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Calculus: Early Transcendentals

William Briggs, Lyle Cochran, Bernard Gillet

Chapter 10

Sequences and Infinite Series - all with Video Answers

Educators

Section 1

An Overview

00:48

Problem 1

Define sequence and give an example.

Linh Vu
Linh Vu
Numerade Educator
00:32

Problem 2

Suppose the sequence $\left\{a_{n}\right\}$ is defined by the explicit formula $a_{n}=1 / n,$ for $n=1,2,3, \ldots .$ Write out the first five terms of the sequence.

Linh Vu
Linh Vu
Numerade Educator
00:46

Problem 3

Suppose the sequence $\left\{a_{n}\right\}$ is defined by the recurrence relation $a_{n+1}=n a_{n},$ for $n=1,2,3, \ldots,$ where $a_{1}=1 .$ Write out the first five terms of the sequence.

Linh Vu
Linh Vu
Numerade Educator
00:55

Problem 4

Find two different explicit formulas for the sequence $\{-1,1,-1,1,-1,1, \ldots .\}$

Linh Vu
Linh Vu
Numerade Educator
01:04

Problem 5

Find two different explicit formulas for the sequence $\{1,-2,3,-4,-5, \dots\}$

Linh Vu
Linh Vu
Numerade Educator
00:59

Problem 6

The first ten terms of the sequence $\left\{2 \tan ^{-1} 10^{n}\right\}_{n=1}^{\infty}$ are rounded to 8 digits right of the decimal point (see table). Make a conjecture about the limit of the sequence.
$$\begin{array}{|r|c|} \hline n & a_{n} \\ \hline 1 & 2.94225535 \\ \hline 2 & 3.12159332 \\ \hline 3 & 3.13959265 \\ \hline 4 & 3.14139265 \\ \hline 5 & 3.14157265 \\ \hline 6 & 3.14159065 \\ \hline 7 & 3.14159245 \\ \hline 8 & 3.14159263 \\ \hline 9 & 3.14159265 \\ \hline 10 & 3.14159265 \\ \hline \end{array}$$

Linh Vu
Linh Vu
Numerade Educator
00:56

Problem 7

The first ten terms of the sequence $\left\{\left(1+\frac{1}{10^{n}}\right)^{10^{\circ}}\right\}_{n=1}^{\infty}$ are rounded to 8 digits right of the decimal point (see table). Make a conjecture about the limit of the sequence.
$$\begin{array}{|r|c|} \hline n & a_{n} \\ \hline 1 & 2.59374246 \\ \hline 2 & 2.70481383 \\ \hline 3 & 2.71692393 \\ \hline 4 & 2.71814593 \\ \hline 5 & 2.71826824 \\ \hline 6 & 2.71828047 \\ \hline 7 & 2.71828169 \\ \hline 8 & 2.71828179 \\ \hline 9 & 2.71828204 \\ \hline 10 & 2.71828203 \\ \hline \end{array}$$

Linh Vu
Linh Vu
Numerade Educator
00:43

Problem 8

Does the sequence $\{1,2,1,2,1,2, \ldots\}$ converge? Explain.

Linh Vu
Linh Vu
Numerade Educator
01:07

Problem 9

The terms of a sequence of partial sums are defined by $S_{n}=\sum_{k=1}^{n} k^{2},$ for $n=1,2,3, \ldots .$ Evaluate the first four terms of the sequence.

Narayan Hari
Narayan Hari
Numerade Educator
00:35

Problem 10

Given the series $\sum_{k=1}^{2} k,$ evaluate the first four terms of its sequence of partial sums $S_{n}=\sum_{k=1}^{n} k$

Linh Vu
Linh Vu
Numerade Educator
00:26

Problem 11

Use sigma notation to write an infinite scries whose first four successive partial sums are $10,20,30,$ and 40

Linh Vu
Linh Vu
Numerade Educator
01:00

Problem 12

Consider the infinite series $\sum_{k=1}^{\infty} \frac{1}{k} .$ Evaluate the first four terms of the sequence of partial sums.

