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. (4) Suppose you knew that, for every n and every I, (* ) Use (E) to show that lim f(z) - Pr(I)| = 0. (Hint: for any e, [e[" ! + 0 as n + co) (ii ) For which I does the Taylor series below converge: POD) = f(k) (0) K! *-0 State your answer as an interval of convergence, a radius of convergence, or simply the set of a for which there is convergence. You may find it useful to compare with the series Poo(z) = K! (iti) Is there anything special about a in (*) ? In other words, if the bound () held with K" in place of oh, would anything change in questions (i) and (ii) ? (tv) Give an example of a function that satisfies () and is not a polynomial