Calculus: Early Transcendentals

Michael Sullivan, Kathleen Miranda

Chapter 7

Sequences and Series - all with Video Answers

Educators

Section 1

Sequences

Problem 1

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.
(a) True or False: Every sequence is a function.
(b) True or False: The third term of the sequence $\{k+1\}_{k=1}^{\infty}$ is $4 .$
(c) True or False: The third term of the sequence $\left\{k^{2}\right\}_{k=2}^{\infty}$ is $9 .$
(d) True or False: Every sequence of real numbers is either increasing or decreasing.
(e) True or False: Every sequence of numbers has a smallest term.
(f) True or False: Every recursively defined sequence has an infinite number of distinct outputs.
(g) True or False: Every sequence has an upper bound, a lower bound, or both an upper bound and a lower bound.
(h) True or False: Every monotonic sequence has an upper bound, a lower bound, or both an upper bound and a lower bound.

Lucas Finney

Lucas Finney

Numerade Educator

Problem 2

Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.
(a) A strictly increasing sequence with an upper bound.
(b) A decreasing sequence without a lower bound.
(c) A sequence of real numbers that is neither increasing nor decreasing.

Lucas Finney

Numerade Educator

Problem 3

What is a sequence?

James Kiss

Numerade Educator

Problem 4

What is a term of a sequence?

James Kiss

Numerade Educator

Problem 5

What is meant by the index of a term of a sequence?

Lauren Shelton

Numerade Educator

Problem 6

Give a recursive definition for $K !$ for integers $k \geq 0 .$ Be sure you define $0 !$ as part of your answer.

Lauren Shelton

Numerade Educator

Problem 7

Give the first five terms of the following recursively defined sequence:
$$a_{1}=1, \text { and } a_{k}=a_{k-1}+2 \text { for } k \geq 2$$
Also, give a closed formula for the sequence.

James Kiss

Numerade Educator

Problem 8

Give the first five terms of the following recursively defined sequence:
$$a_{1}=2, \text { and } a_{k}=a_{k-1}+2 \text { for } k \geq 2$$
Also, give a closed formula for the sequence.

James Kiss

Numerade Educator

Problem 9

Give a recursive definition for the sequence $1,2,3,4, \ldots$ of positive integers. (Hint: Let $a_{1}=1$.)

James Kiss

Numerade Educator

Problem 10

The Lucas numbers are defined recursively as follows:
$$L_{1}=1, L_{2}=3, \text { and } L_{k}=L_{k-2}+L_{k-1} \text { for } k \geq 3$$
What are $L_{3}, L_{4}, L_{5}$, and $L_{6} ?$

James Kiss

Numerade Educator

Problem 11

Define what it means for a sequence $\left\{a_{k}\right\}$ to be eventually strictly increasing.

Lauren Shelton

Numerade Educator

Problem 12

Define what it means for a sequence $\left\{a_{k}\right\}$ to be eventually decreasing.

Lauren Shelton

Numerade Educator

Problem 13

Define what it means for a sequence $\left\{a_{k}\right\}$ to be eventually strictly decreasing.

Lauren Shelton

Numerade Educator

Problem 14

Define what it means for a sequence $\left\{a_{k}\right\}$ to be eventually monotonic.

Lauren Shelton

Numerade Educator

Problem 15

What does it mean for a sequence $\left\{a_{k}\right\}$ to be bounded above? Bounded below? Bounded?

Lauren Shelton

Numerade Educator

Problem 16

Explain why a sequence that is bounded above has infinitely many upper bounds.

Lauren Shelton

Numerade Educator

Problem 17

Give an example of a sequence with neither an upper bound nor a lower bound.

Lauren Shelton

Numerade Educator

Problem 18

Explain why every monotonic sequence has an upper bound, a lower bound, or both an upper bound and a lower bound.

Lauren Shelton

Numerade Educator

Problem 19

Explain why we require the terms of the sequence $\left\{a_{k}\right\}$ to be positive when we use the ratio test from Theorem $7.6$.

James Kiss

Numerade Educator

Problem 20

State a variation of the ratio test from Theorem $7.6$ that would allow you to use ratios to test a sequence $\left\{a_{k}\right\}$ for monotonicity when each $a_{k}<0$.

