• Home
  • Textbooks
  • Calculus
  • Sequences and Series

Calculus

Deborah Hughes-Hallett, Andrew M. Gleason, William G. McCallum

Chapter 9

Sequences and Series - all with Video Answers

Educators

Section 1

Sequences

01:37

Problem 1

Find a formula for $s_{n}, n \geq 1.$
$$4,8,16,32,64, \dots$$

Angela Guo
Angela Guo
Numerade Educator
01:48

Problem 2

Find a formula for $s_{n}, n \geq 1.$
$$1,3,7,15,31, \dots$$

Angela Guo
Angela Guo
Numerade Educator
01:02

Problem 3

Find a formula for $s_{n}, n \geq 1.$
$$2,5,10,17,26, \dots$$

Angela Guo
Angela Guo
Numerade Educator
02:25

Problem 4

Find a formula for $s_{n}, n \geq 1.$
$$1,-3,5,-7,9, \dots$$

Angela Guo
Angela Guo
Numerade Educator
01:05

Problem 5

Find a formula for $s_{n}, n \geq 1.$
$$1 / 3,2 / 5,3 / 7,4 / 9,5 / 11, \ldots$$

Angela Guo
Angela Guo
Numerade Educator
00:52

Problem 6

Find a formula for $s_{n}, n \geq 1.$
$$1 / 2,-1 / 4,1 / 6,-1 / 8,1 / 10, \ldots$$

Angela Guo
Angela Guo
Numerade Educator
00:53

Problem 7

Find the first five terms of the sequence from the formula for $s_{n}, n \geq 1.$
$$2^{n}+1$$

Angela Guo
Angela Guo
Numerade Educator
01:03

Problem 8

Find the first five terms of the sequence from the formula for $s_{n}, n \geq 1.$
$$n+(-1)^{n}$$

Angela Guo
Angela Guo
Numerade Educator
02:22

Problem 9

Find the first five terms of the sequence from the formula for $s_{n}, n \geq 1.$
$$\frac{2 \pi}{2 \pi-1}$$

Farnood Ensan
Farnood Ensan
Numerade Educator
01:13

Problem 10

Find the first five terms of the sequence from the formula for $s_{n}, n \geq 1.$
$$(-1)^{n}\left(\frac{1}{2}\right)^{n}$$

Angela Guo
Angela Guo
Numerade Educator
01:20

Problem 11

Find the first five terms of the sequence from the formula for $s_{n}, n \geq 1.$
$$(-1)^{n+1}\left(\frac{1}{2}\right)^{n-1}$$

Angela Guo
Angela Guo
Numerade Educator
01:56

Problem 12

Find the first five terms of the sequence from the formula for $s_{n}, n \geq 1.$
$$\left(1-\frac{1}{n+1}\right)^{n+1}$$

Angela Guo
Angela Guo
Numerade Educator
02:06

Problem 13

Converge or diverge? If a sequence converges, find its limit.
$$2^{n}$$

Farnood Ensan
Farnood Ensan
Numerade Educator
02:33

Problem 14

Converge or diverge? If a sequence converges, find its limit.
$$(0.2)^{n}$$

Farnood Ensan
Farnood Ensan
Numerade Educator
01:15

Problem 15

Converge or diverge? If a sequence converges, find its limit.
$$3+e^{-2 n}$$

Farnood Ensan
Farnood Ensan
Numerade Educator
02:04

Problem 16

Converge or diverge? If a sequence converges, find its limit.
$$(-0.3)^{n}$$

Farnood Ensan
Farnood Ensan
Numerade Educator
01:15

Problem 17

Converge or diverge? If a sequence converges, find its limit.
$$\frac{n}{10}+\frac{10}{n}$$

Farnood Ensan
Farnood Ensan
Numerade Educator
01:45

Problem 18

Converge or diverge? If a sequence converges, find its limit.
$$\frac{2^{n}}{3^{n}}$$

Farnood Ensan
Farnood Ensan
Numerade Educator
01:27

Problem 19

Converge or diverge? If a sequence converges, find its limit.
$$\frac{2 n+1}{n}$$

Farnood Ensan
Farnood Ensan
Numerade Educator
01:05

Problem 20

Converge or diverge? If a sequence converges, find its limit.
$$\frac{(-1)^{n}}{n}$$

Farnood Ensan
Farnood Ensan
Numerade Educator
01:45

Problem 21

Converge or diverge? If a sequence converges, find its limit.
$$\frac{2^{n}}{n^{3}}$$

