# Ratio test

The ratio test is a convergence test for infinite series: for a series \( \sum a_n \) with nonzero terms, it examines the limit of the ratio of the absolute values of consecutive terms, \( \rho = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \), and concludes absolute convergence when \( \rho < 1 \) and divergence when \( \rho > 1 \). The test is also called d'Alembert's ratio test, after Jean le Rond d'Alembert, who established it in 1768.<sup>[1](https://encyclopediaofmath.org/wiki/D%27Alembert_criterion_%28convergence_of_series%29)</sup> It requires no comparison series, which distinguishes it from comparison and integral tests, and it is especially useful for series involving factorials or exponentials.<sup>[2](https://openstax.org/books/calculus-volume-2/pages/5-6-ratio-and-root-tests)</sup>

| Key fact | Detail |
|---|---|
| Quantity computed | \( \rho = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \) for a series with nonzero terms<sup>[2](https://openstax.org/books/calculus-volume-2/pages/5-6-ratio-and-root-tests)</sup> |
| \( \rho < 1 \) | The series converges absolutely<sup>[2](https://openstax.org/books/calculus-volume-2/pages/5-6-ratio-and-root-tests)</sup> |
| \( \rho > 1 \) or \( \rho = \infty \) | The series diverges<sup>[2](https://openstax.org/books/calculus-volume-2/pages/5-6-ratio-and-root-tests)</sup> |
| \( \rho = 1 \) | Inconclusive: the harmonic series \( \sum 1/n \) diverges and \( \sum 1/n^2 \) converges, yet both give ratio limit 1<sup>[1](https://encyclopediaofmath.org/wiki/D%27Alembert_criterion_%28convergence_of_series%29)</sup> |
| Origin | Established by J. d'Alembert in 1768; also known as the d'Alembert criterion<sup>[1](https://encyclopediaofmath.org/wiki/D%27Alembert_criterion_%28convergence_of_series%29)</sup> |
| Typical use | Series with factorials or exponentials; finding the radius of convergence of power series<sup>[2](https://openstax.org/books/calculus-volume-2/pages/5-6-ratio-and-root-tests)</sup><sup> • </sup><sup>[3](https://mathsessentials.com/series-and-sequences/power-series/)</sup> |

## Statement and meaning

For a series \( \sum_{n=1}^{\infty} a_n \) with nonzero terms, define

\[ \rho = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|. \]

The three outcomes are:<sup>[2](https://openstax.org/books/calculus-volume-2/pages/5-6-ratio-and-root-tests)</sup>

1. If \( 0 \le \rho < 1 \), the series converges absolutely.
2. If \( \rho > 1 \) or \( \rho = \infty \), the series diverges.
3. If \( \rho = 1 \), the test is inconclusive.

The intuition is that the ratio measures whether the series behaves approximately like a geometric series.<sup>[4](https://mathbooks.unl.edu/Calculus/sec-7-5-ratio-alternating.html)</sup> A geometric series with ratio \( r \) converges exactly when \( r < 1 \); the ratio test asks whether the successive-term ratio settles below or above 1 in the limit. Convergence of the terms to zero alone is not enough: both \( \sum 1/n \) and \( \sum 1/n^2 \) have terms tending to zero, but only the second converges, because its terms approach zero fast enough.<sup>[2](https://openstax.org/books/calculus-volume-2/pages/5-6-ratio-and-root-tests)</sup>

The absolute values in the definition are required, since the test concludes absolute convergence.<sup>[5](https://tutorial.math.lamar.edu/classes/calcii/ratiotest.aspx)</sup>

## The inconclusive case

When \( \rho = 1 \), the test gives no conclusion, and this is not a defect of the proof but a genuine boundary: the harmonic series \( \sum 1/n \), which diverges, and \( \sum 1/n^2 \), which converges, both have \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = 1 \).<sup>[1](https://encyclopediaofmath.org/wiki/D%27Alembert_criterion_%28convergence_of_series%29)</sup> In that situation one turns to other tools, such as the p-series test for rational powers, the comparison test, or the limit comparison test.<sup>[6](https://seriesconvergencecalculator.com/ratio-test/)</sup> Series whose terms are polynomials in \( n \) typically give \( \rho = 1 \), so comparison or p-series tests are the better first choice for them.<sup>[7](https://mathsessentials.com/series-and-sequences/ratio-and-root-tests/)</sup>