Linh Vu
Linh Vu
Numerade Educator
00:36

Problem 13

Explicit formulas Write the first four terms of the sequence $\left {a_{n}\right\}_{n=1}^{\infty}$
$$a_{n}=\frac{1}{10^{n}}$$

Linh Vu
Linh Vu
Numerade Educator
00:28

Problem 14

Explicit formulas Write the first four terms of the sequence $\left {a_{n}\right\}_{n=1}^{\infty}$
$$a_{n}=3 n+1$$

Linh Vu
Linh Vu
Numerade Educator
00:37

Problem 15

Explicit formulas Write the first four terms of the sequence $\left {a_{n}\right\}_{n=1}^{\infty}$
$$a_{n}=\frac{(-1)^{n}}{2^{n}}$$

Linh Vu
Linh Vu
Numerade Educator
00:33

Problem 16

Explicit formulas Write the first four terms of the sequence $\left {a_{n}\right\}_{n=1}^{\infty}$
$$a_{n}=2+(-1)^{n}$$

Linh Vu
Linh Vu
Numerade Educator
00:44

Problem 17

Explicit formulas Write the first four terms of the sequence $\left {a_{n}\right\}_{n=1}^{\infty}$
$$a_{n}=\frac{2^{n+1}}{2^{n}+1}$$

Linh Vu
Linh Vu
Numerade Educator
00:31

Problem 18

Explicit formulas Write the first four terms of the sequence $\left {a_{n}\right\}_{n=1}^{\infty}$
$$a_{n}=n+\frac{1}{n}$$

Linh Vu
Linh Vu
Numerade Educator
00:51

Problem 19

Explicit formulas Write the first four terms of the sequence $\left {a_{n}\right\}_{n=1}^{\infty}$
$$a_{n}=1+\sin \frac{\pi n}{2}$$

Linh Vu
Linh Vu
Numerade Educator
00:29

Problem 20

$a_{n}=n !$ (Hint: Recall that $n !=n(n-1)(n-2) \cdots 2 \cdot 1$ )

Linh Vu
Linh Vu
Numerade Educator
00:35

Problem 21

Recurrence relations Write the first four terms of the sequence $\left {a_{n}\right\}$ defined by the following recurrence relations.
$$a_{n+1}=2 a_{n}: a_{1}=2$$

Linh Vu
Linh Vu
Numerade Educator
00:36

Problem 22

Recurrence relations Write the first four terms of the sequence $\left {a_{n}\right\}$ defined by the following recurrence relations.
$$a_{n+1}=\frac{a_{n}}{2} ; a_{1}=32$$

Linh Vu
Linh Vu
Numerade Educator
00:58

Problem 23

Recurrence relations Write the first four terms of the sequence $\left {a_{n}\right\}$ defined by the following recurrence relations.
$$a_{n+1}=3 a_{n}-12 ; a_{1}=10$$

Linh Vu
Linh Vu
Numerade Educator
00:36

Problem 24

Recurrence relations Write the first four terms of the sequence $\left {a_{n}\right\}$ defined by the following recurrence relations.
$$a_{n+1}=a_{n}^{2}-1 ; a_{1}=1$$

Linh Vu
Linh Vu
Numerade Educator
00:51

Problem 25

Recurrence relations Write the first four terms of the sequence $\left {a_{n}\right\}$ defined by the following recurrence relations.
$$a_{n+1}=\frac{1}{1+a_{n}} ; a_{0}=1$$

Linh Vu
Linh Vu
Numerade Educator
00:32

Problem 26

Recurrence relations Write the first four terms of the sequence $\left {a_{n}\right\}$ defined by the following recurrence relations.
$$a_{n+1}=a_{n}+a_{n-1} ; a_{1}=1, a_{0}=1$$

