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Calculus for Scientists and Engineers: Early Transcendental

William Briggs, Lyle Cochran, Bernard Gillett

Chapter 9

Sequences and Infinite Series - all with Video Answers

Educators

Section 1

An Overview

00:42

Problem 1

Define sequence and give an example.

Linh Vu
Linh Vu
Numerade Educator
00:27

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
00:41

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
00:47

Problem 4

Define finite sum and give an example.

Linh Vu
Linh Vu
Numerade Educator
01:06

Problem 5

Define infinite series and give an example.

Linh Vu
Linh Vu
Numerade Educator
00:39

Problem 6

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

Linh Vu
Linh Vu
Numerade Educator
00:58

Problem 7

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.

Linh Vu
Linh Vu
Numerade Educator
01:30

Problem 8

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
00:37

Problem 9

$$a_{n}=1 / 10^{n}$$

Linh Vu
Linh Vu
Numerade Educator
00:32

Problem 10

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
00:41

Problem 11

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
00:42

Problem 12

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
00:56

Problem 13

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
00:33

Problem 14

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

Linh Vu
Linh Vu
Numerade Educator
00:50

Problem 15

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

Linh Vu
Linh Vu
Numerade Educator
01:07

Problem 16

Write the first four terms of the sequence $\left\{a_{n}\right\}_{n=1}^{\infty}$
$$a_{n}=2 n^{2}-3 n+1$$

Linh Vu
Linh Vu
Numerade Educator
00:27

Problem 17

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

Linh Vu
Linh Vu
Numerade Educator
00:30

Problem 18

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

Linh Vu
Linh Vu
Numerade Educator
00:47

Problem 19

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

Linh Vu
Linh Vu
Numerade Educator
00:31

Problem 20

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

Linh Vu
Linh Vu
Numerade Educator
01:13

Problem 21

Write the first four terms of the sequence $\left\{a_{n}\right\}$ defined by the following recurrence relations.
$$a_{n+1}=3 a_{n}^{2}+n+1 ; \quad a_{1}=0$$

Linh Vu
Linh Vu
Numerade Educator
00:32

Problem 22

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

Linh Vu
Linh Vu
Numerade Educator
01:12

Problem 23

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 general nth 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
01:27

Problem 24

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 general nth term of the
sequence.
$$\{1,-2,3,-4,5, \dots\}$$

Linh Vu
Linh Vu
Numerade Educator
01:00

Problem 25

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 general nth term of the
sequence.
$$\{-5,5,-5,5, \dots\}$$

Linh Vu
Linh Vu
Numerade Educator
01:00

Problem 26

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 general nth term of the
sequence.
$$\{2,5,8,11, \dots\}$$

Linh Vu
Linh Vu
Numerade Educator
00:51

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 general nth term of the
sequence.
$$\{1,2,4,8,16, \dots\}$$

Linh Vu
Linh Vu
Numerade Educator
01:05

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 general nth term of the
sequence.
$$\{1,4,9,16,25, \dots\}$$

Linh Vu
Linh Vu
Numerade Educator
01:28

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 general nth term of the
sequence.
$$\{1,3,9,27,81, \ldots\}$$

Linh Vu
Linh Vu
Numerade Educator
01:09

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 general nth term of the
sequence.
$$\{64,32,16,8,4, \dots\}$$

Linh Vu
Linh Vu
Numerade Educator
00:40

Problem 31

Write the terms $a_{1}, a_{2}, a_{3},$ and $a_{4}$ of the following sequences. If the sequence appears to converge, 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
00:50

Problem 32

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

Linh Vu
Linh Vu
Numerade Educator
00:39

Problem 33

Write the terms $a_{1}, a_{2}, a_{3},$ and $a_{4}$ of the following sequences. If the sequence appears to converge, 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
00:39

Problem 34

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

Linh Vu
Linh Vu
Numerade Educator
00:40

Problem 35

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

Linh Vu
Linh Vu
Numerade Educator
01:23

Problem 36

Write the terms $a_{1}, a_{2}, a_{3},$ and $a_{4}$ of the following sequences. If the sequence appears to converge, 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
00:45

Problem 37

Write the terms $a_{1}, a_{2}, a_{3},$ and $a_{4}$ of the following sequences. If the sequence appears to converge, 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
00:53

