Exercise 6.6.9 (Cauchy's Remainder Theorem). Let f be differentiable N+1 times on (-R, R). For each a E (-R, R), let SN (x, a) be the partial sum of the Taylor series for f centered at a; in other words, define N TA SN (x, a) = Σ cn (x - a)" where n=0 Cn = f(n) (a) n! Let EN (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'N (x, a) = (x - a)N. N! f(N+1)(c) EN(x) = EN(x, 0) = = (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 f 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).