Lucas Finney

Numerade Educator

Problem 21

The Fibonacci numbers may be computed with the formula
$$f_{k}=\frac{(1+\sqrt{5})^{k}-(1-\sqrt{5})^{k}}{2^{k} \sqrt{5}}$$
Use this formula to compute $f_{1}, f_{2}, f_{3}, f_{4}$, and $f_{5}$. (Imagine computing $f_{100} .$ )

James Kiss

Numerade Educator

Problem 22

What is the least upper bound property for nonempty subsets of real numbers? Does the least upper bound property hold for subsets of the rational numbers? Does it hold for subsets of the integers?

Lucas Finney

Numerade Educator

Problem 23

Make a statement expressing a property analogous to the least upper bound property for nonempty subsets of real numbers that are bounded below.

Lucas Finney

Numerade Educator

Problem 24

Let $\left\{a_{k}\right\}$ be the sequence $a_{1}=3, a_{2}=3.1, a_{3}=3.14$, $a_{4}=3.141, \ldots$ That is, each term $a_{k}$ contains the first $k$ decimal digits of $\pi$.
(a) Explain why $a_{k}$ is a rational number for each positive integer $k$
(b) Explain why the sequence $\left\{a_{k}\right\}$ is increasing.
(c) Provide an upper bound for the sequence $\left\{a_{k}\right\}$.
(d) What is the least upper bound of the sequence $\left\{a_{k}\right\}$ ?
(e) Use this sequence to explain why the Least Upper Bound Axiom does not apply to the set of rational numbers.

Lucas Finney

Numerade Educator

Problem 25

Find a plausible formula for the genera term of the given sequence.
$$
\{0,1,0,1,0,1, \ldots\}
$$

Lauren Shelton

Numerade Educator

Problem 26

Find a plausible formula for the genera term of the given sequence.
$$
\{1,7,13,19,25, \ldots\}
$$

James Kiss

Numerade Educator

Problem 27

Find a plausible formula for the genera term of the given sequence.
$$
\left\{\frac{1}{3}, \frac{2}{9}, \frac{1}{9}, \frac{4}{81}, \frac{5}{243}, \ldots\right\}
$$

James Kiss

Numerade Educator

Problem 28

Find a plausible formula for the genera term of the given sequence.
$$
\left\{5,-\frac{5}{2}, \frac{5}{4},-\frac{5}{8}, \frac{5}{16}, \ldots\right\}
$$

Lauren Shelton

Numerade Educator

Problem 29

Find a plausible formula for the genera term of the given sequence.
$$
\left\{\frac{2}{3}, \frac{3}{5}, \frac{4}{7}, \frac{5}{9}, \ldots\right\}
$$

James Kiss

Numerade Educator

Problem 30

Find a plausible formula for the genera term of the given sequence.
$$
\left\{1, \frac{1}{2}, \frac{1}{6}, \frac{1}{24}, \frac{1}{120}, \ldots\right\}
$$

Lauren Shelton

Numerade Educator

Problem 31

Provide the first five terms of the given sequence. Unless specified, assume that the first term has index 1.
$$
\left\{\frac{1-(-1)^{k}}{k}\right\}
$$

James Kiss

Numerade Educator

Problem 32

Provide the first five terms of the given sequence. Unless specified, assume that the first term has index 1.
$$
\left\{\frac{n}{2 n+1}\right\}
$$

James Kiss

Numerade Educator

Problem 33

Provide the first five terms of the given sequence. Unless specified, assume that the first term has index 1.
$$
a_{k}=\frac{\cos (k x)}{x^{k}+k^{2}}
$$

Lauren Shelton

Numerade Educator

Problem 34

Provide the first five terms of the given sequence. Unless specified, assume that the first term has index 1.
$$
\left\{\frac{(-1)^{k-1} x^{2 k}}{2 k !}\right\}
$$

Lauren Shelton

Numerade Educator

Problem 35

Provide the first five terms of the given sequence. Unless specified, assume that the first term has index 1.
$$
\left\{\frac{\sqrt{k}}{k}\right\}
$$

