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

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August 21, 2014
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Direct comparison test | Series | AP Calculus BC | Khan Academy

TL;DR

The comparison test states that if a smaller series converges or diverges, a larger series with corresponding terms at least as large or small as the smaller series will have the same convergence or divergence.

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Questions & Answers

Q: What is the purpose of the comparison test?

The comparison test is used to determine the convergence or divergence of a series by comparing it to another series with known convergence or divergence.

Q: How does the comparison test work for convergent series?

If the terms of a series are non-negative and each term is less than or equal to the corresponding term of a larger convergent series, then the smaller series also converges.

Q: What happens if the smaller series diverges?

If the smaller series diverges, the larger series with corresponding terms at least as large as the smaller series also diverges.

Q: Can the comparison test be used with series that have negative terms?

No, the comparison test only applies to series with non-negative terms, as the terms cannot go to negative infinity or oscillate between positive and negative values.

Summary & Key Takeaways

  • The comparison test helps determine if a series converges or diverges by comparing it to a known convergent or divergent series.

  • If the terms of a series are non-negative and each term is less than or equal to the corresponding term of a larger series, and the larger series converges, then the smaller series must also converge.

  • Conversely, if the smaller series diverges, the larger series must also diverge.

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