Deck 9: Sequences and Series

Question 1

Determine whether the sequence $6+2.5 \times(-1)^{n}$ converges or diverges.

A)It converges.
B)It diverges.

Answer 1

It diverges.

Question 2

A radioactive isotope is released into the air as an industrial by-product.This isotope is not very stable due to radioactive decay.Two-thirds of the original radioactive material loses its radioactivity after each month.If 13 grams of this isotope are released into the atmosphere at the end of the first and every subsequent month and the situation goes on ad infinitum, how many grams of radioactive material are in the atmosphere at the end of each month in the long run?

Answer 2

19.5

Question 3

Stock prices for Abercrombie and Fitch fell steadily by an average of $0.94 per day from a high of $83.67 per share on December 24, 2007 to $70.05 on January 15, 2008.Let be the price of a share of stock on December 24, 2007.Write a formula for , the price of a share on the day after December 24.

Answer 3

Question 4

Let $P_{n}$ be the number of people visiting an amusement park on the n^th day after it opens.Suppose $P_{n}=2000-4 n$ .How many people visit the amusement park in its first week?

A)13888
B)13916
C)224
D)14000

Question 5

Let $P_{n}$ be the number of people visiting a zoo on the n^th day after it opens.Suppose $P_{n}=2000-4 n$ You find out that the museum must close if is has fewer than 300 visitors per day.How long will it remain open?

A)It will be closed on day 426.
B)It will be closed on day 501.
C)It will be closed on day 7.
D)It will close on day 1997.

Question 6

Find the value of to 2 decimal places.

Question 7

A radioactive isotope is released into the air as an industrial by-product.This isotope is not very stable due to radioactive decay.Two-thirds of the original radioactive material loses its radioactivity after each month.If 15 grams of this isotope are released into the atmosphere at the end of the first and every subsequent month, how many grams of radioactive material are in the atmosphere at the end of the twelfth month? Round to 2 decimal places.

Question 8

Let $P_{n}$ be the number of people visiting an amusement park on the n^th day after it opens.What does $P_{10}$ represent?

A)The average number of people visiting the amusement park over the first 10 days it was open.
B)The number of days it takes for 10 people to visit the amusement park.
C)The number of people visiting the amusement park on the 10^th day after it opens.
D)The total number of people who have visited the amusement park the first 10 days it was open.

Question 9

Let $P_{n}$ be the number of people visiting a zoo on the n^th day after it opens.What does $\sum_{n=1}^{25} P_{n}$ represent?

A)The average number of people visiting the zoo over the first 25 days it was open.
B)The number of days it takes for 25 people to visit the zoo.
C)The number of people visiting the zoo on the 25^th day it is open.
D)The total number of people who visited the zoo in the first 25 days after it opened.

Question 10

Find the value of the infinite product to 2 decimal places.

Question 11

Find a formula for $s_{n}$ , n $\ge$ 1, for the sequence $7,-\frac{7}{2}, \frac{7}{3},-\frac{7}{4}, \frac{7}{5}$

A) $s_{n}=\frac{7(-1)^{n+1}}{n}$
B) $s_{n}=\frac{7(-1)^{n}}{n}$
C) $s_{n}=\frac{7(-1)^{n+1}}{n+1}$
D) $s_{n}=\frac{7(-1)^{n}}{n+1}$

Question 12

Let $P_{n}$ be the number of people visiting an aquarium on the n^th day after it opens.What does it mean in terms of aquarium attendance if $P_{n}$ > $P_{n+1}$ for all n?

A)The number of people visiting the aquarium goes down each day.
B)The number of people visiting the aquarium goes up each day.
C)The number of people visiting the aquarium goes up some days and down some days.

Question 13

Select the appropriate word to fill in the blank: A convergent sequence is ________________.

