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We are tasked with determining the values of \( m \) for which the improper integral 

\[
\int_1^\infty \sin\left(\frac{1}{x^m}\right) dx
\]

is convergent, where \( m \) is a real number.

### Step 1: Analyze the behavior of the integral as \( x \to \infty \)

As \( x \to \infty \), \( \frac{1}{x^m} \to 0 \) for any \( m > 0 \), and we know that \( \sin(0) = 0 \). So, the integrand tends to zero as \( x \to \infty \), and there is no issue of divergence at the upper limit of the integral.

### Step 2: Analyze the behavior as \( x \to 1 \)

When \( x \to 1 \), the behavior of the integrand depends on \( m \):

- If \( m > 0 \), \( \frac{1}{x^m} \to 1 \), and the integral near \( x = 1 \) behaves well.
- For negative \( m \), as \( x \to 1 \), \( \frac{1}{x^m} \to \infty \), which means the sine function oscillates rapidly, potentially causing issues with convergence.

### Step 3: Analyze the behavior as \( x \to \infty \)

For large values of \( x \), \( \frac{1}{x^m} \) becomes very small, and \( \sin\left( \frac{1}{x^m} \right) \approx \frac{1}{x^m} \) for small values of the argument. The key here is determining whether the integral

\[
\int_1^\infty \frac{1}{x^m} dx
\]

converges. We know this integral converges if and only if \( m > 1 \).

### Conclusion

For the integral to converge, we need \( m > 1 \). Therefore, the correct option is:

**(D) \( m > 1 \)**.

Would you like further explanation or details?

Here are 5 related questions:
1. How do we determine the convergence of improper integrals in general?
2. Why does the integral behave differently for different values of \( m \)?
3. How would the integral change if the sine function were replaced with a different function?
4. What happens if \( m \) is negative in this problem?
5. Can you explain the behavior of oscillating functions in improper integrals?

**Tip**: Always examine the behavior of the integrand at both ends of the integration interval when dealing with improper integrals.

Find the values of m, for which the improper integral ∫₁^∞ sin(1/x^m) dx is convergent, where m is a real number.

Explore the convergence of the improper integral ∫₁^∞ sin(1/x^m) dx and determine the values of m that ensure convergence. This problem involves analysis of improper integrals, oscillating functions, and convergence tests. Suitable for students with knowledge of calculus.

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