Convergence of Alternating Series2

Determine if the series $\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{4n-1}$ converges using the alternating series test.

The series in question involves alternating terms, as indicated by the factor of negative one raised to the power of n minus one. This indicates that the series changes sign with each successive term, which is a hallmark of an alternating series. The alternating series test provides a strategy for determining whether such a series converges. To apply it, we focus on two main conditions. The first condition is that the absolute value of the terms should decrease steadily as n increases, which is known as the condition of monotonicity. The second condition is that the terms approach zero in the limit as n approaches infinity.

In this problem, one would typically begin by examining the general term of the series and verifying that the terms do indeed satisfy these conditions. For the alternating series test to conclude that a series converges, it must satisfy both conditions. It is also crucial to ensure that each step of this verification is clearly laid out, providing a clear picture of why each condition is met. This process also involves understanding the behavior of the series' terms asymptotically. By visualizing or computing the sequence of partial sums, one can further solidify the intuition behind the alternating series test.