Deck 9: Infinite Series

Question 1

Use the Direct Comparison Test (if possible) to determine whether the series

A) diverges
B) converges

Answer 1

B

Question 2

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

A)
B)
C)
D)
E)

Answer 2

E

Question 3

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

A)
B)
C)
D)
E)

Answer 3

E

Question 4

Identify the most appropriate test to be used to determine whether the series converges or diverges.

A) Ratio Test
B) ρ-Series Test
C) Alternating Series Test
D) Telescoping Series Test
E) Root Test

Question 5

Determine the convergence or divergence of the series using any appropriate test from this chapter. Identify the test used.

A) converges; Integral Test
B) converges; Ratio Test
C) converges; Alternating Series Test
D) diverges; Ratio Test
E) diverges; Integral Test

Question 6

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

A) converges
B) diverges

Question 7

True or false: The series converges.

Question 8

Find the Maclaurin polynomial of degree 5 for the function.

A)
B)
C)
D)
E)

Question 9

Determine the convergence or divergence of the series using any appropriate test from this chapter. Identify the test used. ​ ​

A) diverges; Ratio Test
B) converges; both p-series and Integral Test
C) converges; p-series
D) converges; Integral Test
E) diverges; p-series

Question 10

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 11

Use the Direct Comparison Test to determine the convergence or divergence of the series .

A) The series diverges.
B) The series converges.

Question 12

Match the sequence with its graph.

A)
B)
C)
D)
E)

Question 13

Approximate the sum of the series by using the first six terms.

A) 0.723 < S < 0.743
B) 0.683 < S < 0.763
C) 0.733 < S < 0.738
D) 0.693 < S < 0.733
E) 0.73 < S < 0.736

Question 14

Match the sequence with its graph.

A)
B)
C)
D)
E)

Question 15

Match the sequence with its graph.

A)
B)
C)
D)
E)

Question 16

Use the definition to find the Taylor series centered at for the function .

A)
B)
C)
D)
E)

Question 17

Find all values of x for which the series converges. For these values of x, write the sum of the series as a function of x. ​ ​

A)
B)
C)
D)
E) The series diverges for all x.​

Question 18

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

A) Root Test inconclusive
B) diverges
C) converges

Question 19

True or false. The series is convergent.

Question 20

Write the first three terms of the sequence.

A)
B)
C)
D)
E)

Question 21

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

A)
B)
C)
D)
E)

Question 22

Determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit.

A) The sequence converges to 0.
B) The sequence diverges to -2.
C) The sequence converges to 1.
D) The sequence converges to -1.
E) The sequence diverges.

Question 23

Find a power series for the function centered at 0.

A)
B)
C)
D)
E)

Question 24

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

A)
B)
C)
D)
E)

Question 25

Approximate the sum of the series by using the first six terms.

A) 0.783 < S < 4.630
B) 2.066 < S < 3.348
C) 1.745 < S < 3.669
D) 0.569 < S < 4.844
E) 0.302 < S < 5.111

Question 26

Write the first five terms of the sequence of partial sums.

A)
B)
C)
D)
E)

Question 27

Determine the convergence or divergence of the series.

A) cannot be determined from the methods in the chapter
B) Diverges
C) Converges

Question 28

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

A)
B)
C)
D)
E)

Question 29

If the rate of inflation is per year and the average price of a car is currently $35,000, the average price after n years is . Compute the average price after 8 years. Round your answer to two decimal places.

A) $22,259.86
B) $53,714.03
C) $295,400.00
D) $46,685.40
E) $264,600.00

Question 30

Use the polynomial test to determine whether the series converges or diverges.

A) The series converges.
B) The series diverges.

Question 31

A government program that currently costs taxpayers $3.5 billion per year is cut back by 40 percent per year. Compute the budget after the fourth year. Round your answer to the nearest integer.

A) $1,433,600,000
B) $165,995,062
C) $89,600,000
D) $1,012,217,284
E) $453,600,000

Question 32

Determine the convergence or divergence of the series.

A) diverges
B) converges
C) cannot be determined

Question 33

Determine whether the series converges conditionally or absolutely, or diverges.

A) The series converges conditionally but does not converge absolutely.
B) The series converges absolutely but does not converge conditionally.
C) The series diverges.
D) The series converges absolutely.

Question 34

Consider the sequence whose nth term is given by where P is the principal, is the account balance in dollars after n months, and r is the interest rate compounded annually. Find the sixth term of the sequence if and . Round your answer to two decimal places. ​

A) $9,518.26
B) $10,061.16
C) $9,696.45
D) $9,342.82
E) $9,412.67

Question 35

Determine whether the series converges absolutely, converges conditionally, or diverges.

A) converges conditionally
B) diverges
C) converges absolutely

Question 36

Suppose the winner of a $10,000,000 sweepstakes will be paid $200,000 per year for 50 years, starting a year from now. The money earns 6% interest per year. The present value of the winnings is Compute the present value using the formula for the nth partial sum of a geometric series. Round your answer to two decimal places.

A) $10,542,883.62
B) $3,323,509.25
C) $7,028,589.08
D) $2,556,671.23
E) $3,152,372.13

Question 37

Identify the interval of convergence of a power series .

