MATH 1B Lecture 19: The Accuracy of Taylor Series
Document Summary
Math 1b: calculus - lecture 19: the accuracy of taylor series. Consider a function f: that has derivatives of all orders in an open interval containing a. Recall that the taylor series for f at x=a is given by the sum of n=0 to of [ {(f(n))(a)(x- a)n} / n! Often it is the case that f(x) = [ {(f(n))(a)(x-a)n} / n! ] for all x. For example, if f is a function that is exactly 0 near x=0 one must be careful. Let"s put pn(x) = the sum from n=0 to of [ {(f(n))(a)(x-a)n} / n! ] for n. This is the nth order taylor polynomial for f at x =a. Note that [f(x) - pn(x)] / (x-a) = f"(c) for some c between a and x. So if we can bound f"(c), then we can bound r0(x) as a function of x.