Problem 38

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

Linh Vu
Linh Vu
Numerade Educator
01:11

Problem 39

Write the terms $a_{1}, a_{2}, a_{3},$ and $a_{4}$ of the following sequences. If the sequence appears to converge, make a conjecture about its limit. If the sequence diverges, explain why.
$$a_{n+1}=0.5 a_{n}+50 ; a_{0}=100$$

Linh Vu
Linh Vu
Numerade Educator
00:55

Problem 40

Write the terms $a_{1}, a_{2}, a_{3},$ and $a_{4}$ of the following sequences. If the sequence appears to converge, 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
01:44

Problem 41

Consider the formulas for the following sequences. Using a calculator, make a table with at least 10 terms and determine a plausible value for the limit of the sequence or state that it does not exist.
$$\cot ^{-1} 2^{n} ; n=1,2,3, \ldots$$

Dale Sanford
Dale Sanford
Numerade Educator
01:46

Problem 42

Consider the formulas for the following sequences. Using a calculator, make a table with at least 10 terms and determine a plausible value for the limit of the sequence or state that it does not exist.
$$a_{n}=2 \tan ^{-1}(1000 n) ; n=1,2,3, \ldots$$

Linh Vu
Linh Vu
Numerade Educator
01:10

Problem 43

Consider the formulas for the following sequences. Using a calculator, make a table with at least 10 terms and determine a plausible value for the limit of the sequence or state that it does not exist.
$$a_{n}=n^{2}-n ; n=1,2,3, \ldots$$

Linh Vu
Linh Vu
Numerade Educator
01:00

Problem 44

Consider the formulas for the following sequences. Using a calculator, make a table with at least 10 terms and determine a plausible value for the limit of the sequence or state that it does not exist.
$$a_{n}=\frac{100 n-1}{10 n}, n=1,2,3, \ldots$$

Linh Vu
Linh Vu
Numerade Educator
01:28

Problem 45

Consider the formulas for the following sequences. Using a calculator, make a table with at least 10 terms and determine a plausible value for the limit of the sequence or state that it does not exist.
$$a_{n}=\frac{(n-1)^{2}}{\left(n^{2}-1\right)^{\prime}} n=2,3,4, \dots$$

Linh Vu
Linh Vu
Numerade Educator
01:48

Problem 46

Consider the formulas for the following sequences. Using a calculator, make a table with at least 10 terms and determine a plausible value for the limit of the sequence or state that it does not exist.
$$a_{n}=2^{n} \sin \left(2^{-n}\right) ; n=1,2,3, \ldots$$

Linh Vu
Linh Vu
Numerade Educator
01:10

Problem 47

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.
(GRAPH CAN'T COPY)
$$a_{n}=2+2^{-n} ; n=1,2,3, \ldots$$

Linh Vu
Linh Vu
Numerade Educator
01:14

Problem 48

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.
(GRAPH CAN'T COPY)
$$a_{n}=\frac{n^{2}}{n^{2}-1}, n=2,3,4, \ldots$$

Linh Vu
Linh Vu
Numerade Educator
01:30

Problem 49

Consider the following recurrence relations. Using a calculator, make a table with at least 10 terms and determine a plausible value for the limit of the sequence or state that it does not exist.
$$a_{n+1}=\frac{1}{2} a_{n}+2 ; a_{0}=3$$

Linh Vu
Linh Vu
Numerade Educator
01:44

Problem 50

Consider the following recurrence relations. Using a calculator, make a table with at least 10 terms and determine a plausible value for the limit of the sequence or state that it does not exist.
$$a_{n}=\frac{1}{4} a_{n-1}-3 ; a_{0}=1$$

Linh Vu
Linh Vu
Numerade Educator
01:07

Problem 51

Consider the following recurrence relations. Using a calculator, make a table with at least 10 terms and determine a plausible value for the limit of the sequence or state that it does not exist.
$$a_{n+1}=2 a_{n}+1 ; a_{0}=0$$

Linh Vu
Linh Vu
Numerade Educator
00:55

Problem 52

Consider the following recurrence relations. Using a calculator, make a table with at least 10 terms and determine a plausible value for the limit of the sequence or state that it does not exist.
$$a_{n+1}=\frac{a_{n}}{2} ; a_{0}=32$$

