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.

Transcript

• [Voiceover] So let's get a basic understanding of the comparison test when we are trying to decide whether a series is converging or diverging. So let's think of two series. So let's say that I have this magenta series here. It's an infinite series from n equals one to infinity of a sub n. We're speaking in generalities here, and let's have anoth... Read More

Key Insights

• 🏆 The comparison test is a useful tool for determining the convergence or divergence of a series by comparing it to another series with known behavior.
• 🍉 Non-negative terms in both series ensure that the series will either converge to a finite value or diverge to positive infinity.
• 🌥️ If a series is larger than another series with corresponding terms that are at least as large, and the larger series converges, the smaller series also converges.
• 🛩️ Conversely, if the smaller series diverges, the larger series with corresponding terms at least as large as the smaller series also diverges.
• 🍉 The comparison test can only be used for series with non-negative terms and does not apply to series with negative terms or oscillating behavior.
• ❓ By finding a known convergent or divergent series to compare with, the convergence or divergence of a series can be determined.
• 👍 The comparison test provides a helpful tool for proving the convergence or divergence of a series when individual terms cannot be easily evaluated.

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.