Download Chapter 9 Infinite Series

Chapter 9 Infinite Series
Chapter Summary
Section Topics
9.1
Sequences—List the terms of a sequence. Determine whether a sequence converges or
diverges. Write a formula for the nth term of a sequence. Use properties of monotonic
sequences and bounded sequences.
9.2
Series and Convergence—Understand the definition of a convergent infinite series.
Use properties of infinite geometric series. Use the nth Term Test for Divergence of
an infinite series.
9.3
The Integral Test and p-Series—Use the Integral Test to determine whether an infinite
series converges or diverges. Use properties of p-series and harmonic series.
9.4
Comparisons of Series—Use the Direct Comparison Test to determine whether a series
converges or diverges. Use the Limit Comparison Test to determine whether a series
converges or diverges.
9.5
Alternating Series—Use the Alternating Series Test to determine whether an infinite
series converges. Use the Alternating Series Remainder to approximate the sum of an
alternating series. Classify a convergent series as absolutely or conditionally convergent.
Rearrange an infinite series to obtain a different sum.
9.6
The Ratio and Root Tests—Use the Ratio Test to determine whether a series converges
or diverges. Use the Root Test to determine whether a series converges or diverges. Review
the tests for convergence and divergence of an infinite series.
9.7
Taylor Polynomials and Approximations—Find polynomial approximations of
elementary functions and compare them with the elementary functions. Find Taylor and
Maclaurin polynomial approximations of elementary functions. Use the remainder of a
Taylor polynomial.
9.8
Power Series—Understand the definition of a power series. Find the radius and interval
of convergence of a power series. Determine the endpoint convergence of a power series.
Differentiate and integrate a power series.
9.9
Representation of Functions by Power Series—Find a geometric power series that
represents a function. Construct a power series using series operations.
9.10
Taylor and Maclaurin Series—Find a Taylor or Maclaurin series for a function. Find a
binomial series. Use a basic list of Taylor series to find other Taylor series.
Chapter Comments
You may want to think of this chapter as two parts. Part I (Sections 9.1 through 9.6) covers
sequences and series of constant terms and Part II (Sections 9.7 through 9.10) covers series with
variable terms. Part I should be covered quickly so that most of your time in this chapter is spent
in Part II.
In Sections 9.1 through 9.6 there are many different kinds of series and many different tests for
convergence or divergence. Be sure to go over each of these carefully. It is a good idea to review
the basic facts of each test each day before covering the new material for that day. This provides
a review for the students and also allows them to see the similarities and differences among tests.
The table on page 646 in Section 9.6 is a good way to compare the various tests. Be sure to go
over with your students the guidelines for choosing the appropriate test found on page 645.
70
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The nth-Term Test for Divergence, Theorem 9.9 on page 612, is frequently misunderstood. Your
students need to know that it proves divergence only and that it says absolutely nothing about
convergence.
Sections 9.7 through 9.10 often seem difficult for students, so allow extra time for these sections.
You will need to go over the material slowly and do lots of examples. Students should be able to
find the coefficients of a Taylor or Maclaurin polynomial, write a Taylor Series, derive a Taylor
Series from a basic list, and find the radius of convergence and the interval of convergence.
Checking the endpoints should be a matter of recalling Sections 9.2 through 9.6.
Section 9.1 Sequences
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Exercises 19–22 (New)
To give students more practice with recognizing the correct expression for a sequence’s nth term,
we added Exercises 19–22.
Exercises 45 –72 (47– 68 in Calculus 8/e)
To give students more practice, we added Exercises 45–48, 68, and 70.
Capstone
Page 606, Exercise 110 You can use this exercise to review the following concepts.
•
Monotonic sequence
•
Bounded sequence
•
Unbounded sequence
•
Theorem 9.5
Understanding these concepts is important for students so that they can successfully complete
this chapter.
Review the definitions of the concepts listed above and then go over the solution. Note that
examples are not unique, so students may write different sequences.
Solution
(a) an = 10 −
1
n
(b) Impossible. The sequence converges by Theorem 9.5.
(c) an =
3n
4n + 1
(d) Impossible. An unbounded sequence diverges.
Section 9.2 Series and Convergence
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Exercises 1– 6 (1–6 in Calculus 8/e)
We rewrote the direction line to clarify that students should find the sequence of partial sums
S1 , S2 , S3 , S4 , and S5 .
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71
Exercises 25 – 30 (23– 28 in Calculus 8/e)
We reordered the exercises to improve the grading.
