What Is a Ratio Test Calculator?
A ratio test calculator determines whether an infinite series converges or diverges by examining the limit of the ratio of consecutive terms. The ratio test (also called d'Alembert's test) is one of the most powerful and commonly used convergence tests in calculus, especially effective for series involving factorials and exponentials. Enter the general term of the series and the calculator computes the limit and states the conclusion.
How to Use the Ratio Test Calculator
- Enter the general term aₙ of the series.
- Calculate — see the limit L and whether the series converges, diverges, or is inconclusive.
How the Ratio Test Works
The ratio test evaluates the limit of the absolute value of the ratio of successive terms:
L = limₙ→∞ |aₙ₊₁ ÷ aₙ|
| Limit L | Conclusion |
|---|---|
| L < 1 | The series converges absolutely |
| L > 1 (or infinite) | The series diverges |
| L = 1 | Inconclusive — use another test |
Worked Example
For the series with aₙ = 1/n!, the ratio aₙ₊₁/aₙ = n!/(n+1)! = 1/(n+1), whose limit as n → ∞ is 0. Since L = 0 < 1, the series converges absolutely.
When to Use the Ratio Test
- Factorials: series with n! terms simplify beautifully under the ratio test.
- Exponentials: terms like xⁿ are well-suited to it.
- Power series: the ratio test finds the radius of convergence.
When the ratio test gives L = 1, it is inconclusive, and tests like the root test, comparison test, or integral test may help instead.
Frequently Asked Questions
What is the ratio test?
The ratio test examines the limit of |aₙ₊₁/aₙ|. If that limit is less than 1 the series converges, greater than 1 it diverges, and equal to 1 the test is inconclusive.
When does a series converge by the ratio test?
A series converges absolutely when the limit of the ratio of consecutive terms is less than 1.
What does it mean if the limit equals 1?
When L = 1 the ratio test is inconclusive, and you must use a different convergence test such as the root, comparison, or integral test.
What series is the ratio test best for?
It is most effective for series involving factorials and exponential terms, and for finding the radius of convergence of power series.
Is this ratio test calculator free?
Yes — it is completely free, requires no signup, and returns the limit and convergence conclusion.