Linh Vu
Linh Vu
Numerade Educator
01:29

Problem 27

Working with sequences Several terms of a sequence $\left {a_{n}\right\}_{n=1}^{\infty}$ are given.
a. Find the next two terms of the sequence.
b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence).
c. Find an explicit formula for the nith term of the sequence.
$$\left\{1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \ldots\right\}$$

Linh Vu
Linh Vu
Numerade Educator
01:26

Problem 28

Working with sequences Several terms of a sequence $\left {a_{n}\right\}_{n=1}^{\infty}$ are given.
a. Find the next two terms of the sequence.
b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence).
c. Find an explicit formula for the nith term of the sequence.
$$\{2,5,8,11, \dots\}$$

Linh Vu
Linh Vu
Numerade Educator
01:25

Problem 29

Working with sequences Several terms of a sequence $\left {a_{n}\right\}_{n=1}^{\infty}$ are given.
a. Find the next two terms of the sequence.
b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence).
c. Find an explicit formula for the nith term of the sequence.
$$\{1,2,4,8,16, \dots\}$$

Linh Vu
Linh Vu
Numerade Educator
01:44

Problem 30

Working with sequences Several terms of a sequence $\left {a_{n}\right\}_{n=1}^{\infty}$ are given.
a. Find the next two terms of the sequence.
b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence).
c. Find an explicit formula for the nith term of the sequence.
$$\{64,32,16,8,4, \dots\}$$

Linh Vu
Linh Vu
Numerade Educator
01:10

Problem 31

Working with sequences Several terms of a sequence $\left {a_{n}\right\}_{n=1}^{\infty}$ are given.
a. Find the next two terms of the sequence.
b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence).
c. Find an explicit formula for the nith term of the sequence.
$$\{1,3,9,27,81, \dots\}$$

Linh Vu
Linh Vu
Numerade Educator
01:33

Problem 32

Working with sequences Several terms of a sequence $\left {a_{n}\right\}_{n=1}^{\infty}$ are given.
a. Find the next two terms of the sequence.
b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence).
c. Find an explicit formula for the nith term of the sequence.
$$\{1,4,9,16,25, \dots\}$$

Linh Vu
Linh Vu
Numerade Educator
01:00

Problem 33

Working with sequences Several terms of a sequence $\left {a_{n}\right\}_{n=1}^{\infty}$ are given.
a. Find the next two terms of the sequence.
b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence).
c. Find an explicit formula for the nith term of the sequence.
$$\{-5,5,-5,5, \dots\}$$

Linh Vu
Linh Vu
Numerade Educator
01:30

Problem 34

Working with sequences Several terms of a sequence $\left {a_{n}\right\}_{n=1}^{\infty}$ are given.
a. Find the next two terms of the sequence.
b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence).
c. Find an explicit formula for the nith term of the sequence.
$$\{1,0,1,0,1,0,1, \ldots\}$$

Linh Vu
Linh Vu
Numerade Educator
00:48

Problem 35

Limits of sequences Write the terms $a_{1}, a_{2}, a_{3},$ and $a_{4}$ of the following sequences. If the sequence appears to comerge, make a conjecture about its limit. If the sequence diverges, explain why.
$$a_{n}=10^{n}-1 ; n=1,2,3, \ldots$$

Linh Vu
Linh Vu
Numerade Educator
01:29

Problem 36

Limits of sequences Write the terms $a_{1}, a_{2}, a_{3},$ and $a_{4}$ of the following sequences. If the sequence appears to comerge, make a conjecture about its limit. If the sequence diverges, explain why.
$$a_{n+1}=\frac{10}{a_{n}} ; a_{1}=1$$

Linh Vu
Linh Vu
Numerade Educator
00:42

Problem 37

Limits of sequences Write the terms $a_{1}, a_{2}, a_{3},$ and $a_{4}$ of the following sequences. If the sequence appears to comerge, make a conjecture about its limit. If the sequence diverges, explain why.
$$a_{n}=\frac{1}{10^{n}} ; n=1,2,3, \ldots$$

