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.
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.