Convergence of Alternating Series with Logarithmic Denominator

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

The given series is an example of an alternating series, where the terms alternate in sign due to the factor $(-1)^n$. The key to determining the convergence of such series lies in the application of the Alternating Series Test, which provides conditions under which an alternating series will converge. According to this test, for a series in the form of $(-1)^n \cdot a_n$ to converge, the sequence $a_n$ must be positive, decreasing, and approach zero as $n$ approaches infinity.

In this problem, $a_n$ is given by $\frac{n}{\ln(n)}$, which is indeed positive for $n$ greater than or equal to 3. However, analyzing whether the sequence is decreasing and approaches zero is less straightforward due to the presence of the $\frac{n}{\ln(n)}$ factor.

Additionally, the logarithmic function $\ln(n)$ grows without bound but at a slower rate compared to linear functions, adding complexity to determining the nature of the sequence. It's essential to evaluate the behavior of $a_n$ carefully under these conditions.

Exploring how $n$ compares to $\ln(n)$ and considering L'Hôpital's Rule could provide insights into the series' behavior as $n$ increases. Ultimately, understanding the relative growth rates of the numerator and denominator in the context of the convergence test is pivotal in this analysis.

This problem also allows for a deeper understanding of how alternating series function and what conditions lead them to converge or diverge, giving insight into the broader topic of series convergence.