To solve this problem using the Limit Comparison Test, let's go through it step-by-step.

Step 1: Identify the series $a_n$

The series given is: $\sum_{n=1}^{\infty} a_n = \sum_{n=1}^{\infty} \frac{9n^2 + 9n - 1}{(n+2)^4}.$

Step 2: Choose a comparison series $b_n$

To apply the Limit Comparison Test, we need to find a suitable comparison series $b_n$. We should choose $b_n$ to be a simpler series that has a similar growth rate to $a_n$ as $n \to \infty$.

Notice that the highest power term in the numerator of $a_n$ is $9n^2$ and in the denominator is $(n+2)^4$, which approximates to $n^4$ as $n$ becomes large. Thus, a good choice for $b_n$ would be: $b_n = \frac{1}{n^2}.$

This choice simplifies the comparison because $\frac{1}{n^2}$ is a known $p$-series with $p = 2$.

Step 3: Calculate $\lim_{n \to \infty} \frac{a_n}{b_n}$

Now, we calculate the limit: $\lim_{n \to \infty} \frac{a_n}{b_n} = \lim_{n \to \infty} \frac{\frac{9n^2 + 9n - 1}{(n+2)^4}}{\frac{1}{n^2}} = \lim_{n \to \infty} \frac{9n^2 + 9n - 1}{(n+2)^4} \cdot n^2.$

Simplifying further: $= \lim_{n \to \infty} \frac{9n^4 + 9n^3 - n^2}{(n+2)^4}.$

As $n \to \infty$, the highest power terms dominate, so we approximate by focusing on $\frac{9n^4}{n^4} = 9$. Therefore: $\lim_{n \to \infty} \frac{a_n}{b_n} = 9.$

Since this limit is a positive finite number, the Limit Comparison Test applies.

Step 4: Determine convergence of $b_n$

The comparison series $\sum b_n = \sum \frac{1}{n^2}$ is a $p$-series with $p = 2$, which is known to converge (since $p > 1$).

Conclusion

Since $\sum b_n$ converges and $\lim_{n \to \infty} \frac{a_n}{b_n}$ is finite and positive, the original series $\sum a_n$ also converges by the Limit Comparison Test.

Answer

1. Comparison series $b_n = \frac{1}{n^2}$.
2. Value of $p$ in the comparison series: $p = 2$.
3. Limit $\lim_{n \to \infty} \frac{a_n}{b_n} = 9$.
4. $\sum b_n$ is a convergent $p$-series.
5. Therefore, $\sum a_n$ converges by the Limit Comparison Test.

Would you like more details on any of these steps or have additional questions?

Related Questions

1. How do you choose a comparison series for the Limit Comparison Test?
2. What are the conditions required for a $p$-series to converge?
3. Can the Limit Comparison Test be applied if the limit $\lim_{n \to \infty} \frac{a_n}{b_n}$ is zero or infinity?
4. What is the difference between the Limit Comparison Test and the Direct Comparison Test?
5. How do you determine the dominant terms in a series to select an appropriate $b_n$?

Tip

For the Limit Comparison Test, always focus on the highest power terms in the numerator and denominator of $a_n$ to find an appropriate $b_n$.