9.3 Taylor Series

General procedure for finding an infinite polynomial series that converges to a "non-polynomial" function.

Review Factorials  -

                             where  0! = 1

Taylor Series at 0

The Taylor series of f(x) about x = 0 is the power series

where

The Taylor series of degree n is:

provided f has n derivatives at 0.

Example

COMPARE the graph of e^x and the Taylor approximation to increasing values of n on the calculator & discuss "interval of convergence"

Example

Taylor Series about x = a

The Taylor series of f(x) about x = a is the power series where

The nth degree Taylor polynomial for f at x = a is

provided f has n derivatives at a.

at 1

at x = 1