To analyze the expression $\frac{(-1)^n \ln(n)}{n}$, we can look into its behavior as $n$ approaches infinity and determine its convergence.

1. Alternating Nature: The term $(-1)^n$ indicates that the expression alternates in sign, being positive for even $n$ and negative for odd $n$.

2. Growth of $\ln(n)$: The natural logarithm $\ln(n)$ grows slowly compared to polynomial functions. Specifically, as $n$ increases, $\ln(n)$ increases without bound, but at a much slower rate than $n$.

3. Dominating Factor: As $n$ grows larger, the denominator $n$ will dominate the behavior of the fraction. The entire expression tends toward zero since: $\lim_{n \to \infty} \frac{\ln(n)}{n} = 0.$

4. Convergence: Given that the terms alternate and the absolute values $\left| \frac{\ln(n)}{n} \right|$ converge to 0, the series converges by the Alternating Series Test.

Conclusion

The series $\sum_{n=1}^{\infty} \frac{(-1)^n \ln(n)}{n}$ converges.

If you would like more details or have questions about specific aspects of this series, feel free to ask!

Further Questions

1. What are some examples of similar alternating series?
2. Can you explain the Alternating Series Test in more detail?
3. How does the convergence of this series compare to non-alternating series?
4. What are other functions that behave similarly to $\ln(n)$ in terms of growth?
5. Would you like to see a numerical evaluation of the first few terms of the series?

Tip: Always consider the growth rates of functions when analyzing convergence, as they can significantly influence the behavior of series.