Chapter 9
Major theorems, figures, and student response questions
Calculus 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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Definition 9.1.1 (p. 598)
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Figure 9.1.2 (p. 599)
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Definition 9.1.2 (p. 600)
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Figure 9.1.3 (p. 600)
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Theorem 9.1.3 (p. 600)
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Figure 9.1.4 (p. 601)
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Figure 9.1.5 (p. 602)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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Theorem 9.1.5 (p. 603)
The Squeezing Theorem for Sequences
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Theorem 9.1.6 (p. 603)
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Table 9.1.5 (p. 604)
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Definition 9.2.1 (p. 607)
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Figure 9.2.1 (p. 608)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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Table 9.2.2 (p. 608)
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Figure 9.2.3 (p. 610)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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Theorems 9.2.3 and 9.2.4
(p. 611)
Theorems 9.2.3 and 9.2.4 (p. 611)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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Axiom 9.2.5 (p. 612)
The Completeness Axiom
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Definition 9.3.1 (p. 614)
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Definition 9.3.2 (p. 616)
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Theorem 9.3.3 (p. 617)
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Figure 9.3.4 (p. 620)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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Theorem 9.4.1 (p. 624)
The Divergence Test
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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Theorem 9.4.2 (p. 624)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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Theorem 9.4.3 (p. 625)
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Theorem 9.4.4 (p. 626)
The Integral Test
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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Figure 9.4.1 (p. 626)
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Theorem 9.4.5 (p. 627)
Convergence of p-Series
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Theorem 9.5.1 (p. 631)
The Comparison Test
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Figure 9.5.1 (p. 632)
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Theorem 9.5.4 (p. 661)
The Limit Comparison Test
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Theorem 9.5.5 (p. 634)
The Ratio Test
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Theorem 9.5.6 (p. 635)
The Root Test
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Figure 9.6.1 (p. 638)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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Theorem 9.6.1 (p. 638)
The Alternating Series Test
Equations (1) and (2) (p. 638)
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Theorem 9.6.2 (p. 639)
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Figure 9.6.2 (p. 640)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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Figure 9.6.3 (p. 640)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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Definition 9.6.3 (p. 641)
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Theorem 9.6.4 (p. 642)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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Theorem 9.6.5 (p. 644)
Ratio Test for Absolute Convergence
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Page 645
Summary of Convergence Tests
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Page 645
Summary of Convergence Tests
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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Definitions 9.7.2 and 9.7.3 (p. 650)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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Figure 9.7.3 (p. 651)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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Figure 9.7.4 (p. 652)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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Figure 9.7.5 (p. 652)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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Theorem 9.7.4 (p. 655)
The Remainder Estimation Theorem
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Equations (12) and (13) (p. 655)
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Definition 9.8.1 (p. 660)
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Theorem 9.8.2 (p. 662)
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Theorem 9.8.3 (p. 664)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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Figure 9.8.2 (p. 664)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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9.9.1 (p. 669)
Theorem 9.9.2 (p. 669)
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Equation (3) (p. 669)
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Figure 9.9.1 (p. 670)
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Table 9.9.1 (p. 675)
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Theorem 10.10.2 (p. 678)
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Theorem 9.10.4 (p. 680)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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9.10.5 (p. 681)
Theorem 9.10.6 (p. 681)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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Select the best answer for the following question.
1. Find a formula for the general term of the sequence,
2 3 4
1, , ,
, L starting with n 1.
5 25 125
a) a n
n
5n 1
b) a n
n
5n
c) an n
5n1
d) an n 1
5n
Question 1
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Select the best answer for the following question.
2. Determine whether the sequence converges, and if so,
find its limit.
2n 2
2
n 4n 1n1
a) L 0
b) L 2
c) L 1
2
Question 2
d) Does not converge
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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Select the best answer for the following question.
3. Use the ratio an 1 to determine whether the sequence is
an
strictly increasing, strictly decreasing, or neither.
2n
3n 5 n1
a) strictly increasing
b) strictly decreasing
c) neither
Question 3
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Select the best answer for the following question.
4. Determine whether the series converges, and if so find its
sum.
1
k k 1
k 2
a) Sum 0
b) Sum 1
2
c) Sum 1
d) the series diverges
Question 4
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Select the best answer for the following question.
5. Determine whether the series converges, and if so find its
sum.
ln k 2
k1
a) Sum e
b) Sum 1
2
c) Sum ln 2
d) the series diverges
Question 5
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Select the best answer for the following question.
6. Determine whether the series converges or diverges.
k
3
k1
a) the series converges
b) the series diverges
Question 6
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Select the best answer for the following question.
7. Use the limit comparison test to determine whether the
series converges or diverges.
k1
x3
3x 2 6x 1
a) the series converges
b) the series diverges
Question 7
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Select the best answer for the following question.
8. Use the ratio test to determine whether the series
converges or diverges.
k1
4k
k!
a) the series converges
b) the series diverges
Question 8
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Select the best answer for the following question.
9. Classify the series as absolutely convergent, conditionally
convergent, or divergent.
1 k
k
k1
a) absolutely convergent
b) conditionally convergent
c) divergent
Question 9
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Select the best answer for the following question.
10. Find the Taylor polynomial of order n = 2 about x = 1 for
f x ln x
a) p x 1 x 2 x 3
2
2
2
b) p x 1 x 2 1
2
2
2
c) p x 1 x 2 2x 1
2
2
2
d) p x 1 x 2 2x 3
2
2
2
Question 10
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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Select the best answer for the following question.
11. Find the interval of convergence of
k 0
a) 1,1
3k
1
x
.
k!
k
b) 3, 3
c) ,
d) x x 0
Question 11
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Select the best answer for the following question.
12. Find the first four nonzero terms of the Maclaurin series
for the function f x sin 3x .
a) 3x 9 x 3
2
81 x 5
40
b) 3x 1 x 3
2
1
40
243
560
x7
1 x7
x 5 1680
c) 3 x 1 x 3 1 x 5 1 x 7
6
120
5040
d) 3x 9 x 3
2
81 x 5
40
243
560
x
7
Question 12
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Answers
1. c
6. a
11. c
2. b
7. b
12. d
3. a
8. a
4. c
9. a
5. d
10. d
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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