James Kiss

Numerade Educator

Problem 36

Provide the first five terms of the given sequence. Unless specified, assume that the first term has index 1.
$$
\left\{\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots+\frac{1}{2^{k}}\right\}
$$

James Kiss

Numerade Educator

Problem 37

Find the least upper bound of the sequences in
$$
\left\{2-\frac{1}{k^{2}}\right\}
$$

Lauren Shelton

Numerade Educator

Problem 38

Find the least upper bound of the sequences in
$$
\left\{\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \ldots\right\}
$$

Lauren Shelton

Numerade Educator

Problem 39

Find the least upper bound of the sequences in
$$
\{-k\}
$$

Lucas Finney

Numerade Educator

Problem 40

Find the least upper bound of the sequences in
$$
\{0,1,0,1,0,1, \ldots\}
$$

Lauren Shelton

Numerade Educator

Problem 41

Find the least upper bound of the sequences in
$$
\left\{\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots+\frac{1}{2^{k}}\right\}
$$

James Kiss

Numerade Educator

Problem 42

Find the least upper bound of the sequences in
$$
\{2,2.7,2.71,2.718,2.7182, \ldots\}
$$

Lucas Finney

Numerade Educator

Problem 43

Give the first five terms for a geometric sequence $\left\{c r^{k}\right\}_{k=0}^{\infty}$ with the specified values of $c$ and $r$.
$$
c=3, r=\frac{1}{2}
$$

Lauren Shelton

Numerade Educator

Problem 44

Give the first five terms for a geometric sequence $\left\{c r^{k}\right\}_{k=0}^{\infty}$ with the specified values of $c$ and $r$.
$$
c=-2, r=-\frac{1}{3}
$$

Lauren Shelton

Numerade Educator

Problem 45

Give the first five terms for a geometric sequence $\left\{c r^{k}\right\}_{k=0}^{\infty}$ with the specified values of $c$ and $r$.
$$
c=-2, r=-3
$$

Lauren Shelton

Numerade Educator

Problem 46

Give the first five terms for a geometric sequence $\left\{c r^{k}\right\}_{k=0}^{\infty}$ with the specified values of $c$ and $r$.
$$
c=-1, r=-\frac{1}{2}
$$

Lauren Shelton

Numerade Educator

Problem 47

Write each of the arithmetic sequences in the form $\{c+d k\}_{k=0}^{\infty}$.
$$
-3,7,17,27, \ldots
$$

Lauren Shelton

Numerade Educator

Problem 48

Write each of the arithmetic sequences in the form $\{c+d k\}_{k=0}^{\infty}$.
$$
-6,-7.1,-8.2,-9.3, \ldots
$$

Lauren Shelton

Numerade Educator

Problem 49

Write each of the arithmetic sequences in the form $\{c+d k\}_{k=0}^{\infty}$.
$$
1,-1, \ldots
$$

Lauren Shelton

Numerade Educator

Problem 50

Write each of the arithmetic sequences in the form $\{c+d k\}_{k=0}^{\infty}$.
$$
5, \pi, \ldots
$$

Lucas Finney

Numerade Educator

Problem 51

Use the difference test in Theorem $7.6$ to analyze the monotonicity of the given sequence.
$$
\left\{k^{2}-5 k\right\}
$$

James Kiss

Numerade Educator

Problem 52

Use the difference test in Theorem $7.6$ to analyze the monotonicity of the given sequence.
$$
\{\sqrt{k}-\sqrt{k+1}\}
$$

Lucas Finney

Numerade Educator

Problem 53

Use the difference test in Theorem $7.6$ to analyze the monotonicity of the given sequence.
$$
\left\{\frac{k}{k+2}\right\}
$$

James Kiss

Numerade Educator

Problem 54

Use the difference test in Theorem $7.6$ to analyze the monotonicity of the given sequence.
$$
\left\{\frac{1}{k !}\right\}
$$

Lucas Finney

Numerade Educator

Problem 55

Use the ratio test in Theorem $7.6$ to analyze the monotonicity of the given sequence.
$$
\left\{\frac{k^{2}}{k !}\right\}
$$