Exercises 37 – 44 (35– 42 in Calculus 8/e)
We reordered the exercises to improve the grading.
Exercises 59–70 (57– 66 in Calculus 8/e)
We reordered the exercises to improve the grading. To give students more practice, we added
Exercises 64 and 66.
Capstone
Page 615, Exercise 94 You can use this exercise to review the following concepts.
•
Theorem 9.8 and its converse
•
Theorem 9.9
Go over Theorem 9.8 and note that its converse is not true (see note on page 612 near Theorem 9.8).
In both parts (a) and (b) the given information (the expressions approaches 0 as n approach ∞) is
irrelevant and Theorem 9.9 does not apply. So, neither statement is true.
∞
Using a CAS, you can find that
1
∑ n4
=
π4
∞
and that
∑4
1
diverges. Note that each series in
90
n
n =1
parts (a) and (b) is a p-series, which will be covered in the next section (see Section 9.3).
n =1
Solution
(a) False. The fact that
(b) False. The fact that
∞
∑4
n =1
1
n
1
→ 0 is irrelevant to the convergence of
n4
1
4
n
∞
∞
1
1
∑ n4 . Furthermore, ∑ n4
n =1
→ 0 is irrelevant to the convergence of
∞
∑4
n =1
≠ 0.
n =1
1
n
. In fact,
diverges.
Section 9.3 The Integral Test and p-Series
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Exercises 1–24 (1–18 in Calculus 8/e)
We revised the direction line to instruct students to confirm that the Integral Test can be applied
to the series. To give students more practice, we added Exercises 3, 4, 19, 20, 23, and 24.
Capstone
Page 624, Exercise 56 You can use this exercise to review the following concepts.
•
Theorem 9.10
•
p-series
•
Theorem 9.11
Go over Theorem 9.10 and its proof. Then go over the solutions on the next page. The graphs are
provided on transparencies. The solutions can be confirmed by noting that each series is a p-series
and applying Theorem 9.11.
72
© 2010 Cengage Learning. All rights reserved.
Solution
(a)
y
1
x
1
∞
∑
2
1
4
1
∞
∫1
>
n
n =1
3
dx
x
The area under the rectangles is greater than the area under the curve.
(b)
1
∞
dx = ⎡⎣2
x
∫1
Because
∞
x ⎤⎦ 1 = ∞, diverges,
∞
∑
1
n
n =1
diverges.
y
1
x
1
∞
1
∑ n2
2
<
n=2
3
∞
∫1
4
1
dx
x2
The area under the rectangles is less than the area under the curve.
Because
∞
∫1
∞
1
⎡ 1⎤
dx
=
−
⎢ x ⎥ = 1, converges,
x2
⎣ ⎦1
⎛
⎜ and so does
⎝
∞
∞
1
∑ n2
converges
n=2
1⎞
∑ n2 ⎟.
n =1
⎠
Section 9.4 Comparisons of Series
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Although we made changes to the section exercises, no changes were made based on the data.
Capstone
Page 631, Exercise 54 You can use this exercise to review the following concepts.
•
Comparison of series
•
p-series
•
harmonic series
The main point here is that the tests in this section compare series, not just a few terms. Also, you
can review p-series and harmonic series.
© 2010 Cengage Learning. All rights reserved.
73
Solution
This is not correct. The beginning terms do not affect the convergence or divergence of a series.
In fact,
∞
1
1
+
+" =
1000 1001
and 1 +
1
diverges (harmonic)
n =1000 n
∑
1 1
+ +" =
4 9
∞
1
∑ n2
converges (p-series).
n =1
Section 9.5 Alternating Series
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Exercises 11–36 (11– 32 in Calculus 8/e)
To give students more practice, we added Exercises 13–16.
Exercises 37–40 (33–36 in Calculus 8/e)
We reordered the exercises to improve the grading.
Exercises 51–70 (47– 62 in Calculus 8/e)
To give students more practice, we added Exercises 51–54.
Capstone
Page 640, Exercise 76 You can use this exercise to review the following concepts.
•
Theorem 9.16
•
Absolute convergence
•
Conditional convergence
Review the above concepts and then go over the solution. In part (a), ask students to classify an as
absolutely or conditionally convergent.
Solution
(a) False. For example, let an =
Then
But,
∑ an =
∑
∑
∑ n diverges.
an =
(b) True. For if
74
(−1)n
(−1)n
∑
n
n
.
converges and
∑ (− an )
=
∑
(−1)n +1
n
converges.
