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  • Calculus: Early Transcendentals
  • Infinite Sequences and Series

Calculus: Early Transcendentals

James Stewart

Chapter 11

Infinite Sequences and Series - all with Video Answers

Educators

JH

Section 1

Sequences

01:22

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? $

Gabriel Rhodes
Gabriel Rhodes
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02:56

Problem 2

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

Gabriel Rhodes
Gabriel Rhodes
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01:13

Problem 3

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

Gabriel Rhodes
Gabriel Rhodes
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01:42

Problem 4

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

Gabriel Rhodes
Gabriel Rhodes
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01:44

Problem 5

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

Gabriel Rhodes
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01:06

Problem 6

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

Gabriel Rhodes
Gabriel Rhodes
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01:49

Problem 7

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

Gabriel Rhodes
Gabriel Rhodes
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02:01

Problem 8

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

Gabriel Rhodes
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01:12

Problem 9

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

Gabriel Rhodes
Gabriel Rhodes
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00:56

Problem 10

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

Gabriel Rhodes
Gabriel Rhodes
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01:30

Problem 11

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

Gabriel Rhodes
Gabriel Rhodes
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01:40

Problem 12

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

Gabriel Rhodes
Gabriel Rhodes
Numerade Educator
01:23

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\} $

Gabriel Rhodes
Gabriel Rhodes
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01:27

Problem 14

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

Gabriel Rhodes
Gabriel Rhodes
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01:23

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\} $

JH
J Hardin
Numerade Educator
00:58

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\} $

Gabriel Rhodes
Gabriel Rhodes
Numerade Educator
01:53

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\} $

Gabriel Rhodes
Gabriel Rhodes
Numerade Educator
02:15

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\} $

Gabriel Rhodes
Gabriel Rhodes
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04:44

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} $

Gabriel Rhodes
Gabriel Rhodes
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05:15

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} $

Gabriel Rhodes
Gabriel Rhodes
Numerade Educator
04:08

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 $

Gabriel Rhodes
Gabriel Rhodes
Numerade Educator
03:24

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} $

JH
J Hardin
Numerade Educator
00:59

Problem 23

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

Gabriel Rhodes
Gabriel Rhodes
Numerade Educator
01:10

Problem 24

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

Gabriel Rhodes
Gabriel Rhodes
Numerade Educator
01:20

Problem 25

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

Gabriel Rhodes
Gabriel Rhodes
Numerade Educator
00:59

Problem 26

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

Gabriel Rhodes
Gabriel Rhodes
Numerade Educator
01:08

Problem 27

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

Gabriel Rhodes
Gabriel Rhodes
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01:00

Problem 28

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

Gabriel Rhodes
Gabriel Rhodes
Numerade Educator
01:12

Problem 29

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

Gabriel Rhodes
Gabriel Rhodes
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01:38

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}} $

Gabriel Rhodes
Gabriel Rhodes
Numerade Educator
02:23

Problem 31

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

Gabriel Rhodes
Gabriel Rhodes
Numerade Educator
01:21

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) $

Gabriel Rhodes
Gabriel Rhodes
Numerade Educator
02:49

Problem 33

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

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

Gabriel Rhodes
Gabriel Rhodes
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01:05

Problem 34

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

Gabriel Rhodes
Gabriel Rhodes
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01:35

Problem 35

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

Gabriel Rhodes
Gabriel Rhodes
Numerade Educator
01:54

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} $

Gabriel Rhodes
Gabriel Rhodes
Numerade Educator
01:36

Problem 37

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

Gabriel Rhodes
Gabriel Rhodes
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01:28

Problem 38

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

Gabriel Rhodes
Gabriel Rhodes
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04:59

Problem 39

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

Gabriel Rhodes
Gabriel Rhodes
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02:26

Problem 40

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

Gabriel Rhodes
Gabriel Rhodes
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02:01

Problem 41

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

Gabriel Rhodes
Gabriel Rhodes
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01:43

