1. Carlos Xudiera November 30, 2011
Taylor Series
Carlos Xudiera
November 30, 2011

2. Content Why use Taylor Series Definition Taylor’s Theorem
Lagrange Remainder
Applications

3. Why use Taylor Series?
In many problems we face functions which are far more complicated than the standard functions from classical analysis. If we could represent them as series of polynomials then some properties of the functions would be easier to study.

4. Definition
If f(x) is a complex or real function infinitely differentiable at x = a then the series: is called the Taylor Series for f(x) centered at a.

5. Taylor’s Theorem
Let , differentiable m ≥ 1 times at . Then, such that, where

6. Lagrange Remainder
The remainder Rn after n terms is given by with x* between a and x.

7. Applications
sin( ) = ?

9. Applications
If we make x = t2,

10. Applications

11. Other Applications Find solutions to differential equations.
Newton’s method to find the root of a function.
Can be extended to functions in several variables.

12. Resources
Apostol, Tom. Calculus (Volume I). New York: John Wiley & Sons, Inc.

13. Q & A