Farnood Ensan
Farnood Ensan
Numerade Educator
01:04

Problem 22

Converge or diverge? If a sequence converges, find its limit.
$$\frac{2 n+(-1)^{n} 5}{4 n-(-1)^{n} 3}$$

Farnood Ensan
Farnood Ensan
Numerade Educator
01:23

Problem 23

Converge or diverge? If a sequence converges, find its limit.
$$\frac{\sin \pi}{\pi}$$

Farnood Ensan
Farnood Ensan
Numerade Educator
01:18

Problem 24

Converge or diverge? If a sequence converges, find its limit.
$$\cos (\pi n)$$

Farnood Ensan
Farnood Ensan
Numerade Educator
02:06

Problem 25

Match formulas (a)-(d) with graphs ( 1 )-(1V).
(a) $\quad s_{n}=1-1 / n$
(b) $\quad s_{n}=1+(-1)^{n} / n$
(c) $s_{n}=1 / n$
(d) $s_{n}=1+1 / n$
(FIGURE CANNOT COPY)

Farnood Ensan
Farnood Ensan
Numerade Educator
08:00

Problem 26

Match formulas (a)-(e) with descriptions ( $1)-(\mathrm{V})$ of the behavior of the sequence as $n \rightarrow \infty$
(a) $s_{n}=n(n+1)-1$
(b) $s_{n}=1 /(n+1)$
(c) $s_{n}=1-n^{2}$
(d) $s_{n}=\cos (1 / n)$
(e) $s_{n}=(\sin n) / n$
(I) Diverges to $-\infty$
(II) Diverges to $+\infty$
(III) Converges to 0 through positive numbers
(IV) Converges to 1
(V) Converges to 0 through positive and negative numbers

Dorcas Attuabea Addo
Dorcas Attuabea Addo
Numerade Educator
14:17

Problem 27

Match formulas (a)-(e) with graphs ( 1 )-(V).
(a) $s_{n}=2-1 / n$
(b) $s_{n}=(-1)^{n} 2+1 / n$
(c) $s_{n}=2+(-1)^{n} / n$
(d) $s_{n}=2+1 / n$
(e) $s_{n}=(-1)^{n} 2+(-1)^{n} / n$
(FIGURE CANNOT COPY)

Dorcas Attuabea Addo
Dorcas Attuabea Addo
Numerade Educator
01:00

Problem 28

Find the first six terms of the recursively defined sequence.
$$s_{n}=2 s_{n-1}+3 \text { for } n>1 \text { and } s_{1}=1$$

Angela Guo
Angela Guo
Numerade Educator
00:51

Problem 29

Find the first six terms of the recursively defined sequence.
$$s_{n}=s_{n-1}+n \text { for } n>1 \text { and } s_{1}=1$$

Angela Guo
Angela Guo
Numerade Educator
01:46

Problem 30

Find the first six terms of the recursively defined sequence.
$$s_{n}=s_{n-1}+\left(\frac{1}{2}\right)^{n-1} \text { for } n>1 \text { and } s_{1}=0$$

Angela Guo
Angela Guo
Numerade Educator
01:27

Problem 31

Find the first six terms of the recursively defined sequence.
$$s_{n}=s_{n-1}+2 s_{n-2} \text { for } n>2 \text { and } s_{1}=1, s_{2}=5$$

Angela Guo
Angela Guo
Numerade Educator
01:55

Problem 32

In electrical enginecring, a continuous function like $f(t)=$ sin $t,$ where $t$ is time in seconds, is referred to as an analog signal. To digitize the signal, we sample $f(t)$ every $\Delta t$ seconds to form the sequence $s_{n}=f(n \Delta t) .$ For example, sampling $f$ every $1 / 10$ second produces the sequence $\sin (1 / 10), \sin (2 / 10), \sin (3 / 10), \ldots .$ In Problems $32-34$ give the first 6 terms of a sampling of the signal cvery $\Delta t$ seconds.
$$f(t)=(t-1)^{2}, \Delta t=0.5$$

Farnood Ensan
Farnood Ensan
Numerade Educator
02:16

Problem 33

In electrical enginecring, a continuous function like $f(t)=$ sin $t,$ where $t$ is time in seconds, is referred to as an analog signal. To digitize the signal, we sample $f(t)$ every $\Delta t$ seconds to form the sequence $s_{n}=f(n \Delta t) .$ For example, sampling $f$ every $1 / 10$ second produces the sequence $\sin (1 / 10), \sin (2 / 10), \sin (3 / 10), \ldots .$ In Problems $32-34$ give the first 6 terms of a sampling of the signal cvery $\Delta t$ seconds.
$$f(t)=\cos 5 t, \Delta t=0.1$$

