Deck 9: Infinite Series

Question 1

Let be the Taylor expansion for​ about .Find the general term for and find .

Answer 1

​

Question 2

If ​,the first three terms of the Taylor series expansion about for are ​

A)​
B)
C)​
D)​

Answer 2

​

Question 3

​Which of the following series is conditionally convergent? ​

A)​
B)​
C)​
D)​

Answer 3

​

Question 4

​Let represent the Taylor Polynomial of degree n about for .If is used to approximate the value of ,then which of the following expressions is NOT less than ? ​

A)​
B)​
C)​
D)​

Question 5

Find a geometric power series for the function centered at 0. ​

A)
B)
C)
D)
E)

Question 6

The radius of convergence of the power series ​. ​

A)​3
B)​5
C)​4
D)​6

Question 7

The sixth degree term of the Taylor series expansion for about has coefficient​ ​

A)​
B)​
C)​
D)​

Question 8

Which of the following series diverge? ​ ​

A)​I only
B)​II only
C)​I and II only
D)​I and III only

Question 9

The Maclaurin series for ​.What is the Maclaurin series for ? ​

A)​
B)​
C)​
D)​

Question 10

For function ​ Using the Taylor series expansion for about ,the second degree estimate for is ​

A)​4.335
B)​4.750
C)​4.420
D)​5.240

Question 11

​The power series converges for which values of x? ​

A)​
B)​
C)​
D)​​ is any real number.

Question 12

Which of the following series is absolutely convergent? ​

A)​
B)​
C)​
D)​

Question 13

The Maclaurin series represents which expression below?​ ​

A)​
B)​
C)​
D)​

Question 14

Which of the following series is the power series expansion for ​.

A)​
B)​
C)​
D)​

Question 15

​What are all values of a for which the series converges? ​

A)​
B)​
C)​
D)​

Question 16

Find a geometric power series for the function centered at 0. ​

A)
B)
C)
D)
E)

Question 17

The Maclaurin series represents which function below? ​

A)​
B)​
C)​
D)​

Question 18

Which value is smaller, ​ Give a reason for your answer.

Question 19

Let be a function that is differentiable for all .The Taylor expansion for about is given by The first four nonzero terms of are given by
Show that converges for all .​​

Question 20

If the first five terms of the Taylor expansion for about are ,then ​

A)​
B)​
C)​
D)​

Question 21

Find a power series for the function centered at 0. ​

A)
B)
C)
D)
E)

Question 22

Find a power series for the function centered at 0. ​

A)
B)
C)
D)
E)

Question 23

Find a power series for the function centered at 0. ​

A)
B)
C)
D)
E)

Question 24

Find a power series for the function centered at 1. ​

A)
B)
C)
D)
E)

Question 25

Use the power series to determine a power series for the function . ​

A)
B)
C)
D)
E)

Question 26

Find the radius of convergence of the power series. ​ ​

A)
B)
C)8
D)∞
E)64

Question 27

Find a geometric power series for the function centered at 0, (i)by the technique shown in Examples 1 and 2 and (ii)by long division. ​ ​

A)
B)
C)
D)
E)

Question 28

Explain how to use the geometric series to find the series for the function . ​

A)replace x with
B)replace x with and multiply the series by

C)replace x with and divide the series by

D)replace x with and divide the series by

E)replace x with

Question 29

Use the power series to determine a power series centered at 0 for the function . .

A)
B)
C)
D)
E)

Question 30

Identify the interval of convergence of a power series . ​

A)
B)
C)
D)
E)

Question 31

Find the sum of the convergent series by using a well-known function.Round your answer to four decimal places. ​

A)0.1419
B)0.2450
C)0.8961
D)0.1651
E)0.1974

Question 32

Find the sum of the convergent series by using a well-known function.Round your answer to four decimal places. ​

A)0.3365
B)0.8755
C)1.6487
D)0.1823
E)1.6094

Question 33

State where the power series is centered. ​

A)0
B)-2
C)7
D)2
E)-7

Question 34

Identify the interval of convergence of a power series . ​

A)
B)
C)
D)
E)

Question 35

Use the series for to approximate the value of using .Round your answer to three decimal places. ​

A)0.245
B)0.195
C)0.328
D)0.161
E)0.281

Question 36

Find the interval of convergence of the power series.(Be sure to include a check for convergence at the endpoints of the interval. ) ​ ​

A)
B)
C)
D)
E)

Question 37

Find a power series for the function centered at 0. ​

A)
B)
C)
D)
E)

Question 38

Identify the interval of convergence of a power series . ​

A)
B)
C)
D)
E)

Question 39

Find the radius of convergence of the power series. ​ ​

A)
B)8
C)16
D)0
E)4

Question 40

Use the power series to determine a power series centered at 0 for the function . ​

A)
B)
C)
D)
E)

Question 41

Find the interval of convergence of the power series.(Be sure to include a check for convergence at the endpoints of the interval. ) ​ ​

A)
B)
C)
D)
E)

Question 42

Identify the graph of the first 10 terms of the sequence of partial sum of the series for . ​

A)​

B)​

C)​

D)​

E)​

Question 43

Consider the function given by .Find the interval of convergence for . ​

A)
B)
C)
D)
E)

Question 44

Consider the function given by .Find the interval of convergence for . ​

A)
B)
C)
D)
E)

Question 45

Write an equivalent series of the series with the index of summation beginning at . ​

A)
B)
C)
D)
E)

Question 46

Find the differential equation having the solution . ​

A)
B)
C)
D)
E)

Question 47

What is P​₁ a first-degree polynomial function whose value and slope agree with the value and slope of at ? ​

A)P​₁ is the tangent line to the curve at the point
.
B)P​₁ is the tangent line to the curve at the point
.
C)P​₁ is the tangent line to the curve at the point
.
D)P​₁ is the tangent line to the curve at the point
.
E)P​₁ is the tangent line to the curve at the point
.

