Convergence of the Alternating Harmonic Series

Consider the series that starts from one goes to infinity: $(-1)^n \cdot \frac{1}{n}$. Will the alternating harmonic series converge or diverge?

The problem centers on determining whether the alternating harmonic series converges or diverges. In analyzing such series, it's essential to understand some basic properties of alternating series in general. An alternating series is simply a series whose terms alternate in sign. To determine the convergence of such a series, one can often apply the Alternating Series Test. This test states that if the absolute value of the terms in the series decreases monotonically (i.e., each term is less than the one before it) and if the limit of the terms is zero as they approach infinity, then the series converges.

For the alternating harmonic series, you need to observe that the sequence 1/n does indeed decrease monotonically and approaches zero as n tends to infinity. These two observations satisfy the conditions of the Alternating Series Test, hence the series converges. This convergence is not absolute, however. Absolute convergence would require the series of absolute values, in this case, the harmonic series which diverges, to converge. This example illustrates the subtleties involved in working with infinite series and emphasizes the importance of tests for convergence, which are crucial in both pure and applied mathematical analysis. Understanding these fundamental aspects allows one to tackle more complex problems involving the convergence of various types of series in calculus.