Lucas Finney

Numerade Educator

Problem 56

Use the ratio test in Theorem $7.6$ to analyze the monotonicity of the given sequence.
$$
\left\{\sqrt{1-\frac{1}{k}}\right\}
$$

Lucas Finney

Numerade Educator

Problem 57

Use the ratio test in Theorem $7.6$ to analyze the monotonicity of the given sequence.
$$
\left\{\frac{3^{k}}{2 \cdot 4 \cdot 6 \cdots(2 k)}\right\}
$$

Lucas Finney

Numerade Educator

Problem 58

Use the ratio test in Theorem $7.6$ to analyze the monotonicity of the given sequence.
$$
\left\{\frac{(k !)^{2}}{(2 k) !}\right\}
$$

Lucas Finney

Numerade Educator

Problem 59

Use the derivative test in Theorem $7.6$ to analyze the monotonicity of the given sequence.
$$
\left\{\frac{\sqrt{k+1}}{k}\right\}
$$

Lucas Finney

Numerade Educator

Problem 60

Use the derivative test in Theorem $7.6$ to analyze the monotonicity of the given sequence.
$$
\left\{k^{2}-k\right\}
$$

Lucas Finney

Numerade Educator

Problem 61

Use the derivative test in Theorem $7.6$ to analyze the monotonicity of the given sequence.
$$
\left\{\frac{\sin k}{k}\right\}
$$

Lucas Finney

Numerade Educator

Problem 62

Use the derivative test in Theorem $7.6$ to analyze the monotonicity of the given sequence.
$$
\left\{\frac{k !}{(k+1) !}\right\}
$$

Lucas Finney

Numerade Educator

Problem 63

Determine whether the sequences in exercises are monotonic or not. Also determine whether the given sequence is bounded or unbounded.
$$
\left\{2-\frac{k-1}{10}\right\}
$$

Lucas Finney

Numerade Educator

Problem 64

Determine whether the sequences in exercises are monotonic or not. Also determine whether the given sequence is bounded or unbounded.
$$
\left\{\frac{1}{2 k}\right\}
$$

Lucas Finney

Numerade Educator

Problem 65

Determine whether the sequences in exercises are monotonic or not. Also determine whether the given sequence is bounded or unbounded.
$$
a_{k}=\frac{(-1)^{k}}{k}
$$

Lucas Finney

Numerade Educator

Problem 66

Determine whether the sequences in exercises are monotonic or not. Also determine whether the given sequence is bounded or unbounded.
$$
a_{k}=\frac{k^{2}}{k \mid}
$$

Lucas Finney

Numerade Educator

Problem 67

Determine whether the sequences in exercises are monotonic or not. Also determine whether the given sequence is bounded or unbounded.
$$
\{1,-1,1,-1,1, \ldots\}
$$

Lauren Shelton

Numerade Educator

Problem 68

Determine whether the sequences in exercises are monotonic or not. Also determine whether the given sequence is bounded or unbounded.
$$
\left\{1, \frac{3}{4}, \frac{8}{9}, \frac{15}{16}, \frac{24}{25}, \ldots\right\}
$$

Lauren Shelton

Numerade Educator

Problem 69

Determine whether the sequences in exercises are monotonic or not. Also determine whether the given sequence is bounded or unbounded.
$$
\left\{\frac{\cos k}{k}\right\}
$$

Lucas Finney

Numerade Educator

Problem 70

Determine whether the sequences in exercises are monotonic or not. Also determine whether the given sequence is bounded or unbounded.
$$
\left\{\frac{\cos (2 \pi k)}{k}\right\}
$$

Lucas Finney

Numerade Educator

Problem 71

Determine whether the sequences in exercises are monotonic or not. Also determine whether the given sequence is bounded or unbounded.
$$
\left\{\frac{e^{k}}{k !}\right\}
$$

Lucas Finney

Numerade Educator

Problem 72

Determine whether the sequences in exercises are monotonic or not. Also determine whether the given sequence is bounded or unbounded.
$$
\left\{\frac{(2 k) !}{(k !)^{2}}\right\}
$$

