The problem asks to apply the Limit Comparison Test to the series:

$\sum_{n=1}^{\infty} \frac{1}{n^p}$ where $p$ is a specific value, and we are comparing it with the given terms for $a_n$ and $b_n$.

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Step 1: Define $a_n$ and $b_n$

Given:

• $a_n = 6n^{\frac{3}{2}} + 6n^{\frac{5}{2}} + 3n - 3$
• $b_n = \frac{1}{n}$ (which represents a typical harmonic series, i.e., a p-series with $p = 1$).

Step 2: Apply the Limit Comparison Test

We need to compute the following limit:

$L = \lim_{n \to \infty} \frac{a_n}{b_n}$

Substitute the expressions for $a_n$ and $b_n$:

$L = \lim_{n \to \infty} \frac{6n^{\frac{3}{2}} + 6n^{\frac{5}{2}} + 3n - 3}{\frac{1}{n}}$

This can be rewritten as:

$L = \lim_{n \to \infty} \left( n \cdot (6n^{\frac{3}{2}} + 6n^{\frac{5}{2}} + 3n - 3) \right)$

Simplifying the expression:

$L = \lim_{n \to \infty} \left( 6n^{\frac{5}{2}} + 6n^{\frac{7}{2}} + 3n^2 - 3n \right)$

For large $n$, the term with the highest power of $n$ dominates, which is $6n^{\frac{7}{2}}$. So the limit simplifies to:

$L = \lim_{n \to \infty} 6n^{\frac{7}{2}} = \infty$

Step 3: Conclusion

Since the limit is infinite, the Limit Comparison Test tells us that the series:

$\sum_{n=1}^{\infty} \left( 6n^{\frac{3}{2}} + 6n^{\frac{5}{2}} + 3n - 3 \right)$

diverges, because it behaves similarly to the harmonic series, $\sum_{n=1}^{\infty} \frac{1}{n}$, which is known to diverge.

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Questions for Further Exploration:

1. How do the powers of $n$ in the numerator and denominator affect the convergence of a series?
2. Can the Limit Comparison Test be applied when the limit is zero or infinite? What would those cases imply?
3. How does the p-series test help determine convergence for specific values of $p$?
4. What would happen if the exponent of $n$ in $b_n$ were changed to a value less than 1?
5. How can you use the Ratio Test to analyze the convergence of the series?

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Tip:

When performing the Limit Comparison Test, always focus on the terms with the highest powers of $n$ in both the numerator and denominator, as they dominate the behavior of the series as $n$ tends to infinity.