A refinement sharpens the boundary cases. The d'Alembert criterion is stated with the [limit superior and limit inferior](https://www.edgepedia.org/limit-superior-and-limit-inferior): if \( \limsup_{n \to \infty} \frac{|a_{n+1}|}{|a_n|} < 1 \) the series converges absolutely, and if \( \liminf_{n \to \infty} \frac{|a_{n+1}|}{|a_n|} > 1 \) it diverges. Neither conclusion extends to the equality case.<sup>[1](https://encyclopediaofmath.org/wiki/D%27Alembert_criterion_%28convergence_of_series%29)</sup>

## Typical applications

**Factorials and exponentials.** The ratio of consecutive terms of a factorial-containing series simplifies cleanly, since \( (n+1)! = (n+1) \cdot n! \), so the test is the standard first choice when a factorial appears and there are no nth-power terms.<sup>[7](https://mathsessentials.com/series-and-sequences/ratio-and-root-tests/)</sup><sup> • </sup><sup>[8](https://scienceinsights.org/what-is-the-ratio-test-series-convergence-explained/)</sup> The same applies to series built from exponentials such as \( b^n \), where the ratio is constant or approaches a constant.<sup>[2](https://openstax.org/books/calculus-volume-2/pages/5-6-ratio-and-root-tests)</sup>

**Radius of convergence.** For a power series \( \sum c_n (x-a)^n \), applying the ratio test to the terms gives

\[ L = \lim_{n \to \infty} \frac{|c_{n+1}(x-a)^{n+1}|}{|c_n (x-a)^n|} = |x-a| \cdot \lim_{n \to \infty} \left| \frac{c_{n+1}}{c_n} \right|, \]

so the series converges absolutely when \( |x-a| < R \) with

\[ R = \lim_{n \to \infty} \left| \frac{c_n}{c_{n+1}} \right|. \]

The endpoints \( x = a \pm R \) must then be tested separately, because the ratio test returns \( L = 1 \) there.<sup>[3](https://mathsessentials.com/series-and-sequences/power-series/)</sup> When the limit is infinite, the power series converges only at \( x = a \) and the radius of convergence is \( R = 0 \).<sup>[9](https://mathbooks.unl.edu/Calculus/sec-7-7-powerseries.html)</sup> When coefficients are explicitly raised to the nth power, the root test, with \( R = 1/\lim_{n \to \infty} |c_n|^{1/n} \), is often cleaner.<sup>[3](https://mathsessentials.com/series-and-sequences/power-series/)</sup>

## Relation to the root test

The root test uses \( \rho = \lim_{n \to \infty} \sqrt[n]{|a_n|} \) with the same three outcomes, and is often used for series where \( |a_n| = b_n^n \).<sup>[2](https://openstax.org/books/calculus-volume-2/pages/5-6-ratio-and-root-tests)</sup> The two tests are closely related: mathematical work on their connection has introduced intermediate tests, called power mean tests, that are stronger than the ratio test and weaker than the root test.<sup>[10](https://arxiv.org/abs/1804.10056)</sup> In practice the ratio test is applied more often because ratios of consecutive factorial or exponential terms are easier to compute than nth roots.<sup>[7](https://mathsessentials.com/series-and-sequences/ratio-and-root-tests/)</sup>

## References

1. [D'Alembert criterion (convergence of series), Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/D%27Alembert_criterion_%28convergence_of_series%29)
2. [5.6 Ratio and Root Tests, Calculus Volume 2, OpenStax](https://openstax.org/books/calculus-volume-2/pages/5-6-ratio-and-root-tests)
3. [Power Series: Radius & Interval of Convergence, MathsEssentials](https://mathsessentials.com/series-and-sequences/power-series/)
4. [Ratio Test and Alternating Series, Active Calculus (University of Nebraska–Lincoln)](https://mathbooks.unl.edu/Calculus/sec-7-5-ratio-alternating.html)
5. [Calculus II – Ratio Test, Paul's Online Math Notes](https://tutorial.math.lamar.edu/classes/calcii/ratiotest.aspx)
6. [Ratio Test for Series: Rules, Examples, When It Fails](https://seriesconvergencecalculator.com/ratio-test/)
7. [Ratio Test & Root Test for Series, MathsEssentials](https://mathsessentials.com/series-and-sequences/ratio-and-root-tests/)
8. [What Is the Ratio Test? Series Convergence Explained, ScienceInsights](https://scienceinsights.org/what-is-the-ratio-test-series-convergence-explained/)
9. [Power Series, Active Calculus (University of Nebraska–Lincoln)](https://mathbooks.unl.edu/Calculus/sec-7-7-powerseries.html)
10. [A connection between tests for absolute convergence of infinite series, arXiv](https://arxiv.org/abs/1804.10056)

---
*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Analysis and mathematical models › Real analysis*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*

License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license