A)bounded
B)monotonic
C)finite

Question 14

A couple puts $500,000 for their retirement into an account paying 5% annual interest.They estimate that they will need to withdraw $60,000 each year to live on.Assume that the $60,000 is withdrawn on the last day of the year.Find a recursive formula for , the amount of money left in the account at the end of n years, and use it to determine how many years the money will last (how many years until there is less than $60,000 in the account).

Question 15

A radioactive isotope is released into the air as an industrial by-product.This isotope is not very stable due to radioactive decay.Two-thirds of the original radioactive material loses its radioactivity after each month.If 12 grams of this isotope are released into the atmosphere at the end of the first and every subsequent month, identify the closed form sum that gives the amount of the isotope in the atmosphere at the end of the n^th month.

A) $S_{n}=\frac{12-\left(\frac{1}{3}\right)^{n} \cdot 12}{1-\frac{1}{3}}$
B) $S_{n}=\frac{12+\left(\frac{1}{3}\right)^{n} \cdot 12}{\frac{1}{3}}$
C) $S_{n}=\frac{12 \cdot\left(\frac{1}{3}\right)^{n}}{1-\frac{1}{3}}$

Question 16

Compute the first 8 terms of the sequence on plot them on a number line.To what number does it appear the sequence converges, if any?

Question 17

Stock prices for Abercrombie and Fitch fell steadily by an average of $0.94 per day from a high of $83.67 per share on December 24, 2007 to $70.05 on January 15, 2008.Let be the price of a share of stock on December 24, 2007 and be the price of a share on the day after December 24.Write a formula for and then write a sentence to interpret the meaning of your formula.

Question 18

Consider the finite sequence $A_{n}$ given in the graph below.Find $\sum_{n=1}^{5} 3 A_{n}$ .

A)60
B)15
C)75
D)56
E)None of the above

Question 19

Which one of the following sequences diverges to positive infinity as n $\rightarrow$ $\infty$ ?

A) $s_{n}=1+\frac{1}{n}$
B) $s_{n}=\frac{1-n^{2}}{n}$
C) $s_{n}=\frac{\cos n \pi}{n}$
D) $s_{n}=n+n \cos n \pi$

Question 20

Find a recursive formula for $s_{n}$ , n $\ge$ 1, for the sequence 1, 6, 41, 286, 2001, ...

A) $s_{n}=7-s_{n-1}$ with $s_{1}=1$
B) $s_{n}=7 s_{n-1}-1$ with $s_{1}=1$
C) $s_{n}=7-s_{n-1}$ with $s_{1}=7$
D) $s_{n}=7 s_{n-1}-1$ with $s_{1}=7$

Question 21

Does the infinite series converge or diverge?

Question 22

Find the sum .Round to 2 decimal places.

Question 23

Use the integral test to decide whether the series $\sum_{n=1}^{\infty} \frac{(\ln n)^{3}}{n}$ converges or diverges.

A)It converges
B)It diverges

Question 24

Suppose the government spends $3.5 million on highways.Some of this money is earned by the highway workers who in turn spend $1,750,000 on food, travel, and entertainment.This causes $875,000 to be spent by the people who work in the food, travel, and entertainment industries.This $875,000 causes another $437,500 to be spent; the $437,500 causes another $218,750 to be spent, and so on.(Notice that each expenditure is half the previous one.)Assuming that this process continues forever, how many million dollars in total spending is generated by the original $3.5 million expenditure (including the original $3.5 million)?

Question 25

Is a geometric series?

Question 26

Jamie was born in May.In August, her grandparents started a "Go to College in France" fund with $2200, earning a fixed annual interest rate of 7%.They added an additional $2200 each year in August until the last deposit in the year Jamie turned 18.Jamie estimated that she needed $90,000 to go start college in France.How much did she have in her "Go to College in France" fund? Did she have enough?

A) $\sum_{n=0}^{18} 2200(1.07)^{n}=\$ 82233.72$ , no
B) $\sum_{n=0}^{18} 2200(1.07)^{n}=\$ 74797.87$ , no
C)$91533.97, yes
D)$87533.83, no
E) $\sum_{n=0}^{18} 2200(1.07)^{n+2}=\$ 74797.87$ , no

Question 27

Does converge or diverge?

