When to use the limit comparison test?
Understand the Problem
The question is asking for guidance on the appropriate situations in which to apply the limit comparison test, a method used in calculus to determine the convergence or divergence of an infinite series.
Answer
The limit comparison test is appropriate for determining convergence/divergence of series when both series are positive and when comparing the limit $\lim_{n \to \infty} \frac{a_n}{b_n}$.
Answer for screen readers
The limit comparison test is applied when you want to determine the convergence or divergence of an infinite series $\sum a_n$ compared to a known series $\sum b_n$, particularly focusing on the calculation of the limit $\lim_{n \to \infty} \frac{a_n}{b_n}$.
Steps to Solve

Identify the Series Begin by selecting the infinite series you want to analyze for convergence or divergence. This series can be denoted as $\sum a_n$.

Choose a Comparison Series Next, select a known series $\sum b_n$ that is similar in form to your series $\sum a_n$. Typically, this series should be one whose behavior (convergence or divergence) is already established.

Check the Conditions Ensure that both series are positive for sufficiently large values of $n$. This is a crucial step because the limit comparison test can only be applied to positive series.

Calculate the Limit Compute the following limit: $$ L = \lim_{n \to \infty} \frac{a_n}{b_n} $$

Analyze the Limit Determine the value of the limit $L$:
 If $0 < L < \infty$, then both series either converge or both diverge.
 If $L = 0$, then if $\sum b_n$ converges, $\sum a_n$ converges as well.
 If $L = \infty$, then if $\sum b_n$ diverges, $\sum a_n$ diverges.
 Draw a Conclusion Based on the value of $L$, conclude whether the original series $\sum a_n$ converges or diverges by using the established behaviors of the comparison series $\sum b_n$.
The limit comparison test is applied when you want to determine the convergence or divergence of an infinite series $\sum a_n$ compared to a known series $\sum b_n$, particularly focusing on the calculation of the limit $\lim_{n \to \infty} \frac{a_n}{b_n}$.
More Information
The Limit Comparison Test is particularly useful when the series $a_n$ and $b_n$ share similar asymptotic behavior as $n$ approaches infinity. This test can simplify the analysis of complex series by reducing the problem to comparing it with simpler, wellknown series.
Tips
 Choosing an inappropriate comparison series that does not have an established convergence or divergence.
 Forgetting to ensure that both series are positive for sufficiently large $n$.
 Miscalculating the limit or interpreting its value incorrectly, which can lead to wrong conclusions.