## Educators

GR
JS

### Problem 1

(a) What is a sequence?
(b) What does it mean to say that $\lim_{n \to \infty} a_n = 8?$
(c) What does it mean to say that $\lim_{n \to \infty} a_n = \infty?$

GR
Gabriel R.

### Problem 2

(a) What is a convergent sequence? Give two examples.
(b) What is a divergent sequence? Give two examples.

GR
Gabriel R.

### Problem 3

List the first five terms of the sequence.
$a_n = \frac {2^n}{2n + 1}$

GR
Gabriel R.

### Problem 4

.List the first five terms of the sequence.
$a_n = \frac {n^2 - 1}{n^2 + 1}$

GR
Gabriel R.

### Problem 5

List the first five terms of the sequence.
$a_n = \frac {(-1)^{n-1}}{5^n}$

GR
Gabriel R.

### Problem 6

List the first five terms of the sequence.
$$a_n = \cos {n \pi}{2}$$

GR
Gabriel R.

### Problem 7

List the first five terms of the sequence.
$a_n = \frac {1}{n + 1}!$

GR
Gabriel R.

### Problem 8

List the first five terms of the sequence.
$a_n = \frac {(-1)^nn}{n! + 1}$

GR
Gabriel R.

### Problem 9

List the first five terms of the sequence.
$a_1 = 1, a_{n+1} = 5a_n - 3$

GR
Gabriel R.

### Problem 10

List the first five terms of the sequence.
$a_1 = 6, a_{n+1} = \frac {a_n}{n}$

GR
Gabriel R.

### Problem 11

List the first five terms of the sequence.
$a_1 = 2, a_{n+1} = \frac {a_n}{1 + a_n}$

GR
Gabriel R.

### Problem 12

List the first five terms of the sequence.
$a_1 = 2, a_2 = 1, a_{a+1} = a_n - a_{n-1}$

GR
Gabriel R.

### Problem 13

Find a formula for the general term $a_n$ of the sequence, assuming that the pattern of the first few terms continues.
$\left\{\begin{array} \frac {1}{2}, \frac {1}{4}, \frac {1}{6}, \frac {1}{8}, \frac {1}{10}, . . . .\end{array}\right\}$

GR
Gabriel R.

### Problem 14

Find a formula for the general term $a_n$ of the sequence, assuming that the pattern of the first few terms continues.
$$\left\{\begin{array} 4, -1, \frac {1}{4}, - \frac {1}{16}, \frac {1}{64}, . . . . .\end{array}\right\}$$

GR
Gabriel R.

### Problem 15

Find a formula for the general term $a_n$ of the sequence, assuming that the pattern of the first few terms continues.
$\left\{ -3, 2, - \frac {4}{3}, {8}{9}, - \frac {16}{27}, . . .\right\}$

JS
Joseph S.

### Problem 16

Find a formula for the general term $a_n$ of the sequence, assuming that the pattern of the first few terms continues.
$\left\{\begin{array} 5, 8, 11, 14, 17, . . . . .\end{array}\right\}$

GR
Gabriel R.

### Problem 17

Find a formula for the general term $a_n$ of the sequence, assuming that the pattern of the first few terms continues.
$\left\{\begin{array} \frac {1}{2}, - \frac {4}{3}, \frac {9}{4}, - \frac {16}{5}, \frac {25}{6}, . . . . .\end{array}\right\}$

GR
Gabriel R.

### Problem 18

Find a formula for the general term $a_n$ of the sequence, assuming that the pattern of the first few terms continues.
$\left\{\begin{array} 1, 0, -1, 0, 1, 0, -1, 0, . . . .\end{array}\right\}$

GR
Gabriel R.

### Problem 19

Calculate, to four decimal places, the first ten terms of the sequence and use them to plot the graph of the sequence by hand. Does the sequence appear to have a limit? If so, calculate it. If not, explain why.
$a_n = \frac {3n}{1 + 6n}$

GR
Gabriel R.

### Problem 20

Calculate, to four decimal places, the first ten terms of the sequence and use them to plot the graph of the sequence by hand. Does the sequence appear to have a limit? If so, calculate it. If not, explain why.
$a_n = 2 + \frac {(-1)^n}{n}$

GR
Gabriel R.

### Problem 21

Calculate, to four decimal places, the first ten terms of the sequence and use them to plot the graph of the sequence by hand. Does the sequence appear to have a limit? If so, calculate it. If not, explain why.
$a_n = 1 + (- \frac {1}{2})^n$

GR
Gabriel R.

