Alternating Series Convergence Check

Determine if the series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{2n-1}$ converges or diverges using the alternating series test.

The given series is an example of an alternating series, which is characterized by terms that alternate in sign. The alternating series test, sometimes known as the Leibniz test, provides a way to determine the convergence of such series. According to this test, an alternating series converges if the absolute value of the terms decreases monotonically to zero, and each term is positive when the negative sign is ignored.

This problem is particularly interesting because it challenges you to apply a specific convergence test which requires checking two essential conditions. The first condition demands that the sequence of absolute values of the terms, disregarding their sign, must be monotonically decreasing. The second condition is that the limit of the sequence of absolute values must approach zero as n approaches infinity.

This problem also introduces the idea that not all convergence tests are applicable to every series. In this case, recognizing the presence of alternating terms narrows down the list of potential tests, highlighting the importance of series classification in real analysis. Successfully solving this problem not only develops your analytical capabilities but also prepares you to handle more complex series in future exercises and courses.