Deck 8: Infinite Sequences and Series

Question 1

Write the fourth-degree Taylor polynomial centered about the origin for the function .

Answer 1

Question 2

Which of the following is the degree 4 Taylor polynomial centered at $a = 0$ for $f ( x ) = \cos ( 2 x )$ ?
1) $1 - ( 2 x ) ^ { 2 } + 16 x ^ { 4 }$ 2) $1 - 2 x ^ { 2 } + \frac { 2 } { 3 } x ^ { 4 }$ 3) $1 - \frac { ( 2 x ) ^ { 2 } } { 2 ! } + \frac { ( 2 x ) ^ { 4 } } { 4 ! }$

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

Answer 2

2, 3

Question 3

Find the second-degree Taylor polynomial of the function .

Answer 3

Question 4

According to Taylor's Formula, what is the maximum error possible in the use of the sum $\sum _ { n = 0 } ^ { 4 } \frac { x ^ { n } } { n ! }$ to approximate $e ^ { x }$ in the interval $- 1 \leq x \leq 1$ ?

A) $\frac { e } { 240 }$
B) $\frac { e } { 48 }$
C) $\frac { e } { 480 }$
D) $\frac { e } { 24 }$
E) $\frac { e } { 20 }$
F) $\frac { e } { 120 }$
G) $\frac { e } { 12 }$
H) $\frac { e } { 60 }$

Question 5

Estimate the range of values of x for which the approximation $\frac { 1 } { x } = 1 - ( x - 1 ) + ( x - 1 ) ^ { 2 }$ is accurate to within 0.01.

A)[0.68, 1.41]
B)[0.61, 1.54]
C)[0.995, 1.005]
D)[1.51, 2.59]
E)[0.95, 1.05]
F)[0.80, 1.23]
G)[0.980, 1.023]
H)[0.89, 1.14]

Question 6

Give the 4th-degree Taylor polynomial for about the point . Using this polynomial, approximate . Give the maximum error for this approximation.

Question 7

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

Question 8

Which of the following is the degree 2 Taylor polynomial centered at $a = - 1$ for $f ( x ) = \frac { 1 } { x }$ ?
1) $- 1 - x - x ^ { 2 }$ 2) $- 1 - ( x + 1 ) - ( x + 1 ) ^ { 2 }$ 3) $- 1 - ( x + 1 ) - 2 ( x + 1 ) ^ { 2 }$

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

Question 9

Which of the following is the degree 2 Taylor polynomial centered at $a = 2$ for $f ( x ) = \ln x$ ?
1) $\ln 2 - \frac { x - 2 } { 2 } - \frac { ( x - 2 ) ^ { 2 } } { 8 }$ 2) $\ln 2 + \frac { x - 2 } { 2 } - \frac { ( x - 2 ) ^ { 2 } } { 4 }$ 3) $\ln 2 + \frac { x - 2 } { 2 } - \frac { ( x - 2 ) ^ { 2 } } { 8 }$

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

Question 10

Which of the following is the degree 2 Taylor polynomial centered at $a = 3$ for $f ( x ) = \ln x$ ?
1) $\ln 3 + \frac { x - 3 } { 3 } - \frac { ( x - 3 ) ^ { 2 } } { 18 }$ 2) $\ln 3 + \frac { x - 3 } { 3 } - \frac { ( x - 3 ) ^ { 2 } } { 9 }$ 3) $\ln 3 - \frac { x - 3 } { 3 } + \frac { ( x - 3 ) ^ { 2 } } { 18 }$

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

Question 11

What is the smallest value of n that will guarantee (according to Taylor's Formula) that the Taylor polynomial $T _ { n }$ at the number 0 will be within 0.0001 of $e ^ { x }$ for $0 \leq x \leq 1$ ?

A)4
B)5
C)8
D)6
E)7
F)2
G)3
H)9

Question 12

Estimate the range of values of x for which the approximation is accurate to within 0.001.

Question 13

Estimate the range of values of x for which the approximation is accurate to within 0.0002.

Question 14

Estimate the range of values of x for which the approximation $\ln x = \ln 2 + \frac { 1 } { 2 } ( x - 2 ) - \frac { 1 } { 8 } ( x - 2 ) ^ { 2 }$ is accurate to within 0.01.

A)[1.08, 3.20]
B)[1.80, 2.20]
C)[0.89, 3.56]
D)[1.43, 2.66]
E)[0.45, 1.78]
F)[0.71, 1.33]
G)[1.90, 2.10]
H)[1.99, 2.01]

Question 15

Consider the function .(a) Find the fourth-degree Taylor polynomial of f at .(b) What is the remainder?
(c) What is the absolute minimum value of f, and where does it occur?

