Convergence of Alternating Series with Polynomial Terms

Determine if the series $\frac{(-1)^{n}n^3}{n^3 + 5}$ is absolutely convergent, conditionally convergent, or divergent.

This problem is a great opportunity to delve into the study of series, focusing on the concept of convergence, particularly in the context of alternating series. When analyzing a series like this, you first want to consider its absolute convergence. In this case, this involves taking the absolute value of each term in the series and then analyzing the resulting non-alternating series for convergence. If this series does not converge, then the next step is to consider the original series for conditional convergence using the Alternating Series Test, which may indicate that the series converges due to its alternating nature even if the absolute series does not.

Understanding the difference between absolute and conditional convergence is critical in real analysis. Absolute convergence implies that the series will remain convergent under any rearrangement of its terms, which is not necessarily true for conditionally convergent series. For this problem, it’s essential to compare the given series with known convergent series, often through application of convergence tests such as the Ratio Test or the Comparison Test, which can provide insights into the behavior of the series as n approaches infinity. Additionally, considering the terms' behavior, especially their polynomial nature, as n grows, allows you to establish asymptotic characteristics which influence convergence properties. This problem will help reinforce the understanding of these high-level conceptual tools and improve problem-solving strategies in analyzing series.