The Label Is Decided by a Series You Were Not Handed

You'll learnto ask the two questions in the order that saves work, to read the label off a pair of verdicts rather than off a single test, to transfer convergence upward from the absolute series and never downward, to check both conditions of the alternating series test before using its answer, to tell a failed test apart from a divergent series, and to say why an absolutely convergent sum survives reordering when a conditional one does not.

This topic is not a new test but a name, and the name is settled by a series nobody gave you: strip every sign off the terms and ask about that one first. If it converges you are finished. If it does not, exactly one question remains, and the commonest error in the whole unit is answering it by looking at the series you already have.

8 STEPS · 6 QUICK CHECKS · TWO QUESTIONS, ASKED IN ORDER · v1

strip the signs and ask about that series first

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 a running total in almost every step, so a sequence of dots that flattens onto a lime rule is a series converging to that value, and one that keeps climbing is a series with no sum.

Two hues, two questions

Pink is the series itself and teal is the same series with every sign stripped off. Those are the two questions this topic asks, in that order of importance and the reverse order of asking. Violet is a third object built out of the first two — a comparison in step 4 and a reordering in step 7.

Watch for

At n = 1 the pink and teal totals are the same number, because the first term is positive; they separate as soon as a negative term arrives, and the small pink dot sits inside the larger teal one until then. No drawn dot is a limit — step 3 ends 0.04 short of ln 2 and reaches it in the end.

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