Start From One You Know, Then Ask What Changed

You'll learnto build a series by manipulating a known one instead of differentiating fifteen times, to catch a misremembered series with a single evaluation at a point whose answer is obvious, to substitute into the condition as well as into the series, to keep the radius straight through differentiation and integration, and to check the endpoints again after every operation because integration can gain one.

Differentiating a function fifteen times is rarely the intended route. The intended route is to start from a series you already know and substitute, differentiate or integrate. Every one of those is legal inside the interval of convergence, and every one of them changes something about that interval — which is the half that gets dropped.

8 STEPS · 6 QUICK CHECKS · EVERY OPERATION CHANGES SOMETHING · v1

substitute into the condition, not just the series

Before you start

What you're looking at

A function plane. Pink is the function, teal is the series that represents it, and violet is what an operation turned that series into. A partial sum leaving its function is the edge of the interval of convergence arriving.

The bar under the axis

Where the series converges, drawn as a number line draws an interval, with a filled circle for an endpoint that is included. Watch it in steps 3 and 6: substitution shortens it, and integration can add an end to it.

Watch for

A partial sum is a polynomial, so it is defined everywhere and always finite. Where it leaves its function it has not blown up — it has simply stopped representing anything. A unit of y is drawn about half as tall as a unit of x, so do not read slopes off the picture.

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