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 recognize that all four interval shapes occur so no default bracket is safe, and to handle the two special radii.

A power series has an x inside it, so whether it adds up to anything depends on which x you put in. One line of the ratio test tells you how far from the center you can go, and it tells you nothing at all about the two points at exactly that distance. Those you have to test one at a time, as ordinary numerical series, and all four answers really occur. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · THE ENDPOINTS ARE A SEPARATE QUESTION · v2

The interval is where the V sits under the rule, ends excluded

Before you start

What you're looking at

The bottom of the panel is a number line of x values. The violet V rising from the center is the number the ratio test hands back at each x, and the lime rule across the panel is the value one.

The question

Under the rule the series converges and above it the series diverges. What happens at the two places where the V sits exactly on the rule?

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