Three Cases, and the Middle One Is a Single Point

You'll learnto form the ratio by canceling rather than by dividing two computed terms, to read its limit against the divide at one, to treat the value one as an absence of information rather than as divergence, to see why the test can never settle a p-series, and to keep the absolute values that stop a negative limit reading as convergence.

The ratio test has three outcomes and only two of them are verdicts. Below one the series converges, above one it diverges, and exactly at one the test has told you nothing whatever. That third case is not a near miss or a weak result: two of the most familiar series in the course both land on it, and one of them diverges while the other converges. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · L = 1 IS AN ABSENCE, NOT A VERDICT · v2

Under 1 it converges, over 1 it diverges, at 1 it says nothing

Before you start

What you're looking at

The height of a dot is the ratio of one term to the one before it, so the lime rule at 1 is the divide: dots settling below it mean a sum exists and above it mean none.

The question

What happens to a series whose ratio settles on the rule itself, neither below it nor above?

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