Farnood Ensan
Farnood Ensan
Numerade Educator
02:13

Problem 34

In electrical enginecring, a continuous function like $f(t)=$ sin $t,$ where $t$ is time in seconds, is referred to as an analog signal. To digitize the signal, we sample $f(t)$ every $\Delta t$ seconds to form the sequence $s_{n}=f(n \Delta t) .$ For example, sampling $f$ every $1 / 10$ second produces the sequence $\sin (1 / 10), \sin (2 / 10), \sin (3 / 10), \ldots .$ In Problems $32-34$ give the first 6 terms of a sampling of the signal cvery $\Delta t$ seconds.
$$f(t)=\frac{\sin t}{t}, \Delta t=1$$

Farnood Ensan
Farnood Ensan
Numerade Educator
02:19

Problem 35

To smooth a sequence, $s_{1}, s_{2}, s_{3}, \ldots,$ we replace each term $s_{n}$ by $t_{n},$ the average of $s_{n}$ with its neighboring terms
$$ t_{n}=\frac{\left(s_{n-1}+s_{n}+s_{n+1}\right)}{3} \text { for } n>1 $$
We start with $t_{1}=\left(s_{1}+s_{2}\right) / 2,$ since $s_{1}$ has only one neighbor. For Problems $35-37$, smooth the sequence once and then smooth the resulting sequence. What do you notice?
$$18,-18,18,-18,18,-18,18 \ldots$$

Farnood Ensan
Farnood Ensan
Numerade Educator
01:50

Problem 36

To smooth a sequence, $s_{1}, s_{2}, s_{3}, \ldots,$ we replace each term $s_{n}$ by $t_{n},$ the average of $s_{n}$ with its neighboring terms
$$ t_{n}=\frac{\left(s_{n-1}+s_{n}+s_{n+1}\right)}{3} \text { for } n>1 $$
We start with $t_{1}=\left(s_{1}+s_{2}\right) / 2,$ since $s_{1}$ has only one neighbor. For Problems $35-37$, smooth the sequence once and then smooth the resulting sequence. What do you notice?
$$0,0,0,18,0,0,0,0 \dots$$

Farnood Ensan
Farnood Ensan
Numerade Educator
03:15

Problem 37

To smooth a sequence, $s_{1}, s_{2}, s_{3}, \ldots,$ we replace each term $s_{n}$ by $t_{n},$ the average of $s_{n}$ with its neighboring terms
$$ t_{n}=\frac{\left(s_{n-1}+s_{n}+s_{n+1}\right)}{3} \text { for } n>1 $$
We start with $t_{1}=\left(s_{1}+s_{2}\right) / 2,$ since $s_{1}$ has only one neighbor. For Problems $35-37$, smooth the sequence once and then smooth the resulting sequence. What do you notice?
$$1,2,3,4,5,6,7,8 \dots$$

Farnood Ensan
Farnood Ensan
Numerade Educator
00:50

Problem 38

Find a recursive definition for the sequence.
$$1,3,5,7,9, \dots$$

Angela Guo
Angela Guo
Numerade Educator
00:33

Problem 39

Find a recursive definition for the sequence.
$$2,4,6,8,10, \dots$$

Angela Guo
Angela Guo
Numerade Educator
00:44

Problem 40

Find a recursive definition for the sequence.
$$3,5,9,17,33, \dots$$

Angela Guo
Angela Guo
Numerade Educator
01:06

Problem 41

Find a recursive definition for the sequence.
$$1,5,14,30,55, \dots$$

Angela Guo
Angela Guo
Numerade Educator
01:02

Problem 42

Find a recursive definition for the sequence.
$$1,3,6,10,15, \dots$$

Angela Guo
Angela Guo
Numerade Educator
01:34

Problem 43

Find a recursive definition for the sequence.
$$1,2, \frac{3}{2}, \frac{5}{3}, \frac{8}{5}, \frac{13}{8}, \ldots$$

Angela Guo
Angela Guo
Numerade Educator
01:03

Problem 44

Show that the sequence $s_{n}$ satisfies the recurrence relation.
$$\begin{aligned} &s_{n}=3 n-2\\ &s_{n}=s_{n-1}+3 \text { for } n>1 \text { and } s_{1}=1 \end{aligned}$$

