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## Taylor's Theorem

Theorem.(Taylor) Every functional in has the representation

for some between and , where is the gradientvector evaluated at , and is the Hessian matrix of at , i.e.,

(52) (53)

Proof.See Goldstein [Goldstein 1967] p. 119. The Taylor infinite series is treated in Williamson and Crowell [Williamsonet al.1972]. The present form is typically more useful for computing bounds on the error incurred by neglecting higher order terms in the Taylor expansion.