Series Convergence Using Comparison Test

Prove that the sum from $N=1$ to $\infty$ of $\frac{1}{N^3 + 4}$ is convergent or divergent using the comparison test.

The comparison test is a powerful tool in real analysis for determining the convergence or divergence of infinite series. In this problem, we are tasked with analyzing the convergence of the series where the general term is given by one over N cubed plus four. The comparison test involves finding another series with known convergence properties, which can effectively serve as an upper or lower bound to our given series. The critical idea is to compare the given series to a more familiar p-series or geometric series.

As we investigate the convergence or divergence of this series, it is important to recognize the behavior of the terms as N approaches infinity. A well-structured approach involves simplifying the expression to a form where a comparison with a p-series, specifically one over N to the power of p, becomes apparent. For this specific problem, identifying a suitable p-value is necessary to make a valid comparison. For p-series, if p is greater than one, the series converges, whereas it diverges for p less than or equal to one.

Furthermore, key to using the comparison test is rigorously justifying the choice of the series used for comparison and ensuring the inequality between the terms holds for all relevant N. This provides a conceptual understanding that while numerical values and inequalities are pivotal, the underlying analytical reasoning assures the integrity of the results. Determining convergence through the comparison test not only illustrates the behavior of infinite series but also deepens understanding of various convergence criteria within the broader context of real analysis.