### Problem 22

Calculate, to four decimal places, the first ten terms of the sequence and use them to plot the graph of the sequence by hand. Does the sequence appear to have a limit? If so, calculate it. If not, explain why.
$a_n = 1 + \frac{10^n}{9^n}$

JS
Joseph S.

### Problem 23

Determine whether the sequence converges or diverges. If it converges, find the limit.
$a_n = \frac {3 + 5n^2}{n + n^2}$

GR
Gabriel R.

### Problem 24

Determine whether the sequence converges or diverges. If it converges, find the limit.
$a_n = \frac {3 + 5n^2}{1 + n}$

GR
Gabriel R.

### Problem 25

Determine whether the sequence converges or diverges. If it converges, find the limit.
$a_n = \frac {n^4}{n^3 - 2n}$

GR
Gabriel R.

### Problem 26

Determine whether the sequence converges or diverges. If it converges, find the limit.
$a_n = 2 + (0.86)^n$

GR
Gabriel R.

### Problem 27

Determine whether the sequence converges or diverges. If it converges, find the limit.
$a_n = 3^n 7^{-n}$

GR
Gabriel R.

### Problem 28

Determine whether the sequence converges or diverges. If it converges, find the limit.
$a_n = \frac {3 \sqrt {n}}{\sqrt {n} + 2}$

GR
Gabriel R.

### Problem 29

Determine whether the sequence converges or diverges. If it converges, find the limit.
$a_n = e^{-1/ \sqrt n}$

GR
Gabriel R.

### Problem 30

Determine whether the sequence converges or diverges. If it converges, find the limit.
$a_n = \sqrt { \frac {1 + 4n^2}{1 + n^2}}$

GR
Gabriel R.

### Problem 31

Determine whether the sequence converges or diverges. If it converges, find the limit.
$a_n = \frac {4^n}{1 + 9^n}$

GR
Gabriel R.

### Problem 32

Determine whether the sequence converges or diverges. If it converges, find the limit.
$a_n = \cos \left( \frac {n \pi}{n + 1} \right)$

GR
Gabriel R.

### Problem 33

Determine whether the sequence converges or diverges. If it converges, find the limit.

$a_n = \frac {n^2}{\sqrt {n^3 + 4n}}$

GR
Gabriel R.

### Problem 34

Determine whether the sequence converges or diverges. If it converges, find the limit.
$a_n = e^{2n/(n + 2)}$

GR
Gabriel R.

### Problem 35

Determine whether the sequence converges or diverges. If it converges, find the limit.
$a_n = \frac {(-1)^n}{2 \sqrt n}$

GR
Gabriel R.

### Problem 36

Determine whether the sequence converges or diverges. If it converges, find the limit.
$a_n = \frac {(-1)^{n + 1}n}{n + \sqrt n}$

GR
Gabriel R.

### Problem 37

Determine whether the sequence converges or diverges. If it converges, find the limit.
$\left \{ \frac {(2n - 1)!}{(2n + 1)!}\right \}$

GR
Gabriel R.

### Problem 38

Determine whether the sequence converges or diverges. If it converges, find the limit.
$\left \{ \frac {\ln n}{\ln 2n} \right \}$

GR
Gabriel R.

### Problem 39

Determine whether the sequence converges or diverges. If it converges, find the limit.
$\{ \sin n \}$

GR
Gabriel R.

### Problem 40

Determine whether the sequence converges or diverges. If it converges, find the limit.
$a_n = \frac {\tan^{-1}n}{n}$

GR
Gabriel R.

### Problem 41

Determine whether the sequence converges or diverges. If it converges, find the limit.
$\{ n^2e^{-n}\}$

GR
Gabriel R.

### Problem 42

Determine whether the sequence converges or diverges. If it converges, find the limit.
$a_n = \ln (n + 1) - \ln n$

GR
Gabriel R.

### Problem 43

Determine whether the sequence converges or diverges. If it converges, find the limit.
$a_n = \frac { \cos^2 n}{2^n}$

GR
Gabriel R.

### Problem 44

Determine whether the sequence converges or diverges. If it converges, find the limit.
$a_n = \sqrt [n]{2^{1 + 3n}}$

GR
Gabriel R.

### Problem 45

Determine whether the sequence converges or diverges. If it converges, find the limit.
$a_n = n \sin (1/n)$

GR
Gabriel R.

### Problem 46

Determine whether the sequence converges or diverges. If it converges, find the limit.
$a_n = 2^{-n} \cos n \pi$

GR
Gabriel R.