Problem 42

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

Gabriel Rhodes
Gabriel Rhodes
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01:59

Problem 43

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

Gabriel Rhodes
Gabriel Rhodes
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01:27

Problem 44

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

Gabriel Rhodes
Gabriel Rhodes
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02:46

Problem 45

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

Gabriel Rhodes
Gabriel Rhodes
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01:58

Problem 46

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

Gabriel Rhodes
Gabriel Rhodes
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00:55

Problem 47

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

Gabriel Rhodes
Gabriel Rhodes
Numerade Educator
02:02

Problem 48

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

Gabriel Rhodes
Gabriel Rhodes
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01:42

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) $

Gabriel Rhodes
Gabriel Rhodes
Numerade Educator
01:52

Problem 50

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

Gabriel Rhodes
Gabriel Rhodes
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03:42

Problem 51

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

Gabriel Rhodes
Gabriel Rhodes
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07:35

Problem 52

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

Gabriel Rhodes
Gabriel Rhodes
Numerade Educator
04:36

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 \} $

Gabriel Rhodes
Gabriel Rhodes
Numerade Educator
01:59

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 \} $

Gabriel Rhodes
Gabriel Rhodes
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02:38

Problem 55

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

Gabriel Rhodes
Gabriel Rhodes
Numerade Educator
05:35

Problem 56

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

Gabriel Rhodes
Gabriel Rhodes
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01:23

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} $

JH
J Hardin
Numerade Educator
03:21

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} $

Gabriel Rhodes
Gabriel Rhodes
Numerade Educator
03:14

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) $

Gabriel Rhodes
Gabriel Rhodes
Numerade Educator
03:37

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} $

Gabriel Rhodes
Gabriel Rhodes
Numerade Educator
01:31

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} $

JH
J Hardin
Numerade Educator
02:17

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!} $

JH
J Hardin
Numerade Educator
03:48

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} $

JH
J Hardin
Numerade Educator
01:59

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 $ ?

Gabriel Rhodes
Gabriel Rhodes
Numerade Educator
03:28

Problem 65

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.

JH
J Hardin
Numerade Educator
02:00

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?

JH
J Hardin
Numerade Educator
03:26

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?

JH
J Hardin
Numerade Educator
08:21

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.

JH
J Hardin
Numerade Educator
06:51

Problem 69

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

JH
J Hardin
Numerade Educator
06:28

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.

JH
J Hardin
Numerade Educator
01:52

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?

JH
J Hardin
Numerade Educator
02:04

Problem 72

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

JH
J Hardin
Numerade Educator
02:19

Problem 73

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

JH
J Hardin
Numerade Educator
04:38

Problem 74

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

JH
J Hardin
Numerade Educator
02:05

Problem 75

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

JH
J Hardin
Numerade Educator
02:37

Problem 76

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

JH
J Hardin
Numerade Educator
04:46

Problem 77

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

JH
J Hardin
Numerade Educator
01:50

Problem 78

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

JH
J Hardin
Numerade Educator
02:20

Problem 79

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

JH
J Hardin
Numerade Educator
08:54

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. $

JH
J Hardin
Numerade Educator
10:32

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.

JH
J Hardin
Numerade Educator
09:49

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.

JH
J Hardin
Numerade Educator
08:25

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.

JH
J Hardin
Numerade Educator
02:43

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.

JH
J Hardin
Numerade Educator
03:11

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.

JH
J Hardin
Numerade Educator
03:18

Problem 86

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

JH
J Hardin
Numerade Educator
01:58

Problem 87

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

JH
J Hardin
Numerade Educator
01:45

Problem 88

Prove Theorem 7.

JH
J Hardin
Numerade Educator
02:58

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. $

JH
J Hardin
Numerade Educator
16:33

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.)

JH
J Hardin
Numerade Educator
12:51

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. $

JH
J Hardin
Numerade Educator
07:41

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}} $

JH
J Hardin
Numerade Educator
21:30

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 $.

JH
J Hardin
Numerade Educator

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