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
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Identify the Series Begin by selecting the infinite series you want to analyze for convergence or divergence. This series can be denoted as $\sum a_n$.
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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.
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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.
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Calculate the Limit Compute the following limit: $$ L = \lim_{n \to \infty} \frac{a_n}{b_n} $$
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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, well-known 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.
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