Linh Vu
Linh Vu
Numerade Educator
00:50

Problem 38

Limits of sequences Write the terms $a_{1}, a_{2}, a_{3},$ and $a_{4}$ of the following sequences. If the sequence appears to comerge, make a conjecture about its limit. If the sequence diverges, explain why.
$$a_{n+1}=\frac{a_{n}}{10} ; a_{0}=1$$

Linh Vu
Linh Vu
Numerade Educator
00:56

Problem 39

Limits of sequences Write the terms $a_{1}, a_{2}, a_{3},$ and $a_{4}$ of the following sequences. If the sequence appears to comerge, make a conjecture about its limit. If the sequence diverges, explain why.
$$a_{n}=3+\cos \pi n ; n=1,2,3, \ldots$$

Hoan Nguyen
Hoan Nguyen
Numerade Educator
01:09

Problem 40

Limits of sequences Write the terms $a_{1}, a_{2}, a_{3},$ and $a_{4}$ of the following sequences. If the sequence appears to comerge, make a conjecture about its limit. If the sequence diverges, explain why.
$$a_{n}=1-10^{-n} ; n=1,2,3, \ldots$$

Linh Vu
Linh Vu
Numerade Educator
00:49

Problem 41

Limits of sequences Write the terms $a_{1}, a_{2}, a_{3},$ and $a_{4}$ of the following sequences. If the sequence appears to comerge, make a conjecture about its limit. If the sequence diverges, explain why.
$$a_{n+1}=1+\frac{a_{n}}{2} ; a_{0}=2$$

Linh Vu
Linh Vu
Numerade Educator
00:59

Problem 42

Limits of sequences Write the terms $a_{1}, a_{2}, a_{3},$ and $a_{4}$ of the following sequences. If the sequence appears to comerge, make a conjecture about its limit. If the sequence diverges, explain why.
$$a_{n+1}=1-\frac{a_{n}}{2} ; a_{0}=\frac{2}{3}$$

Linh Vu
Linh Vu
Numerade Educator
02:34

Problem 43

Limits of sequences Write the terms $a_{1}, a_{2}, a_{3},$ and $a_{4}$ of the following sequences. If the sequence appears to comerge, make a conjecture about its limit. If the sequence diverges, explain why.
$$a_{n+1}=\frac{a_{n}}{11}+50 ; a_{0}=50$$

Linh Vu
Linh Vu
Numerade Educator
01:03

Problem 44

Limits of sequences Write the terms $a_{1}, a_{2}, a_{3},$ and $a_{4}$ of the following sequences. If the sequence appears to comerge, make a conjecture about its limit. If the sequence diverges, explain why.
$$a_{n+1}=10 a_{n}-1 ; a_{0}=0$$

Linh Vu
Linh Vu
Numerade Educator
02:19

Problem 45

Explicit formulas for sequences Consider the formulas for the following sequences $\left\{a_{n}\right\}_{n=1}^{\infty}$ Make a table with at least ten terms and determine a plausible limit of the sequence or state that the sequence diverges.
$$a_{n}=\frac{5^{n}}{5^{n}+1}$$

Linh Vu
Linh Vu
Numerade Educator
02:59

Problem 46

Explicit formulas for sequences Consider the formulas for the following sequences $\left\{a_{n}\right\}_{n=1}^{\infty}$ Make a table with at least ten terms and determine a plausible limit of the sequence or state that the sequence diverges.
$$a_{n}=2^{n} \sin \left(2^{-n}\right)$$

Linh Vu
Linh Vu
Numerade Educator
01:39

Problem 47

Explicit formulas for sequences Consider the formulas for the following sequences $\left\{a_{n}\right\}_{n=1}^{\infty}$ Make a table with at least ten terms and determine a plausible limit of the sequence or state that the sequence diverges.
$$a_{n}=n^{2}+n$$

