Question

Determine whether the sequence with th given \(n\) th term is monotonic. Discuss the boundedness of th sequence. Use a graphing utility to confirm your results. \(a_{n}=(-1)^{n}\left(\frac{1}{n}\right)\)

Short answer

The sequence \(a_{n}=(-1)^{n}\left(\frac{1}{n}\right)\) is not monotonic because it alternates in sign. However, it is bounded as no term in the sequence will have a magnitude greater than 1 or less than -1. The graph of the sequence will confirm this.

Step-by-step solution

Understand the Sequence

The sequence \(a_{n}=(-1)^{n}\left(\frac{1}{n}\right)\) alternates in sign because of the factor \((-1)^n\), and it also decreases in magnitude because \(1/n\) gets smaller as n increases. Thus, it's not monotonic because it does not purely increase or decrease.

Check for Boundedness

To determine the boundedness of the sequence, observe that regardless of the value of n, the maximum magnitude of any term in the sequence is 1 (when n = 1) and the minimum magnitude is 0 (as n approaches infinity). Hence, the sequence is bounded above by 1 and below by -1.

Use Graphing Utility for Verification

Deploy a graphing calculator or software to plot several terms of the sequence. It will clearly show that the sequence alternates sign and shrinks in magnitude, which verifies the results of the previous steps.