Angela Guo
Angela Guo
Numerade Educator
01:39

Problem 45

Show that the sequence $s_{n}$ satisfies the recurrence relation.
$$\begin{array}{l} s_{n}=n(n+1) / 2 \\ s_{n}=s_{n-1}+n \text { for } n>1 \text { and } s_{1}=1 \end{array}$$

Angela Guo
Angela Guo
Numerade Educator
02:17

Problem 46

Show that the sequence $s_{n}$ satisfies the recurrence relation.
$$\begin{aligned} &s_{n}=2 n^{2}-n\\ &x_{n}=s_{n-1}+4 n-3 \text { for } n>1 \text { and } s_{1}=1 \end{aligned}$$

Farnood Ensan
Farnood Ensan
Numerade Educator
02:46

Problem 47

Define a sequence recursively by $x_{n}=$ $f\left(x_{n-1}\right)$ for $n>1$ and $x_{1}=a .$ Depending on $f$ and the starting value $a$, this sequence may converge to a limit $L$. If $L$ exists, it has the property that $f(L)=L$. For the functions and starting values in Problems $47-50$, use a calculator to see if the sequence converges. [To obtain the terms of the sequence, repeatedly push the function button. $]$
$$f(x)=\cos x, a=0$$

Dorcas Attuabea Addo
Dorcas Attuabea Addo
Numerade Educator
02:19

Problem 48

Define a sequence recursively by $x_{n}=$ $f\left(x_{n-1}\right)$ for $n>1$ and $x_{1}=a .$ Depending on $f$ and the starting value $a$, this sequence may converge to a limit $L$. If $L$ exists, it has the property that $f(L)=L$. For the functions and starting values in Problems $47-50$, use a calculator to see if the sequence converges. [To obtain the terms of the sequence, repeatedly push the function button. $]$
$$f(x)=e^{-x}, a=0$$

Dorcas Attuabea Addo
Dorcas Attuabea Addo
Numerade Educator
03:03

Problem 49

Define a sequence recursively by $x_{n}=$ $f\left(x_{n-1}\right)$ for $n>1$ and $x_{1}=a .$ Depending on $f$ and the starting value $a$, this sequence may converge to a limit $L$. If $L$ exists, it has the property that $f(L)=L$. For the functions and starting values in Problems $47-50$, use a calculator to see if the sequence converges. [To obtain the terms of the sequence, repeatedly push the function button. $]$
$$f(x)=\sin x, a=1$$

Dorcas Attuabea Addo
Dorcas Attuabea Addo
Numerade Educator
02:10

Problem 50

Define a sequence recursively by $x_{n}=$ $f\left(x_{n-1}\right)$ for $n>1$ and $x_{1}=a .$ Depending on $f$ and the starting value $a$, this sequence may converge to a limit $L$. If $L$ exists, it has the property that $f(L)=L$. For the functions and starting values in Problems $47-50$, use a calculator to see if the sequence converges. [To obtain the terms of the sequence, repeatedly push the function button. $]$
$$f(x)=\sqrt{x}, a=0.5$$

Dorcas Attuabea Addo
Dorcas Attuabea Addo
Numerade Educator
01:36

Problem 51

Let $V_{n}$ be the number of new SUVs sold in the US in month $n,$ where $n=1$ is January 2004 . In terms of SUVs, what do the following represent?
(a) $V_{10}$
(b) $V_{n}-V_{n-1}$
(c) $\sum_{i=1}^{12} V_{i}$ and $\sum_{i=1}^{n} V_{i}$

Farnood Ensan
Farnood Ensan
Numerade Educator
02:06

Problem 52

(a) Let $s_{n}$ be the number of ancestors a person has $n$ generations ago. What is $s_{1} ? s_{2} ?$ Find a formula for $s_{n}$
(b) For which $n$ is $s_{n}$ greater than 6 billion, the current world population? What does this tell you about your ancestors?

Farnood Ensan
Farnood Ensan
Numerade Educator
02:49

Problem 53

For $1 \leq n \leq 10,$ find a formula for $p_{n},$ the payment in year $n$ on a loan of $\$ 100,000 .$ Interest is $5 \%$ per year, compounded annually, and payments are made at the end of each year for ten years. Each payment is $\$ 10,000$ plus the interest on the amount of money outstanding.

Angela Guo
Angela Guo
Numerade Educator
06:35

Problem 54

World oil consumption was 82.459 million barrels per day in 2005 and is increasing by about $1.3 \%$ per year. Let $c_{n}$ be daily world oil consumption $n$ years after 2005
(a) Find a formula for $c_{n}$
(b) Find and interpret $c_{n}-c_{n}-1$
(c) What does the sum $\sum_{n=1}^{18} 365 c_{n}$ represent? (You do
not need to compute this sum.)

