Direct comparison test  Series  AP Calculus BC  Khan Academy  Summary and Q&A
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.
Key Insights
 đ 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.
 đ Nonnegative terms in both series ensure that the series will either converge to a finite value or diverge to positive infinity.
 đĨī¸ 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.
 đŠī¸ Conversely, if the smaller series diverges, the larger series with corresponding terms at least as large as the smaller series also diverges.
 đ The comparison test can only be used for series with nonnegative terms and does not apply to series with negative terms or oscillating behavior.
 â By finding a known convergent or divergent series to compare with, the convergence or divergence of a series can be determined.
 đ The comparison test provides a helpful tool for proving the convergence or divergence of a series when individual terms cannot be easily evaluated.
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
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 nonnegative 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 nonnegative 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 nonnegative 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.