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The problem asks us to test the convergence of the series

\[
\sum_{n=1}^{\infty} \frac{\sin(n)}{n!}
\]

using the **Ratio Test**.

### Solution Using the Ratio Test

The Ratio Test states that for a series \(\sum a_n\), if

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

then:
- If \(L < 1\), the series converges absolutely.
- If \(L > 1\), the series diverges.
- If \(L = 1\), the test is inconclusive.

In this problem, we have:
\[
a_n = \frac{\sin(n)}{n!}.
\]

Let's apply the Ratio Test by finding \(\left| \frac{a_{n+1}}{a_n} \right|\):

\[
\left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{\sin(n+1)}{(n+1)!} \cdot \frac{n!}{\sin(n)} \right| = \left| \frac{\sin(n+1)}{\sin(n)} \cdot \frac{1}{n+1} \right|.
\]

#### Step-by-Step Analysis:
1. As \(n \to \infty\), \(\sin(n)\) oscillates between -1 and 1, so \(\sin(n)\) and \(\sin(n+1)\) do not grow without bound.
2. However, the factorial in the denominator, \((n+1)!\), grows very rapidly as \(n\) increases.

Because the factorial growth in the denominator (from \(\frac{1}{n+1}\)) dominates the oscillations of \(\sin(n)\), we expect this limit to approach zero as \(n \to \infty\).

Thus:
\[
\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = 0 < 1.
\]

### Conclusion
Since \(L = 0 < 1\), the Ratio Test confirms that the series \(\sum_{n=1}^{\infty} \frac{\sin(n)}{n!}\) **converges absolutely**.

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

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### Additional Related Questions
1. What happens if we test the convergence of \(\sum_{n=1}^{\infty} \frac{\cos(n)}{n!}\) instead?
2. Can the Ratio Test be used on series with only trigonometric terms (e.g., \(\sum_{n=1}^{\infty} \sin(n)\))?
3. How does the factorial in the denominator influence convergence in general?
4. What is an example of a divergent series that fails the Ratio Test?
5. How would we test convergence if we had \(\sum_{n=1}^{\infty} \frac{\sin(n)}{n^2}\) instead?

### Tip
The Ratio Test is especially useful when dealing with factorials or exponential terms in a series, as these terms tend to grow (or decay) very quickly, making it easier to apply the test effectively.

Test for convergence of the following series using the ratio test: \( \sum_{n=1}^{\infty} \frac{\sin(n)}{n!} \)

Explore the convergence of the series \( \sum_{n=1}^{\infty} \frac{\sin(n)}{n!} \) with a step-by-step solution using the Ratio Test. Learn how the factorial term in the denominator affects convergence and how the Ratio Test can confirm absolute convergence.

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