**Mathematical Concept Explained****Topic: Sequences and Convergence****Explanation:**Consider a sequence \(\{x_n\}\). There can be situations where the absolute values of the terms in the sequence, \(\{|x_n|\}\), converge, but the sequence \(\{x_n\}\) itself does not converge. **Example Statement:**"A sequence such that \(\{|x_n|\}\) converges but \(\{x_n\}\) does not converge."**Details:**1. **Convergence of \(\{|x_n|\}\)**: - This implies that the magnitudes of the terms in the sequence approach a finite limit as \(n\) tends to infinity. Mathematically, \(\lim_{{n \to \infty}} |x_n| = L\), where \(L\) is some finite number.2. **Non-convergence of \(\{x_n\}\)**: - This means that the sequence itself does not approach a finite limit as \(n\) tends to infinity. In other words, \(\lim_{{n \to \infty}} x_n\) does not exist.**Conclusion:**Such situations typically arise in sequences where the terms oscillate with increasing or constant magnitude but do not settle to a single value. An example of such a sequence is \(x_n = (-1)^n \frac{1}{n}\), where the absolute values \(\left|\frac{(-1)^n}{n}\right| = \frac{1}{n}\) converge to 0, but the sequence \(\frac{(-1)^n}{n}\) does not converge since it continually alternates between positive and negative values.

Name: **Mathematical Concept Explained****Topic: Sequences and Convergence*
Uploaded: 2024-03-20T06:38:58.000Z
Channel: Bartleby
Description: **Mathematical Concept Explained****Topic: Sequences and Convergence****Explanation:**Consider a sequence \(\{x_n\}\). There can be situations where the absolute values of the terms in the sequence, \(\{|x_n|\}\), converge, but the sequence \(\{x_n\