Question 16

Find the second-degree Taylor polynomial for , centered about . Also obtain a bound for the error in using this polynomial to approximate .

Question 17

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

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

Question 18

Find the Taylor polynomial for the function at the point .

Question 19

Which of the following is the degree 3 Taylor polynomial centered at $a = 0$ for $f ( x ) = \cos ( 2 x )$ ?
1) $1 - ( 2 x ) ^ { 2 } + 16 x ^ { 4 }$ 2) $1 - 2 x ^ { 2 } + \frac { 8 } { 3 } x ^ { 4 }$ 3) $1 - \frac { ( 2 x ) ^ { 2 } } { 2 ! }$

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

Question 20

Which of the following is the degree 3 Taylor polynomial centered at $a = 0$ for $f ( x ) = \frac { 1 - \cos x } { x }$ ?
1) $\frac { x } { 2 } - \frac { x ^ { 3 } } { 24 }$ 2) $- \frac { x } { 2 } + \frac { x ^ { 3 } } { 24 }$ 3) $x - x ^ { 3 }$

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

Question 21

Find the coefficient of $x ^ { 3 }$ in the Maclaurin series for $f ( x ) = \sin 2 x$ .

A) $- \frac { 2 } { 3 }$
B) $- \frac { 4 } { 3 }$
C) $\frac { 4 } { 3 }$
D) $- \frac { 8 } { 3 }$
E) $\frac { 2 } { 3 }$
F) $\frac { 8 } { 3 }$
G) $- \frac { 1 } { 3 }$
H) $\frac { 1 } { 3 }$

Question 22

Find the terms in the Maclaurin series for the function $f ( x ) = e ^ { - x }$ , as far as the term in $x ^ { 3 }$ .

A) $1 - x + \frac { 1 } { 2 } x ^ { 2 } - \frac { 1 } { 6 } x ^ { 3 }$
B) $1 + x + \frac { 1 } { 2 } x ^ { 2 } - \frac { 1 } { 6 } x ^ { 3 }$
C) $1 - x + x ^ { 2 } - x ^ { 3 }$
D) $1 + x + x ^ { 2 } + x ^ { 3 }$
E) $1 - x + \frac { 1 } { 2 } x ^ { 2 } - \frac { 1 } { 3 } x ^ { 3 }$
F) $1 + x + \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 3 } x ^ { 3 }$
G) $- x + x ^ { 3 }$
H) $x - x ^ { 3 }$

Question 23

(a) Find the third-order Taylor polynomial associated with .(b) Use the Taylor polynomial from part (a) to find an approximation of .(c) Compare the value you calculated in part (b) with your calculator's value for

Question 24

Find the third-degree Taylor polynomial of the function .

Question 25

Find the terms in the Maclaurin series for the function $f ( x ) = \ln ( 1 + x )$ , as far as the term in $x ^ { 3 }$ .

A) $1 - x + x ^ { 2 } - x ^ { 3 }$
B) $x - x ^ { 2 } + x ^ { 3 }$
C) $1 - x + \frac { 1 } { 2 } x ^ { 2 } - \frac { 1 } { 6 } x ^ { 3 }$
D) $x - \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 3 } x ^ { 3 }$
E) $1 + \frac { 1 } { 2 } x + \frac { 2 } { 3 } x ^ { 2 } + \frac { 5 } { 6 } x ^ { 3 }$
F) $x + \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 6 } x ^ { 3 }$
G) $1 + \frac { x } { 2 } + \frac { 1 } { 6 } x ^ { 2 } + \frac { 1 } { 24 } x ^ { 3 }$
H) $x - \frac { 1 } { 24 } x ^ { 2 } + \frac { 1 } { 120 } x ^ { 3 }$

Question 26

Find the third-degree Taylor polynomial centered at for . Use this result to approximate .

Question 27

Given ,
(a) calculate .(b) calculate .(c) calculate .

Question 28

Find the radius of convergence of the Maclaurin series for $f ( x ) = \frac { 1 } { 4 + x ^ { 2 } }$ .

A)1
B) $\frac { 1 } { 8 }$
C) $\infty$
D) $\frac { 1 } { 4 }$
E) $\frac { 1 } { 2 }$
F)4

G)8
H)2

Question 29

Find the coefficient of $x ^ { 4 }$ in the Maclaurin series for $f ( x ) = x \cos \left( x ^ { 3 } \right)$ .

A)16
B) $- \frac { 2 } { 3 }$
C)3
D)4
E)-16
F) $\frac { 2 } { 3 }$
G)-3
H)0

Question 30

Find the coefficient of $x ^ { 4 }$ in the Maclaurin series for $f ( x ) = e ^ { - 2 x }$ .

