The Test Runs One Way Only

You'll learnto see where the test comes from by watching a term appear as the gap between two running totals, to state the one conclusion it supports and refuse the other, to recognize the three shapes of term that make it fire, 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. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · ONE CONCLUSION, IN ONE DIRECTION · v2

Terms that miss zero end it; terms that reach zero settle nothing

Before you start

What you're looking at

One panel whose bottom axis counts whole steps. Pink stems are the terms, violet dots are the running total, and the axis itself is the height the terms have to reach.

The question

If the terms do get down to zero, has anything been settled about the total?

Step 1 / 12