Dorcas Attuabea Addo
Dorcas Attuabea Addo
Numerade Educator
04:56

Problem 55

(a) Cans are stacked in a triangle on a shelf. The bottom row contains $k$ cans, the row above contains one can fewer, and so on, until the top row, which has one can. How many rows are there? Find $a_{n},$ the number of cans in the $n^{\text {th }}$ row, $1 \leq n \leq k$ (where the top row is $n=1$ ).
(b) Let $T_{n}$ be the total number of cans in the top $n$ rows. Find a recurrence relation for $T_{n}$ in terms of $T_{n-1}$
(c) Show that $T_{n}=\frac{1}{2} n(n+1)$ satisfies the recurrence relation.

Angela Guo
Angela Guo
Numerade Educator
08:06

Problem 56

You are deciding whether to buy a new or a two-year-old car (of the same make) based on which will have cost you less when you resell it at the end of three years. Your cost consists of two parts: the loss in value of the car and the repairs. A new car costs $\$ 20,000$ and loses $12 \%$ of its value each year. Repairs are $\$ 400$ the first year and increase by $18 \%$ each subsequent year.
(a) For a new car, find the first three terms of the sequence $d_{n}$ giving the depreciation (loss of value) in dollars in year $n .$ Give a formula for $d_{n}$
(b) Find the first three terms of the sequence $r_{n},$ the repair cost in dollars for a new car in year $n$. Give a formula for $r_{n-}$
(c) Find the total cost of owning a new car for three years.
(d) Find the total cost of owning the two-year-old car for three years. Which should you buy?

Angela Guo
Angela Guo
Numerade Educator
06:04

Problem 57

The Fibonacci sequence first studied by the thirteenth century Italian mathematician Leonardo di Pisa, also known as Fibonacci, is defined recursively by
$F_{n}=F_{n-1}+F_{n-2}$ for $n>2$ and $F_{1}=1, F_{2}=1$
The Fibonacei sequence occurs in many branches of mathematics and can be found in patterns of plant growth (phyllotaxis).
(a) Find the first 12 terms.
(b) Show that the sequence of successive ratios $F_{n+1} / F_{n}$ appears to comverge to a number $r$ satisfying the equation $r^{2}=r+1 .$ (The number $r$ was known as the golden ratio to the ancient Greeks.)
(c) Let $r$ satisfy $r^{2}=r+1 .$ Show that the sequence $s_{n}=A r^{n},$ where $A$ is constant, satisfies the Fibonacci equation $s_{n}=s_{n-1}+s_{n-2}$ for $n>2.$

Angela Guo
Angela Guo
Numerade Educator
06:07

Problem 58

This problem defines the Calkin-Wilf-Newman sequence of positive rational numbers. The sequence is remarkable because every positive rational number appears as one of its terms and none appears more than once. Every real number $x$ can be written as an integer $A$ plus a number $B$ where $0 \leq B<1 .$ For example, for $x=12 / 5=2+2 / 5$ we have $A=2$ and $B=2 / 5 .$ For $x=3=3+0$ we have $A=3$ and $B=0 .$ Define the function $f(x)$ by
$$ f(x)=A+(1-B) $$
For example, $f(12 / 5)=2+(1-2 / 5)=13 / 5$ and $f(3)=3+(1-0)=4$
(a) Evaluate $f(x)$ for $x=25 / 8,13 / 9,$ and $\pi$
(b) Find the first six terms of the recursively defined Calkin-Wilf-Newman sequence: $s_{n}=1 / f\left(s_{n-1}\right)$ for $n>1$ and $s_{1}=1$

Angela Guo
Angela Guo
Numerade Educator
00:54

Problem 59

Write a definition for $\lim _{n \rightarrow+\infty} s_{n}=L$ similar to the $\epsilon, \delta$ definition for $\lim _{x \rightarrow a} f(x)=L$ in Section $1.8 .$ Instead of $\delta$ you will need $N,$ a value of $n$

Vincenzo Zaccaro
Vincenzo Zaccaro
Numerade Educator
02:01

Problem 60

The sequence $s_{n}$ is increasing, the sequence $t_{n}$ converges, and $s_{n} \leq t_{n}$ for all $n .$ Show that $s_{n}$ converges.

Farnood Ensan
Farnood Ensan
Numerade Educator