AP Calculus BC glossary
Comparison test
Also called: Direct comparison test
The direct comparison test settles a series of positive terms by bounding it against a series whose behaviour is known. If it is term by term smaller than a convergent series it converges, and if it is larger than a divergent series it diverges.
Only two of the four possible comparisons say anything. Being smaller than a divergent series and larger than a convergent series are both uninformative, which is where most errors happen.
The usual comparison targets are p-series and geometric series, since those are the ones you can classify at sight.
When the inequality goes the wrong way
If you cannot get the inequality you need, switch to the limit comparison test, which only requires the terms to behave alike rather than be ordered.
Appears in: Unit 10: Infinite Sequences and Series (BC)