It looks like the image contains a mathematical problem related to verifying the divergence or convergence of a series using specific tests. Here's the analysis and steps to solve the problem based on the details visible:

Problem Overview

The question involves determining whether a series converges or diverges using the Comparison Test or the Limit Comparison Test. These tests are used to compare the given series with another known series to establish convergence or divergence.

General Strategy:

1. Identify the series given.
2. Choose an appropriate comparison series that you can use to apply the Comparison Test or Limit Comparison Test.
3. Apply the test correctly by comparing the terms and using the limit or inequalities as required.
4. Conclude whether the series converges or diverges based on the comparison.

Detailed Explanation of Steps:

1. Comparison Test:

   • If $0 \leq a_n \leq b_n$ for all $n \geq N$ (some large number N), and if $\sum b_n$ converges, then $\sum a_n$ also converges.
   • Conversely, if $\sum b_n$ diverges and $0 \leq b_n \leq a_n$, then $\sum a_n$ also diverges.

2. Limit Comparison Test:

   • Given two positive term series $\sum a_n$ and $\sum b_n$, if $\lim_{n \to \infty} \frac{a_n}{b_n} = c$, where $0 < c < \infty$, then both series either converge or diverge together.

Steps Specific to the Problem:

• For the given series $a_n = \frac{1}{n^2}$ (example), compare it with the harmonic series $b_n = \frac{1}{n}$.
• For the Limit Comparison Test, find: $\lim_{n \to \infty} \frac{a_n}{b_n}.$
• Fill in the correct answers for each box based on your calculations.

Next Steps:

Would you like me to explain the solution in more detail, or help you fill in specific steps from your work?

Further Questions:

1. What is the comparison series you have used in your problem?
2. Did you encounter any difficulties in applying the limit comparison test?
3. Would you like to understand more about why certain series are suitable for comparison?
4. Do you need clarification on the conditions under which the tests apply?
5. Are there specific steps or boxes you would like me to address?

Tip:

Always ensure that your comparison series is easy to work with and well-known in terms of its convergence behavior to make the tests more straightforward.