The series provided is:

$\sum_{n=2}^{\infty} \frac{7 \sqrt{n}}{\ln(3n)}$

We are tasked with determining whether this series converges or diverges. Let's explore possible methods:

1. Nth-Term Test for Divergence:

The Nth-term test states that if the limit of the sequence does not approach 0 as $n \to \infty$, then the series must diverge.

We look at the general term of the series: $a_n = \frac{7 \sqrt{n}}{\ln(3n)}$

Taking the limit as $n \to \infty$:

$\lim_{n \to \infty} \frac{7 \sqrt{n}}{\ln(3n)} = \lim_{n \to \infty} \frac{7 \sqrt{n}}{\ln(n)}$

Since $\sqrt{n}$ grows much faster than $\ln(n)$ as $n \to \infty$, the ratio $\frac{\sqrt{n}}{\ln(n)}$ tends to infinity. Thus:

$\lim_{n \to \infty} \frac{7 \sqrt{n}}{\ln(3n)} = \infty$

Since the limit of the general term is not 0, the series diverges by the Nth-term test.

Correct Answer:

The correct choice is A. The series diverges. This is revealed by the nth-term test.

Would you like more details or have any questions? Here are five related questions:

1. What other convergence tests could be applied if the Nth-term test was inconclusive?
2. How can we compare series using the limit comparison test?
3. How does the growth rate of logarithmic functions compare to polynomial functions in series analysis?
4. What is the integral test, and how does it apply to series convergence?
5. Could the series be rewritten or approximated using another well-known divergent series?

Tip: Always consider the growth rates of functions when determining the limit of sequences in series analysis.