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To determine the convergence of the series 

\[
\sum_{n=2}^{\infty} (-1)^n \sin\left(\frac{1}{n}\right)
\]

we can apply the **Alternating Series Test** (Leibniz’s Test). According to this test, an alternating series of the form \(\sum (-1)^n a_n\) converges if:

1. \(a_n\) is positive.
2. \(a_n\) is monotonically decreasing.
3. \(\lim_{n \to \infty} a_n = 0\).

### Step-by-Step Analysis

1. **Define \( a_n \):**  
   Here, \( a_n = \sin\left(\frac{1}{n}\right) \). Since \(\sin\left(\frac{1}{n}\right) > 0\) for all \(n \geq 2\), \( a_n \) is positive.

2. **Check Monotonicity:**  
   For large \(n\), \(\frac{1}{n}\) becomes small, and \(\sin\left(\frac{1}{n}\right)\) is approximately \(\frac{1}{n}\) (since \(\sin(x) \approx x\) when \(x\) is close to zero). This implies that \(\sin\left(\frac{1}{n}\right)\) is decreasing as \(n\) increases, so \(a_n\) is monotonically decreasing.

3. **Limit of \( a_n \):**  
   \[
   \lim_{n \to \infty} \sin\left(\frac{1}{n}\right) = \sin(0) = 0.
   \]

### Conclusion

All conditions of the Alternating Series Test are satisfied. Therefore, the series

\[
\sum_{n=2}^{\infty} (-1)^n \sin\left(\frac{1}{n}\right)
\]

**converges** by the Alternating Series Test.

Would you like more details on the convergence, or do you have any further questions?

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Here are some related questions for further exploration:

1. How would the convergence be affected if the terms were not alternating?
2. Can we determine if this series converges absolutely or conditionally?
3. How does the behavior of \( \sin\left(\frac{1}{n}\right) \) relate to \( \frac{1}{n} \) for large \(n\)?
4. Could we use the Integral Test for a non-alternating version of this series?
5. What happens to the series if we start the summation from \( n=1 \) instead of \( n=2 \)?

**Tip:** For alternating series, confirming that the terms go to zero is crucial; if they don’t, the series cannot converge.

Use alternating test on this: \(\sum_{n=2}^{\infty} (-1)^n \sin\left(\frac{1}{n}\right)\)

Explore the convergence of the series \(\sum_{n=2}^{\infty} (-1)^n \sin\left(\frac{1}{n}\right)\) using the Alternating Series Test (Leibniz’s Test). This problem demonstrates key mathematical concepts, including series convergence and the behavior of trigonometric functions for large inputs.

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