Deck 10: Series and Taylor Polynomials

Question 1

Find the limit of the following sequence. ​ ​

A)
B)
C)
D)
E) The limit does not exist.

Answer 1

Question 2

Find the limit of the sequence ​

A)
B) 1
C) 0
D)
E) The limit does not exist.

Answer 2

0

Question 3

​Write the first five terms of the sequence. ​ ​
​

A) ​
B) ​
C) ​
D) ​
E) ​

Answer 3

​

Question 4

Find the next three terms of the geometric sequence 8,40,200,.... ​​

A) ​232, 264, 296
B) ​1,000, 5,000, 25,000
C) ​360, 520, 680
D) ​205, 210, 215
E) 200,1,000,5,000

Question 5

Find the limit of the following sequence. ​ ​

A)
B)
C)
D)
E) The limit does not exist.

Question 6

Write an expression for the ​nth term of the sequence , , , ,.... ​​

A) ​
B) ​
C) ​
D) ​
E)

Question 7

​Give an example of a sequence that converges to ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 8

Find the limit of the following sequence. ​ ​

A)
B)
C)
D)
E) The limit does not exist.

Question 9

What are the next three terms in the arithmetic sequence -2,-6,-10,.... ​​

A) ​-14, -18, -22
B) ​2, 6, 10
C) ​56, -224, 896
D) ​-8, -10, -12
E) -10,-14,-18

Question 10

Find the limit of the following sequence. ​ ​

A)
B)
C)
D)
E) The limit does not exist.

Question 11

Write an expression for the ​nth term of the sequence 4,24,124,624,.... ​​

A) ​
B) ​
C) ​
D) ​
E)

Question 12

Find the limit of the following sequence. ​ ​

A)
B)
C)
D)
E) The limit does not exist.

Question 13

Find the limit of the following sequence. ​ ​

A)
B)
C)
D)
E) The limit does not exist.

Question 14

Write an expression for the ​nth term of the sequence , , , ,.... ​​

A) ​
B) ​
C) ​
D) ​
E)

Question 15

Write the first five terms of the sequence. ​ ​

A) ​
B) ​
C) ​
D) ​
E) ​none of the above

Question 16

Determine the convergence or divergence of the sequence If the sequence converges,use a symbolic algebra utility to find its limit. ​

A)
B)
C)
D)
E) The sequence diverges.

Question 17

Find the limit of the following sequence. ​ ​

A)
B)
C)
D)
E) The limit does not exist.

Question 18

Find the limit of the following sequence. ​ ​

A)
B)
C)
D)
E) The limit does not exist.

Question 19

Find the limit of the following sequence. ​ ​

A)
B)
C)
D)
E) The limit does not exist.

Question 20

​Consider the sequence whose nth term is given by where P is the principal, is the amount of compound interest after n months,and ​r is the annual percentage rate.Write the first four terms of the sequence for and Round your answer to two decimal places. ​

A) ​9,563.33, 9,637.22, 9,716.38, 9,755.88
B) ​​9,564.34, 9,637.22, 9,716.38, 9,789.25
C) ​​9,563.33, 9,627.09, 9,691.27, 9,755.88
D) ​​9,564.34, 9,637.22, 9,691.27, 9,755.88
E) ​​9,564.34,9,627.09,9,691.27,9,789.25

Question 21

​Determine the convergence or divergence of the series Use a symbolic algebra utility to verify your result. ​​

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

Question 22

​Determine the convergence or divergence of the p-series ​

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

Question 23

A ball is dropped from a height of 14 feet,and on each rebound it rises to its preceding height.Write an expression for the height of the nth rebound.

A)
B)
C)
D)
E)

Question 24

The annual spending by tourists in a resort city is 900 million dollars.Approximately 75% of that revenue is again spent in the resort city,and of that amount approximately 75% is again spent in the resort city.If this pattern continues,write the geometric series that gives the total amount of spending generated by the 900 million dollars (including the initial outlay of 900 million dollars)and find the sum of the series.

