Consider the sequence of vectors x0, x1, . . . , where, for k ≥ 1, the vector xk is defined by the Newton’s method recursion formula (6) given an initial “guess” x0 at a root of the equation f(x) = 0. (Here we assume that f: X ⊆ Rn → Rn is a differentiable function.) By imitating the argument in the single-variable case, show that if the sequence {xk } converges to a vector L and Df(L) is an invertible matrix, then L must satisfy f(L) = 0.
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