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Calculus Single Variable

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

Chapter 9

Sequences and Series - all with Video Answers

Educators

Section 1

Sequences

00:53

Problem 1

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

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01:03

Problem 2

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

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Angela Guo
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01:01

Problem 3

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

Angela Guo
Angela Guo
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01:13

Problem 4

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
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01:20

Problem 5

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
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01:56

Problem 6

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
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01:37

Problem 7

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

Angela Guo
Angela Guo
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01:48

Problem 8

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

Angela Guo
Angela Guo
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01:02

Problem 9

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

Angela Guo
Angela Guo
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02:25

Problem 10

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

Angela Guo
Angela Guo
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01:05

Problem 11

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

Angela Guo
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00:52

Problem 12

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

Angela Guo
Angela Guo
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02:04

Problem 13

Match formulas (a)-(d) with graphs (I)-(IV).
(a) $\quad s_{n}=1-1 / n$
(b) $s_{n}=1+(-1)^{n} / n$
(c) $\quad s_{n}=1 / n$
(d) $s_{n}=1+1 / n$

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03:21

Problem 14

Match formulas (a)-(e) with graphs (I)-(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) $s_{n}=(-1)^{n} 2+(-1)^{n} / n$

Angela Guo
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02:43

Problem 15

Match formulas (a)-(e) with descriptions (I)-(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

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00:45

Problem 16

Do the sequences converge or diverge? If a sequence converges, find its limit.
$$2^{n}$$

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01:06

Problem 17

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

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00:49

Problem 18

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

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00:24

Problem 19

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

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01:23

Problem 20

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

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00:39

Problem 21

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

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01:00

Problem 22

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

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00:47

Problem 23

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

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00:52

Problem 24

Do the sequences converge or diverge? If a sequence converges, find its limit.
$$\frac{1}{n}+\ln n$$

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01:47

Problem 25

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

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01:45

Problem 26

Do the sequences converge or diverge? If a sequence converges, find its limit.
$$\frac{\sin n}{n}$$

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00:48

Problem 27

Do the sequences converge or diverge? If a sequence converges, find its limit.
$$\cos (\pi n)$$

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01:00

Problem 28

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

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00:51

Problem 29

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

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01:46

Problem 30

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

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01:27

Problem 31

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

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01:00

Problem 32

Let $a_{1}=8, b_{1}=5,$ and, for $n > 1,$
$$\begin{array}{l}
a_{n}=a_{n-1}+3 n \\
b_{n}=b_{n-1}+a_{n-1}.
\end{array}$$
Give the values of $a_{2}, a_{3}, a_{4}$

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01:15

Problem 33

Let $a_{1}=8, b_{1}=5,$ and, for $n > 1,$
$$\begin{array}{l}
a_{n}=a_{n-1}+3 n \\
b_{n}=b_{n-1}+a_{n-1}.
\end{array}$$
Give the values of $b_{2}, b_{3}, b_{4}, b_{5}$.

Angela Guo
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02:00

Problem 34

Let $a_{1}=8, b_{1}=5,$ and, for $n > 1,$
$$\begin{array}{l}
a_{n}=a_{n-1}+3 n \\
b_{n}=b_{n-1}+a_{n-1}.
\end{array}$$
Suppose $s_{1}=0, s_{2}=0, s_{3}=1,$ and that $s_{n}=$ $s_{n-1}+s_{n-2}+s_{n-3}$ for $n \geq 4 .$ The members of the resulting sequence are called tribonacci numbers. ${ }^{3}$ Find
$s_{4}, s_{5}, \ldots, s_{10}$

Angela Guo
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00:50

Problem 35

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

Angela Guo
Angela Guo
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00:33

Problem 36

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

Angela Guo
Angela Guo
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00:44

Problem 37

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

Angela Guo
Angela Guo
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01:06

Problem 38

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

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Angela Guo
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01:02

Problem 39

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

Angela Guo
Angela Guo
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01:34

Problem 40

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
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01:03

Problem 41

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

Angela Guo
Angela Guo
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01:39

Problem 42

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
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02:01

Problem 43

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

Angela Guo
Angela Guo
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00:43

Problem 44

Use the formula for $s_{n}$ to give the third term of the sequence, $s_{3}$.
$$s_{n}=(-1)^{n} 2^{n-1} \cdot n^{2}$$

Angela Guo
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01:28

Problem 45

Use the formula for $s_{n}$ to give the third term of the sequence, $s_{3}$.
$$s_{n}=\sum_{k=0}^{n} 3 \cdot 2^{k}$$

Angela Guo
Angela Guo
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01:42

Problem 46

Use the formula for $s_{n}$ to give the third term of the sequence, $s_{3}$.
$$\begin{array}{l}
s_{n}=2+3 s_{n-1}, \text { where } \\
s_{0}=5
\end{array}$$

