The image contains a set of infinite series for convergence analysis using comparison tests. Let’s start by analyzing one of these series. You can select a specific problem, and I will solve it step by step with an explanation.

For example, do you want to begin with:

• $13.29$: $\sum_{n=1}^\infty \frac{2n}{3n^2 - 5}$,
• $13.31$: $\sum_{n=1}^\infty \frac{\sqrt{n^2 + 7}}{n^5 + 12}$,
• or any other?

Let me know if you'd like me to pick one for demonstration! Would you also like me to solve multiple problems or provide detailed theory for comparison tests?

5 Related Questions:

1. What is the comparison test for convergence, and how does it work?
2. Can we determine convergence using simpler terms or bounds?
3. How do improper fractions behave in infinite series?
4. What are the alternative convergence tests (e.g., ratio, root)?
5. How does the behavior of sine or logarithmic terms affect series convergence?

Tip:

Always simplify terms in the numerator and denominator to approximate the behavior of the series for large $n$.