Convergence of Series with Terms Involving Cubic Denominators

Determine the convergence or the divergence of the series $\sum_{n=1}^{\infty} \frac{1}{n^3 + 5}$.

When determining the convergence or divergence of a series, it's important to identify the characteristics of the series and apply the appropriate convergence tests. In this case, we are examining the infinite series with terms 1 over n cubed plus 5. This series belongs to a broader category of p-series, where you might normally apply tests like the p-test or comparison test to determine convergence. Generally, a p-series of the form 1 over n to the power of p converges if p is greater than 1. However, the presence of an additional constant in the denominator complicates the direct application of these tests.

An effective strategy here would involve transforming or comparing the given series with a known p-series. This can be achieved by using the comparison test, which allows you to compare it with a simpler series that you already know converges or diverges based on its characteristics. Alternatively, the limit comparison test might be useful if finding a precise comparative series. These tests collectively enable the determination of whether the tail terms of your series are behaving similarly to a known divergent or convergent series.

Understanding and selecting the right convergence test is crucial for rigorous mathematical proof. Each test provides insights about different series' behavior depending on their form. Crucially, you need to manipulate the series appropriately, allowing you to extract meaningful comparisons that confirm convergence or divergence based on previously studied examples. Building these skills helps in dealing with more complex infinite series and understanding their behavior over the natural numbers.