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To determine whether the series \( \sum_{n=7}^{\infty} \frac{n!}{n^n} \) is convergent, we can apply the Ratio Test.

The Ratio Test states that for a series \( \sum a_n \), if we compute
\[
L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|
\]
and find that:
1. \( L < 1 \), then the series converges absolutely.
2. \( L > 1 \), then the series diverges.
3. \( L = 1 \), the test is inconclusive.

### Applying the Ratio Test
Let \( a_n = \frac{n!}{n^n} \).

1. **Calculate \( a_{n+1} \):**
   \[
   a_{n+1} = \frac{(n+1)!}{(n+1)^{n+1}}
   \]

2. **Set up \( \frac{a_{n+1}}{a_n} \):**
   \[
   \frac{a_{n+1}}{a_n} = \frac{\frac{(n+1)!}{(n+1)^{n+1}}}{\frac{n!}{n^n}} = \frac{(n+1)! \cdot n^n}{n! \cdot (n+1)^{n+1}}
   \]

3. **Simplify \( \frac{a_{n+1}}{a_n} \):**
   Since \( (n+1)! = (n+1) \cdot n! \), we have:
   \[
   \frac{a_{n+1}}{a_n} = \frac{(n+1) \cdot n^n}{(n+1)^{n+1}} = \frac{n^n}{(n+1)^n}
   \]

4. **Take the limit as \( n \to \infty \):**
   \[
   L = \lim_{n \to \infty} \frac{n^n}{(n+1)^n} = \lim_{n \to \infty} \left( \frac{n}{n+1} \right)^n
   \]

5. **Evaluate the limit:**
   Note that \( \frac{n}{n+1} = 1 - \frac{1}{n+1} \), so
   \[
   L = \lim_{n \to \infty} \left( 1 - \frac{1}{n+1} \right)^n = \lim_{n \to \infty} e^{-\frac{n}{n+1}} = e^{-1} = \frac{1}{e}
   \]

Since \( L = \frac{1}{e} < 1 \), the series converges by the Ratio Test.

### Answer
- **Behavior**: Convergent
- **Primary Reason**: Ratio Test
- **Secondary Reason**: None needed

Would you like further details or have any questions?

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### Follow-Up Questions:
1. How is the Ratio Test generally applied to series with factorial terms?
2. Why does \( \left( \frac{n}{n+1} \right)^n \) tend to \( e^{-1} \) as \( n \to \infty \)?
3. What are the implications of the Ratio Test result if \( L = 1 \)?
4. Can we use other convergence tests for this series, like the Root Test?
5. How would the result change if the series had a different growth rate in the denominator?

**Tip:** When using the Ratio Test, factorials in the terms often simplify the limit calculations due to their rapid growth.

Determine whether the series \( \sum_{n=7}^{\infty} \frac{n!}{n^n} \) is convergent or divergent, and explain using the Ratio Test.

Explore the convergence of the series \( \sum_{n=7}^{\infty} \frac{n!}{n^n} \) by applying the Ratio Test. This problem involves series convergence, factorial notation, and the use of the Ratio Test, ideal for undergraduate calculus students.

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