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The problem is asking about the behavior of the series:

\[
\sum_{n=1}^{\infty} \frac{n^2 - 1}{4n^3 + 1}
\]

as \( n \) approaches infinity. We need to determine whether this series converges, diverges, is indeterminate, or undefined.

### Analyzing the Series:

1. **Leading Terms Approximation:**
   - For large \( n \), the leading term in the numerator \( n^2 - 1 \) is approximately \( n^2 \).
   - Similarly, in the denominator \( 4n^3 + 1 \), the leading term is \( 4n^3 \).

   Therefore, for large \( n \), the general term of the series behaves like:

   \[
   \frac{n^2 - 1}{4n^3 + 1} \approx \frac{n^2}{4n^3} = \frac{1}{4n}
   \]

2. **Comparison with a Known Series:**
   - The expression \( \frac{1}{4n} \) resembles the general term of the harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \), which is known to diverge.

3. **Comparison Test:**
   - Since \( \frac{n^2 - 1}{4n^3 + 1} \) is asymptotically similar to \( \frac{1}{4n} \), and the harmonic series diverges, we can apply the **Comparison Test**. 
   - If a series is compared to a divergent series and the terms are eventually larger (or not significantly smaller), the series itself diverges.

### Conclusion:
The series \( \sum_{n=1}^{\infty} \frac{n^2 - 1}{4n^3 + 1} \) **diverges** as \( n \) approaches infinity.

Would you like any further explanation, or do you have additional questions?

Here are five related questions you might explore:
1. How does the Comparison Test work for determining the convergence of series?
2. What is the harmonic series, and why does it diverge?
3. Can you explain the difference between the Ratio Test and the Comparison Test?
4. How would you handle a series with alternating signs?
5. What are some common series that converge, and how can they be identified?

**Tip:** Always look at the dominant terms of both the numerator and denominator in a series to approximate its behavior for large \( n \). This can help quickly determine whether the series converges or diverges.

Explore the convergence of the series \( \sum_{n=1}^{\infty} \frac{n^2 - 1}{4n^3 + 1} \) as \( n \) approaches infinity. Understand the asymptotic behavior of the series and apply the Comparison Test to determine its convergence or divergence. Suitable for undergraduate-level study of series convergence and mathematical analysis.

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