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

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

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.

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.

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

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

Applications sin(0.47832) = ?

Applications If we make x = t2,

Applications

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.

Resources Apostol, Tom. Calculus (Volume I). New York: John Wiley & Sons, Inc. http://mathworld.wolfram.com/LagrangeRemainder.html http://mathworld.wolfram.com/TaylorSeries.html http://www.ugrad.math.ubc.ca/coursedoc/math101/notes/series/appsTaylor.html http://en.wikipedia.org/wiki/Taylor_series http://en.wikipedia.org/wiki/Taylor's_theorem

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