To show that the sequence \(\frac{9^n}{n!}\) converges, we will use the Ratio Test, which is a common method for determining the convergence of sequences and series. The Ratio Test states that a sequence \(\{a_n\}\) converges if the limit 

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

is less than 1.

Given the sequence \(\frac{9^n}{n!}\), let's denote \(a_n = \frac{9^n}{n!}\). We will compute the ratio \(\left| \frac{a_{n+1}}{a_n} \right|\):

\[
\frac{a_{n+1}}{a_n} = \frac{\frac{9^{n+1}}{(n+1)!}}{\frac{9^n}{n!}} = \frac{9^{n+1}}{(n+1)!} \cdot \frac{n!}{9^n} = \frac{9 \cdot 9^n}{(n+1) \cdot 9^n} = \frac{9}{n+1}
\]

Next, we find the limit as \(n\) approaches infinity:

\[
L = \lim_{n \to \infty} \left| \frac{9}{n+1} \right| = \lim_{n \to \infty} \frac{9}{n+1} = 0
\]

Since \(L = 0 < 1\), the Ratio Test confirms that the sequence \(\frac{9^n}{n!}\) converges.

Therefore, the sequence \(\frac{9^n}{n!}\) converges.

Would you like details or have any questions about this solution? Here are some related questions:

1. What is the Ratio Test, and how is it applied to sequences and series?
2. How does the Factorial function \(n!\) grow compared to exponential functions like \(9^n\)?
3. Can the Ratio Test be used for series as well, and if so, how?
4. What is the general behavior of the sequence \(\frac{k^n}{n!}\) for any constant \(k\)?
5. Are there other tests for convergence that can be used for sequences or series?
6. How does the limit comparison test differ from the ratio test?
7. Why is the factorial function a strong factor in determining the convergence of sequences involving \(n!\)?
8. What would happen if the sequence were \(\frac{9^n}{(n+1)!}\) instead of \(\frac{9^n}{n!}\)?

**Tip:** When determining the convergence of a sequence or series, consider the growth rates of the numerator and the denominator separately. Factorials grow much faster than exponential functions, which often leads to convergence in sequences involving ratios of these two.

Learn how to prove the convergence of the sequence \( \frac{9^n}{n!} \) using the Ratio Test in mathematics. This sequence analysis involves understanding exponential growth, factorial functions, and the application of the Ratio Test to determine convergence.

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