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.