Lucas Finney

Numerade Educator

Problem 73

Determine whether the sequences in exercises are monotonic or not. Also determine whether the given sequence is bounded or unbounded.
$$
\left\{(-1)^{k} k\right\}
$$

Lucas Finney

Numerade Educator

Problem 74

Determine whether the sequences in exercises are monotonic or not. Also determine whether the given sequence is bounded or unbounded.
$$
\left\{5\left(1-\left(\frac{1}{10}\right)^{k}\right)\right\}
$$

Lucas Finney

Numerade Educator

Problem 75

Use Newton's method (see Example 8 ) to approximate a root for the given function with the specified value of $x_{0}$. Terminate your sequence when $\left|x_{n+1}-x_{n}\right|<0.001$.
$$
f(x)=x^{3}-2, x_{0}=1
$$

Lucas Finney

Numerade Educator

Problem 76

Use Newton's method (see Example 8 ) to approximate a root for the given function with the specified value of $x_{0}$. Terminate your sequence when $\left|x_{n+1}-x_{n}\right|<0.001$.
$$
f(x)=e^{x}+\sin x, x_{0}=0
$$

Lucas Finney

Numerade Educator

Problem 77

Use Newton's method (see Example 8 ) to approximate a root for the given function with the specified value of $x_{0}$. Terminate your sequence when $\left|x_{n+1}-x_{n}\right|<0.001$.
$$
f(x)=e^{x}+\sin x, x_{0}=-2
$$

Lucas Finney

Numerade Educator

Problem 78

Use Newton's method (see Example 8 ) to approximate a root for the given function with the specified value of $x_{0}$. Terminate your sequence when $\left|x_{n+1}-x_{n}\right|<0.001$.
$$
f(x)=\sqrt{x+1}-\frac{1}{x}, x_{0}=1
$$

Lucas Finney

Numerade Educator

Problem 79

Use Newton's method to derive the recursion formula
$$x_{k+1}=\frac{1}{2}\left(x_{k}+\frac{a}{x_{k}}\right)$$
for approximating $\sqrt{a} .$ (Hint: Let $f(x)=x^{2}-a$.)

Lucas Finney

Numerade Educator

Problem 80

Use the result of Exercise 79 to approximate the square roots in exercises. In each case, start with $x_{0}=1$ and stop when $\left|x_{k+1}-x_{k}\right|<0.001$.
$$
\sqrt{2}
$$

Lucas Finney

Numerade Educator

Problem 81

Use the result of Exercise 79 to approximate the square roots in exercises. In each case, start with $x_{0}=1$ and stop when $\left|x_{k+1}-x_{k}\right|<0.001$.
$$
\sqrt{3}
$$

Lucas Finney

Numerade Educator

Problem 82

Use the result of Exercise 79 to approximate the square roots in exercises. In each case, start with $x_{0}=1$ and stop when $\left|x_{k+1}-x_{k}\right|<0.001$.
$$
\sqrt{4}
$$

Lucas Finney

Numerade Educator

Problem 83

Use the result of Exercise 79 to approximate the square roots in exercises. In each case, start with $x_{0}=1$ and stop when $\left|x_{k+1}-x_{k}\right|<0.001$.
$$
\sqrt{101}
$$

Lucas Finney

Numerade Educator

Problem 84

Explain why Newton's method will fail if you choose a value of $x_{0}$ for which $f^{\prime}\left(x_{0}\right)=0$.

Lucas Finney

Numerade Educator

Problem 85

Newton's method will also fail when the difference of successive approximations, $\left|x_{k+1}-x_{k}\right|$, does not decrease as $k$ increases.
(a) Show that this happens for the function $f(x)=\sqrt[3]{x-2}$ when you choose $x_{0}=1$.
(b) What is the root of $f(x)=\sqrt[3]{x-2}$ ?

