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The problem provided asks to determine whether the given infinite series is convergent or divergent. The series is:

\[
\sum_{n=0}^{\infty} \frac{4n^2 - n^3}{10 + 2n^3}
\]

### Step-by-step Analysis:

1. **Simplifying the General Term**:  
   The general term of the series is given by:

   \[
   a_n = \frac{4n^2 - n^3}{10 + 2n^3}
   \]

   As \(n \to \infty\), the highest power of \(n\) in both the numerator and denominator dominates. The leading term in the numerator is \(-n^3\), and the leading term in the denominator is \(2n^3\). So for large \(n\), the general term behaves as:

   \[
   a_n \sim \frac{-n^3}{2n^3} = -\frac{1}{2}
   \]

2. **Test for Convergence**:  
   - For a series to converge, the terms \(a_n\) must approach 0 as \(n \to \infty\). Here, the general term approaches \(-\frac{1}{2}\), which does **not** approach 0.
   
   - Since the terms do not tend to 0, the series **diverges** by the **n-th term test for divergence**.

### Conclusion:

The given series is **divergent**.

---

Would you like further clarification or details on any part of the explanation?

Here are 5 related questions:
1. What is the n-th term test for divergence, and how is it applied?
2. How do you determine the leading term when analyzing the behavior of a series?
3. What are other common tests for convergence of series?
4. Can a series diverge even if some of its terms decrease in value?
5. What is the importance of limits in analyzing the convergence of series?

**Tip**: Always check if the general term of a series approaches zero before applying more advanced tests for convergence.

Determine if the following series is convergent or divergent: \(\sum_{n=0}^{\infty} \frac{4n^2 - n^3}{10 + 2n^3}\).

Analyze whether the infinite series \(\sum_{n=0}^{\infty} \frac{4n^2 - n^3}{10 + 2n^3}\) converges or diverges. Explore key concepts like the N-th term test for divergence. Ideal for college-level calculus students.

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