AP® Calculus BC · Unit 10 · Topic 10.8
Ratio Test for Convergence
The ratio test looks at how each term compares in size to the one before it: if that ratio shrinks toward a number less than 1, the series converges absolutely, because the terms are eventually behaving like a shrinking geometric sequence.
Take the ratio of consecutive terms, a_(n+1)/an, and look at what happens as n gets huge. If that limit L is less than 1, the terms are eventually shrinking at least as fast as a geometric sequence with ratio L, so the series converges absolutely. If L is greater than 1, the terms are eventually growing, so the series diverges. The test works because far enough out, a_(n+1)/an close to a constant L means the series behaves just like sum of Ln — and you already know exactly when that converges.
This makes the ratio test the tool of choice whenever a series has factorials, exponentials, or both, since those are exactly the terms where consecutive ratios collapse into something clean. Compare sum of 2n/n! : the ratio simplifies to 2/(n+1), which goes to 0, so it converges no matter how large 2n looks. Compare sum of n2/3n: the ratio simplifies to (1/3) times ((n+1)/n)2, which goes to 1/3, again less than 1, so it converges.
The test sits next to the root test as the two tools built for terms with variable exponents, and next to comparison and integral tests as backup when the limit equals 1. It also sets up power series convergence, where you apply the ratio test to the general term and solve for which x values make the limit less than 1 — that is exactly how radius of convergence gets found.
Key terms
- Ratio test
- A convergence test that examines the limit of a_(n+1)/an to decide if a series converges absolutely, diverges, or stays inconclusive.
- Absolute convergence
- Convergence that holds even when every term is replaced by its absolute value.
- Inconclusive case
- The situation where the ratio test limit equals 1 and gives no information about convergence either way.
- Geometric-like behavior
- The tendency of a sequence of ratios approaching a constant L to make the series behave like a geometric series with that ratio.
Watch it explained
- How it worksHow does the limit of the ratio of consecutive terms decide convergence?
- Worked exampleShow me what the ratio test says about the sum of 2n divided by n factorial.
- Worked exampleFind the ratio test limit for the sum of n2/3n and what it concludes.
- Common mistakeWhy doesn't the ratio test decide anything when its limit comes out to exactly 1?
Check yourself
End of topic.