Evaluate the following proposed proof that 1. According to Example 2.1.4, √n² +1 lim nxn+ (cos n)² = 0. 2. Since {1/(n + 1)} is a tail of the sequence {1/n}1, Proposition 2.1.15 implies that 1 n+1 Step 4 is faulty. Step 5 is faulty. Step 2 is faulty. The proof is valid. Step 3 is faulty. Step 1 is faulty. = 1 lim = 0. n→∞ n 1 lim n→∞ n + 1 3. Since 0≤ (cos n)² ≤ 1, the order properties of the real numbers imply that n≤n+ (cos n)² ≤ n + 1, and 0. 1 n+ (cos n)² 4. The preceding three steps, together with Lemma 2.2.1 (the squeeze lemma), imply that 1 lim n→x n + (cos n)² 5. According to Proposition 2.2.5, the limit of a product is the product of the limits, so 1 0. n √n² +1 lim n→→∞ n + (cos n)² Which of the following statements best describes this proposed proof? = 0. lim √√n² + 1 = 0. n→∞0
Evaluate the following proposed proof that 1. According to Example 2.1.4, √n² +1 lim nxn+ (cos n)² = 0. 2. Since {1/(n + 1)} is a tail of the sequence {1/n}1, Proposition 2.1.15 implies that 1 n+1 Step 4 is faulty. Step 5 is faulty. Step 2 is faulty. The proof is valid. Step 3 is faulty. Step 1 is faulty. = 1 lim = 0. n→∞ n 1 lim n→∞ n + 1 3. Since 0≤ (cos n)² ≤ 1, the order properties of the real numbers imply that n≤n+ (cos n)² ≤ n + 1, and 0. 1 n+ (cos n)² 4. The preceding three steps, together with Lemma 2.2.1 (the squeeze lemma), imply that 1 lim n→x n + (cos n)² 5. According to Proposition 2.2.5, the limit of a product is the product of the limits, so 1 0. n √n² +1 lim n→→∞ n + (cos n)² Which of the following statements best describes this proposed proof? = 0. lim √√n² + 1 = 0. n→∞0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Evaluate the following proposed proof that
1. According to Example 2.1.4,
√n² +1
lim
nxn+ (cos n)²
= 0.
2. Since {1/(n + 1)} is a tail of the sequence {1/n}1, Proposition 2.1.15 implies that
1
n+1
Step 4 is faulty.
Step 5 is faulty.
Step 2 is faulty.
The proof is valid.
Step 3 is faulty.
Step 1 is faulty.
=
1
lim = 0.
n→∞ n
1
lim
n→∞ n + 1
3. Since 0≤ (cos n)² ≤ 1, the order properties of the real numbers imply that
n≤n+ (cos n)² ≤ n + 1, and
= = 0.
1
<
n+ (cos n)² n
4. The preceding three steps, together with Lemma 2.2.1 (the squeeze lemma), imply that
1
lim
n→∞ n + (cos n)²
5. According to Proposition 2.2.5, the limit of a product is the product of the limits, so
0.
√n² +1
lim
nx n + (cos n)²
Which of the following statements best describes this proposed proof?
= 0. lim √n² + 1 = 0.
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