When to use the limit comparison test?

Understand the Problem

The question is asking about the circumstances or criteria under which the limit comparison test is employed in mathematical analysis, specifically in the context of determining the convergence or divergence of infinite series.

Answer

The limit comparison test determines the convergence or divergence of series by comparing them to a known series.
Answer for screen readers

The limit comparison test is used to determine the convergence or divergence of infinite series by comparing them to a known series.

Steps to Solve

  1. Identify Infinite Series Determine the infinite series you want to analyze. Let’s denote the series in question as $ \sum a_n $.

  2. Select a Comparison Series Choose a benchmark series $ \sum b_n $ that is known to converge or diverge. The series $ b_n $ should resemble $ a_n $ in terms of growth. A common choice is a $ p$-series, which has the form $ b_n = \frac{1}{n^p} $ where $ p $ is a positive constant.

  3. Establish Conditions for Comparison The limit comparison test requires that $ a_n, b_n > 0 $ for all $ n $ (or large enough $ n $).

  4. Calculate the Limit Calculate the limit: $$ L = \lim_{n \to \infty} \frac{a_n}{b_n} $$ If $ L $ is a positive finite number ($ 0 < L < \infty $), you can make conclusions about the convergence or divergence of the series.

  5. Conclude Based on the Limit

  • If $ L > 0 $ and finite, both series either converge or both diverge.
  • If $ L = 0$ and $ b_n $ converges, then $ a_n $ also converges.
  • If $ L = \infty$ and $ b_n $ diverges, then $ a_n $ also diverges.

The limit comparison test is used to determine the convergence or divergence of infinite series by comparing them to a known series.

More Information

The limit comparison test is useful because it allows you to handle difficult series by relating them to simpler ones. It’s often applied when a series is difficult to assess directly, and it works well in many practical scenarios involving series.

Tips

  • Assuming $ a_n $ is positive for all $ n $. Make sure $ a_n $ and $ b_n $ are positive for the limit comparison test to apply properly.
  • Miscalculating the limit $ L $. Double-check your limit calculation as small errors can lead to incorrect conclusions.

AI-generated content may contain errors. Please verify critical information

Thank you for voting!