Lucas Finney

Numerade Educator

Problem 86

Use the result of Example 7 to approximate the levels of the drug Excellenté during the first week, assuming the dosages and decay rates in exercises.
$$
L_{1}=200, p=50
$$

Faizanullah Kazmi

Faizanullah Kazmi

Numerade Educator

Problem 87

Use the result of Example 7 to approximate the levels of the drug Excellenté during the first week, assuming the dosages and decay rates in exercises.
$$
L_{1}=100, p=25
$$

Faizanullah Kazmi

Faizanullah Kazmi

Numerade Educator

Problem 88

Use the result of Example 7 to approximate the levels of the drug Excellenté during the first week, assuming the dosages and decay rates in exercises.
$$
L_{1}=300, p=60
$$

Faizanullah Kazmi

Faizanullah Kazmi

Numerade Educator

Problem 89

Suppose you invest $\$ 100.00$ in a bank that pays you $5 \%$ interest compounded annually. The balance in the account after $k$ years is given by $a_{k}=100(1+0.05)^{k} .$ To the nearest cent, determine the first five terms of the sequence, starting at $k=0 .$ What does $k=0$ mean in practical terms? Determine whether the sequence is bounded. Determine whether the sequence is increasing, decreasing, or not monotonic.

Faizanullah Kazmi

Faizanullah Kazmi

Numerade Educator

Problem 90

Prove that the ratio of successive terms of a nonzero
geometric sequence is constant.

Lucas Finney

Numerade Educator

Problem 91

Prove that a sequence $\left\{a_{k}\right\}$ that is both increasing and decreasing is constant.

Lucas Finney

Numerade Educator

Problem 92

Prove that every sequence of the form $\left\{a_{k}\right\}_{k=n}^{\infty}$ can be rewritten as a sequence of the form $\left\{\alpha_{k}\right\}_{k=1}^{\infty}$.

Lucas Finney

Numerade Educator

Problem 93

Prove that if $\left\{a_{k}\right\}_{k=1}^{\infty}$ is a sequence of positive real numbers, then the sequence $\left {S_{n}\right\}_{n=1}^{\infty}$, where the sequence $S_{n}=a_{1}+a_{2}+\cdots+a_{n}$, is an increasing sequence.

Lucas Finney

Numerade Educator

Problem 94

Let $\left\{a_{k}\right\}$ be a sequence. Prove Theorem $7.6$ (a) along with the following variations:
(a) Show that when $a_{k+1}-a_{k} \geq 0$ for every $k \geq 1$, the sequence is increasing.
(b) Show that when $a_{k+1}-a_{k}>0$ for every $k \geq 1$, the sequence is strictly increasing.
(c) Show that when $a_{k+1}-a_{k} \leq 0$ for every $k \geq 1$, the sequence is decreasing.
(d) Show that when $a_{k+1}-a_{k}<0$ for every $k \geq 1$, the sequence is strictly decreasing.

Lucas Finney

Numerade Educator

Problem 95

Let $\left\{a_{k}\right\}$ be a sequence of positive terms. Prove Theorem $7.6$ (b) along with the following variations:
(a) Show that when $\frac{a_{k+1}}{a_{k}} \geq 1$ for every $k \geq 1$, the sequence is increasing.
(b) Show that when $\frac{a_{k+1}}{a_{k}}>1$ for every $k \geq 1$, the sequence is strictly increasing.
(c) Show that when $\frac{a_{k+1}}{a_{k}} \leq 1$ for every $k \geq 1$, the sequence is decreasing.
(d) Show that when $\frac{a_{k+1}}{a_{k}}<1$ for every $k \geq 1$, the sequence is strictly decreasing.

Lucas Finney

Numerade Educator

Problem 96

Let $a(x)$ be a differentiable function on the interval $[1, \infty)$, and let $a_{k}=a(k)$ for every positive integer $k$. Prove Theorem $7.6$ (c) along with the following variations:
(a) Show that when $a^{\prime}(x) \geq 0$ for $x>1$, the sequence $\left\{a_{k}\right\}$ is increasing.
(b) Show that when $a^{\prime}(x)>0$ for $x>1$, the sequence $\left\{a_{k}\right\}$ is strictly increasing.
(c) Show that when $a^{\prime}(x) \leq 0$ for $x>1$, the sequence $\left\{a_{k}\right\}$ is decreasing.
(d) Show that when $a^{\prime}(x)<0$, for $x>1$, the sequence $\left\{a_{k}\right\}$ is strictly decreasing.

Lucas Finney

Numerade Educator