The ratio test is one of the most reliable tools an IB Mathematics: Analysis and Approaches (AA) Higher Level candidate can carry into Paper 2. It is also one of the most mistrusted, because students are taught the rule but not the judgement that surrounds it. A series converges or diverges, the limit is computed, and the candidate writes a single letter on the answer line — but the work that earns the marks lives in the line above, and that work is where most of the lost points happen.
This article unpacks the ratio test as it actually appears in the IB Math AA Paper 2 setting. It is written for HL candidates who have seen the theorem in class, can state the three cases, and want to convert mechanical familiarity into a Paper 2 score that holds up against an IB examiner. We will look at the precise form of the test, the limit step that decides the mark, the decision tree that tells you when the ratio test is the wrong tool, and the way a markscheme typically rewards the conclusion line. By the end, the ratio test should feel less like a recipe and more like a deliberate choice on the answer page.
What the ratio test actually says, in the form IB examiners use
The theorem itself is short. For a series of positive terms an, form the ratio of consecutive terms an+1} / an and take the limit as n tends to infinity. Call that limit L. Three cases follow: if L is strictly less than 1, the series converges absolutely; if L is strictly greater than 1, the series diverges; if L equals 1 exactly, the test is inconclusive.
IB examiners will not test whether you can recite this statement. They will test whether you can apply it to a series whose terms involve factorials, powers of n, and exponential growth in the numerator or denominator. The most common Paper 2 series on which the ratio test lands cleanly look like n! in either the numerator or denominator, because factorials collapse under the ratio in a way that produces a clean algebraic limit. A series like sum (n! / 3n) is the canonical example: the ratio of consecutive terms is (n+1) / 3, which diverges to infinity, and the test returns divergence in a single line of algebra.
The other class where the ratio test shines is geometric-style series with an extra polynomial factor. Consider sum (n2 / 2n). The ratio of an+1} to an is ((n+1)2 / 2n+1) · (2n / n2), which simplifies to ((n+1)2 / 2n2). As n grows, the polynomial ratio tends to 1/2, and 1/2 is less than 1, so the series converges absolutely. The mark is not in the answer; the mark is in showing the simplification of that ratio with no sign errors and a clean limit statement.
Two micro-conventions matter in the IB markscheme. First, the conclusion line must name the case — 'since L is less than 1, the series converges' — and not merely write the numerical value of L. Second, when the limit is 0, the same conclusion holds, and students often write 0 as if it were a failure case. It is not. L = 0 is a strongly convergent signal. Examiners are trained to credit the conclusion when the limit is anywhere strictly below 1.
Why the limit step is where most of the marks are lost
A typical IB Paper 2 part (c) on series convergence offers three or four marks, and the ratio itself is rarely the line that breaks a candidate. What breaks them is the algebra between forming the ratio and quoting L. Three recurring mistakes dominate.
The first is failing to write |an+1} / an|. For series with positive terms this absolute value is decorative, but IB examiners will sometimes slip a sign into the question — for example, terms of the form (-1)n · n! / 2n — and the markscheme requires the absolute value inside the limit. If the candidate omits it, the conclusion about absolute convergence is not properly justified, and the final mark can be withheld even when the limit has been correctly computed. Build the absolute value into the first line of working every time; it costs nothing and insures the mark.
The second mistake is sloppy cancellation in factorials. Students write (n+1)! as n! + 1, drop a factor of n, or treat n! as if it grows polynomially. The ratio of (n+1)! to n! is exactly n+1, no exceptions, and that identity should be written explicitly. The markscheme reads this cancellation step for signs of fluency, and a confused line of factorial algebra is often the difference between a 4 and a 5 on the part.
The third mistake is treating the limit of the polynomial ratio as if it always required L'Hôpital's rule. In the IB syllabus, L'Hôpital's rule is on the AA HL syllabus, and a candidate may legitimately invoke it. But for a ratio like ((n+1)2 / 2n2), the limit is 1/2 by the standard result that nk grows slower than any exponential base greater than 1. Writing 'as n tends to infinity, the dominant term is 1/2' is faster, cleaner, and is the form the markscheme prefers. Reserve L'Hôpital's rule for limits of functions of a continuous variable, and keep the ratio-test limits in discrete form.
When the ratio test is the wrong tool: building a decision tree
Choosing the ratio test reflexively is one of the costliest habits a candidate can develop. Three common Paper 2 series defeat it cleanly, and recognising them before reaching for the ratio is a tactical skill that separates band 6 from band 5 work.
Series with only polynomial growth — sum (1/np) — give a ratio limit of 1 for any value of p, and the ratio test returns nothing useful. The p-series test is the right tool, and the candidate who defaults to the ratio test here wastes a line of working and reaches an inconclusive result. Series with logarithmic growth, like sum (1/(n · (log n)p)), fall into the same trap. The ratio collapses to 1, the test is silent, and the candidate has to switch to the integral test or the comparison test, having lost two or three minutes of paper time.
Alternating series that are conditionally convergent are a different kind of trap. The ratio test on an alternating series whose terms are decreasing in absolute value will give a limit of 1 in the inconclusive case, even when the alternating series test would have produced a clean 'converges' verdict. For the alternating harmonic series, for instance, the ratio of consecutive absolute terms is n / (n+1), which tends to 1, and the ratio test is silent. The alternating series test, applied to the original series, is the right tool, and an examiner awarding method marks will not give credit for the ratio test here.
The decision tree worth memorising is short. If the series involves factorials, exponentials in n, or a closed-form recurrence that makes the ratio telescoping, take the ratio test. If the series is a p-series, a log-series, or a clean alternating series, use the test that matches its structure. If the series is something unfamiliar, the comparison test against a known series is usually safer than the ratio test. The ratio test is sharp; it is not universal.
Worked example: a series that hides its factorial
Consider the series sum (2n · n! / (n+1)n). A candidate reading this on Paper 2 has to make a judgement in the first ten seconds: does the ratio test simplify? Let us work it through.
Form the ratio of an+1} to an:
- an+1} = 2n+1 · (n+1)! / (n+2)n+1
- an = 2n · n! / (n+1)n
The ratio is:
(2n+1 / 2n) · ((n+1)! / n!) · ((n+1)n / (n+2)n+1)