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 behavior, to repair an inequality that points the useless way, and to replace the inequality with a limit when the terms are a quotient.

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. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · A TRUE INEQUALITY IS NOT AN ARGUMENT · v2

Under something convergent, or over something divergent, and nothing else

Before you start

What you're looking at

An index panel. Pink stems are the terms of the series being tested, teal stems belong to the series it is being compared with, and violet dots are a running total.

The question

If one list of terms sits under another, has anything been settled about their sums?

Step 1 / 12