Limit comparison test | Series | AP Calculus BC | Khan Academy | Summary and Q&A

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September 2, 2014
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Limit comparison test | Series | AP Calculus BC | Khan Academy

TL;DR

The Comparison Test and Limit Comparison Test are useful tools for determining the convergence or divergence of infinite series.

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Key Insights

  • 👻 The Comparison Test allows us to determine the convergence or divergence of an infinite series by comparing it to another series with known convergence behavior.
  • 📁 The Limit Comparison Test is a more flexible version of the Comparison Test and can be used when direct comparison is not possible.
  • 🍉 Both tests require that the series have non-negative terms and involve taking the limit of the ratio of corresponding terms.
  • ↗️ The Comparison Test provides an upper bound on the convergence of the series being analyzed.
  • 👻 The Limit Comparison Test allows us to establish the convergence or divergence of a series by finding a "similar" series with known behavior.
  • 🥳 The limit of the ratio of terms plays a crucial role in both tests, as it determines the convergence or divergence of the series.
  • 💨 The Limit Comparison Test is a more formal way of determining if two series have similar convergence or divergence behavior.

Questions & Answers

Q: How does the Comparison Test work?

The Comparison Test is used to determine the convergence or divergence of an infinite series by comparing its terms to another series. By showing that the terms of the second series are greater or equal to the terms of the original series, if the second series converges, the original series also converges.

Q: When can the Limit Comparison Test be applied?

The Limit Comparison Test is useful when the terms of two infinite series cannot be directly compared. It requires that both series have non-negative terms and the limit as n approaches infinity of the ratio of their corresponding terms is positive and finite.

Q: What happens if the limit of the ratio of terms is zero?

If the limit of the ratio of terms is zero, it indicates that the behavior of the two series is similar. In this case, the convergence or divergence of one series implies the same for the other series.

Q: Can the Comparison Test be used for any infinite series?

The Comparison Test can be applied to series with non-negative terms. However, it may not be applicable if the terms do not satisfy the necessary conditions for comparison.

Summary & Key Takeaways

  • The Comparison Test involves comparing the terms of a given infinite series to another series, where each term is greater or equal to the corresponding term of the original series.

  • The Limit Comparison Test is applied when the Comparison Test cannot be directly used. It involves taking the limit as n approaches infinity of the ratio of corresponding terms of two infinite series.

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