Start From One You Know, Then Ask What Changed

You'll learnto build a series by reshaping one you already know, 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.

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. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · EVERY OPERATION CHANGES SOMETHING · v2

Every operation is legal inside the interval and changes the interval

Before you start

What you're looking at

A function plane. Pink is the function, teal is a partial sum of the series that represents it, and the bar under the axis is the stretch of x where the two agree, with a filled end where that endpoint is included.

The question

A series is only a function on the stretch where it converges. If you substitute, differentiate or integrate, what happens to that stretch?

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