MTH 253 Calculus (Other Topics) Chapter 11 – Infinite Sequences and Series Section 11.8 –Taylor and Maclaurin Series Copyright © 2009 by Ron Wallace, all rights reserved.

Back to the goal of the chapter …  Approximating transcendental functions through the use of algebraic functions.  Solution.. Transcendental function must be differentiable of ALL orders. Find a power series equal to the transcendental function. Consider the interval of convergence. Use a portion of the series to approximate the function.

Local Linear Approximations Review of Section 3.10 Find a linear equation that approximates a function around a point given the derivative of the function at that point. Tangent Line f(x)

Local Quadratic Approximations Find a quadratic equation that approximates a function around a point given the first & second derivatives of the function at that point. f(x) y(x)

Local Cubic Approximations Find a Cubic equation that approximates a function around a point given the first, second, and third derivatives of the function at that point. f(x) y(x)

Local Cubic Approximations Find a Cubic equation that approximates a function around a point given the first, second, and third derivatives of the function at that point. f(x) y(x)

Example Find the Linear, Quadratic, and Cubic equations that approximate the above function around a = 1.

Example Find the Linear, Quadratic, and Cubic equations that approximate the above function around x 0 = 1.

Generalization … n th degree Polynomial Approximation If f(x) can be differentiated n times at x = a, then the n th Taylor Polynomial for f(x) about x = a is …

Taylor Series NOTE: This assumes that f (n) (a) exists for all n.

Maclaurin Series NOTE: This assumes that f (n) (0) exists for all n. Taylor Series with a = 0

Question …  For a function f(x) that is differentiable for all orders at x = a, will the Taylor Series converge to f(x) for each value of the domain of the function? Maybe!  For some … yes  For some … no  For some … for part of the domain  For all … yes, when x = a

Example …  Find the Taylor Series for f(x) = ln x at x = 1. Determine the interval of convergence.

Three Important Maclaurin Series All three of these converge for all values of x.

One more … ln t ? Begin with x= ! (-1) n+1 (n-1)!

One more … ln t ? Subtracting

One more … ln t ?

Example …