# Taylor series operator commutes with composition

From Calculus

## Statement

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