Question 28

Which of the following series are geometric? (1) $5+5 a+5 a^{2}+5 a^{3}+\ldots$
(2) $5+7 a+9 a^{2}+11 a^{3}+\ldots$
(3) $5+5 a k+5 a^{2} k^{2}+5 a^{3} k^{3}+\ldots$

A)(1)and (2)
B)(1)and (3)
C)(2)and (3)
D)(1)only
E)(2)only

Question 29

Does the series converge or diverge?

Question 30

Find the 6th partial sum of the series $\sum_{i=0}^{\infty}\left(\frac{5}{3}\right)^{i}$ (to two decimal places).

A)26.12
B) $\infty$
C)83.28
D)248.66

Question 31

A ball is dropped from a height of 18 feet and bounces.Each bounce is of the height of the bounce before.Find the total vertical feet the ball has traveled when it hits the floor for the 4^th time.Round to 1 decimal place.

Question 32

A tennis ball is dropped from a height of 15 feet and bounces.Each bounce is $\frac{1}{2}$ the height of the bounce before.A superball has a bounce $\frac{3}{4}$ the height of the bounce before, and is dropped from a height of 5 feet.Which ball bounces a greater total vertical distance?

A)The tennis ball
B)The superball

Question 33

Find the sum of the first 6 terms of the series .Round to 2 decimal places.

Question 34

Find the sum of the series to 2 decimal places.

Question 35

Does the series converge or diverge?

Question 36

A ball is dropped from a height of 11 feet and bounces.Each bounce is $\frac{2}{3}$ of the height of the bounce before.Find an expression for the height to which the ball rises after it hits the floor for the n^th time.

A) $11\left(\frac{2}{3}\right) \cdot n$
B) $11\left(\frac{2}{3}\right)^{n}$
C) $11\left(1-\left(\frac{2}{3}\right)^{n}\right)$
D) $\frac{11\left(1-\left(\frac{2}{3}\right)^{n}\right)}{1-\frac{2}{3}}$

Question 37

Use the integral test, if applicable, to determine whether the series $\sum_{n=1}^{\infty} \frac{n+2}{n^{2}+n}$ converges or diverges.

A)It converges.
B)It diverges.
C)The integral test does not apply.

Question 38

Use the integral test to decide whether the series converges or diverges.

Question 39

Does the series converge or diverge.Explain.

Question 40

Consider the series:
(a)Find a formula for the general term .
(b)Find the partial sums .
(c)Use your result from part (b)to predict the limit of the partial sums, .Does this indicate that the series converges or diverges?

Question 41

If $\sum_{n=1}^{\infty} a_{n}$ converges then $\sum_{n=1}^{\infty} k a_{n}$ converges (k $\neq$ 0).

Question 42

Determine whether the following series converge or diverge:
a) $\sum_{n=1}^{\infty} \frac{12}{n^{12}}$
b) $\sum_{n=1}^{\infty} \frac{\sqrt{n}}{12 n}$

A)(a)converges and (b)diverges
B)(a)diverges and (b)converges
C)both series diverge
D)both series converge

Question 43

Find the interval of convergence for $\sum_{n=0}^{\infty} \frac{x^{n}}{\sqrt{n+7}}$ .

A) $-7<$$x<1$
B) $-1 \leq x<1$
C) $-1 \leq x<7$
D) $-7<$$x<7$

Question 44

If a power series $\sum a_{k} x^{k}$ converges at x = c then it also converges at x = -c.

Question 45

Use the alternating series test to decide if $\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{8 n^{4}}$ converges.

A)It converges.
B)It diverges.

Question 46

If a power series $\sum a_{k} x^{k}$ converges at x = 6 and x = 7 then it converges at x = -6.

Question 47

Use the comparison test to determine whether $\sum_{n=1}^{\infty} \frac{1}{5 n^{2}+e^{n}}$ converges.

