Suppose *₁ = 409, and n+1 = (1 − n)¹/3 when n E N. Consider the following proposed proof that the sequence {n} diverges. Seeking a contradiction, suppose that the sequence {n}=1 converges to a limit *. 1. For an arbitrary natural number n, the inequality > 1 is equivalent to the inequality 1 - <0, which is equivalent to the inequality (1 — än)¹/³ < 0, or *n+1 <0. Similarly, n < 0 if and only if n+1 > 1. 2. Applying the preceding observation twice shows that if > 1, then n+2 > 1. 3. Since ₁ > 1, the preceding step implies, by mathematical induction, that In> 1 whenever n is an odd natural number. 4. Combining the first and third steps shows that n < 0 whenever n is an even natural number. 5. Apply the definition of limit with & equal to 1/3. There exists a natural number M such that n - x < 1/3 when n ≥ M. The triangle inequality implies that |*M = *M+1| ≤ ™M − x| + |x − xM+1| <2/3. By steps 3 and 4, one of the numbers and M+1 is greater than 1, and the other is less than 0, so M - *M+1| > 1, contradiction. O Which of the following statements best describes this proposed proof? Step 5 is faulty. Step 4 is faulty. The proof is valid. Step 1 is faulty. Step 2 is faulty. O Step 3 is faulty.
Suppose *₁ = 409, and n+1 = (1 − n)¹/3 when n E N. Consider the following proposed proof that the sequence {n} diverges. Seeking a contradiction, suppose that the sequence {n}=1 converges to a limit *. 1. For an arbitrary natural number n, the inequality > 1 is equivalent to the inequality 1 - <0, which is equivalent to the inequality (1 — än)¹/³ < 0, or *n+1 <0. Similarly, n < 0 if and only if n+1 > 1. 2. Applying the preceding observation twice shows that if > 1, then n+2 > 1. 3. Since ₁ > 1, the preceding step implies, by mathematical induction, that In> 1 whenever n is an odd natural number. 4. Combining the first and third steps shows that n < 0 whenever n is an even natural number. 5. Apply the definition of limit with & equal to 1/3. There exists a natural number M such that n - x < 1/3 when n ≥ M. The triangle inequality implies that |*M = *M+1| ≤ ™M − x| + |x − xM+1| <2/3. By steps 3 and 4, one of the numbers and M+1 is greater than 1, and the other is less than 0, so M - *M+1| > 1, contradiction. O Which of the following statements best describes this proposed proof? Step 5 is faulty. Step 4 is faulty. The proof is valid. Step 1 is faulty. Step 2 is faulty. O Step 3 is faulty.
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:Suppose #₁ = 409, and n+1 = (1-xn)1/3 when n € N. Consider the following
proposed proof that the sequence {n} diverges.
Seeking a contradiction, suppose that the sequence {n}1 converges to a limit .
1. For an arbitrary natural number n, the inequality n > 1 is equivalent to the
inequality 1 - <0, which is equivalent to the inequality (1-x₂)¹/³ <0, or
*n+1 <0. Similarly, n < 0 if and only if *n+1 > 1.
2. Applying the preceding observation twice shows that if > 1, then n+2 > 1.
3. Since ₁ > 1, the preceding step implies, by mathematical induction, that
In > 1 whenever n is an odd natural number.
4. Combining the first and third steps shows that n < 0 whenever n is an even
natural number.
5. Apply the definition of limit with & equal to 1/3. There exists a natural number M
such that nx < 1/3 when n ≥ M. The triangle inequality implies that
|*M = XM+1| ≤ |*M − x | + |x − xM+1| <2/3.
By steps 3 and 4, one of the numbers M and M+1 is greater than 1, and the
other is less than 0, so XM - M+1| > 1, contradiction. O
Which of the following statements best describes this proposed proof?
Step 5 is faulty.
Step 4 is faulty.
The proof is valid.
Step 1 is faulty.
Step 2 is faulty.
Step 3 is faulty.
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