Alternating Series Convergence Test for Series Involving Fractions

Apply the alternating series test to different series to determine convergence or divergence: $(-1)^n \cdot \frac{3n-1}{2n+1}$ and $(-1)^{n+1} \cdot \frac{n^2}{n^3+4}$.

The alternating series test is a fundamental tool in analyzing infinite series that alternate in sign. When handling series such as these, where terms oscillate between positive and negative, our primary goal is to determine whether the series converges or diverges. This test is particularly useful because, for an alternating series to converge, we only need to check two conditions: the absolute value of the terms must be decreasing, and the limit of the terms as n approaches infinity must be zero.

For the series in this problem, focus on analyzing the behavior of the fractions involved. It's crucial to understand how the numerator and denominator influence the sequence. In these cases, recognize the roles of polynomial degrees: the degree of the polynomial in the numerator versus the denominator will largely dictate the term's behavior as n approaches infinity. Understanding dominance of terms in this context is key.

Beyond just applying the test mechanically, appreciating the convergence process offers deeper insights into series behavior and helps in identifying more complex situations where convergence might not be decided simply. By thoroughly examining series within the framework of the alternating series test, learners develop a rigorous understanding of sequence convergence that extends to broader analyses in series theory.