1. 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.

2. 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.

3. 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)

4. 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)

5. 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)

6. 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)

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

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

9. 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 …

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

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

12. 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

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

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

15. One more … ln t ? Begin with … @ x=0 1 2 -3! (-1) n+1 (n-1)!

16. One more … ln t ? Subtracting

17. One more … ln t ?

18. Example …