For a non-negative integers n, let the function fn be given by fn(x) = ∑nk=0 = 1 + x + x2/2 + x3/6 + ... + xn/n! where n! = 1 * 2 * ... * n, with the convention that 0! = 1 so that f0(x) = 1, f1(x) = 1 + x, etc. Hint: fn+1(x) = fn(x) + (xn+1)/(n+1)! and f'n+1(x) = fn(x) (why?) (A) Show by mathematical induction that for non-negative integers n that f2n(x) is a positive integer for all real numbers. (B) By using the result from (A) (incl. the hint); why can we conclude that f9: R -> R has an inverse function? Does f10 have an inverse function? (C) Given n >= 0; show that limx->∞ fn(x)/ex=0

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