This involves a proof to Taylor's Theorem. Assume and let be defined on where and suppose the n-th derivative exists on . Then for each nonzero there is some between and such that .

Let be the unique solution of and let

Show that is differentiable on (I think this part is straightforward since is differentiable on and the other terms are simply polynomials). Show that (not really sure how this works out, not great with combining series), show that (again, don't know), and finally show that there is such that .

My professor showed an alternate proof in class which seems at least somewhat easier, but I am absolutely lost here. Thanks.