Now that we have discussed the concept of series, we can return to our discussion on Taylor polynomials of innite degree. Recall that a Taylor polynomial for f (3:) at a matches all derivatives of f at a. If we take the limit as the number of terms in the polynomial approaches innity, we obtain a special type of power series, called Taylor Series _ Mm where the en's are given by CH | n The Taylor Series of f at a is given. by 00 (")0 la "a ma \"gaff; )(mia)\"=f(a)($ia)0+flak\"? a)1+f2(!)(:c a)2+f3(! )(a: 603+... In the special case that a = 0, this series is known as the Maclaurin Series of f 00 (n) I h' m 00 Recall that a power series for function f at a is of the form 2 cn(a: * a)\". 21:0~ 12.0.1 Approximating Functions with Taylor Series We will use the nth degree Taylor polynomial, Tn(m), of a function f (1') to approximate the function itself. This is especially useful for approximating diicult function values, denite integrals or limits. We would like to get an estimate for the remainder Rn(:), where Rum) = at) , Tn($) Then, if we approximate f($0) by Tn($0) for 930 near a, we would like to gure out how big is an (9:0)I = lf(:c0) i Tn(:r0)l? If the remainder is close to 0, then the Taylor polynomial will be a good approximation for the function itself. Taylor's Inequality Suppose that f is n + 1 times differentiable and let Rn(ac) denote the difference between f (as) and the Taylor polynomial of degree n for f(9') centered at (L. Then if |f(\"+1)(c)| S M for all c between a and :r, the error RT, (:3) = f (as) i T 7r,(:t:) satises the inequality M n+1 (71+ 1)! I IRAQI S |ma To use the Taylor's Inequality to bound the error in the approximation f(9') m Tn(m), we do the following: 1. Find an upper bound M on the absolute value of the 'n + 1st derivative of f between a (the center) and m (the value you want to sub into your function and approximate). M (11+ 1)! 2. The error IRn(:r)| is at most |:r a|\"+1.