To determine whether the series

$\sum_{n=1}^{\infty} \frac{7n^{3/2} + 4n}{9n^2 - 3n + 1}$

converges or diverges, we will use the Limit Comparison Test by comparing this series with a simpler series. The Limit Comparison Test is particularly useful when the terms of the series have a similar form to a basic $p$-series or a geometric series.

Step 1: Analyzing the Dominant Terms

The terms in both the numerator and denominator have leading powers that dominate the behavior of the fraction as $n \to \infty$.

• In the numerator, the dominant term is $7n^{3/2}$.
• In the denominator, the dominant term is $9n^2$.

Thus, for large $n$, the term

$\frac{7n^{3/2} + 4n}{9n^2 - 3n + 1} \approx \frac{7n^{3/2}}{9n^2} = \frac{7}{9} \cdot \frac{1}{n^{1/2}}.$

So, we will compare the given series to the series

$\sum \frac{1}{n^{1/2}}.$

Step 2: Choosing the Comparison Series

The series $\sum \frac{1}{n^{1/2}}$ is a $p$-series with $p = \frac{1}{2}$. Since $p \leq 1$, the $p$-series $\sum \frac{1}{n^{1/2}}$ diverges.

Step 3: Applying the Limit Comparison Test

To apply the Limit Comparison Test, we find the limit

$\lim_{n \to \infty} \frac{\frac{7n^{3/2} + 4n}{9n^2 - 3n + 1}}{\frac{1}{n^{1/2}}} = \lim_{n \to \infty} \frac{7n^{3/2} + 4n}{9n^2 - 3n + 1} \cdot n^{1/2}.$

Simplifying this expression:

$= \lim_{n \to \infty} \frac{7n^{3/2} \cdot n^{1/2} + 4n \cdot n^{1/2}}{9n^2 - 3n + 1} = \lim_{n \to \infty} \frac{7n^2 + 4n^{3/2}}{9n^2 - 3n + 1}.$

Now, divide the numerator and denominator by $n^2$:

$= \lim_{n \to \infty} \frac{7 + \frac{4}{n^{1/2}}}{9 - \frac{3}{n} + \frac{1}{n^2}}.$

As $n \to \infty$, the terms $\frac{4}{n^{1/2}}$, $\frac{3}{n}$, and $\frac{1}{n^2}$ approach zero. Thus, we have

$= \frac{7}{9}.$

Step 4: Conclusion

Since $\frac{7}{9}$ is a positive, finite constant, the Limit Comparison Test tells us that the behavior of our series is the same as that of $\sum \frac{1}{n^{1/2}}$, which diverges. Therefore, the given series

$\sum_{n=1}^{\infty} \frac{7n^{3/2} + 4n}{9n^2 - 3n + 1}$

diverges.

Would you like further details or have any questions?

Here are some additional questions to deepen your understanding:

1. What is the Limit Comparison Test, and how does it apply to series with polynomial terms?
2. How do we determine the dominant term in a rational expression for series comparison?
3. What is a $p$-series, and how does its convergence depend on $p$?
4. Why is the Limit Comparison Test more effective than the Direct Comparison Test in some cases?
5. How can you identify the convergence behavior of a series by just looking at the dominant terms?

Tip: When applying the Limit Comparison Test, focus on the highest power terms in both the numerator and the denominator to simplify your analysis.