 1 , - \frac { 12 } { 10 } , \frac { 19 } { 15 } , - \frac { 26 } { 20 } , \frac { 33 } { 25 } , \ldots \right\}$ .

Find the Maclaurin series expansion with $n = 5$ for $f ( x ) = 2 ^ { x }$ . Use this expansion to approximate $2 ^ { 0.1 }$ .

Consider the power series $\sum _ { k = 1 } ^ { \infty } \frac { ( - 1 ) ^ { k } } { k } x ^ { k }$ .(a) Find the radius of convergence.(b) Determine what happens at the end points (absolute or conditional convergence, or divergence).

Find the limit of the sequence $a _ { n } = 1 + \left( - \frac { 4 } { 5 } \right) ^ { n }$ .

Find an approximation for $\sin ( 0.1 )$ accurate to 6 decimal places.(Note: sin's argument is measured in radians.)

Find the terms of the Maclaurin series for $\frac { 1 } { \sqrt { 1 + 2 x } }$ , as far as the term in $x ^ { 3 }$ .

Find the radius of convergence of $\sum _ { n = 1 } ^ { \infty } \frac { 2 \cdot 5 \cdot 8 \cdots ( 3 n - 1 ) } { 3 \cdot 7 \cdot 11 \cdots ( 4 n - 1 ) } ( x + 1 ) ^ { n }$ .

Find the Taylor polynomial $T _ { 3 } ( x )$ for the function $f ( x ) = \frac { 5 x } { 2 + 4 x }$ at the point $x _ { 0 } = 0$ .

Find the Maclaurin series expansion for $f ( x ) = \ln ( 1 - x )$ and determine the interval of convergence.

If $f ( x ) = e ^ { x ^ { 2 } }$ , compute $f ^ { ( 11 ) } ( 0 )$ .

Evaluate $\int \frac { 1 } { 1 + x ^ { 5 } } d x$ as a power series.