10.8 Ratio Test for Convergence

The Ratio Test

For a series Σaₙ, compute L = lim(n→∞) |a_{n+1}/aₙ|. If L < 1, the series converges absolutely. If L > 1 (or L = ∞), the series diverges. If L = 1, the test is inconclusive — try another test.

When to use it

The Ratio Test shines on series with factorials, exponentials, or nⁿ terms, since these simplify nicely when you divide consecutive terms. It's the go-to test for power series to find the radius of convergence, since ratios of xⁿ terms collapse cleanly.

Handling the inconclusive case

If L = 1, the Ratio Test tells you nothing — the series could converge or diverge. Common examples: Σ1/n (diverges) and Σ1/n² (converges), both giving L = 1. Switch to the p-series test, integral test, or comparison test instead.

Key terms

Ratio Test formula

L = lim(n→∞) |a_{n+1}/aₙ|

Ratio Test: L < 1

Series converges absolutely.

Ratio Test: L > 1 (or ∞)

Series diverges.

Ratio Test: L = 1

Inconclusive — use another test (e.g., p-series, comparison, integral test).

Best use cases for Ratio Test

Series with factorials, exponentials, or nⁿ terms, and finding radius of convergence for power series.

Try an AP-style question

What does the ratio test say about Σ n/2ⁿ?

1. Converges, since the ratio approaches 0
2. Converges, since the ratio approaches 1/2
3. Inconclusive, since the ratio approaches 1
4. Diverges, since the ratio approaches 2