Alternating Harmonic Series Convergence

Consider the series $\sum_{k=1}^{\infty} (-1)^{k+1} \frac{1}{k}$. Prove that this alternating harmonic series converges using the Alternating Series Test.

In this problem, you are tasked with showing the convergence of the alternating harmonic series, which is a classic example in real analysis. The series is defined by an alternating set of positive and negative terms, decreasing in absolute value. To establish the convergence, you will use the Alternating Series Test, which provides a straightforward set of conditions that, if satisfied, guarantee the convergence of an alternating series. The test requires that the absolute value of the terms of the series is monotonically decreasing and that the limit of these terms as they approach infinity is zero. This example highlights the importance of understanding behavior at infinity and monotonicity—core concepts in real analysis that are crucial for mastering series and their convergence properties.