AP Calculus BC glossary

Limit comparison test

The limit comparison test compares two series of positive terms by taking the limit of their ratio. If that limit is finite and positive, both series converge or both diverge together.

It is easier to apply than direct comparison because no inequality is needed, only that the terms grow at comparable rates. Choose the comparison series by keeping the dominant power in the numerator and denominator.

For $\sum \frac{3n^2 + 1}{n^4 - 2}$ , the dominant behaviour is $\frac{n^2}{n^4} = \frac{1}{n^2}$ , so compare with that convergent p-series.

The limit must be finite and positive

A limit of zero or infinity makes the standard version inconclusive, so pick a comparison series that matches the growth rate more closely.

Appears in: Unit 10: Infinite Sequences and Series (BC)