Angela Guo
Angela Guo
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00:50

Problem 47

Use the formula for $s_{n}$ to give the third term of the sequence, $s_{3}$.
$$s_{n}=\int_{1 / n}^{1} \frac{1}{x^{2}} d x$$

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01:37

Problem 48

Concern analog signals in electrical engineering, which are continuous functions $f(t),$ where $t$ is time. To digitize the signal, we sample $f(t)$ every $\Delta t$ to form the sequence $s_{n}=f(n \Delta t) .$ For example, if $f(t)=\sin t$ with $t$ in seconds, sampling $f$ every $1 / 10$ second produces the sequence $\sin (1 / 10), \sin (2 / 10), \sin (3 / 10), \ldots .$ Give the first 6 terms of a sampling of the signal every $\Delta t$ seconds.
$$f(t)=(t-1)^{2}, \Delta t=0.5$$

Angela Guo
Angela Guo
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02:24

Problem 49

Concern analog signals in electrical engineering, which are continuous functions $f(t),$ where $t$ is time. To digitize the signal, we sample $f(t)$ every $\Delta t$ to form the sequence $s_{n}=f(n \Delta t) .$ For example, if $f(t)=\sin t$ with $t$ in seconds, sampling $f$ every $1 / 10$ second produces the sequence $\sin (1 / 10), \sin (2 / 10), \sin (3 / 10), \ldots .$ Give the first 6 terms of a sampling of the signal every $\Delta t$ seconds.
$$f(t)=\cos 5 t, \Delta t=0.1$$

Angela Guo
Angela Guo
Numerade Educator
02:00

Problem 50

Concern analog signals in electrical engineering, which are continuous functions $f(t),$ where $t$ is time. To digitize the signal, we sample $f(t)$ every $\Delta t$ to form the sequence $s_{n}=f(n \Delta t) .$ For example, if $f(t)=\sin t$ with $t$ in seconds, sampling $f$ every $1 / 10$ second produces the sequence $\sin (1 / 10), \sin (2 / 10), \sin (3 / 10), \ldots .$ Give the first 6 terms of a sampling of the signal every $\Delta t$ seconds.
$$f(t)=\frac{\sin t}{t}, \Delta t=1$$

Angela Guo
Angela Guo
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02:55

Problem 51

We smooth a sequence, $s_{1}, s_{2}, s_{3}, \ldots,$ by replacing each term $s_{n}$ by $t_{n},$ the average of $s_{n}$ with its neighboring terms
\[t_{n}=\frac{\left(s_{n-1}+s_{n}+s_{n+1}\right)}{3} \text { for } n>1.\]
Start with $t_{1}=\left(s_{1}+s_{2}\right) / 2,$ since $s_{1}$ has only one neighbor. Smooth the given sequence once and then smooth the resulting sequence. What do you notice?
$$18,-18,18,-18,18,-18,18 \ldots$$

Angela Guo
Angela Guo
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02:26

Problem 52

We smooth a sequence, $s_{1}, s_{2}, s_{3}, \ldots,$ by replacing each term $s_{n}$ by $t_{n},$ the average of $s_{n}$ with its neighboring terms
\[t_{n}=\frac{\left(s_{n-1}+s_{n}+s_{n+1}\right)}{3} \text { for } n>1.\]
Start with $t_{1}=\left(s_{1}+s_{2}\right) / 2,$ since $s_{1}$ has only one neighbor. Smooth the given sequence once and then smooth the resulting sequence. What do you notice?
$$0,0,0,18,0,0,0,0 \ldots$$

Angela Guo
Angela Guo
Numerade Educator
02:29

Problem 53

We smooth a sequence, $s_{1}, s_{2}, s_{3}, \ldots,$ by replacing each term $s_{n}$ by $t_{n},$ the average of $s_{n}$ with its neighboring terms
\[t_{n}=\frac{\left(s_{n-1}+s_{n}+s_{n+1}\right)}{3} \text { for } n>1.\]
Start with $t_{1}=\left(s_{1}+s_{2}\right) / 2,$ since $s_{1}$ has only one neighbor. Smooth the given sequence once and then smooth the resulting sequence. What do you notice?

