Alternating Series Test for Cosine Series

Consider the series $\frac{\cos(n \pi)}{n}$. Can we apply the alternating series test to it?

This problem concerns the series with terms given by the expression involving the cosine of n times pi divided by n. To tackle this problem, one must consider the nature of the cosine function, especially when its argument involves multiples of pi. It's important to notice that the cosine of n pi results in alternating values, depending on whether n is odd or even. This alternating behavior is key when applying the alternating series test, which is used to determine the convergence of series whose terms alternate in sign.

The alternating series test requires two conditions for convergence: the absolute value of the terms in the series must be decreasing, and the limit of the terms as n approaches infinity should be zero. In this context, addressing the question means analyzing the behavior of the terms in the given series as n increases. A thoughtful understanding of how cosine interacts with integer multiples of pi and how these interact with the division by n sheds light on whether the series satisfies these conditions.

Therefore, the problem is a practical exercise in identifying the proper application of convergence tests for series, particularly focusing on those with alternating sign, a common topic in calculus courses. It highlights the importance of being able to discern structural characteristics of series that can be exploited to determine their convergence, a fundamental aspect when studying series and the behavior of functions represented by them.