Watch the Running Total, Not the Terms

You'll learnto keep the terms, the running totals and the sum of a series apart, to state convergence as a fact about the running totals alone, to see why terms shrinking to zero can rule a series out but never rule one in, and to recognize the two different ways a series can fail to have a sum.

A series question always has three objects in it, and almost every mistake in this unit is one of them wearing another's name. There are the terms, there are the running totals you get by adding those terms up, and there is the sum. Only the middle one has a limit worth taking. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · THREE OBJECTS, AND ONLY ONE OF THEM IS THE SUM · v2

The terms are a list, the running totals are a list, the sum is a number

Before you start

What you're looking at

The bottom axis counts whole steps, so everything on it is a list of dots. Pink stems are the numbers being added; violet or teal dots are the running total.

The question

Do those running totals settle on one height, or do they keep moving forever?

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