A)16
B) $- \frac { 2 } { 3 }$
C)3
D)4
E)-16
F) $\frac { 2 } { 3 }$
G)-3
H)0

Question 31

Use the 3rd-degree Taylor polynomial of about to approximate . Use the remainder term to give an upper bound for the error in this approximation.

Question 32

Find the coefficient of $x ^ { 5 }$ in the Maclaurin series for $f ( x ) = \int \cos \left( x ^ { 2 } \right) d x$ .Note: The series is unique except for the constant of integration.

A) $- \frac { 1 } { 10 }$
B) $\frac { 1 } { 15 }$
C) $- \frac { 1 } { 5 }$
D) $\frac { 2 } { 5 }$
E) $- \frac { 2 } { 5 }$
F) $- \frac { 1 } { 15 }$
G) $\frac { 1 } { 5 }$
H) $\frac { 1 } { 10 }$

Question 33

Write the Taylor polynomial at 0 of degree 4 for .

Question 34

Find the second-degree Taylor polynomial of the function .

Question 35

(a) Use series to compute correct to three decimal places.(b) Use integration by parts to compute .(c) Compare your answers in parts (a) and (b) above.

Question 36

Use series to compute correct to four decimal places.

Question 37

Find the third Taylor polynomial associated with . What is the remainder?

Question 38

Find the first four terms in the Maclaurin series for $f ( x ) = x e ^ { - x }$ .

A) $x - x ^ { 2 } + x ^ { 3 } - x ^ { 4 }$
B) $x - \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 3 } x ^ { 3 } - \frac { 1 } { 4 } x ^ { 4 }$
C) $x - x ^ { 2 } + \frac { 1 } { 2 } x ^ { 3 } - \frac { 1 } { 6 } x ^ { 4 }$
D) $x - 2 x ^ { 2 } + 3 x ^ { 3 } - 4 x ^ { 4 }$
E) $x + \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 6 } x ^ { 3 } + \frac { 1 } { 24 } x ^ { 4 }$
F) $x + x ^ { 2 } + \frac { 1 } { 3 } x ^ { 3 } + \frac { 1 } { 8 } x ^ { 4 }$
G) $x + x ^ { 2 } + \frac { 1 } { 2 } x ^ { 3 } + \frac { 1 } { 6 } x ^ { 4 }$
H) $\frac { 1 } { 2 } x - \frac { 1 } { 6 } x ^ { 2 } + \frac { 1 } { 24 } x ^ { 3 } - \frac { 1 } { 120 } x ^ { 4 }$

Question 39

The first three derivatives of are , and .(a) Give the first four terms of the Taylor series associated with f at .(b) Give the second-order Taylor polynomial, , associated with f at .(c) Suppose that and that from part (b) is used to approximate . Prove that the error in this approximation does not exceed .

Question 40

Find the third-degree Taylor polynomial of the function .

Question 41

Give the Taylor series expansion of about the point .

Question 42

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

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 43

(a) Express as a Maclaurin series.(b) Evaluate as a series.

Question 44

Express as a Maclaurin Series.

Question 45

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

A) $1 - x + x ^ { 2 } - x ^ { 3 }$
B) $1 - \frac { 1 } { 2 } x + \frac { 1 } { 4 } x ^ { 2 } + \frac { 1 } { 8 } x ^ { 3 }$
C) $1 + \frac { 1 } { 2 } x + \frac { 3 } { 8 } x ^ { 2 } + \frac { 5 } { 16 } x ^ { 3 }$
D) $1 - \frac { 1 } { 2 } x + \frac { 3 } { 4 } x ^ { 2 } - \frac { 5 } { 8 } x ^ { 3 }$
E) $1 + \frac { 1 } { 2 } x + \frac { 1 } { 4 } x ^ { 2 } + \frac { 1 } { 6 } x ^ { 3 }$
F) $1 - \frac { 1 } { 2 } x + \frac { 1 } { 6 } x ^ { 2 } + \frac { 1 } { 24 } x ^ { 3 }$
G) $1 + \frac { 1 } { 2 } x + \frac { 3 } { 4 } x ^ { 2 } + \frac { 15 } { 16 } x ^ { 3 }$
H) $1 - \frac { 1 } { 2 } x + \frac { 3 } { 8 } x ^ { 2 } + \frac { 7 } { 24 } x ^ { 3 }$

Question 46

(a) Express as a Maclaurin series.(b) Evaluate as a series.

Question 47

Find the coefficient of $x ^ { 3 }$ in the binomial series for $\frac { 1 } { ( 1 + x ) ^ { 4 } }$ .

