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To analyze whether the given series

\[
\sum_{n=1}^{\infty} \frac{\sqrt{3n^3 + 15n + 1}}{4n^9 + 4n}
\]

is convergent or divergent, we need to choose an appropriate convergence test. Here’s a structured approach to evaluating this series:

1. **Identify the Dominant Terms**:
   - The dominant term in the numerator, \(\sqrt{3n^3 + 15n + 1}\), is \(\sqrt{3n^3} = n^{3/2}\).
   - The dominant term in the denominator, \(4n^9 + 4n\), is \(4n^9\).

   This simplifies the terms to \(\frac{n^{3/2}}{n^9} = \frac{1}{n^{15/2}}\) for large \(n\).

2. **Choose a Convergence Test**:
   - Since the series resembles a \(p\)-series, we can use the **Limit Comparison Test** with \(b_n = \frac{1}{n^{15/2}}\).

3. **Applying the Limit Comparison Test**:
   - Calculate \(\lim_{n \to \infty} \frac{a_n}{b_n}\), where \(a_n = \frac{\sqrt{3n^3 + 15n + 1}}{4n^9 + 4n}\) and \(b_n = \frac{1}{n^{15/2}}\).
   - If this limit is finite and positive, then the convergence behavior of \(\sum a_n\) will match that of \(\sum b_n\), which converges for \(p = 15/2 > 1\).

Let me know if you would like to see the calculations for the limit or further breakdown of the steps. 

### Follow-Up Questions
1. What are the steps in finding the dominant term in a series?
2. How does the Limit Comparison Test work in identifying convergence?
3. Why does the \(p\)-series converge for \(p > 1\)?
4. Can this series be tested with the Integral Test instead?
5. How can you apply the Ratio Test to identify the behavior of a series?

#### Tip
The Limit Comparison Test is particularly useful when the terms of the series have similar forms, especially for complex expressions involving roots and powers.

Determine whether the series \(\sum_{n=1}^{\infty} \frac{\sqrt{3n^3 + 15n + 1}}{4n^9 + 4n}\) is convergent or divergent.

Learn how to determine the convergence or divergence of the series \(\sum_{n=1}^{\infty} \frac{\sqrt{3n^3 + 15n + 1}}{4n^9 + 4n}\) using various tests such as the Limit Comparison Test and the Integral Test. Suitable for undergraduate calculus students.

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