### Problem 47

Determine whether the sequence converges or diverges. If it converges, find the limit.
$a_n = \left( 1+ \frac {2}{n} \right)^n$

GR
Gabriel R.

### Problem 48

Determine whether the sequence converges or diverges. If it converges, find the limit.
$a_n = \sqrt[n]{n}$

GR
Gabriel R.

### Problem 49

Determine whether the sequence converges or diverges. If it converges, find the limit.
$a_n = \ln(2n^2 + 1) - \ln(n^2 + 1)$

GR
Gabriel R.

### Problem 50

Determine whether the sequence converges or diverges. If it converges, find the limit.
$a_n = \frac { (\ln n)^2}{n}$

GR
Gabriel R.

### Problem 51

Determine whether the sequence converges or diverges. If it converges, find the limit.
$a_n = \arctan (\ln n)$

GR
Gabriel R.

### Problem 52

Determine whether the sequence converges or diverges. If it converges, find the limit.
$a_n = n - \sqrt {n + 1} \sqrt {n + 3}$

GR
Gabriel R.

### Problem 53

Determine whether the sequence converges or diverges. If it converges, find the limit.
$\left \{ 0, 1, 0, 0, 1, 0, 0, 0, 1, . . . \right \}$

GR
Gabriel R.

### Problem 54

Determine whether the sequence converges or diverges. If it converges, find the limit.
$\left \{ \frac {1}{1}, \frac {1}{3}, \frac {1}{2}, \frac {1}{4}, \frac {1}{3}, \frac {1}{5}, \frac {1}{4}, \frac {1}{6}, . . . \right \}$

GR
Gabriel R.

### Problem 55

Determine whether the sequence converges or diverges. If it converges, find the limit.
$a_n = \frac {n!}{2^n}$

GR
Gabriel R.

### Problem 56

Determine whether the sequence converges or diverges. If it converges, find the limit.
$a_n = \frac {(-3)^n}{n!}$

GR
Gabriel R.

### Problem 57

Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. (See the margin note on page 699 for advice on graphing sequence .)
$a_n = (-1)^n \frac {n}{n + 1}$

JS
Joseph S.

### Problem 58

Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. (See the margin note on page 699 for advice on graphing sequence .)
$a_n = \frac { \sin n}{n}$

GR
Gabriel R.

### Problem 59

Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. (See the margin note on page 699 for advice on graphing sequence .)
$a_n = \arctan \left( \frac {n^2}{n^2 + 4} \right)$

GR
Gabriel R.

### Problem 60

Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. (See the margin note on page 699 for advice on graphing sequence .)
$a_n = \sqrt[n]{3^n + 5^n}$

GR
Gabriel R.

### Problem 61

Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. (See the margin note on page 699 for advice on graphing sequence .)
$a_n = \frac {n^2 \cos n}{1 + n^2}$

JS
Joseph S.

### Problem 62

Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. (See the margin note on page 699 for advice on graphing sequence .)
$a_n = \frac { 1 \cdot 3 \cdot 5 \cdot \cdot \cdot \cdot \cdot (2n - 1)}{n!}$

JS
Joseph S.

### Problem 63

Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. (See the margin note on page 699 for advice on graphing sequence .)
$a_n = \frac {1 \cdot 3 \cdot 5 \cdot \cdot \cdot \cdot \cdot (2n - 1)}{(2n)^n}$

JS
Joseph S.

### Problem 64

(a) Determine whether the sequence defined as follows is convergent or divergent:
$a_1 = 1$ $a_{n + 1} = 4 - a_n$ for $n \ge 1$

(b) What happens if the first term is $a_1 = 2$ ?

GR
Gabriel R.

If $\$ $1000 is invested at$ 6 \% $interest, compounded annually, then after$ n $years the investment is worth$ a_n = 1000(1.06)^n $dollars. (a) Find the first five terms of the sequence$ \{ a_n\}. $(b) Is the sequence convergent or divergent? Explain. JS Joseph S. Numerade Educator ### Problem 66 If you deposit$ \100 at the end of every month into an account that pays $3 \%$ interest per year compounded monthly, the amount of interest accumulated after $n$ months is given by the sequence
$I_n = 100 \left( \frac {1.0025^n - 1}{0.0025} - n\right)$
(a) Find the first six terms of the sequence.
(b) How much interest will you have earned after two years?

JS
Joseph S.

