The given series is (7n^3-2)/(7n^2+4n^5+10)

To determine if the series \(\sum_{n=1}^\infty \frac{7n^3 - 2}{7n^2 + 4n^5 + 10}\) converges or diverges, we'll use the Limit Comparison Test.

### Given Series:
\[
a_n = \frac{7n^3 - 2}{7n^2 + 4n^5 + 10}
\]

### Step-by-Step Solution:
1. **Simplify the Series:**
   Identify the dominant terms in the numerator and denominator for large \(n\). Here, the dominant term in the numerator is \(7n^3\) and in the denominator is \(4n^5\).

   So, for large \(n\):
   \[
   a_n \approx \frac{7n^3}{4n^5} = \frac{7}{4n^2}
   \]

2. **Choose a Comparison Series:**
   Based on the simplification, a natural choice for the comparison series is:
   \[
   b_n = \frac{1}{n^2}
   \]

3. **Apply the Limit Comparison Test:**
   Compute the limit of \(\frac{a_n}{b_n}\):
   \[
   \lim_{n \to \infty} \frac{a_n}{b_n} = \lim_{n \to \infty} \left( \frac{7n^3 - 2}{7n^2 + 4n^5 + 10} \cdot n^2 \right) = \lim_{n \to \infty} \frac{7n^5 - 2n^2}{7n^2 + 4n^5 + 10}
   \]

4. **Simplify the Limit Expression:**
   Divide the numerator and denominator by \(n^5\):
   \[
   = \lim_{n \to \infty} \frac{7 - \frac{2}{n^3}}{\frac{7}{n^3} + 4 + \frac{10}{n^5}} = \frac{7 - 0}{0 + 4 + 0} = \frac{7}{4}
   \]

5. **Interpret the Result:**
   The limit \(\frac{7}{4}\) is a finite, positive number. Since \(0 < \frac{7}{4} < \infty\), the Limit Comparison Test tells us that both series \(\sum a_n\) and \(\sum b_n\) either both converge or both diverge.

6. **Determine Convergence or Divergence:**
   The series \(\sum \frac{1}{n^2}\) is a p-series with \(p = 2 > 1\), and it converges. Therefore, by the Limit Comparison Test, the given series \(\sum \frac{7n^3 - 2}{7n^2 + 4n^5 + 10}\) also converges.

### Final Answer:
The series \(\sum_{n=1}^\infty \frac{7n^3 - 2}{7n^2 + 4n^5 + 10}\) converges.

Would you like further details or have any questions?

### Relative Questions:
1. How do you identify the dominant terms in a series for large \(n\)?
2. Why is \(\sum \frac{1}{n^2}\) a common comparison series?
3. What are the steps to apply the Limit Comparison Test?
4. What other tests can be used to determine the convergence of a series?
5. How does the value of the limit \(\frac{a_n}{b_n}\) affect the conclusion of the Limit Comparison Test?
6. Why is it important to choose a comparison series that is simple?
7. Can the Limit Comparison Test be applied if the terms are not positive?
8. What are the characteristics of a p-series and how does its convergence depend on the value of \(p\)?

### Tip:
When simplifying the terms in a series for the Limit Comparison Test, always focus on the highest power terms in both the numerator and the denominator, as they will dominate the behavior of the series for large \(n\).

Explore the convergence of the series ∑(7n^3 - 2)/(7n^2 + 4n^5 + 10) using the Limit Comparison Test. Understand step-by-step how to apply the test, simplify terms, and interpret results effectively.

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