A) The geometric series is The sum of the series is $3600 million.
B) The geometric series is The sum of the series is $67,500 million.
C) The geometric series is The sum of the series is 3600 million.
D) The geometric series is The sum of the series is $67,500 million.
E) The geometric series is The sum of the series is $675 million.

Question 25

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

A) ​5 , , , ,
B) ​​5 , , , ,
C) ​​5 , , , ,
D) ​​5 , , , 321 ,
E) ​​5 , , , ,

Question 26

​Determine the convergence or divergence of the following series Use a symbolic algebra utility to verify your result. ​​

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

Question 27

​Determine the convergence or divergence of the p-series ​

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

Question 28

A factory is polluting a river such that at every mile down river from the factory an environmental expert finds 15% less pollutant than at the preceding mile.If the pollutant's concentration is 600 ppm (parts per million)at the factory,what is its concentration 12 miles down river? ​

A) 180 ppm
B) ​90 ppm
C) ​85.35 ppm
D) 705.88 ppm
E) ​100.41 ppm

Question 29

Express the value of the given repeating decimal as a fraction.[Hint: Write as an infinite series.] ​ ​​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 30

​Determine the convergence or divergence of the series Use a symbolic algebra utility to verify your result. ​​

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

Question 31

​Determine whether the series is a p-series. ​

A) ​ is not a p-series.
B) ​​ is a p-series.

Question 32

​Determine the convergence or divergence of the series Use a symbolic algebra utility to verify your result. ​​

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

Question 33

​Find the sum of the convergent series. ​
​ ​

A) ​16
B) ​8
C) ​12
D) ​4
E) ​3

Question 34

The repeating decimal is expressed as a geometric series Write the decimal as the ratio of two integers. ​​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 35

​Find the sum of the convergent series. ​
​ ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 36

A deposit of $800 is made each month in an account that earns 8.4% interest,compounded monthly.The balance in the account after n months is given by Find the balance after 20 years by computing the 240th term of the sequence.Round your answer to two decimal places. ​

A) $696,946.54
B) $1,018,546.54
C) $24,073.84
D) $343,954.07
E) $1,125.60

Question 37

A company produces a new product for which it estimates the annual sales to be 5000 units.Suppose that in any given year 10% of the units (regardless of age)will become inoperative.How many units will be in use after n years? ​

A)
B)
C)
D)
E)

Question 38

​Bouncing Ball.A ball dropped from a height of 39 feet bounces to 0.6666666667 of its former height with each bounce.Find the total vertical distance that the ball travels. ​

A) ​195 feet
B) ​82 feet
C) ​234 feet
D) ​111 feet
E) ​117 feet

Question 39

​You accept a job that pays a salary of $60,000 the first year.During the next 49 years,you will receive a 6% raise each year.What would be your total compensation over the 50-year period? Round your answer to the nearest integer. ​

A) ​$17,420,154
B) ​$1,000,000
C) ​$56,400
D) ​$360,000
E) ​$3,600

Question 40

​Determine the convergence or divergence of the series Use a symbolic algebra utility to verify your result. ​​

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

Question 41

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

A) ​-8
B) ​1
C) ​8
D) ​-1
E) ​0

Question 42

​Determine the convergence or divergence of the p-series ​

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

Question 43

​Determine the convergence or divergence of the following series. ​ ​

A) ​diverges
B) ​converges

Question 44

Find the radius of convergence of the series ​ ​

A) ​16
B) ​18
C) ​1
D) ​9
E) ​8

Question 45

Determine the convergence or divergence of the following series. ​

A) diverges
B) converges
C) inconclusive

Question 46

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

A) ​Diverges.
B) Converges.
C) ​Ratio Test is inconclusive.

