Convergence of an Alternating Series

Consider the series: $\frac{(-1)^{n+1}}{5^n}$. Will the series converge or diverge?

The given series can be analyzed using the Alternating Series Test, which stipulates certain conditions under which an alternating series converges. These series are integral to understanding advanced calculus, especially when considering functions expressed as infinite sums. The series in question, with its terms decreasing steadily and tending to zero, suggests convergence, which reflects a fundamental behavior of alternating series that can counterbalance divergence tendencies present in other series types. Recognizing the conditions for convergence—such as ensuring the terms decrease steadily and approach zero—is crucial for mastering series in calculus and understanding the interplay between series terms and their limits. Furthermore, investigating the absolute convergence by examining whether a series' absolute values converge is a related concept that often arises in discussions around alternating series. This approach differentiates itself from the absolute convergence criterion, providing a different path for converging series understanding and application.