Determining Convergence of Trigonometric Series

Determine if the series $\frac{\cos(n\pi)}{n^3}$ is absolutely convergent, conditionally convergent, or divergent.

This problem involves determining the convergence of a series with trigonometric terms. In this case, the series involves cosine of n times pi divided by n cubed. When addressing series in real analysis, the strategy typically involves checking absolute convergence first, as absolute convergence implies convergence. For the series in question, the terms cosine of n pi alternate between positive and negative one. This is a key property of the cosine function for multiples of pi and influences whether the series can be classified as absolutely convergent or conditionally convergent.

Analyzing the series's absolute convergence involves examining the absolute values of the series terms, which in this case simplifies to one over n cubed. You can use the p-series test for convergence, which states that a series of the form one over n to the power of p, with p greater than one, is convergent. Given that three is indeed greater than one, the p-series test confirms that the series formed by taking the absolute values of the terms is convergent.

Thus, this series is absolutely convergent, because the p-series convergence criterion is satisfied. Exploring this solution deepens understanding of the mechanisms that dictate series behavior in real analysis, offering insight not only into the convergence of individual series but also into broader concepts about boundedness, absolute convergence, and the interplay of trigonometric functions within mathematical sequences.