Ratio Test Calculator
Ratio Test Calculator
Slug: `/ratio-test-calculator/`
Primary keyword: ratio test calculator
Meta title: Ratio Test Calculator: Limit of Consecutive Terms, Step by Step
Meta description: Apply the ratio test to any series. See the simplified ratio, the limit L, and the verdict, with the inconclusive case explained properly.
Word count: approximately 1,000
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The statement
Take a series with non-zero terms. Compute L as the limit, as n grows large, of the absolute value of a(n+1) divided by a(n).
Three outcomes:
- L under 1: the series converges absolutely.
- L above 1: the series diverges.
- L exactly 1: the test tells you nothing.
That third line is not a formality. It happens constantly, and knowing what to do next is most of the skill.
Why it works
Compare against a geometric series. If consecutive terms eventually shrink by a factor of roughly L each time, the tail of your series behaves like a geometric series with ratio L. Geometric series converge when the ratio is under 1, and the comparison carries that verdict across.
When L exceeds 1 the terms are growing, so they cannot approach zero, and the divergence test finishes the job.
At L equal to 1 the comparison collapses. The terms are shrinking, but not geometrically, and geometric intuition no longer applies.
When to reach for it
Factorials. Anything raised to the power n. Products of both.
Ratio testing shines on those because the ratio cancels enormous amounts of structure. Consider two to the n over n factorial. Form the ratio of consecutive terms and almost everything disappears, leaving two over n plus one. That heads to zero, so L equals 0 and the series converges.
Try the same series with a comparison test and you will work considerably harder for the same answer.
Worked examples
Series: n factorial over n to the n
Form the ratio and simplify. What remains is n over n plus one, all raised to the power n. That expression approaches one over e as n grows, roughly 0.3679.
L is 0.3679, under 1, so the series converges.
Series: one over n squared
The ratio simplifies to n squared over n plus one squared, which approaches 1.
L equals 1 exactly. Inconclusive. The ratio test has failed, and you switch to the p-series test, which settles it in one step: p equals 2, greater than 1, so it converges.
Series: three to the n over n
Ratio simplifies to three times n over n plus one, approaching 3.
L equals 3, above 1, so the series diverges.
The inconclusive case
Any series whose terms are a ratio of polynomials returns L equal to 1. So does every p-series. That covers a large fraction of exam questions, which is why the ratio test alone is never enough.
When L comes back as 1, move to:
- the p-series test, when the term is a power of n
- limit comparison, when the term merely resembles a p-series
- the integral test, when the term comes from a function you can integrate
- the alternating series test, when signs flip
Shortest possible summary
Factorial in the term? Use this test.
On numerical estimates
Our calculator computes the ratio limit numerically when it cannot match a closed form, and slow convergence is a genuine problem there.
Take the coefficients one over n factorial. After 120 terms the numerical limit sits near 0.009 rather than 0. The verdict is right, the number is not exact. Take one over n and the numerical limit reads about 0.9867 rather than exactly 1, which risks reporting a false verdict on a borderline case.
For that reason the engine reports a symbolic result whenever it recognises the form, and clearly labels everything else as estimated.
Errors people make
- Forgetting absolute values. The test uses the magnitude of the ratio. Alternating series otherwise produce negative ratios and nonsense.
- Reading L equal to 1 as divergence. It means unknown, not divergent.
- Applying it at a power series endpoint. L equals 1 there by construction.
- Cancelling factorials incorrectly. Remember that n plus one factorial equals n plus one times n factorial. Almost every factorial ratio reduces through that identity.
- Stopping at absolute convergence. If the test gives L under 1, convergence is absolute, which is a stronger statement worth writing down.
What the test does not give you
A verdict, and nothing else. The ratio test never produces the sum.
Series of two to the n over n factorial converges by ratio test, and its sum from n equal to one is e squared minus one, about 6.389. That value comes from recognising the exponential series, not from the test. Convergence and evaluation are separate questions, and most convergent series have no closed form at all.
Questions
Does the ratio test prove absolute convergence?
Yes. L under 1 gives absolute convergence, which is stronger than plain convergence.
Can I use it on a series with zero terms?
Not directly, since the ratio needs a non-zero denominator. Split the series or use the root test.
Ratio test or root test?
Ratio for factorials. Root for whole terms raised to the nth power. When the root test settles a case, the ratio test usually can too, but with heavier algebra.
Why does it fail for p-series?
Because polynomial growth is not geometric, and the comparison the test rests on needs geometric behaviour.
Is L equal to 1 common?
Very. Every p-series and every rational term produces it.