Given a function f : R - R and its Taylor polynomial Pr expanded around a =0, it is a fact that there is c in between' 0 and I such that f() - PC = (E) (n + 1)! ere f(+1) is the (n + 1)th derivative of f, and recall that PRO = I = f(O) + f(0)=+ F(0) 2 f() (0) + - + n k! 2 n! k-0 bu may recognize the n = 0 case as the "Mean Value Theorem."Problem 2. (i) Suppose you knew that, for every n and every I, (* ) Use ( E) to show that lim If (z) - Pr(z)| = 0. 7100 (Hint: for any 8, [e[" ! -+ 0 as n + co) (ii) For which I does the Taylor series below converge: 18 f(k) (0) K! *-0 State your answer as an interval of convergence, a radius of convergence, or simply the set of I for which there is convergence. You may find it useful to compare with the series Poo(z) = > K! (ini) Is there anything special about a in (*) ? In other words, if the bound () held with K in place of wh, would anything change in questions (i) and (it)? (tv) Give an example of a function that satisfies (*) and is not a polynomial