To determine the convergence or divergence of the series

$\sum_{n=7}^{\infty} \frac{7n^3 - 7n^2 + 7}{3 + 3n^4}$

using the limit comparison test, we proceed as follows:

1. Identify the Dominant Terms: We need to choose a simpler series $b_n$ for comparison, typically by identifying the dominant terms in the numerator and denominator of $a_n = \frac{7n^3 - 7n^2 + 7}{3 + 3n^4}$ as $n \to \infty$.

   • In the numerator $7n^3 - 7n^2 + 7$, the term $7n^3$ is dominant.
   • In the denominator $3 + 3n^4$, the term $3n^4$ is dominant.

   Therefore, as $n \to \infty$, $a_n \approx \frac{7n^3}{3n^4} = \frac{7}{3n}$.

   Thus, we choose $b_n = \frac{1}{n}$, which is a divergent $p$-series (since $\sum \frac{1}{n}$ diverges).

2. Apply the Limit Comparison Test: According to the limit comparison test, we compute

   $\lim_{n \to \infty} \frac{a_n}{b_n} = \lim_{n \to \infty} \frac{\frac{7n^3 - 7n^2 + 7}{3 + 3n^4}}{\frac{1}{n}} = \lim_{n \to \infty} \frac{n(7n^3 - 7n^2 + 7)}{3 + 3n^4}.$

3. Simplify the Expression: Expanding this,

   $\lim_{n \to \infty} \frac{7n^4 - 7n^3 + 7n}{3 + 3n^4}.$

   Dividing both the numerator and the denominator by $n^4$ to simplify, we get

   $\lim_{n \to \infty} \frac{7 - \frac{7}{n} + \frac{7}{n^3}}{\frac{3}{n^4} + 3} = \frac{7}{3}.$

4. Interpret the Result: The limit is a positive finite constant ($\frac{7}{3}$), so by the limit comparison test, $\sum a_n$ and $\sum b_n$ will either both converge or both diverge.

Since $\sum \frac{1}{n}$ diverges, it follows that $\sum a_n$ also diverges.

Conclusion

The series $\sum_{n=7}^{\infty} \frac{7n^3 - 7n^2 + 7}{3 + 3n^4}$ diverges.

Would you like a more detailed breakdown of any steps?

Here are 5 related questions to explore further:

1. How does the limit comparison test differ from the direct comparison test?
2. What are $p$-series, and why does $\sum \frac{1}{n}$ diverge?
3. Can the ratio test be used to determine convergence for this series?
4. What other series tests could be applied to this series, if any?
5. How would the solution change if the numerator and denominator terms were different?

Tip: In the limit comparison test, always choose a comparison term that matches the behavior of the dominant terms in $a_n$ as $n \to \infty$.