(a) Consider the following statement. The direct comparison test can be used to show that the first series converges by comparing it to the second series 00 n +5 n=2 n=2 1 The following is a proposed proof for the statement. n n 1. We have < = for all n ≤ 2. +5 2. The summation 1 converges because it is a p-series with n=2 p = 2 > 1. n 3. So converges by part (i) of the direct comparison test. n + 5 n=2 Identify the error(s) in the proposed proof. (Select all that apply.) n n 1 n n 1 The first sentence should say. instead of < n³ + 5 3 The first sentence should say n ≥ 2 instead of n ≤ 2. The second sentence states that the summation is a p-series when this is not the case. The second sentence should say diverges instead of converges. The third sentence should say diverges instead of converges. (b) Consider the following statement. The limit comparison test can be used to show that the first series converges by comparing it to the second series. n - 5 n=2 n=2 The following is a proposed proof for the statement. 1. Use the limit comparison test with an n3 5 and bn n³ 1 an n n² 2. We take lim = lim . = lim 5 q༤-༥ 3 nn 5 1 n→ n³ 1 n 3. Since = lim n→ ∞ 1 n=2 n 1 5 = -1 < 0. - 3 1 is a convergent (partial) p-series [p = 2 > 1], the series converges. n3 5 n=2 Identify the error(s) in the proposed proof. (Select all that apply.) The first sentence should say an n 1 3 instead of an 3 n + 5 n - 5 1 m² The first sentence should say b = 2 instead of b 1 n The second sentence should say n→ - instead of n→ ∞. The second sentence should conclude with the statement = 1 > 0 instead of = −1 < 0. The third sentence claims that is convergent when it is really divergent. n=2
(a) Consider the following statement. The direct comparison test can be used to show that the first series converges by comparing it to the second series 00 n +5 n=2 n=2 1 The following is a proposed proof for the statement. n n 1. We have < = for all n ≤ 2. +5 2. The summation 1 converges because it is a p-series with n=2 p = 2 > 1. n 3. So converges by part (i) of the direct comparison test. n + 5 n=2 Identify the error(s) in the proposed proof. (Select all that apply.) n n 1 n n 1 The first sentence should say. instead of < n³ + 5 3 The first sentence should say n ≥ 2 instead of n ≤ 2. The second sentence states that the summation is a p-series when this is not the case. The second sentence should say diverges instead of converges. The third sentence should say diverges instead of converges. (b) Consider the following statement. The limit comparison test can be used to show that the first series converges by comparing it to the second series. n - 5 n=2 n=2 The following is a proposed proof for the statement. 1. Use the limit comparison test with an n3 5 and bn n³ 1 an n n² 2. We take lim = lim . = lim 5 q༤-༥ 3 nn 5 1 n→ n³ 1 n 3. Since = lim n→ ∞ 1 n=2 n 1 5 = -1 < 0. - 3 1 is a convergent (partial) p-series [p = 2 > 1], the series converges. n3 5 n=2 Identify the error(s) in the proposed proof. (Select all that apply.) The first sentence should say an n 1 3 instead of an 3 n + 5 n - 5 1 m² The first sentence should say b = 2 instead of b 1 n The second sentence should say n→ - instead of n→ ∞. The second sentence should conclude with the statement = 1 > 0 instead of = −1 < 0. The third sentence claims that is convergent when it is really divergent. n=2
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
Please answer parts a and b to the following problem
![(a) Consider the following statement.
The direct comparison test can be used to show that the first series converges by comparing it to the second series
00
n
+5
n=2
n=2
1
The following is a proposed proof for the statement.
n
n
1. We have
<
=
for all n ≤ 2.
+5
2. The summation
1
converges because it is a p-series with
n=2
p = 2 > 1.
n
3. So
converges by part (i) of the direct comparison test.
n + 5
n=2
Identify the error(s) in the proposed proof. (Select all that apply.)
n
n
1
n
n
1
The first sentence should say.
instead of
<
n³ + 5
3
The first sentence should say n ≥ 2 instead of n ≤ 2.
The second sentence states that the summation is a p-series when this is not the case.
The second sentence should say diverges instead of converges.
The third sentence should say diverges instead of converges.
(b) Consider the following statement.
The limit comparison test can be used to show that the first series converges by comparing it to the second series.
n
- 5
n=2
n=2
The following is a proposed proof for the statement.
1. Use the limit comparison test with an
n3
5
and bn
n³
1
an
n
n²
2. We take lim
= lim
.
= lim
5
q༤-༥
3
nn
5
1 n→ n³ 1
n
3. Since
= lim
n→ ∞ 1
n=2
n
1
5
= -1 < 0.
-
3
1
is a convergent (partial) p-series [p = 2 > 1], the series
converges.
n3
5
n=2
Identify the error(s) in the proposed proof. (Select all that apply.)
The first sentence should say an
n
1
3
instead of an
3
n
+ 5
n
- 5
1
m²
The first sentence should say b
=
2
instead of b
1
n
The second sentence should say n→ - instead of n→ ∞.
The second sentence should conclude with the statement = 1 > 0 instead of = −1 < 0.
The third sentence claims that
is convergent when it is really divergent.
n=2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F17c4b70c-f4d2-42b7-8112-84748cd90a73%2F5135b0f9-776c-40d2-b6bf-11bda500beb4%2Fuj0f3pq_processed.png&w=3840&q=75)
Transcribed Image Text:(a) Consider the following statement.
The direct comparison test can be used to show that the first series converges by comparing it to the second series
00
n
+5
n=2
n=2
1
The following is a proposed proof for the statement.
n
n
1. We have
<
=
for all n ≤ 2.
+5
2. The summation
1
converges because it is a p-series with
n=2
p = 2 > 1.
n
3. So
converges by part (i) of the direct comparison test.
n + 5
n=2
Identify the error(s) in the proposed proof. (Select all that apply.)
n
n
1
n
n
1
The first sentence should say.
instead of
<
n³ + 5
3
The first sentence should say n ≥ 2 instead of n ≤ 2.
The second sentence states that the summation is a p-series when this is not the case.
The second sentence should say diverges instead of converges.
The third sentence should say diverges instead of converges.
(b) Consider the following statement.
The limit comparison test can be used to show that the first series converges by comparing it to the second series.
n
- 5
n=2
n=2
The following is a proposed proof for the statement.
1. Use the limit comparison test with an
n3
5
and bn
n³
1
an
n
n²
2. We take lim
= lim
.
= lim
5
q༤-༥
3
nn
5
1 n→ n³ 1
n
3. Since
= lim
n→ ∞ 1
n=2
n
1
5
= -1 < 0.
-
3
1
is a convergent (partial) p-series [p = 2 > 1], the series
converges.
n3
5
n=2
Identify the error(s) in the proposed proof. (Select all that apply.)
The first sentence should say an
n
1
3
instead of an
3
n
+ 5
n
- 5
1
m²
The first sentence should say b
=
2
instead of b
1
n
The second sentence should say n→ - instead of n→ ∞.
The second sentence should conclude with the statement = 1 > 0 instead of = −1 < 0.
The third sentence claims that
is convergent when it is really divergent.
n=2
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