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 centred where it was asked for rather than where the algebra is easiest.

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.

8 STEPS · 6 QUICK CHECKS · SUBSTITUTE IT BACK AND CHECK · v1

substitute the first few n and see what comes out

Before you start

What you're looking at

A function plane. Pink is the function being expanded, teal is the series that is correct for it, and violet is a rival built from a wrong general term or a wrong centre. A rival that leaves the pink curve is a general term that does not reproduce the function.

Why sine

Half of sine's coefficients at the origin are zero, which forces the series index and the derivative order apart: term number n carries the power 2n+1. On a function like the exponential the two indices coincide and a wrong general term would still reproduce every term.

Watch for

In steps 6 and 7 the panel's left half is empty. That is not a drawing choice — the logarithm is not defined there, which is exactly why it has no series centred at 0 and has to be centred at 1 instead. A unit of y is drawn about twice as tall as a unit of x, so do not read slopes off the picture.

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