Ratio test is one of 6 series formulas in the calculus section of this library, and it is used at ap · university level.
Why ratio test works
If consecutive terms settle into a fixed ratio, then far out the series behaves like a geometric series with that ratio. Geometric series converge exactly when the ratio is under 1 in size, and the comparison carries that verdict across, which is also why a limit of exactly 1 decides nothing.
What each symbol means
$a_n$ is the series term and $L$ the limiting ratio.
Ratio test: when it holds
The series converges if $L<1$, diverges if $L>1$, and the test is inconclusive if $L=1$.
When it stops applying
The ratio has to settle on a limit. For a series whose terms follow two alternating patterns the ratios can oscillate forever without approaching anything, and then the test simply does not apply, though the root test often still does.
Ratio test: a worked example
For $\sum1/n!$, the ratio is $1/(n+1)\to0$, so it converges.
The mistake to avoid
What people do: Declaring divergence when the limiting ratio turns out to be 1.
Why it goes wrong: That value is the inconclusive case: the sum of 1 over n diverges while the sum of 1 over n squared converges, and both have ratio limit 1.
Do this instead: Switch to a different test, such as comparison, the integral test, or the p-series rule, the moment the ratio limit turns out to be 1.
Ratio test: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $a_n$ is the series term and $L$ the limiting ratio.
- Check the conditions before substituting. The series converges if $L<1$, diverges if $L>1$, and the test is inconclusive if $L=1$.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most calculus slips.
Where this formula fits
- Subject
- Calculus formulas — 42 entries in this library
- Topic
- Series
- Level
- AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where ratio test comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Sequences and Series — the lesson behind this formula: recognize arithmetic, geometric, and recursive patterns.
- Calculus Calculator — check your substitution and the value it produces.
- Study calculus — the subject guide that explains the ideas these formulas compress.
- Calculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 42 calculus formulas — the full grouped reference, or the complete formula library.
Questions about ratio test
Do I need the absolute values?
Yes. The test is about absolute convergence, so take the size of each ratio before letting the index grow.
Which series is this test best suited to?
Anything containing factorials or with the index up in an exponent, since consecutive terms then cancel almost completely when divided.
What does it tell me about a power series?
Solving the ratio condition for the variable gives the radius of convergence directly. The two endpoints then have to be tested separately by other means.
Is a ratio limit of 0 a problem?
Not at all, since 0 is comfortably under 1 and the series converges absolutely. That is exactly what happens for the sum of one over n factorial.