### Problem 67

A fish has 5000 catfish in his pond. The number of catfish by $8 \%$ per month and the farmer harvests 300 catfish per month.
(a) Show that the catfish population $P_n$ after $n$ months is given recursively by
$P_n = 1.08 P_{n-1} - 300$
$P_0 = 5000$
(b) How many catfish are in the pond after six months?

JS
Joseph S.

### Problem 68

Find the first 40 terms of the sequence defined by
$a_{n + 1} =\left\{ \begin{array}{ll} \frac{1}{2} a_n & \text{if } a_n \text{ is an even number} \\ 3a_n + 1 & \text{if } a_n \text{ is an odd number } \end{array} \right.$
and $a_1 = 11.$ Do the same if $a_1 = 25.$ Make a conjecture about this type of sequence.

JS
Joseph S.

### Problem 69

For what values of $r$ is the sequence $\left\{ nr^n \right\}$ convergent?

JS
Joseph S.

### Problem 70

(a) If $\left \{ a_n \right\}$ is convergent, show that
$\displaystyle\lim_{n\to\infty} a_{n+1} = \displaystyle\lim_{n\to\infty} a_n$
(b) A sequence $\left\{ a_n \right\}$ is defined by $a_1 = 1$ and $a_{n + 1} = 1/(1 + a_n)$ for $n \ge 1.$ Assuming that $\left\{ a_n \right\}$ is convergent, find its limit.

JS
Joseph S.

### Problem 71

Suppose you know that $\left\{ a_n \right\}$ is a decreasing sequence and all its terms lie between the numbers 5 and 8. Explain why the sequence has a limit. What can you say about the value of the limit?

JS
Joseph S.

### Problem 72

Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?
$a_n = \cos n$

JS
Joseph S.

### Problem 73

Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?
$a_n = \frac{1}{2n + 3}$

JS
Joseph S.

### Problem 74

Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?
$a_n = \frac{1 - n}{2 +n}$

JS
Joseph S.

### Problem 75

Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?
$a_n = n(-1)^n$

JS
Joseph S.

### Problem 76

Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?
$a_n = 2 + \frac{(-1)^n}{n}$

JS
Joseph S.

### Problem 77

Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?
$a_n = 3 - 2ne^{-n}$

JS
Joseph S.

### Problem 78

Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?
$a_n = n^3 - 3n + 3$

JS
Joseph S.

### Problem 79

Find the limit of the sequence
$\left\{ \sqrt 2, \sqrt{2\sqrt2}, \sqrt{2\sqrt{2\sqrt2}}, \cdot \cdot \cdot \right\}$

JS
Joseph S.

### Problem 80

A sequence $\left\{ a_n \right\}$ is given by $a_1 = \sqrt 2, a_{n + 1} = \sqrt {2 + a_n}.$
(a) By induction or otherwise, show that $\left\{ a_n \right\}$ is increasing and bounded above by 3. Apply the Monotonic Sequence Theorem to show that $\lim_{n\to\infty} a_n$ exists.
(b) Find $\lim_{n\to\infty} a_n.$

JS
Joseph S.

### Problem 81

Show that the sequence defined by
$a_1 = 1$
$a_{n + 1} = 3 - \frac{1}{a_n}$
is increasing and $a_n < 3$ for all $n.$ Deduce that $\{ a_n \}$ is convergent and find its limit.

JS
Joseph S.

### Problem 82

Show that the sequence defined by
$a_1 = 2$
$a_{n + 1} = \frac {1}{3 - a_n}$
satisfies $0 < a_n \le 2$ and is decreasing. Deduce that the sequence is convergent and find its limit.

JS
Joseph S.

### Problem 83

(a) Fibonacci posed the following : Suppose that rabbits live forever and that every month each pair produces a new pair which becomes productive at age 2 months. If we start with one newborn pair, how many pairs or rabbits will we have in the $n$th month? Show that the answer is $f_n$ where $\{ f_n \}$ is the Fibonacci sequence defined in Example 3(c).
(b) Let $a_n = f_{n + 1} / f_n$ and show that $a_{n - 1} = 1 + 1/a_{n - 2}.$ Assuming that $\{ a_n \}$ is convergent, find its limit.

JS
Joseph S.

### Problem 84

(a) Let $a_1 =a, a_2 = f(a), a_3 = f(a_2) = f( f(a)), . . . , a_{n + 1} = f(a_n),$ where $f$ is a continuous function. If $lim_{n \to\infty} a_n = L,$ show that $f(L) = L.$
(b) Illustrate part (a) by taking $f(x) = \cos x, a = 1,$ and estimating the value of $L$ to five decimal places.