Linh Vu
Linh Vu
Numerade Educator
01:48

Problem 53

Consider the following recurrence relations. Using a calculator, make a table with at least 10 terms and determine a plausible value for the limit of the sequence or state that it does not exist.
$$a_{n+1}=\frac{1}{2} \sqrt{a_{n}}+3 ; a_{0}=1000$$

Linh Vu
Linh Vu
Numerade Educator
02:23

Problem 54

Consider the following recurrence relations. Using a calculator, make a table with at least 10 terms and determine a plausible value for the limit of the sequence or state that it does not exist.
$$a_{n+1}=\sqrt{1+a_{n}} ; a_{0}=1$$

Linh Vu
Linh Vu
Numerade Educator
01:09

Problem 55

Suppose a ball is thrown upward to a height of $h_{0}$ meters. Each time the ball bounces, it rebounds to a frac-
tion $r$ of its previous height. Let $h_{n}$ be the height after the nth 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
01:38

Problem 56

Suppose a ball is thrown upward to a height of $h_{0}$ meters. Each time the ball bounces, it rebounds to a frac-
tion $r$ of its previous height. Let $h_{n}$ be the height after the nth 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
01:40

Problem 57

Suppose a ball is thrown upward to a height of $h_{0}$ meters. Each time the ball bounces, it rebounds to a frac-
tion $r$ of its previous height. Let $h_{n}$ be the height after the nth 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
01:26

Problem 58

Suppose a ball is thrown upward to a height of $h_{0}$ meters. Each time the ball bounces, it rebounds to a frac-
tion $r$ of its previous height. Let $h_{n}$ be the height after the nth 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
00:58

Problem 59

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.
$$0.3+0.03+0.003+\cdots$$

Linh Vu
Linh Vu
Numerade Educator
00:50

Problem 60

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.
$$0.6+0.06+0.006+\cdots$$

Linh Vu
Linh Vu
Numerade Educator
00:48

Problem 61

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.
$$4+0.9+0.09+0.009+\cdots$$

Linh Vu
Linh Vu
Numerade Educator
01:45

Problem 62

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.
$$1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots$$

Linh Vu
Linh Vu
Numerade Educator
05:24

Problem 63

Consider the following infinite series.
a. Find the first four terms of the sequence of partial sums.
b. Use the results of part (a) to find a formula for $S_{n}$
c. Find the value of the series.
$$\sum_{k=1}^{\infty} \frac{2}{(2 k-1)(2 k+1)}$$

Dale Sanford
Dale Sanford
Numerade Educator
03:29

Problem 64

Consider the following infinite series.
a. Find the first four terms of the sequence of partial sums.
b. Use the results of part (a) to find a formula for $S_{n}$
c. Find the value of the series.
$$\sum_{k=1}^{\infty} \frac{1}{2^{k}}$$

Dale Sanford
Dale Sanford
Numerade Educator
04:45

Problem 65

Consider the following infinite series.
a. Find the first four terms of the sequence of partial sums.
b. Use the results of part (a) to find a formula for $S_{n}$
c. Find the value of the series.
$$\sum_{k=1}^{\infty} \frac{1}{4 k^{2}-1}$$

Dale Sanford
Dale Sanford
Numerade Educator
04:03

Problem 66

Consider the following infinite series.
a. Find the first four terms of the sequence of partial sums.
b. Use the results of part (a) to find a formula for $S_{n}$
c. Find the value of the series.
$$\sum_{k=1}^{\infty} \frac{2}{3^{k}}$$

Dale Sanford
Dale Sanford
Numerade Educator
04:07

Problem 67

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 terms of the sequence must decrease in size.
c. If the terms of the sequence $\left\{a_{n}\right\}$ are positive and increase in size, then the sequence of partial sums for the series $\sum_{k=1}^{\infty} a_{k}$ diverges.