Question 48

Find the Maclaurin polynomial of degree 4 for the function. ​ ​

A)
B)
C)
D)
E)

Question 49

Consider the function and its second-degree polynomial at .Compute the value of and Round your answer to four decimal places. ​

A)
B)
C)
D)
E)

Question 50

Consider the function given by .Find the interval of convergence for . ​

A)
B)
C)
D)
E)

Question 51

Find a first-degree polynomial function P₁ whose value and slope agree with the value and slope of at . ​

A)
B)
C)​
D)
E)

Question 52

Use a graphing utility to graph and P₁,a first-degree polynomial function whose value and slope agree with the value and slope of f at . ​

A)​

​
B)​

C)​

D)​

E)​ ​

Question 53

Find the interval of convergence of the power series.(Be sure to include a check for convergence at the endpoints of the interval. ) ​ ​

A)
B)
C)
D)
E)

Question 54

Consider the function given by .Find the interval of convergence for . ​

A)
B)
C)
D)
E)

Question 55

Find the interval of convergence of the power series .(Be sure to include a check for convergence at the endpoints of the interval. ) ​

A)
B)
C)
D)
E)

Question 56

Find a first-degree polynomial function P₁ whose value and slope agree with the value and slope of f at .What is P₁ called? ​ ​

A) ;tangent line to at

B) ;secant line to at

C) ;tangent line to at

D) ;differential of at

E) ;tangent line to at

Question 57

Find the interval of convergence of the power series .(Be sure to include a check for convergence at the endpoints of the interval. ) ​

A)
B)
C)
D)
E)

Question 58

Write an equivalent series of the series with the index of summation beginning at . ​

A)
B)
C)
D)
E)

Question 59

Find the interval of convergence of the power series.(Be sure to include a check for convergence at the endpoints of the interval. ) ​ ​

A)
B)
C)
D)
E)

Question 60

Find a first-degree polynomial function P₁ whose value and slope agree with the value and slope of f at .What is P₁ called? ​ ​

A) tangent line to at

B) secant line to ​
C) secant line to
D) secant line to
E) tangent line to

Question 61

Find the fourth degree Taylor polynomial centered at for the function. ​ ​

A)
B)
C)
D)
E)

Question 62

Use the Root Test to determine the convergence or divergence of the series . ​

A)converges
B)diverges

Question 63

Find the fourth degree Maclaurin polynomial for the function. ​ ​

A)
B)
C)
D)
E)

Question 64

Find the third Taylor polynomial for expanded about . ​

A)
B)
C)
D)
E)

Question 65

Use the Root Test to determine the convergence or divergence of the series. ​ ​

A)converges
B)diverges
C)Root Test inconclusive

Question 66

Determine the degree of the Maclaurin polynomial required for the error in the approximation of the function to be less than 0.001. ​

A)3
B)2
C)5
D)1
E)4

Question 67

Find the Maclaurin polynomial of degree two for the function . ​

A)
B)
C)
D)
E)

Question 68

Find the Maclaurin polynomial of degree 5 for the function. ​ ​

A)
B)
C)
D)
E)

Question 69

Determine the values of x for which the function can be replaced by the Taylor polynomial if the error cannot exceed 0.004.Round your answer to four decimal places. ​

A)
B)
C)
D)
E)

Question 70

Use the Ratio Test to determine the convergence or divergence of the series. ​ ​

A)Ratio Test inconclusive
B)converges
C)diverges

Question 71

Use the Root Test to determine the convergence or divergence of the series. ​ ​

A)Root Test inconclusive
B)converges
C)diverges

Question 72

Use the Root Test to determine the convergence or divergence of the series . ​

A)converges
B)diverges

Question 73

Use the Ratio Test to determine the convergence or divergence of the series . ​

A)converges
B)diverges

Question 74

Use the Ratio Test to determine the convergence or divergence of the series. ​ ​

A)converges
B)diverges
C)Ratio Test inconclusive

Question 75

Use the Root Test to determine the convergence or divergence of the series. ​ ​

A)Root Test inconclusive
B)converges
C)diverges

Question 76

Find the third degree Taylor polynomial centered at for the function. ​ ​

A)
B)
C)
D)
E)

Question 77

Use the Ratio Test to determine the convergence or divergence of the series . ​

A)diverges
B)converges

Question 78

Find the Maclaurin polynomial of degree 3 for the function. ​ ​

A)
B)
C)
D)
E)

Question 79

Use the Ratio Test to determine the convergence or divergence of the series. ​ ​

A)converges
B)Ratio Test inconclusive
C)diverges

Question 80

Find the Maclaurin polynomial of degree 4 for the function. ​ ​

A)
B)
C)
D)
E)