Four Right Terms and a Wrong General Term

You'll learnto turn a Taylor polynomial into a series by finding a general term, to read a derivative cycle without letting its zeros confuse the index, to match the factorial to the exponent rather than to the term number, to check a general term by substituting the first few values back into it, and to keep a series centered where it was asked for.

A Taylor series is the polynomial that never stops, and it needs one thing the polynomial did not: a general term. Getting four terms right and the general term wrong is the characteristic error here, and it has a mechanical fix — put the first few values of n back into the general term and see whether the terms you already wrote come out. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · SUBSTITUTE IT BACK AND CHECK · v2

Put n back into the general term and see if your terms come out

Before you start

What you're looking at

A function plane. Pink is the function being expanded, teal is a partial sum of the series that is correct for it, and violet is a rival built from a wrong general term or a wrong center. A unit of height is drawn taller than a unit of width, so nothing here is a claim about slope.

The question

You can write the first four terms of a series by hand. What do you write for the term that stands for all of them at once?

Step 1 / 12