Alternating Series Convergence Test for Harmonic Series

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

In this problem, we are tasked with determining the convergence of a specific alternating series. The given series is the alternating harmonic series. One of the primary tools for analyzing the convergence of alternating series is the Alternating Series Test, sometimes known as the Leibniz Test. This test provides a straightforward way to conclude convergence by checking two conditions: that the terms decrease in absolute value, and that the limit of the terms as n approaches infinity is zero. The application of this test highlights crucial concepts in real analysis such as convergence criteria and behavior of infinite series.

Understanding alternating series requires familiarity with sequences, particularly the behavior of terms as n increases. We are essentially checking whether the "oscillations" caused by the alternating signs are counterbalancing decreases in the terms' magnitudes sufficiently to ensure convergence. The harmonic series itself, without alternation, is famously divergent, providing a neat contrast to explore how alternating signs change this characteristic.

This exploration into the convergence of alternating series opens up further considerations in the series and convergence tests subject area, including comparisons with other tests like the Ratio or Root Test, and the behavior of absolutely and conditionally convergent series, an important distinction when extending real analysis concepts towards complex analysis.