1
an converged, then so would
∑ an by Theorem 9.16.
© 2010 Cengage Learning. All rights reserved.
Section 9.6 The Ratio and Root Tests
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Exercises 13–34 (13–32 in Calculus 8/e)
To give students more practice, we added Exercises 13 and 14.
Exercises 35–50 (37–50 in Calculus 8/e)
To give students more practice, we added Exercises 35 and 36.
Capstone
Page 649, Exercise 98 You can use this exercise to review the following concepts.
•
Ratio Test
•
Root Test
Review the Ratio and Root Tests (Theorems 9.17 and 9.18, respectively). Then apply the Ratio
Test to parts (a)–(c) and the Root Test to parts (d)–(f ).
Solution
(a) Converges
(Ratio Test)
(b) Inconclusive (See Ratio Test)
(c) Diverges
(Ratio Test)
(d) Diverges
(Root Test)
(e) Inconclusive (See Root Test)
(f ) Diverges
(Root Test, e > 1 )
Section 9.7 Taylor Polynomials and Approximations
Tips and Tools for Problem Solving
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Although we made changes to the section exercises, no changes were made based on the data.
Capstone
Page 659, Exercise 34 You can use this exercise to review the following concepts.
•
Taylor polynomials
•
Natural exponential function
Go over the definition of nth Taylor polynomial. You can also review the natural exponential
function and its derivative. The graph in part (b) is provided on a transparency. The conclusions
in part (c) are also discussed in Example 2 and the Technology note on page 651.
© 2010 Cengage Learning. All rights reserved.
75
Solution
(a) f ( x) = e x
f (1) = e
f ′( x ) = e x
f ′(1) = e
f ′′( x) = f ′′′( x) = f (4) ( x) = e x and f ′′(1) = f ′′′(1) = f (4) (1) = e
P1 ( x ) = e + e( x − 1)
e
( x − 1)2
2
e
e
e
2
3
P4 ( x) = e + e( x − 1) + ( x − 1) + ( x − 1) +
( x − 1)4
2
6
24
P2 ( x) = e + e( x − 1) +
x
1.00
1.25
1.50
1.75
2.00
ex
e
3.4093
4.4817
5.7546
7.3891
P1 ( x )
e
3.3979
4.0774
4.7570
5.4366
P2 ( x )
e
3.4828
4.4172
5.5215
6.7957
P4 ( x )
e
3.4903
4.4809
5.7485
7.3620
(b)
7
P4
P2
P1
y = ex
−6
6
−1
(c) As the degree increases, the accuracy increases. As the distance from x to 1 increases,
the accuracy decreases.
Section 9.8 Power Series
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Although we made changes to the section exercises, no changes were made based on the data.
Capstone
Page 669, Exercise 62 You can use this exercise to review the following concepts.
•
Power series
•
Radius and interval of convergence
Review the above terms and then go over the solution. Be sure that your students understand that
there are many possible answers.
Solution
Many answers possible.
(a)
∞
n =1
76
⎛ x⎞
∑ ⎜⎝ 2 ⎟⎠
n
Geometric:
x
< 1⇒ x < 2
2
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(b)
(c)
∞
(−1)n x n
n =1
n
∑
converges for −1 < x ≤ 1
∞
∑ (2 x + 1)
n
Geometric:
n =1
2 x + 1 < 1 ⇒ −1 < x < 0
∞
(d)
∑
(x
− 2)
n
n 4n
n =1
converges for − 2 ≤ x < 6
Section 9.9 Representation of Functions by Power Series
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Exercises 5 –16 (5 –16 in Calculus 8/e)
We reordered the exercises to improve the grading. We also revised Exercises 5–9, 11–14,
and 16.
Capstone
Page 677, Exercise 64 You can use this exercise to review the following concepts.
