Alternating Series Test for Convergence2

Consider the series $\sum_{k=1}^{\infty} (-1)^{k+1} \frac{k+3}{k(k+1)}$. Determine if this series converges using the Alternating Series Test.

In this problem, we explore the convergence of a series using the Alternating Series Test. This test is a valuable tool when dealing with series where the terms alternate in sign. It sets out two conditions that need to be met for a series to converge: the absolute value of the terms must be decreasing, and the limit of the terms as they go to infinity must be zero. By ensuring these conditions are satisfied, we can confirm the convergence of series that might otherwise appear complex or divergent from an initial glance.

Understanding the Alternating Series Test is crucial as it's one of the most straightforward convergence tests. It gives insight into how alternating signs affect the sum of an infinite series. Furthermore, this test is foundational in analyzing series with alternating terms and preparing students for more advanced topics like absolute and conditional convergence. The problem at hand also brings attention to the behavior of rational terms and their impact on convergence, emphasizing the analytical techniques required for successful examination of series beyond simple numerical assessments.

When tackling this type of problem, it's important to systematically check the necessary conditions of the test. Establish whether the sequence of absolute values is decreasing, and compute the limit of terms as they approach infinity to ensure convergence. Such problems reinforce the importance of methodical analytical strategies and the need for precision in mathematical justification, guiding students to deepen their understanding of infinite series.