Linh Vu
Linh Vu
Numerade Educator
01:26

Problem 48

Explicit formulas for sequences Consider the formulas for the following sequences $\left\{a_{n}\right\}_{n=1}^{\infty}$ Make a table with at least ten terms and determine a plausible limit of the sequence or state that the sequence diverges.
$$a_{n}=\frac{100 n-1}{10 n}$$

Linh Vu
Linh Vu
Numerade Educator
01:29

Problem 49

Limits from graphs Consider the following sequences.
a. Find the first four terms of the sequence.
b. Based on part (a) and the figure, determine a plausible limit of the sequence
$$a_{n}=2+2^{-n} ; n=1,2,3, \ldots$$
(graph cannot copy)

Linh Vu
Linh Vu
Numerade Educator
00:54

Problem 50

Limits from graphs Consider the following sequences.
a. Find the first four terms of the sequence.
b. Based on part (a) and the figure, determine a plausible limit of the sequence
$$a_{n}=\frac{n^{2}}{n^{2}-1} ; n=2,3,4, \dots$$
(graph cannot copy)

Linh Vu
Linh Vu
Numerade Educator
01:56

Problem 51

Recurrence relations Consider the following recurrence relations. Make a table with at least ten terms and determine a plausible limit of the sequence or state that the sequence diverges.
$$a_{n+1}=\frac{1}{2} a_{n}+2 ; a_{1}=3$$

Linh Vu
Linh Vu
Numerade Educator
02:21

Problem 52

Recurrence relations Consider the following recurrence relations. Make a table with at least ten terms and determine a plausible limit of the sequence or state that the sequence diverges.
$$a_{n}=\frac{1}{4} a_{n-1}-3 ; a_{0}=1$$

Linh Vu
Linh Vu
Numerade Educator
01:23

Problem 53

Recurrence relations Consider the following recurrence relations. Make a table with at least ten terms and determine a plausible limit of the sequence or state that the sequence diverges.
$$a_{n+1}=4 a_{n}+1 ; a_{0}=1$$

Linh Vu
Linh Vu
Numerade Educator
01:32

Problem 54

Recurrence relations Consider the following recurrence relations. Make a table with at least ten terms and determine a plausible limit of the sequence or state that the sequence diverges.
$$a_{n+1}=\frac{a_{n}}{10}+3 ; a_{0}=10$$

Linh Vu
Linh Vu
Numerade Educator
01:40

Problem 55

Recurrence relations Consider the following recurrence relations. Make a table with at least ten terms and determine a plausible limit of the sequence or state that the sequence diverges.
$$a_{n+1}=\frac{1}{2} \sqrt{a_{n}}+3 ; a_{1}=8$$

Linh Vu
Linh Vu
Numerade Educator
01:28

Problem 56

Recurrence relations Consider the following recurrence relations. Make a table with at least ten terms and determine a plausible limit of the sequence or state that the sequence diverges.
$$a_{n+1}=\sqrt{8 a_{n}+9} ; a_{1}=10$$

Linh Vu
Linh Vu
Numerade Educator
02:05

Problem 57

Heights of bouncing balls A ball is thrown upwand to a height of $h_{0}$ meters. After each bounce, the ball rebounds to a fraction r of its previous height. Let $h_{n}$ be the height after the nath bounce. Consider the following values of $h_{0}$ and $r$
a. Find the first four terms of the sequence of heights $\left\{h_{n}\right\}$.
b. Find an explicit formula for the nith term of the sequence $\left\{h_{n}\right\}$.
$$h_{0}=20, r=0.5$$

Linh Vu
Linh Vu
Numerade Educator
01:41

Problem 58

Heights of bouncing balls A ball is thrown upwand to a height of $h_{0}$ meters. After each bounce, the ball rebounds to a fraction r of its previous height. Let $h_{n}$ be the height after the nath bounce. Consider the following values of $h_{0}$ and $r$
a. Find the first four terms of the sequence of heights $\left\{h_{n}\right\}$.
b. Find an explicit formula for the nith term of the sequence $\left\{h_{n}\right\}$.
$$h_{0}=10, r=0.9$$

