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Taylor's Theorem

  • Definition: Let f be a function having finite nth derivative f^(n) everywhere in an open interval (a, b) and assume that f^(n-1) is continuous on the closed interval [a, b]. Assume that c is in [a, b]. Then, for every x in [a, b], x not equal to c, there exists a point x_1 interior to the interval joining x and c such that f(x) = f(c) + sum[[f^(k)(x-c)^k]/k!] + [f^(n)(x_1)(x-c)^N]/n! as k ranges from 1 to n-1.

    Source: Mathematical Analysis, second edition by Tom M. Apostol

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