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The given series is:

\[
\sum_{k=2}^{\infty} \sqrt{\frac{k^2 - k}{k^4}}
\]

We are asked to determine whether this series converges or diverges using the **Limit Comparison Test**.

### Step 1: Simplify the General Term

Let's simplify the general term \( \sqrt{\frac{k^2 - k}{k^4}} \):

\[
\frac{k^2 - k}{k^4} = \frac{k^2(1 - \frac{1}{k})}{k^4} = \frac{1 - \frac{1}{k}}{k^2}
\]

Taking the square root of this expression:

\[
\sqrt{\frac{1 - \frac{1}{k}}{k^2}} = \frac{\sqrt{1 - \frac{1}{k}}}{k}
\]

For large values of \( k \), \( 1 - \frac{1}{k} \) approaches 1, so the general term behaves like:

\[
\frac{1}{k}
\]

### Step 2: Use the Limit Comparison Test

We will compare this series with the simpler series \( \sum \frac{1}{k} \), which is the **harmonic series**. The harmonic series is known to **diverge**.

Now, let's apply the **Limit Comparison Test**:

\[
\lim_{k \to \infty} \frac{\frac{\sqrt{1 - \frac{1}{k}}}{k}}{\frac{1}{k}} = \lim_{k \to \infty} \sqrt{1 - \frac{1}{k}} = 1
\]

Since this limit is a finite, positive number (specifically, 1), the Limit Comparison Test tells us that the given series behaves like the harmonic series. Since the harmonic series diverges, the given series also **diverges**.

### Conclusion

The series \( \sum_{k=2}^{\infty} \sqrt{\frac{k^2 - k}{k^4}} \) **diverges**.

---

Would you like more details on any step of the solution, or do you have any questions?

Here are 5 related questions to explore further:

1. What is the Limit Comparison Test, and how is it applied to different series?
2. Why does the harmonic series diverge, and how is this used in series comparisons?
3. Can the Root Test or Ratio Test be applied to this series? How would they work?
4. How do you recognize when a series can be compared to the harmonic series?
5. What are other common divergent series, and how do they compare to the harmonic series?

**Tip:** When using the Limit Comparison Test, always choose a series to compare that you know well—like geometric series or p-series. This makes it easier to reach conclusions about convergence or divergence.

Use the Limit Comparison Test to determine whether the series \( \sum_{k=2}^{\infty} \sqrt{\frac{k^2 - k}{k^4}} \) converges or diverges.

Learn how to determine whether the series \( \sum_{k=2}^{\infty} \sqrt{\frac{k^2 - k}{k^4}} \) converges or diverges using the Limit Comparison Test. This solution breaks down the series' asymptotic behavior and compares it to the harmonic series, which diverges.

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