•
Operations with power series
•
Interval of convergence
Review the operations with power series on page 673 and go over the solution. You can also
use this exercise to review interval of convergence. Ask students to determine the interval of
convergence for each series,
∞
∑
x n and
n=0
n
∞
⎛ x⎞
∑ ⎜⎝ 5 ⎟⎠ . The intervals of convergence are (−1, 1)
n=0
∞
∞
n
⎛ x⎞
∑
∑ ⎜⎝ 5 ⎟⎠ ?
n=0
n=0
The answer is the intersection of the intervals of convergence of the two original series, ( −1, 1).
and ( − 5, 5), respectively. What is the interval of convergence for the sum
xn +
Solution
You can verify that the statement is incorrect by calculating the constant terms of each side:
∞
∑ xn
+
n=0
∞
⎛
∑ ⎜⎝1 +
n=0
∞
⎛ x⎞
∑ ⎜⎝ 5 ⎟⎠
n=0
n
x⎞
⎛
= (1 + 1) + ⎜ x + ⎟ + "
5⎠
⎝
1⎞ n
1⎞ ⎛
1⎞
⎛
⎟ x = ⎜1 + ⎟ + ⎜1 + ⎟ x + "
5⎠
5⎠ ⎝
5⎠
⎝
The formula should be
∞
∑
n=0
xn +
∞
n
⎛ x⎞
∑ ⎜⎝ 5 ⎟⎠ =
n=0
© 2010 Cengage Learning. All rights reserved.
n
⎡
⎛1⎞ ⎤ n
+
1
⎢
∑
⎜ ⎟ ⎥x .
⎝ 5 ⎠ ⎦⎥
n=0 ⎣
⎢
∞
77
Section 9.10 Taylor and Maclaurin Series
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Exercises 1–12 (1–10 in Calculus 8/e)
To give students more practice, we added Exercises 5 and 6.
Exercises 17–26 (15–20 in Calculus 8/e)
To give students more practice, we added Exercises 18, 20, 22, and 23.
Exercises 27–40 (21– 30 in Calculus 8/e)
To give students more practice, we added Exercises 29, 30, 32, and 34.
Exercises 63–66 (53 and 54 in Calculus 8/e)
To give students more practice, we added Exercises 65 and 66.
Exercises 67–74 (55–58 in Calculus 8/e)
We reordered the exercises to improve the grading. To give students more practice, we added
Exercises 67, 70, 72, and 74.
Capstone
Page 688, Exercise 86 You can use this exercise to review the following concepts.
•
Power series
•
Natural exponential function
Review the power series for the natural exponential function. Then go over the solution.
Solution
(a) Replace x with ( − x ).
(b) Replace x with 3x.
(c) Multiply series by x.
(d) Replace x with 2 x, then replace x with − 2 x, and add the two together.
78
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Chapter 9 Project
Chasing A Pot of Gold
You are standing at the beginning of a sidewalk, 1000 meters from a pot of gold. You walk
toward the pot of gold at a rate of 1 meter per second. After each second, the sidewalk stretches
uniformly and instantaneously, increasing its length by 1000 meters.
Exercises
1. What is the total length of the sidewalk after 1 second?
2. How far are you from the pot of gold after 1 second?
3. What is the total length of the sidewalk after 2 seconds?
4. How far are you from the pot of gold after 2 seconds?
In Exercises 5–11, consider the sequence
{d n} where
d n is your distance from the pot of
gold after n seconds but before the road stretches.
5. Find d 0 , d1 , d 2 , d3 , and d 4 .
6. Find an expression for d1 in terms of d 0 . Similarly, find an expression for d 2 , d3 , and d 4
in terms of d1 , d 2 , and d3 , respectively.
7. Use the expressions you wrote in Exercise 6 to show that d1 = 1( d 0 − 1),
(
(
1
2
d 2 = 2 d0 − 1 +
)), and d
3
(
= 3 ⎡d 0 − 1 +
⎣
1
2
+
1
3
)⎤⎦.
8. Use the results from Exercise 7 to write an expression for d n in terms of d 0 .
∞
9. Does the series
1
∑k
converge or diverge? What theorem did you use?
k =1
10. What does your answer to Exercise 9 imply about the value of d n for large values of n?
What does this mean in the context of the problem?
11. Use the integral
n +1
∫1
1
dx to approximate how long it will take to reach the pot of gold.
x
In Exercises 12 and 13, suppose the sidewalk is still 1000 meters long, but you walk toward
the pot of gold at a speed of 0.25 meter per second.
12. Write an expression for d n in terms of d 0 .
13. Will you reach the pot of gold? If so, approximately how long will it take?
In Exercises 14 and 15, suppose the sidewalk is 5000 meters long, you walk toward the
pot of gold at a speed of 1 meter per second, and the sidewalk stretches 5000 meters after
every second.
14. Write an expression for d n in terms of d 0 .
15. Will you reach the pot of gold? If so, approximately how long will it take?
16. Does the length of the sidewalk or the speed at which you walk impact whether you reach
the pot of gold? Explain.
© 2010 Cengage Learning. All rights reserved.
79