The Radius Comes Free; the Endpoints Never Do

You'll learnto get the radius from the ratio test in one line, to see why that same test is silent at both endpoints rather than merely unhelpful, to test each endpoint as its own numerical series, to recognise that all four interval shapes occur so no default bracket is safe, to keep a radius and an interval apart as answers to different questions, and to handle the two special radii.

A power series converges on an interval centred at its centre. The radius comes straight from the ratio test. The endpoints do not, and cannot: at each endpoint the ratio limit is exactly one, which 10.8 established decides nothing at all. Each endpoint is a separate question, answered by a different test, and all four combinations of answers really occur.

8 STEPS · 6 QUICK CHECKS · THE ENDPOINTS ARE A SEPARATE QUESTION · v1

the interval is where the V is under the rule

Before you start

What you're looking at

Not a graph of the series. The horizontal axis is x and the vertical axis is the ratio-test limit at that x. Convergence needs that limit under 1, so the interval of convergence is the stretch of x where the V sits below the lime rule.

The bar under the axis

That is the interval itself, drawn the way a number line draws one. A filled circle means the endpoint is included and an open one means it is not — and which it is cannot be read off the V, which is the entire point of the topic.

Watch for

Steps 3 to 5 draw the same V four times. Four different series produce it, their radius is the same, and their intervals are four different shapes. Nothing about the picture above the axis changes; everything about the bar below it does.

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