10.14 Finding Taylor or Maclaurin Series for a Function

Recall the definition of a Taylor series:

There are some Maclaurin series that are so common that they are better to be memorized than derived for the AP test. They are as follows.

Note that since sine and inverse tangent are odd functions, all the terms have odd powers, which is a contrast to the cosine function, which is even, and thus have even powers. The first four are variations of the geometric series, with the multiplying factor being various powers of x. The series (1 + x)^a is also known as the binomial series with any arbitrary value of a.

IMPORTANT: These formulas only work for Maclaurin series and we cannot substitute (x-a) into these formulas for x to make them a Taylor series!