The Test Runs One Way Only

You'll learnto derive the test from the fact that a term is a difference of consecutive partial sums, to state the one conclusion it supports and refuse the other, to recognise the three shapes of term that make it fire, to see why a vanishing term settles nothing by watching three such series end three different ways, and to spend one line on it before reaching for anything harder.

This test has divergence in its name, and the name is the whole lesson. If the terms fail to approach zero the series diverges, and that is the only conclusion available. If the terms do approach zero the test is silent — not encouraging, not suggestive, silent. Reading it in the other direction is the most consequential error in this unit.

8 STEPS · 6 QUICK CHECKS · ONE CONCLUSION, IN ONE DIRECTION · v1

terms not to zero → diverges · terms to zero → nothing at all

Before you start

What you're looking at

One index panel. Pink stems standing on the axis are the terms, violet dots above them are the running totals, and teal is whatever is being compared against the violet sequence. A lime rule is a limit that exists.

The line at zero is the whole test

The horizontal axis is where the terms have to arrive. In steps 3 and 7 they do not, and the series is finished on the spot. In step 4 all three sets of stems reach it and the test has nothing to say about any of them.

Watch for

Step 1 draws each term as the gap between two consecutive totals, because that is what a term is. Watch the gaps close as the totals settle — that closing is the proof, and the test is just the same sentence read backwards.

Step 1 / 8