Math 2300 Section 005 – Calculus II – Fall 2010

Notes on Taylor Series – October 18, 2010

Definition of a Taylor Series

The Taylor series for centered at is

Similarly, the Taylor series for centered at is

Examples

Since a Taylor series is the same as a Taylor polynomial, but is taken to infinite degree, we already know many Taylor series. From the last section, we know the following.

where

There is a bit of a technicality here. In section 10.1, we wrote between the functions and their Taylor polynomials. Now, we're writing instead. ( means “approximately equal to,” while means “equal to.”) The main point to understand here is detailed in the next portion of these notes.