Exercise 6.6.9 (Cauchy's Remainder Theorem). Let f be differentiable N+1 times on (-R, R). For each a € (-R, R), let SN(x, a) be the partial sum of the Taylor series for f centered at a; in other words, define N SN (x, a) = c(x-a)" where Cn= n=0 f(n) (a) n! Let Ex(x, a) = f(x)-SN(x, a). Now fix x 0 in (-R, R) and consider EN(x, a) as a function of a. (a) Find EN (x, x). (b) Explain why EN (x, a) is differentiable with respect to a, and show (c) Show -f(N+1)(a) E' (x, a) = (x - a)N. N! EN(x)=EN(x, 0): == TAƒ(N+1) (c) (x (x- c) x N! for some c between 0 and x. This is Cauchy's form of the remainder for Taylor series centered at the origin. Exercise 6.6.10. Consider f(x) = 1/√√1-x. (a) Generate the Taylor series for f centered at zero, and use Lagrange's Remainder Theorem to show the series converges to fƒ on [0, 1/2]. (The case x < 1/2 is more straightforward while x = care.) What happens when we attempt this with x > 1/2? 1/2 requires some extra (b) Use Cauchy's Remainder Theorem proved in Exercise 6.6.9 to show the series representation for f holds on [0, 1).