Determine Series Convergence Using Comparison Test

Determine the convergence or divergence of the series $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n^4 + 5}}$.

In this problem, we explore the convergence or divergence of the series with terms defined as one over the square root of the fourth power of n plus a constant. A crucial strategy involves comparing this series with a more familiar or simpler series whose convergence behavior is known. This involves the application of the Comparison Test, a fundamental tool in analyzing series. The Comparison Test states that if the terms of one series are less than or equal to the terms of a convergent series for all n beyond a certain point, the series in question also converges. Conversely, if the terms are greater than or equal to those of a divergent series, divergence follows.

To apply this test effectively, consider the term inside the series. As n grows larger, the $n^4$ term will dominate the constant term, suggesting that the series behaves similarly to a simplified version. An insightful approach might involve comparing it with a p-series, specifically where p equals $1/2$, because it has a well-known divergence behavior. By finding an appropriate comparable series, we can make concrete conclusions about the original series' convergence.

Understanding these strategies not only solidifies one's grasp on series behavior analysis but also illustrates the power of simplifying complex mathematical expressions through appropriate approximations and comparisons, a technique extensively useful in calculus and real analysis. This type of problem sharpens skills in series convergence testing, a core component in higher-level mathematics courses focused on calculus and real numbers.