Question 47

​Determine the convergence or divergence of the following series. ​ ​

A) ​converges
B) ​diverges

Question 48

​Write the first five terms of the power series ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 49

Determine the convergence or divergence of the p-series

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

Question 50

​​Determine the convergence or divergence of the following series. ​ ​

A) ​inconclusive
B) ​converges
C) ​diverges

Question 51

​Test the series for convergence for using any appropriate test. ​

A) ​converges
B) ​diverges

Question 52

​Find the radius of convergence of the series ​​

A) ​10
B) ​2
C) ​4
D) ​13
E) ​9

Question 53

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

A) ​0
B) ​10
C) ​20
D) ​100
E) ​

Question 54

​Approximate the sum of the convergent series using four terms.Estimate the maximum error of your approximation.Round your answer to four decimal places. ​
​

A) ​The approximate value is 1.0286.
The maximum error of your approximation is 0.0084.
B) ​​The approximate value is 1.0788.
The maximum error of your approximation is 0.0052.​
C) ​​The approximate value is 1.0108.
The maximum error of your approximation is 0.0217.​
D) ​​The approximate value is 2.0023.
The maximum error of your approximation is ​1.0023.
E) ​​​The approximate value is 1.0812.
The maximum error of your approximation is ​0.0069.

Question 55

​Determine the convergence or divergence of the following series. ​ ​

A) ​diverges
B) ​converges

Question 56

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

A) ​Diverges.
B) Converges.
C) ​Ratio Test is inconclusive.

Question 57

​Determine the convergence or divergence of the p-series ​

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

Question 58

​Determine the convergence or divergence of the following series. ​ ​

A) ​converges
B) ​diverges

Question 59

​Test the series for convergence or divergence using any appropriate test. ​

A) ​diverges
B) ​converges

Question 60

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

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

Question 61

​Find the third Taylor polynomial at for the given function. ​
​ ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 62

​Find the radius of convergence centered at for the following function. ​ ​

A) ​4
B) ​2
C) ​0
D) ​1
E) ​

Question 63

​Find the Taylor polynomials (centred at zero)of degree (a)1,(b)2,(c)3,and (d)4. ​
​ ​

A) ​ , , ,
B) ​​ , , ,
C) ​​ , , ,
D) ​​ , , ,
E) ​​ , , ,

Question 64

​Find the radius of convergence of the series ​

A) ​1
B) ​2
C) ​3
D) ​7
E) ​9

Question 65

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

A) ​
B) ​
C) ​
D) ​
E) ​

Question 66

​A Taylor polynomial approximation of is given below.Use a graphing utility to graph both functions. ​ ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 67

​Differentiate the series for to find the power series for the function ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 68

​Apply Taylor's Theorem to find the power series centered at for the function ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 69

​A Taylor polynomial approximation of is given below.Use a graphing utility to graph both functions. ​ ​

A) ​
B) ​
C) ​
​
D) ​
​
E) ​

Question 70

​Find the power series for the function using the power series for ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 71

​Find the radius of convergence centered at for the following function. ​ ​

A) ​4
B) ​2
C) ​0
D) ​1
E) ​

Question 72

​Use a symbolic differentiation utility to find the Taylor polynomials (centred at zero)of degrees (a)2,(b)4,(c)6,(d)8. ​ ​

A) ​ , , ,
B) ​​ , , ,
C) ​​ , , ,
D) ​​ , , ,
E) ​​ , , ,

Question 73

​Find the radius of convergence centered at for the following function. ​ ​

A) ​4
B) ​2
C) ​0
D) ​1
E) ​

Question 74

​Find the radius of convergence of where ​

A) ​1
B) ​∞
C) ​
D) ​8
E) ​16

Question 75

​Find the radius of convergence centered at for the following function. ​ ​

A) ​4
B) ​2
C) ​0
D) ​1
E) ​

Question 76

​Apply Taylor's Theorem to find the power series centered at for the function ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 77

Integrate the series for to find the power series for the function ​

A)
B)
C)
D)
E)

Question 78

​Find the power series for the function using the power series for ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 79

​A Taylor polynomial approximation of is given below.Use a graphing utility to graph both functions. ​ ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 80

​A Taylor polynomial approximation of is given below.Use a graphing utility to graph both functions. ​ ​

A) ​
B) ​
C) ​
D) ​
E) ​