Angela Guo
Angela Guo
Numerade Educator
01:29

Problem 54

For the function $f$ 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 given, use a calculator to see if the sequence converges. [To obtain the terms of the sequence, push the function button repeatedly.]
$$f(x)=\cos x, a=0$$

Angela Guo
Angela Guo
Numerade Educator
01:20

Problem 55

For the function $f$ 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 given, use a calculator to see if the sequence converges. [To obtain the terms of the sequence, push the function button repeatedly.]
$$f(x)=e^{-x}, a=0$$

Angela Guo
Angela Guo
Numerade Educator
01:20

Problem 56

For the function $f$ 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 given, use a calculator to see if the sequence converges. [To obtain the terms of the sequence, push the function button repeatedly.]
$$f(x)=\sin x, a=1$$

Angela Guo
Angela Guo
Numerade Educator
01:14

Problem 57

For the function $f$ 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 given, use a calculator to see if the sequence converges. [To obtain the terms of the sequence, push the function button repeatedly.]
$$f(x)=\sqrt{x}, a=0.5$$

Angela Guo
Angela Guo
Numerade Educator
03:19

Problem 58

Let $V_{n}$ be the number of new SUVs sold in the US in month $n,$ where $n=1$ is January 2016 . 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 $\sum_{i=1}^{n} V_{i}$

Angela Guo
Angela Guo
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02:06

Problem 59

(a) Let $s_{n}$ be the number of ancestors a person has $n$ generations ago. (Your ancestors are your parents, grandparents, great-grandparents, etc.) 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?

Angela Guo
Angela Guo
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02:49

Problem 60

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
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04:56

Problem 61

(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
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08:06

Problem 62

You are deciding whether to buy a new or a two-yearold 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
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06:04

Problem 63

The Fibonacci sequence, first studied by the thirteenth century Italian mathematician Leonardo di Pisa, also known as Fibonacci, is defined recursively by
\[F_{n}=F_{n-1}+F_{n-2} \text { for } n>2 \text { and } F_{1}=1, F_{2}=1.\]
The Fibonacci 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 converge 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
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06:07

Problem 64

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
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01:17

Problem 65

Explain what is wrong with the statement.
An increasing sequence that converges to 0.

Angela Guo
Angela Guo
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00:54

Problem 66

Explain what is wrong with the statement.
A convergent sequence consists entirely of terms greater than $2,$ then the limit of the sequence is greater than 2 .

Angela Guo
Angela Guo
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01:07

Problem 67

Explain what is wrong with the statement.
If the convergent sequence has limit $L$ and $s_{n}<2$ for all $n,$ then $L< 2$ .

Angela Guo
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00:52

Problem 68

Give an example of:
An increasing sequence that converges to 0.

Angela Guo
Angela Guo
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00:35

Problem 69

Give an example of:
A monotone sequence that does not converge.

Angela Guo
Angela Guo
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00:42

Problem 70

Decide if the statements are true or false. Give an explanation for your answer.
You can tell if a sequence converges by looking at the first 1000 terms.

Angela Guo
Angela Guo
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00:55

Problem 71

Decide if the statements are true or false. Give an explanation for your answer.
If the terms $s_{n}$ of a convergent sequence are all positive then $\lim _{n \rightarrow \infty} s_{n}$ is positive.

Angela Guo
Angela Guo
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02:14

Problem 72

Decide if the statements are true or false. Give an explanation for your answer.
If the sequence $s_{n}$ of positive terms is unbounded, then the sequence has a term greater than a million.

Angela Guo
Angela Guo
Numerade Educator
01:54

Problem 73

Decide if the statements are true or false. Give an explanation for your answer.
If the sequence $s_{n}$ of positive terms is unbounded, then the sequence has an infinite number of terms greater than a million.

Angela Guo
Angela Guo
Numerade Educator
00:51

Problem 74

Decide if the statements are true or false. Give an explanation for your answer.
If a sequence $s_{n}$ is convergent, then the terms $s_{n}$ tend to zero as $n$ increases.

Angela Guo
Angela Guo
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00:42

Problem 75

Decide if the statements are true or false. Give an explanation for your answer.
A monotone sequence cannot have both positive and negative terms.

Angela Guo
Angela Guo
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01:28

Problem 76

Decide if the statements are true or false. Give an explanation for your answer.
If a monotone sequence of positive terms does not converge, then it has a term greater than a million.

Angela Guo
Angela Guo
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01:06

Problem 77

Decide if the statements are true or false. Give an explanation for your answer.
If all terms $s_{n}$ of a sequence are less than a million, then the sequence is bounded.

Angela Guo
Angela Guo
Numerade Educator
01:07

Problem 78

Decide if the statements are true or false. Give an explanation for your answer.
If a convergent sequence has $s_{n} \leq 5$ for all $n,$ then $\lim _{n \rightarrow \infty} s_{n} \leq 5$.

Angela Guo
Angela Guo
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03:31

Problem 79

Which of the sequences I-IV is monotone and bounded for $n \geq 1 ?$
I. $s_{n}=10-\frac{1}{n}$
II. $s_{n}=\frac{10 n+1}{n}$
III. $s_{n}=\cos n$
IV. $s_{n}=\ln n$
(a) I
(b) $\mathrm{I}$ and $\mathrm{II}$
(c) $\mathrm{II}$ and $\mathrm{IV}$
(d) I, II, and III

Angela Guo
Angela Guo
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