Two Pairings Conclude, and Two Conclude Nothing

You'll learnto pair an inequality with a known series in one of the two directions that concludes anything, to see why the other two are empty rather than weak, to pick a comparison series by reading dominant behaviour, to repair an inequality that points the useless way, and to replace the inequality entirely with a limit whenever the terms are a quotient of polynomials.

A comparison argument has two moving parts: an inequality and a known series. Get the pairing wrong and the algebra is still correct, the inequality is still true, and the argument still proves nothing at all. Only two of the four combinations conclude anything, and the other two are an absence rather than a weaker verdict.

8 STEPS · 6 QUICK CHECKS · A TRUE INEQUALITY IS NOT AN ARGUMENT · v1

under something convergent · over something divergent · nothing else

Before you start

What you're looking at

An index panel. The horizontal axis is n and takes whole-number values only. Pink stems are the terms of the series being tested, teal stems are the terms of the comparison series, and violet dots are a running total. A lime rule is a limit that exists.

Two shorter series, two different fates

In step 2 both teal families sit under the same pink one at every index. The inequality is true both times. One of the two totals settles and the other does not — which is the whole reason that pairing concludes nothing.

Watch for

From step 6 the panel stops drawing terms and starts drawing the ratio of two terms as its own sequence. A flat row of dots at a finite positive height is the limit comparison test passing; a row sinking to the axis or climbing out of the frame is one of the two one-directional cases.

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