A)It converges.
B)It diverges.

Question 48

The harmonic series $\sum \frac{1}{n}$ diverges.We can form a new series from the difference between consecutive terms of the harmonic series obtaining $\sum\left[\frac{1}{n+1}-\frac{1}{n}\right]$ . This series also diverges.

Question 49

Use the limit comparison test to determine whether $\sum_{n=1}^{\infty} \frac{7 n^{5}}{n^{6}+n^{2}+2}$ converges.

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

Question 50

What does the ratio test tell us about the series $\sum_{n=1}^{\infty} \frac{1}{n^{3}}$ ?

A)The series converges.
B)The series diverges.
C)The ratio test doesn't tell us anything about the convergence of the series.

Question 51

Estimate the error in approximating the sum of the alternating series $S=\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{1+2^{n}}$ by the sum of the first 11 terms.

A)0.000024
B)0.00024
C)0.0024
D)0.024

Question 52

If the series has a radius of convergence of 4 and has a radius of convergence of 6, what is the radius of convergence of ?

Question 53

If a series of constants diverges, then must diverge?

Question 54

Leonard Euler found the following series to be noteworthy because it is part of a process that can be used to approximate :
(a)Re-write this series in summation notation.
(b)Use an appropriate test to show that the series converges.

Question 55

If $\sum a_{k}$ is the sum of a series of numbers and $\lim _{k \rightarrow \infty} \alpha_{k}=0$ , then the series converges.

Question 56

Find an expression for the general term of the series $\frac{x}{5}+\frac{x^{2}}{12}+\frac{x^{3}}{31}+\frac{x^{4}}{68}+\cdots$

A) $\frac{x^{n}}{n+4}$ for $n \geq 1$ .
B) $\frac{x^{n}}{7 n-2}$ for $n \geq 1$ .
C) $\frac{x^{n}}{n^{3}+4}$ for $n \geq 1$ .
D) $\frac{x^{n}}{n^{2}+12}$ for $n \geq 1$ .

Question 57

What does the ratio test tell us about the series $\sum_{n=1}^{\infty} \frac{n}{2^{n}}$ ?

A)The series converges.
B)The series diverges.
C)The ratio test doesn't tell us anything about the convergence of the series.

Question 58

Use the ratio test to find the radius of convergence of .

Question 59

If $\lim _{n \rightarrow \infty} a_{n}$ is not zero, then $\sum_{n=1}^{\infty} a_{n}$ does not converge.

Question 60

What does the ratio test tell us about the series $\sum_{n=1}^{\infty} \frac{(-0.125)^{n-1}}{\sqrt{n}}$ ?

A)The series diverges.
B)The series converges.
C)The ratio test doesn't tell us anything about the convergence of the series.

Question 61

If the power series $\sum C_{n} x^{n}$ converges for $x=a, a>0$ , then it converges for $x=\frac{a}{2}$ .

Question 62

Find the radius of convergence of .

Question 63

Does converge?

Question 64

If $\sum C_{n} 6^{n}$ is convergent, then $\sum C_{n}(-6)^{n}$ is also convergent.

Question 65

Find the radius of convergence of

Question 66

The ratio test can be used to determine whether $\sum_{n=1}^{\infty} \frac{1}{n^{2}}$ converges.

Question 67

Find the interval of convergence of the series

Question 68

Does converge?

Question 69

For what values of p does the series $1+9 p+(9 p)^{2}+(9 p)^{3}+\ldots+(9 p)^{n}+$ converge, if any?

A) $\frac{-1}{9}$$<$$p<$$\frac{1}{9}$
B) $\frac{-1}{18}$ $<$$p<$$\frac{1}{18}$
C)-1 < p < 1
D) $-1 \leq p \leq 1$
E)The series diverges for all values of p.

Question 70

If $a_{n}>a_{n+1}>0$ (for all n)and $\sum a_{n}$ converges, then $\sum(-1)^{n} a_{n}$ converges.