Taylor series operator commutes with composition

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Statement

Suppose x_0 is a real number. Suppose g is a function defined on a subset of the reals that is infinitely differentiable at x_0. Suppose f is a function defined on a subset of the reals that is infinitely differentiable at g(x_0). Then, the composite of two functions f \circ g is infinitely differentiable at x_0, and its Taylor series can be computed formally by composing the Taylor series for f at g(x_0) with the Taylor series for g at x_0. Formally, what this means is that we write down the Taylor series for f at g(x_0), then plug in for x the entire expression for the Taylor series of g, then simplify.

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