Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Sequence and Series
A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.
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**Example of a Metric Space and Cauchy Sequence** **Problem:** Provide an example of a metric space \(X, d\) and a sequence \(\{p_n\} \subseteq X\) such that \(\{p_n\}\) is a Cauchy sequence in \(X, d\) but \(\{p_n\}\) does not converge in \(X, d\) (i.e., \(X, d\) is not a complete metric space). Use \(\varepsilon\)-\(N\) arguments to demonstrate that your sequence is Cauchy and that the limit is not in \(X, d\). **Explanation:** To solve this problem, consider constructing a sequence in a specific metric space that illustrates the properties of Cauchy sequences without reaching convergence due to the incompleteness of the space. You will need to detail your reasoning using \(\varepsilon\)-\(N\) arguments, which involves choosing an appropriate metric space and defining a sequence that meets the criteria.
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Transcribed Image Text:**Example of a Metric Space and Cauchy Sequence** **Problem:** Provide an example of a metric space \(X, d\) and a sequence \(\{p_n\} \subseteq X\) such that \(\{p_n\}\) is a Cauchy sequence in \(X, d\) but \(\{p_n\}\) does not converge in \(X, d\) (i.e., \(X, d\) is not a complete metric space). Use \(\varepsilon\)-\(N\) arguments to demonstrate that your sequence is Cauchy and that the limit is not in \(X, d\). **Explanation:** To solve this problem, consider constructing a sequence in a specific metric space that illustrates the properties of Cauchy sequences without reaching convergence due to the incompleteness of the space. You will need to detail your reasoning using \(\varepsilon\)-\(N\) arguments, which involves choosing an appropriate metric space and defining a sequence that meets the criteria.
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