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Cauchy sequence- series (TIFR-2013 problem 11)

Question:

True/False:

Consider the sequences \(x_n=\sum_{1}^{n} \frac{1}{j} \) and \(y_n=\sum_{1}^{n} \frac{1}{j^2} \). Then \(\left\{x_n\right\} \) is Cauchy but \(\left\{y_n \right\} \) is not.

Discussion: We are given sequence of partial sums of a very well known type of series.  \(\left\{x_n\right\}\) is a divergent sequence and \(\left\{y_n\right\}\) is convergent. Also, \(\mathbb{R}\) is complete. So every Cauchy sequence is convergent and any convergent sequence (as always happens in metric spaces) is Cauchy.

The true statement would be \(\left\{x_n\right\}\) is not Cauchy and \(\left\{y_n\right\}\) is Cauchy sequence.

 

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