Linh Vu
Linh Vu
Numerade Educator
01:28

Problem 59

Heights of bouncing balls A ball is thrown upwand to a height of $h_{0}$ meters. After each bounce, the ball rebounds to a fraction r of its previous height. Let $h_{n}$ be the height after the nath bounce. Consider the following values of $h_{0}$ and $r$
a. Find the first four terms of the sequence of heights $\left\{h_{n}\right\}$.
b. Find an explicit formula for the nith term of the sequence $\left\{h_{n}\right\}$.
$$h_{0}=30, r=0.25$$

Linh Vu
Linh Vu
Numerade Educator
01:14

Problem 60

Heights of bouncing balls A ball is thrown upwand to a height of $h_{0}$ meters. After each bounce, the ball rebounds to a fraction r of its previous height. Let $h_{n}$ be the height after the nath bounce. Consider the following values of $h_{0}$ and $r$
a. Find the first four terms of the sequence of heights $\left\{h_{n}\right\}$.
b. Find an explicit formula for the nith term of the sequence $\left\{h_{n}\right\}$.
$$h_{0}=20, r=0.75$$

Linh Vu
Linh Vu
Numerade Educator
01:10

Problem 61

Sequences of partial sums For the following infinite series, find the first four terms of the sequence of partial sums. Then make a conjecture about the value of the infinite series or state that the series diverges.
$$0.3+0.03+0.003+\cdots$$

Linh Vu
Linh Vu
Numerade Educator
01:18

Problem 62

Sequences of partial sums For the following infinite series, find the first four terms of the sequence of partial sums. Then make a conjecture about the value of the infinite series or state that the series diverges.
$$0.6+0.06+0.006+\cdots$$

Linh Vu
Linh Vu
Numerade Educator
00:44

Problem 63

Sequences of partial sums For the following infinite series, find the first four terms of the sequence of partial sums. Then make a conjecture about the value of the infinite series or state that the series diverges.
$$4+0.9+0.09+0.009+\cdots$$

Linh Vu
Linh Vu
Numerade Educator
01:04

Problem 64

Sequences of partial sums For the following infinite series, find the first four terms of the sequence of partial sums. Then make a conjecture about the value of the infinite series or state that the series diverges.
$$\sum_{k=1}^{2} 10^{k}$$

Linh Vu
Linh Vu
Numerade Educator
01:15

Problem 65

Sequences of partial sums For the following infinite series, find the first four terms of the sequence of partial sums. Then make a conjecture about the value of the infinite series or state that the series diverges.
$$\sum_{k=1}^{\infty} \frac{6}{10^{k}}$$

Linh Vu
Linh Vu
Numerade Educator
01:37

Problem 66

Sequences of partial sums For the following infinite series, find the first four terms of the sequence of partial sums. Then make a conjecture about the value of the infinite series or state that the series diverges.
$$\sum_{k=1}^{\infty} \cos \pi k$$

Linh Vu
Linh Vu
Numerade Educator
05:24

Problem 67

Formulas for sequences of partial sums Consider the follow. ing infinite series.
a. Find the first four partial sums $S_{1}, S_{2}, S_{3},$ and $S_{4}$ of the series.
b. Find a formula for the nth partial sum $S_{n}$ of the infinite series. Use this formula to find the next four partial sums $S_{5}, S_{6}, S_{7},$ and $S_{8}$ of the infinite series.
c. Make a conjecture for the value of the series.
$$\sum_{k=1}^{\infty} \frac{2}{(2 k-1)(2 k+1)}$$

