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.

Essential Questions

How are Taylor polynomials constructed?