AP Calculus BC
Direct Comparison vs Limit Comparison Test
Direct comparison requires you to prove a term by term inequality against a known series. Limit comparison only requires the two series to grow at comparable rates, judged by the limit of their ratio, which makes it easier to apply when the inequality is awkward.
Direct comparison
Use when: The inequality is obvious, such as dropping a positive term from a denominator.
Limit comparison
Use when: The terms clearly behave like a known series but the inequality points the wrong way or is hard to establish.
Side by side
| Direct comparison | Limit comparison | |
|---|---|---|
| Requires | A proven inequality | A finite positive limit of the ratio |
| Useful conclusions | Smaller than convergent, larger than divergent | Both series behave the same way |
| Terms must be | Positive | Positive |
| Fails when | The inequality goes the wrong way | The limit is 0 or infinite |
Only two of the four possible direct comparisons say anything. Being smaller than a divergent series and larger than a convergent series are both uninformative, and that is where most errors with this test happen.
Limit comparison sidesteps the problem. Choose the comparison series by keeping the dominant power on top and bottom: for that is , a convergent p-series.
The comparison target
Almost always a p-series or a geometric series, because those are the only ones you can classify at sight.
Frequently asked questions
What if the limit comparison limit is zero?
The standard version is inconclusive. Pick a comparison series whose growth rate matches more closely and try again.
In the CED: Unit 10: Infinite Sequences and Series (BC)