Dale Sanford
Dale Sanford
Numerade Educator
03:27

Problem 68

Suppose a ball is thrown upward to a height of $h_{0}$ meters. Each time the ball bounces,
it rebounds to a fraction $r$ of its previous height. Let $h_{n}$ be the height after the nth bounce and let $S_{n}$ be the total distance the ball has traveled at the moment of the nth 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 69

Suppose a ball is thrown upward to a height of $h_{0}$ meters. Each time the ball bounces,
it rebounds to a fraction $r$ of its previous height. Let $h_{n}$ be the height after the nth bounce and let $S_{n}$ be the total distance the ball has traveled at the moment of the nth 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
01:43

Problem 70

Consider the following infinite series.
a. Write out the first four terms of the sequence of partial sums.
b. Estimate the limit of $\left\{S_{n}\right\}$ or state that it does not exist.
$$\sum_{k=1}^{\infty} \cos (\pi k)$$

Linh Vu
Linh Vu
Numerade Educator
01:10

Problem 71

Consider the following infinite series.
a. Write out the first four terms of the sequence of partial sums.
b. Estimate the limit of $\left\{S_{n}\right\}$ or state that it does not exist.
$$\sum_{k=1}^{\infty} 9(0.1)^{k}$$

Linh Vu
Linh Vu
Numerade Educator
01:22

Problem 72

Consider the following infinite series.
a. Write out the first four terms of the sequence of partial sums.
b. Estimate the limit of $\left\{S_{n}\right\}$ or state that it does not exist.
$$\sum_{k=1}^{\infty} 1.5^{k}$$

Linh Vu
Linh Vu
Numerade Educator
01:48

Problem 73

Consider the following infinite series.
a. Write out the first four terms of the sequence of partial sums.
b. Estimate the limit of $\left\{S_{n}\right\}$ or state that it does not exist.
$$\sum_{k=1}^{\infty} 3^{-k}$$

Linh Vu
Linh Vu
Numerade Educator
00:41

Problem 74

Consider the following infinite series.
a. Write out the first four terms of the sequence of partial sums.
b. Estimate the limit of $\left\{S_{n}\right\}$ or state that it does not exist.
$$\sum_{k=1}^{\infty} k$$

Linh Vu
Linh Vu
Numerade Educator
00:57

Problem 75

Consider the following infinite series.
a. Write out the first four terms of the sequence of partial sums.
b. Estimate the limit of $\left\{S_{n}\right\}$ or state that it does not exist.
$$\sum_{k=1}^{\infty}(-1)^{k}$$

Linh Vu
Linh Vu
Numerade Educator
01:13

Problem 76

Consider the following infinite series.
a. Write out the first four terms of the sequence of partial sums.
b. Estimate the limit of $\left\{S_{n}\right\}$ or state that it does not exist.
$$\sum_{k=1}^{\infty}(-1)^{k} k$$

Linh Vu
Linh Vu
Numerade Educator
01:14

Problem 77

Consider the following infinite series.
a. Write out the first four terms of the sequence of partial sums.
b. Estimate the limit of $\left\{S_{n}\right\}$ or state that it does not exist.
$$\sum_{k=1}^{\infty} \frac{3}{10^{k}}$$

Linh Vu
Linh Vu
Numerade Educator
04:02

Problem 78

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.
When a biologist begins a study, a colony of prairie dogs has a population of $250 .$ Regular measurements reveal that each month the prairie dog population increases by $3 \%$ Let $p_{n}$ be the population (rounded to whole numbers) at the end of the $n$ th month, where the initial population is $p_{0}=250$

Dale Sanford
Dale Sanford
Numerade Educator
03:52

Problem 79

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.
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 $M_{0}=20 \mathrm{g}$

Dale Sanford
Dale Sanford
Numerade Educator
04:33

Problem 80

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.
The Consumer Price Index (the CP1 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 $c_{0}=100$

Dale Sanford
Dale Sanford
Numerade Educator
04:43

Problem 81

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.
Jack took a 200 -mg dose of a strong painkiller 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 $d_{0}=200$ mg.

Dale Sanford
Dale Sanford
Numerade Educator
07:35

Problem 82

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.
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 $$\begin{aligned}
&a_{n+1}=\frac{1}{2}\left(a_{n}+\frac{c}{a_{n}}\right)\\
&\text { for } n=0,1,2,3, \ldots
\end{aligned}$$
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 how many terms of the sequence are needed to approximate $\sqrt{c}$ with an error less than 0.01

Dale Sanford
Dale Sanford
Numerade Educator