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SANGHERA.JAS
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Since I don't know how to use latex I have posed my question in word file.
Yours help is greatly appreciated.
Yours help is greatly appreciated.
A Cauchy sequence is a sequence of numbers in which the terms become arbitrarily close to each other as the sequence progresses. This means that for any small distance, there exists a point in the sequence after which all terms are within that distance of each other.
The convergence of a Cauchy sequence means that the sequence has a limit, or a value that the terms of the sequence approach as the sequence progresses. This is important because it allows us to make precise statements about the behavior of the sequence and its terms.
A Cauchy sequence converges if and only if it is also a convergent sequence, meaning that it has a limit. To determine if a sequence converges, you can use a variety of tests such as the ratio test, the root test, or the direct comparison test.
If a Cauchy sequence does not converge, it is called a divergent sequence. This means that the sequence does not have a limit, and its terms do not approach a specific value as the sequence progresses. Divergent sequences can exhibit a variety of behaviors, such as increasing without bound or oscillating between different values.
No, a Cauchy sequence can only converge to a value that is within its range. This means that the limit of the sequence must be one of the terms in the sequence. If the limit of the sequence is not within its range, then the sequence is not a Cauchy sequence.