JS
Joseph S.

### Problem 85

(a) Use a graph to guess the value of the limit
$\displaystyle \lim_{n \to \infty} \frac {n^5}{n!}$
(b) Use a graph of the sequence in part (a) to find the smallest values of $N$ that correspond to $\varepsilon = 0.1$ and $\varepsilon = 0.001$ in Definition 2.

JS
Joseph S.

### Problem 86

Use Definition 2 directly to prove that $lim_{n \to \infty} r^n = 0$ when $\mid r \mid < 1.$

JS
Joseph S.

### Problem 87

Prove Theorem 6.
[Hint: Use either Definition 2 or the Squeeze Theorem. ]

JS
Joseph S.

Prove Theorem 7.

JS
Joseph S.

### Problem 89

Prove that if $\lim_{n \to \infty} a_n = 0$ and $\left \{ b_n \right \}$ is bounded, then $\lim_{n \to\infty} (a_n b_n) = 0.$

JS
Joseph S.

### Problem 90

Let $a_n = \left ( 1 + \frac {1}{n} \right)^n.$
(a) Show that if $0 \le a < b,$ then
$\frac {b^{n + 1} - a^{n + 1}}{b -a } < (n + 1)b^n$
(b) Deduce that $b^n[(n + 1)a - nb] < a^{n + 1}.$
(c) Use $a = 1 + 1/(n + 1)$ and $b = 1 + 1/n$ in part (b) to show that $\left \{ a_n \right \}$ is increasing.
(d) Use $a = 1$ and $b = 1 + 1/(2n)$ in part (b) to show that $a_{2n} < 4.$
(e) Use parts (c) and (d) to show that $a_n < 4$ for all $n.$
(f) Use Theorem 12 to show that $\lim_{n \to\infty} (1 + 1/n)^n$ exists.
(The limit is $e.$ See Equation 3.6.6.)

JS
Joseph S.

### Problem 91

Let $a$ and $b$ be positive numbers with $a > b.$ Let $a_1$ be their arithmetic mean and $b_1$ their geometric mean:
$a_1 = \frac {a + b}{2}$
$b_1 = \sqrt {ab}$
Repeat this process so that, in general,
$a^{n + 1} = \frac {a_n + b_n}{2}$
$b_{n + 1} = \sqrt {a_n b_n}$
(a) Use mathematical induction to show that
$a_n > a_{n + 1} > b_{n + 1} > b_n$
(b) Deduce that both $\{ a_n \}$ and $\{ b_n \}$ are convergent.
(c) Show that $\lim_{n \to\infty} a_n = \lim_{n \to \infty} b_n$. Gauss called the common value of these limits the arithmetic-geometric mean of the numbers $a$ and $b.$

JS
Joseph S.

### Problem 92

(a) Show that if $\lim_{n \to \infty} a_{2n} = L$ and $\lim_{n \to\infty} a_{2n + 1} = L,$ then $\{ a_n \}$ is convergent and $\lim_{n \to \infty} a_n = L$.
(b) If $a_1 = 1$ and
$a_{n + 1} = 1 + \frac {1}{1 + a_n}$
find the first eight terms of the sequence $\{ a_n \}$. Then use part (a) to show that $\lim_{n \to \infty} a_n = \sqrt{ 2 }$. This gives the continued fraction expansion
$\sqrt{ 2 } = 1 + \frac {1}{ 2 + \frac {1}{2 + \cdot \cdot \cdot}}$

JS
Joseph S.

### Problem 93

The size of an undisturbed fish population has been modeled by the formula
$p_{n + 1} = \frac {bp_n}{a + p_n}$
where $p_n$ is the fish population after $n$ years and $a$ and $b$ are positive constants that depend on the species and its environment. Suppose that the population in year 0 is $p_0 > 0.$
(a) Show that if $\{ p_n \}$ is convergent, then the only possible values for its limit are 0 and $b - a$.
(b) Show that $p_{n + 1} < (b/a)p_n$.
(c) Use part (b) to show that if $a > b,$ then $\lim_{n \to \infty} p_n = 0$; in other words, the population dies out.
(d) Now assume that $a < b$. Show that if $p_0 < b - a$, then $\{ p_n \}$ is increasing and $0 < p_n < b - a$. Show also that if $p_0 > b - a$, then $\{ p_n \}$ is decreasing and $p_n > b - a$. Deduce that if $a < b$, then $\lim_{n \to \infty} p_n = b - a$.

JS
Joseph S.