Dale Sanford
Dale Sanford
Numerade Educator
05:24

Problem 68

Formulas for sequences of partial sums Consider the follow. ing infinite series.
a. Find the first four partial sums $S_{1}, S_{2}, S_{3},$ and $S_{4}$ of the series.
b. Find a formula for the nth partial sum $S_{n}$ of the infinite series. Use this formula to find the next four partial sums $S_{5}, S_{6}, S_{7},$ and $S_{8}$ of the infinite series.
c. Make a conjecture for the value of the series.
$$\sum_{k=1}^{\infty} \frac{1}{2^{k}}$$

Dale Sanford
Dale Sanford
Numerade Educator
05:24

Problem 69

Formulas for sequences of partial sums Consider the follow. ing infinite series.
a. Find the first four partial sums $S_{1}, S_{2}, S_{3},$ and $S_{4}$ of the series.
b. Find a formula for the nth partial sum $S_{n}$ of the infinite series. Use this formula to find the next four partial sums $S_{5}, S_{6}, S_{7},$ and $S_{8}$ of the infinite series.
c. Make a conjecture for the value of the series.
$$\sum_{k=1}^{2} 90(0.1)^{k}$$

Dale Sanford
Dale Sanford
Numerade Educator
04:45

Problem 70

Formulas for sequences of partial sums Consider the follow. ing infinite series.
a. Find the first four partial sums $S_{1}, S_{2}, S_{3},$ and $S_{4}$ of the series.
b. Find a formula for the nth partial sum $S_{n}$ of the infinite series. Use this formula to find the next four partial sums $S_{5}, S_{6}, S_{7},$ and $S_{8}$ of the infinite series.
c. Make a conjecture for the value of the series.
$$\sum_{k=1}^{\infty} \frac{2}{3^{k-1}}$$

Dale Sanford
Dale Sanford
Numerade Educator
01:46

Problem 71

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. The sequence of partial sums for the series $1+2+3+\cdots$ is $\{1,3,6,10, \ldots\}$
b. If a sequence of positive numbers converges, then the sequence
is decreasing.
c. If the terms of the sequence $\left\{a_{n}\right\}$ are positive and increasing. then the sequence of partial sums for the series $\sum_{k=1}^{\infty} a_{k}$ diverges.

Nick Johnson
Nick Johnson
Numerade Educator
02:56

Problem 72

Consider the following situations that generate a sequence.
a. Write out the first five terms of the sequence.
b. Find an explicit formula for the terms of the sequence.
c. Find a recurrence relation that generates the sequence.
d. Using a calculator or a graphing utility, estimate the limit of the sequence or state that it does not exist.

Linh Vu
Linh Vu
Numerade Educator
01:47

Problem 73

A material transmutes $50 \%$ of its mass to another element every 10 years due to radioactive decay. Let $M_{n}$ be the mass of the radioactive material at the end of the $n$ th decade. where the initial mass of the material is $M_{0}=20 \mathrm{g}$

Linh Vu
Linh Vu
Numerade Educator
00:29

Problem 74

The Consumer Price Index (the CPI is a measure of the U.S. cost of living) is given a base value of 100 in the year $1984 .$ Assume the CPI has increased by an average of $3 \%$ per year since $1984 .$ Let $c_{n}$ be the CPI $n$ years after $1984,$ where $c_{0}=100$

Amrita Bhasin
Amrita Bhasin
Numerade Educator
01:40

Problem 75

Jack took a 200 -mg dose of a pain killer at midnight. Every hour, $5 \%$ of the drug is washed out of his bloodstream. Let $d_{n}$ be the amount of drug in Jack's blood $n$ hours after the drug was taken, where $d_{0}=200 \mathrm{mg}$.

