To determine if the series

$\sum_{n=0}^{\infty} \frac{5n^2 + 7}{8n^2 + 2}$

converges or diverges, we can analyze the behavior of the terms as $n \to \infty$.

Step 1: Analyzing the Terms

The general term of the series is: $a_n = \frac{5n^2 + 7}{8n^2 + 2}.$

As $n$ grows large, the terms $5n^2$ and $8n^2$ dominate the numerator and denominator, respectively. We can approximate the term by simplifying it as follows: $a_n \approx \frac{5n^2}{8n^2} = \frac{5}{8}.$

Step 2: Applying the Limit Comparison Test

Since $a_n \approx \frac{5}{8}$ as $n \to \infty$, the series does not tend to zero. This means the terms do not satisfy the necessary condition for convergence of a series, which requires $a_n \to 0$ as $n \to \infty$.

Conclusion

Since the terms do not tend to zero, the series diverges by the Test for Divergence (also known as the nth-term test for divergence).

Would you like further details on the steps or the reasoning behind each one?

Here are some related questions that might help deepen your understanding:

1. How does the Test for Divergence help determine convergence or divergence?
2. Can this series be tested using other convergence tests, like the Ratio Test?
3. Why does the behavior of the term $a_n$ as $n \to \infty$ influence the series’ convergence?
4. What is the difference between absolute convergence and conditional convergence?
5. Can you give an example of a series where the terms tend to zero, but the series still diverges?

Tip: For series involving polynomials in the numerator and denominator, consider the highest powers for large $n$ to get an idea of the term’s behavior.