A)6
B)20
C)-6
D)-10
E)-20
F)-12
G)10
H)12

Question 48

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

A) $1 - x + x ^ { 2 } - x ^ { 3 }$
B) $1 + x - \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 3 } x ^ { 3 }$
C) $1 - x + \frac { 3 } { 2 } x ^ { 2 } - \frac { 5 } { 2 } x ^ { 3 }$
D) $1 + x + 3 x ^ { 2 } + 5 x ^ { 3 }$
E) $1 - x + \frac { 3 } { 2 } x ^ { 2 } - \frac { 7 } { 3 } x ^ { 3 }$
F) $1 + x + \frac { 1 } { 2 } x ^ { 2 } + \frac { 7 } { 3 } x ^ { 3 }$
G) $1 - x + \frac { 5 } { 2 } x ^ { 2 } - \frac { 7 } { 3 } x ^ { 3 }$
H) $1 + x + \frac { 7 } { 2 } x ^ { 2 } + \frac { 11 } { 3 } x ^ { 3 }$

Question 49

Find the Taylor series for at 2.

Question 50

Express as a Maclaurin Series.

Question 51

Find the Maclaurin series expansion with for . Use this expansion to approximate .

Question 52

Find the Maclaurin series expansion for and determine the interval of convergence.

Question 53

Find the Taylor polynomial of degree 4 at 0 for the function defined by . Then compute the value of accurate to as many decimal places as the polynomial of degree 4 allows.

Question 54

Express as a Maclaurin Series.

Question 55

If the Maclaurin series for is , find .

Question 56

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

A)2
B)-1
C)1
D) $- \frac { 1 } { 2 }$
E) $\frac { 1 } { 2 }$
F) $- \sqrt { 2 }$
G)-2

H) $\sqrt { 2 }$

Question 57

How many coefficients in the binomial series expansion of $( 1 + x ) ^ { 7 }$ are divisible by 7?

A)0
B)5
C)7
D)3
E)2
F)6
G)1
H)4

Question 58

Find the coefficient of $x ^ { 3 }$ in the binomial series for $( 1 + x ) ^ { 5 }$ .

A)3
B)6
C)15
D)20
E)10
F)5
G)16
H)12

Question 59

Use the binomial series to expand the function $\sqrt { 4 + x }$ as a power series. Give the coefficient of $x ^ { 2 }$ in that series.

A) $- \frac { 1 } { 8 }$
B) $- \frac { 1 } { 32 }$
C) $- \frac { 1 } { 64 }$
D) $\frac { 1 } { 8 }$
E) $\frac { 1 } { 32 }$
F) $- \frac { 1 } { 16 }$
G) $\frac { 1 } { 64 }$
H) $\frac { 1 } { 16 }$

Question 60

Find the Taylor series for about the origin.

Question 61

Find the coefficient of in the Maclaurin series for .

Question 62

Use the binomial series to expand as a power series. State the radius of convergence.

Question 63

Find the terms in the power series expansion for the function , as far as the term in .

Question 64

Find the sum of the series .

Question 65

Find the sum of the series .

Question 66

Use the binomial series to expand as a power series. State the radius of convergence.

Question 67

Which of the following is the power series centered at $a = 0$ for $f ( x ) = \frac { x } { 1 + 4 x }$ ?
1) $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } 4 ^ { n } x ^ { n }$ 2) $\sum _ { n = 0 } ^ { \infty } 4 ^ { n } x ^ { n + 1 }$ 3) $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } 4 ^ { n } x ^ { n + 1 }$

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

Question 68

Which of the following is the power series centered at $a = 0$ for $f ( x ) = \frac { 1 } { x + 4 }$ ?
1) $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { 4 ^ { n } } x ^ { n }$ 2) $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { 4 ^ { n + 1 } } x ^ { n }$ 3) $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } } { 4 ^ { n } } x ^ { n - 1 }$

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

Question 69

Use the binomial series formula to obtain the Maclaurin series for .

Question 70

Which of the following is the power series centered at $a = 0$ for $f ( x ) = \frac { 1 } { 1 + 4 x }$ ?
1) $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } 4 ^ { n } x ^ { n }$ 2) $\sum _ { n = 0 } ^ { \infty } 4 ^ { n } x ^ { n }$ 3) $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { 4 ^ { n } } x ^ { n }$

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

Question 71

If , compute .

Question 72

Use the binomial series to expand as a power series. State the radius of convergence.

Question 73

Find the sum of the series .

Question 74

Let , compute .

Question 75

Find the terms of the Maclaurin series for , as far as the term in .

Question 76

Find the sum of the series .

Question 77

If , compute .

Question 78

Use the binomial series to expand as a power series. State the radius of convergence.

Question 79

Use the binomial series to expand as a power series. State the radius of convergence.

Question 80

Use the binomial series to expand as a power series. State the radius of convergence.