A)
B)
C)
D)
E)

Question 38

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

A) diverges
B) converges

Question 39

Approximate the sum of the series by using the first six terms.

A) 5.398 < S < 5.404
B) 5.393 < S < 5.405
C) 5.381 < S < 5.416
D) 5.391 < S < 5.391
E) 5.297 < S < 5.501

Question 40

Use a power series to approximate the value of the integral with an error less than 0.001. Round your answer to four decimal places.

A) 0.1063
B) 0.0900
C) 0.0982
D) 0.1071
E) 0.0985

Question 41

Write the next two apparent terms of the sequence .

A)
B)
C)
D)
E)

Question 42

Suppose the annual spending by tourists in a resort city is $100 million. Approximately 75% of that revenue is again spent in the resort city, and of that amount approximately 75% is again spent in the same city, and so on. Summing all of this spending indefinitely, leads to the geometric series . Find the sum of this series.

A) $800 million
B) $401 million
C) $200 million
D) $801 million
E) $400 million

Question 43

Write the first five terms of the sequence of partial sums.

A)
B)
C)
D)
E)

Question 44

True or false: The series converges.

Question 45

Sketch the graph of the sequence of partial sum of the series . ​

A)
B)
C)
D)
E)

Question 46

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 47

Use the Limit Comparison Test to determine the convergence or divergence of the series .

A) The series converges.
B) The series diverges.

Question 48

Determine the minimal number of terms required to approximate the sum of the series with an error of less than 0.007.

A) 5
B) 4
C) 2
D) 6
E) 3

Question 49

Find the sum of the convergent series.

A)
B)
C)
D)
E)

Question 50

Use the binomial series to find the Maclaurian series for the function. ​​

A) ​​
​
B) ​​
​
C) ​​
​
D) ​​
​
E) ​​

Question 51

Use the Limit Comparison Test (if possible) to determine whether the series converges or diverges. ​

A) diverges
B) converges

Question 52

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 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

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

A)
B)
C)
D)
E)

Question 55

Use the polynomial test to determine whether the series converges or diverges.

A) The series diverges.
B) The series converges.

Question 56

Use a power series to approximate the value of the integral with an error of less than 0.01. Round your answer to two decimal places. ​

A) 0.74
B) 0.89
C) 0.88
D) 0.84
E) 0.81

Question 57

Suppose an electronic games manufacturer producing a new product estimates the annual sales to be 7,000 units. Each year 25% of the units that have been sold will become inoperative. So, 7,000 units will be in use after 1 year, units will be in use after 2 years, and so on. How many units will be in use after n years? ​

A) units
B) units
C) units
D) units
E) units

Question 58

Use the Integral Test to determine the convergence or divergence of the series.

A) diverges
B) converges
C) Integral Test inconclusive

Question 59

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

A)
B)
C)
D)
E)

Question 60

State where the power series is centered.

A) 0
B) -3
C) 9
D) 3
E) -9

Question 61

Suppose you go to work at a company that pays $0.02 for the first day, $0.04 for the second day, $0.08 for the third day, and so on. If the daily wage keeps doubling, what would your total income be for working 29 days? Round your answer to two decimal places. ​

A) $5,368,709.12
B) $2,684,354.56
C) $10,737,418.22
D) $42,949,672.88
E) $21,474,836.44

Question 62

Use the Direct Comparison Test (if possible) to determine whether the series converges or diverges. ​

A) converges
B) diverges

Question 63

Use the binomial series to find the Maclaurian series for the function . ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 64

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 65

Find the Maclaurin series for the function .

A)
B)
C)
D)
E)

Question 66

Determine whether the series converges conditionally or absolutely, or diverges.

A) The series converges absolutely.
B) The series diverges.
C) The series converges absolutely but does not converge conditionally.
D) The series converges conditionally but does not converge absolutely.

Question 67

Find the Maclaurin polynomial of degree 4 for the function.

A)
B)
C)
D)
E)

Question 68

Find the radius of convergence of the power series.

A)
B)
C) 8
D)
E) 64

Question 69

Use the definition to find the Taylor series centered at for the function .

A)
B)
C)
D)
E)

Question 70

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

A)
B)
C)
D)
E)

Question 71

Use the Direct Comparison Test to determine the convergence or divergence of the series .

A) The series converges.
B) The series diverges.

Question 72

Determine the values of x for which the function can be replaced by the Taylor polynomial if the error cannot exceed 0.006. Round your answer to four decimal places.

A)
B)
C)
D)
E)

Question 73

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

A)
B)
C)
D)
E)

Question 74

Find the sum of the convergent series

A)
B)
C)
D)
E)

Question 75

Use Theorem 9.11 to determine the convergence or divergence of the series.

A) diverges
B) converges
C) Theorem 9.11 inconclusive

Question 76

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 77

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

A) converges
B) Ratio Test inconclusive
C) diverges

Question 78

The terms of a series are defined recursively. Determine the convergence or divergence of the series. Explain your reasoning. ,

A) diverges; Ratio Test
B) converges; Ratio Test
C) converges; Integral Test
D) diverges; Alternating Series Test
E) diverges; Root Test

Question 79

Identify the interval of convergence of a power series .

A)
B)
C)
D)
E)

Question 80

Find a power series for the function centered at 1.

A)
B)
C)
D)
E)