To determine whether the series \( a_n = \frac{8n!}{8^n} \) is convergent or divergent, we can use the **Ratio Test**. This test is useful for determining the convergence of series with factorials or exponential terms.

### Applying the Ratio Test

The Ratio Test involves the following limit:

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

1. Start with the expression for \( a_n \) and \( a_{n+1} \):

\[
a_n = \frac{8n!}{8^n}, \quad a_{n+1} = \frac{8(n+1)!}{8^{n+1}}
\]

2. Compute the ratio \( \frac{a_{n+1}}{a_n} \):

\[
\frac{a_{n+1}}{a_n} = \frac{8(n+1)!}{8^{n+1}} \div \frac{8n!}{8^n} = \frac{8(n+1)!}{8^{n+1}} \times \frac{8^n}{8n!}
\]

This simplifies to:

\[
\frac{a_{n+1}}{a_n} = \frac{(n+1)!}{8 \cdot n!} = \frac{n+1}{8}
\]

3. Now, take the limit of this expression as \( n \to \infty \):

\[
L = \lim_{n \to \infty} \left| \frac{n+1}{8} \right| = \lim_{n \to \infty} \frac{n+1}{8} = \infty
\]

### Conclusion:
Since the limit \( L = \infty \), which is greater than 1, the Ratio Test tells us that the series **diverges**.

Would you like more details, or do you have any questions?

Here are 5 related questions:
1. What does the Ratio Test imply when \( L < 1 \)?
2. How do factorials affect the convergence of a series?
3. Can a series with factorials ever converge?
4. What is the root test, and how does it compare to the ratio test?
5. How does an exponential term in a series influence convergence?

**Tip:** The Ratio Test is especially useful for series involving factorials and exponentials due to their rapid growth rates.

Explore the convergence or divergence of the series an = 8n!/8^n using the Ratio Test. Understand how factorials and exponential functions impact series behavior and learn to apply the Ratio Test to determine divergence in this advanced calculus problem.

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