Three Cases, and the Middle One Is a Single Point

You'll learnto form the ratio by cancelling 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, to keep the absolute values that stop a negative limit reading as convergence, and to pick the test from the shape of the term.

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.

8 STEPS · 6 QUICK CHECKS · L = 1 IS AN ABSENCE, NOT A VERDICT · v1

under 1 converges · over 1 diverges · exactly 1 says nothing

Before you start

What you're looking at

An index panel. The horizontal axis is n and takes whole-number values only. The vertical axis is the value of the ratio, so the lime rule at 1 is the divide the whole test turns on: dots settling below it mean convergence and dots settling above it mean divergence.

Two shades, two kinds of object

Each series keeps one colour. A deep stem is a term of that series; a light dot is something worked out from those terms — the ratio in most steps, and the running total in the second half of step 3, where the caption and the legend both say so.

Watch for

No drawn dot is the limit. Step 2 opens with a ratio sitting exactly on the rule at n = 1 for a series that converges outright, and step 3 ends with two sequences that have not reached 1 at n = 12 and both reach it in the end.

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