Subhadeepta Sahoo
Subhadeepta Sahoo
Numerade Educator
03:27

Problem 76

76-77. Distance traveled by bouncing balls A ball is thrown upward to a height of $h_{0}$ meters. After each bounce, the ball rebounds to a fraction $r$ of its previous height. Let $h_{m}$ be the height after the nith bounce and let $S_{n}$ be the total distance the ball has traveled at the moment of the nith bounce.
a. Find the first four terms of the sequence $\left\{S_{n}\right\}$
b. Make a table of 20 terms of the sequence $\left\{S_{n}\right\}$ and determine a plausible value for the limit of $\left\{S_{n}\right\}$
$$h_{0}=20, r=0.5$$

Dale Sanford
Dale Sanford
Numerade Educator
03:27

Problem 77

Distance traveled by bouncing balls A ball is thrown upward to a height of $h_{0}$ meters. After each bounce, the ball rebounds to a fraction $r$ of its previous height. Let $h_{m}$ be the height after the nith bounce and let $S_{n}$ be the total distance the ball has traveled at the moment of the nith bounce.
a. Find the first four terms of the sequence $\left\{S_{n}\right\}$
b. Make a table of 20 terms of the sequence $\left\{S_{n}\right\}$ and determine a plausible value for the limit of $\left\{S_{n}\right\}$
$$h_{0}=20, r=0.75$$

Dale Sanford
Dale Sanford
Numerade Educator
07:35

Problem 78

A square root finder A well-known method for approximating
$\sqrt{c}$ for positive real numbers $c$ consists of the following recurrence relation (based on Newton's method; see Section 4.8 ). Let $a_{0}=c$ and
$$a_{n+1}=\frac{1}{2}\left(a_{n}+\frac{c}{a_{n}}\right), \text { for } n=0,1,2,3, \ldots$$
a. Use this recurrence relation to approximate $\sqrt{10}$. How many terms of the sequence are needed to approximate $\sqrt{10}$ with an error less than $0.01 ?$ How many terms of the sequence are needed to approximate $\sqrt{10}$ with an error less than $0.0001 ?$ (To compute the error, assume a calculator gives the exact value.)
b. Use this recurrence relation to approximate $\sqrt{c}$, for $c=2,3, \ldots . .10 .$ Make a table showing the number of terms of the sequence needed to approximate $\sqrt{c}$ with an error less than 0.01

Dale Sanford
Dale Sanford
Numerade Educator
02:00

Problem 79

Fixed-point iteration A method for estimating a solution to the equation $x=f(x)$. known as fixed-point iteration, is based on the following recurrence relation. Let $x_{0}=c$ and $x_{n+1}=f\left(x_{n}\right)$ for $n=1,2,3, \ldots$ and a real number $c .$ lf the sequence $\left\{x_{n}\right\}_{n=0}^{\infty}$ converges to $L$, then $L$ is a solution to the equation $x=f(x)$ and $L$ is called a fixed point of $f .$ To estimate $L$ with $p$ digits of accuracy to the right of the decimal point, we can compute the terms of the sequence $\left\{x_{n}\right\}_{n=0}^{\infty}$ until two successive values agree to $p$ digits of accuracy. Use fixed-point iteration to find a solution to the following equations with $p=3$ digits of accuracy using the given value of $x_{0}$
$$x=\cos x ; x_{0}=0.8$$

Kian Manafi
Kian Manafi
Numerade Educator
02:00

Problem 80

Fixed-point iteration A method for estimating a solution to the equation $x=f(x)$. known as fixed-point iteration, is based on the following recurrence relation. Let $x_{0}=c$ and $x_{n+1}=f\left(x_{n}\right)$ for $n=1,2,3, \ldots$ and a real number $c .$ lf the sequence $\left\{x_{n}\right\}_{n=0}^{\infty}$ converges to $L$, then $L$ is a solution to the equation $x=f(x)$ and $L$ is called a fixed point of $f .$ To estimate $L$ with $p$ digits of accuracy to the right of the decimal point, we can compute the terms of the sequence $\left\{x_{n}\right\}_{n=0}^{\infty}$ until two successive values agree to $p$ digits of accuracy. Use fixed-point iteration to find a solution to the following equations with $p=3$ digits of accuracy using the given value of $x_{0}$
$$x=\frac{\sqrt{x^{3}+1}}{20} ; x_{0}=5$$

Kian Manafi
Kian Manafi
Numerade Educator