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Proof Of Taylor's Formula
PROOF OF TAYLOR'S FORMULA Let F(x) = R_{n}(x), G(x) = (x  c)^{n+1}. Then F(x) = f(x)  P_{n}(x). f(x) and the nth Taylor polynomial P_{n}(x) have the same value and first n derivatives at x = c. Therefore F(c) = F'(c) = F"(c) = ... = F^{(n)}(c) = 0. We also see that G(c) = G(c) = G"(c) = ... = G^{(n)}(c) = 0. Using the Generalized Mean Value Theorem n + 1 times, we have for some t_{0} strictly between c and x; for some t_{1} strictly between c and t_{0}; for some t_{n} strictly between c and t_{n1}. It follows that Either x < t_{0} < t_{1} < ... < t_{n} < c or x > t_{0} > t_{1} > ... > t_{n} > c, so t_{n} is strictly between c and x. The (n + l)st derivatives of F(t) and G(t) are f^{(n+1)}(t) = f^{(n+1)}(t)  0, G^{(n+1)}(t) = (n + 1)! Substituting, we have and Taylor's Formula follows at once.


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