# Ratio test

Find the limit L of |a_(n+1) / a_n|. If L < 1 the series converges, if L > 1 it diverges, and if L = 1 the test says nothing.

Maths (university) · Part 07: Sequences and series · Card 6 of 9 · Last updated 2026-10-19

## Analogy

Checking whether each step of a staircase is shorter than the last by a steady factor.

## Example

**Series of 2^n / n!**

a_(n+1) / a_n = 2 / (n + 1).
As n grows, 2/(n+1) tends to 0.
L = 0, less than 1.
So the series converges.

## In practice

Works very well with factorials and powers, such as n! or 2^n.

## Practice

### Flashcards

- **What does the ratio test conclude when L < 1?** The series converges.

### Interview questions

**Does the series of 2^n / n! converge? Show your test.**

- Use the ratio test
- a_(n+1)/a_n = 2/(n+1)
- Limit as n grows is 0
- 0 is less than 1, so it converges

## Questions people ask

### What if the ratio test gives L = 1?

The test is inconclusive. Both the harmonic series (which diverges) and the series of 1 over n squared (which converges) give a limit of 1. You need a different test, such as comparison or the integral test, to decide. The ratio test cannot separate these two cases.

### Which series suit the ratio test best?

Those with factorials, such as 2 to the n over n factorial, or with exponentials like n over 3 to the n. The ratio of consecutive terms simplifies nicely, because most factors cancel and leave a simple fraction in n. For 2 to the n over n factorial the ratio is just 2 over n plus 1.

## Related concepts

- [Partial sum](https://studyvisually.com/learn/science/maths-university/maths-university-part-7/maths-university-part-7-concept-2)
- [Geometric series](https://studyvisually.com/learn/science/maths-university/maths-university-part-7/maths-university-part-7-concept-3)
- [Divergence test](https://studyvisually.com/learn/science/maths-university/maths-university-part-7/maths-university-part-7-concept-4)
- [p-series](https://studyvisually.com/learn/science/maths-university/maths-university-part-7/maths-university-part-7-concept-5)

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