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To determine the convergence and sum of the given series \(\sum_{n=1}^{\infty} \frac{5^n n!}{3^n (n+1)!}\), let's analyze the series.

First, simplify the term inside the summation:

\[
\frac{5^n n!}{3^n (n+1)!}
\]

Notice that \((n+1)! = (n+1) \cdot n!\), so we can rewrite the term as:

\[
\frac{5^n n!}{3^n (n+1) n!} = \frac{5^n}{3^n (n+1)}
\]

This simplifies to:

\[
\frac{\left(\frac{5}{3}\right)^n}{n+1}
\]

Now, the series becomes:

\[
\sum_{n=1}^{\infty} \frac{\left(\frac{5}{3}\right)^n}{n+1}
\]

To determine convergence, we use the ratio test. Define \(a_n = \frac{\left(\frac{5}{3}\right)^n}{n+1}\). The ratio test is given by:

\[
\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|
\]

Calculate \(a_{n+1}\):

\[
a_{n+1} = \frac{\left(\frac{5}{3}\right)^{n+1}}{(n+2)}
\]

Now, find the ratio:

\[
\left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{\frac{\left(\frac{5}{3}\right)^{n+1}}{n+2}}{\frac{\left(\frac{5}{3}\right)^n}{n+1}} \right| = \left| \frac{\left(\frac{5}{3}\right)^{n+1}}{\left(\frac{5}{3}\right)^n} \cdot \frac{n+1}{n+2} \right| = \left| \frac{5}{3} \cdot \frac{n+1}{n+2} \right|
\]

Simplify the limit:

\[
\lim_{n \to \infty} \left| \frac{5}{3} \cdot \frac{n+1}{n+2} \right| = \left| \frac{5}{3} \cdot \lim_{n \to \infty} \frac{n+1}{n+2} \right| = \left| \frac{5}{3} \cdot 1 \right| = \frac{5}{3}
\]

Since \(\frac{5}{3} > 1\), the ratio test tells us that the series diverges.

Thus, the series \(\sum_{n=1}^{\infty} \frac{5^n n!}{3^n (n+1)!}\) diverges.

Would you like more details or have any questions?

Here are five more questions you can ask:
1. How do you use the root test to determine series convergence?
2. Can you explain how to find the radius of convergence for a power series?
3. What is the difference between absolute and conditional convergence?
4. How do you test for divergence of an infinite series?
5. Can you show an example of a convergent geometric series?

**Tip:** When determining the convergence of a series, always consider applying various tests (ratio test, root test, comparison test, etc.) to confirm the behavior of the series.

Explore the convergence analysis of the series ∑ (5^n n!) / (3^n (n+1)!) using the ratio test. This detailed solution breaks down the simplification of terms and application of the ratio test to determine divergence. Suitable for advanced college students and those interested in advanced mathematical series analysis.

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