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
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Definition 9.1.1 (p. 598)
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Figure 9.1.2 (p. 599)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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Definition 9.1.2 (p. 600)
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Figure 9.1.3 (p. 600)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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Theorem 9.1.3 (p. 600)
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Figure 9.1.4 (p. 601)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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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
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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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)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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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
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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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)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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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
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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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)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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Theorem 9.6.2 (p. 639)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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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)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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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
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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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
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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Equations (12) and (13) (p. 655)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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Definition 9.8.1 (p. 660)
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Theorem 9.8.2 (p. 662)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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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)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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Equation (3) (p. 669)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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Figure 9.9.1 (p. 670)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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Table 9.9.1 (p. 675)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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Theorem 10.10.2 (p. 678)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis
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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
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.



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
5n1
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  1n1
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 n1

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
k1
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
k1
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.


k1
x3
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.


k1
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 
k1

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
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.



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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