A Series Converges When Its Partial Sums Settle

You'll learnto keep the terms, the partial sums and the sum of a series apart, to state the definition of convergence as a statement about one of those three and not the others, to see why terms tending to zero can rule a series out and can never rule one in, to recognise the two different ways a series can diverge, and to take a limit directly whenever the partial sums can be written down.

An infinite series problem always involves three different objects, and almost every error in this unit is one of them wearing another's name. There are the terms, there are the partial sums built by adding those terms up, and there is the sum of the series. Only the middle one has a limit worth taking, and that limit is the whole definition.

8 STEPS · 6 QUICK CHECKS · THREE OBJECTS, AND ONLY ONE OF THEM IS THE SUM · v1

the terms are a list · the partial sums are a list · the sum is a number

Before you start

What you're looking at

One panel, and the horizontal axis is the index n rather than a variable x — it takes whole-number values only, so everything on it is a list of dots and not a curve. Pink stems standing on the axis are the terms. Violet dots above them are the running totals. A lime rule is a limit, and it is drawn only when there is one.

The scale is deliberately uneven

A unit of y is drawn about three times taller than one index step, so a partial sum creeping toward its limit is visible. Every comparison in this file is between two heights at the same index, which is exactly what that stretch preserves. Nothing here claims a length or an area.

Watch for

In step 4 the two series start identically — the first two violet dots are shared, not merely close. Watch which one flattens onto the rule and which one goes through it at n = 4 without slowing down. Both sets of pink stems are shrinking to nothing the whole time.

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