MATH 2B Chapter Notes - Chapter 11.10: Kolmogorov Space, Ratio Test, Linear Algebra

• School: UC-Irvine
• Department: Mathematics
• Course: MATH 2B
• Professor: George Donaldson

Document Summary

The idea is to obtain a good approximation to a function f (x) among all polynomials of degree n. There are many sensible notions of what good approximation" could mean. The notion here is that we want our approximating polynomial to share the value and rst n derivatives with f (x) at a point x = a. Suppose that f is a function which is n-times differentiable at x = a. The nth taylor polynomial of f centered at x = a is the polynomial. 3! (x a)3 + f (n 1)(a) (n 1)! (x a)n 1 + f (n)(a) n! (x a)n. Taylor polynomials are, essentially, higher order versions of the linear approximation. Example let f (x) = e als of f centered at zero are therefore. 2 x, so f (k)(0) = (cid:0) 1. 22 2! x3 x2 = 1 +