# Taylor Series

Power series allow us to represent associated functions on an appropriate interval.

By the end of this section you should be able to do the following:

Represent a function as a Taylor series or a Maclaurin series.

Interpret Taylor series and Maclaurin series.

Represent a function at a point as a Taylor polynomial.

Approximate function values using a Taylor polynomial.

By the end of this section you should know the following:

There is a formula for finding the coefficient of the nth degree term in a Taylor polynomial (see below).

In many cases, as the degree of a Taylor polynomial increases, the nth degree polynomial will approach the original function over some interval.

Taylor polynomials for a function f centered at x = a can be used to approximate function values of f near x = a.

A Taylor polynomial for

*f(x)*is a partial sum of the Taylor series for*f(x)*.The Maclaurin series for 1/(1-x) is a geometric series.

The Maclaurin series for sin x , cos x, and e^x provides the foundation for constructing the Maclaurin series for other functions.