The answer to this question says that a MacLaurin polynomial of a function ff is the unique NthN^\text{th} order polynomial that minimizes the following functional:

L[g]=limx→0(f(x)−g(x)xN)2,

L[g] = \lim_{x\rightarrow 0}\left(\frac{f(x)-g(x)}{x^N}\right)^2,

and that the minimum value attained is 00,

\min_{g} L[g] = 0

\min_{g} L[g] = 0

I’m wondering if anyone has a reference for a proof of this fact in a forward sense, i.e., starting from the above, show that gg must be equal to the formula for the MacLaurin series of ff, without directly invoking some form of Taylor’s theorem to expand ff. Basically, assuming we didn’t know anything about Taylor/MacLaurin series or polynomials, can we derive them from the above minimization problem?

Possible path

We know gg is an N^\text{th}N^\text{th} order polynomial:

g(x) = \sum_{n=0}^N a_n x^n = \vec{X}\cdot\vec{a},

g(x) = \sum_{n=0}^N a_n x^n = \vec{X}\cdot\vec{a},

where

\vec{X} = \left[1\;\; x\;\; x^2\;\;\ldots\;\;x^N\right]^\text{T},

\vec{X} = \left[1\;\; x\;\; x^2\;\;\ldots\;\;x^N\right]^\text{T},

and

\vec{a} = \left[a_0\;\; a_1\;\; a_2\;\;\ldots\;\;a_N\right]^\text{T}.

\vec{a} = \left[a_0\;\; a_1\;\; a_2\;\;\ldots\;\;a_N\right]^\text{T}.

Now I know I’ll need to know the sensitivities of that functional to the parameters:

\nabla_\vec{a} L[g] =

\left[

\frac{\partial L[g]}{\partial a_0}\;\;

\frac{\partial L[g]}{\partial a_1}\;\;

\frac{\partial L[g]}{\partial a_2}\;\;\ldots\;\;

\frac{\partial L[g]}{\partial a_N}

\right]^\text{T}

\nabla_\vec{a} L[g] =

\left[

\frac{\partial L[g]}{\partial a_0}\;\;

\frac{\partial L[g]}{\partial a_1}\;\;

\frac{\partial L[g]}{\partial a_2}\;\;\ldots\;\;

\frac{\partial L[g]}{\partial a_N}

\right]^\text{T}

And that through application of the chain rule, I’ll also need that

\nabla_\vec{a} g = \vec{X}.

\nabla_\vec{a} g = \vec{X}.

Setting the gradient of the functional w.r.t. \vec{a}\vec{a} to zero for each component should give me a way to solve for gg, I just don’t see a clear path forward that results in the formula for a Taylor polynomial. I hope someone can illuminate the way.

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