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For the sequence {xn}, where xn consider the following statements

I. {xn} is a Cauchy sequence

II. {xn} is not convergent

III. {xn} is not bounded

Select the correct answer using the codes given below

  • a)
    (II) and (III)

  • b)
    (I) and (III)

  • c)
    (I) and (II)

  • d)
    (I), (II) and (III)

Correct answer is option 'A'. Can you explain this answer?
Verified Answer
For the sequence {xn}, where xn =consider the followingstatementsI. {x...
The sequence xn​ is defined as the nth harmonic number, which is the sum of the reciprocals of the positive integers up to n:
Let's consider each statement:
I. A Cauchy sequence is a sequence where for every positive real number ε, there is an integer N such that for all m,n>N, the absolute difference ∣xn​−xm​∣ is less than ε. For the harmonic sequence, the difference between terms does not eventually become arbitrarily small because as n grows larger, the terms being added to the sum 1/n get smaller, but there's an infinite number of them, so the sum continues to grow without bound. Therefore, xn​ is not a Cauchy sequence.
II. The harmonic series is divergent, which means that as n approaches infinity, xn​ increases without bound and does not converge to a limit. Therefore, the sequence xn​ is not convergent.
III. A sequence is bounded if there is a real number M such that for all n, ∣xn​∣ ≤ M. The harmonic sequence is not bounded because it increases without limit as n approaches infinity.
Given these points, the correct statements are:
II. xn​ is not convergent. III. xn​ is not bounded.
The sequence xn​ is indeed not a Cauchy sequence, but the statement is not given as an option, so we do not consider it in the multiple-choice answers.
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