How to expand this tensorial Taylor expansion to the $n$th term?

Question

Wikipedia describes the use of the Hessian matrix in a Taylor series expension. I've noted that the first term is written in terms of the original function, the second term uses the gradient of the function, and the third term uses the Hessian of the function. While each term is a scalar, I see a pattern of using higher rank tensor operations as the order of approximation increases.

$$y = f(\vec{x} + \Delta \vec{x}) \approx f(\vec{x}) + \nabla f(\vec{x}) \Delta \vec{x} + \frac{1}{2} \Delta \vec{x}^T H(\vec{x})\Delta \vec{x}$$

How can we expand this approximation to an arbitrary $n$th term? I am expecting, or hoping, there is a way to do this in terms of tensors.

Answer 1

This answer is not 'tensorial', but there is a series expansion given on wikipedia for several variables. Leaving aside notions of vector spaces and their dual spaces when considering contravariant or covariant indices, you can see each term as an array of derivatives.

Answer 2

You could write $$ f(x+Δx)=f(x)+f'(x)Δx+\tfrac12f''(x)[Δx,Δx]+\tfrac16f'''(x)[Δx,Δx,Δx]+... $$ where the derivatives are tensors, specifically vector valued symmetric multi-linear forms. The non-linear arguments are separated from the linear arguments of the tensor.

For instance for the exact solution of $x'=f(x)$ you get by recursively inserting the differential equation into derivatives of itself $$\begin{align} x(t+h)&=x(t)+hf(x)+\tfrac12h^2f'(x)f(x)+\tfrac16h^3\Bigl(f''(x)[f(x),f(x)]+f'(x)^2f(x)\Bigr)\\ &~~~+\tfrac1{24}h^4\Bigl(f'''(x)[f(x),f(x),f(x)]+3f''(x)[f'(x)f(x),f(x)]+f'(x)f''(x)[f(x),f(x)]+f'(x)^3f(x)\Bigr)+.. \end{align}$$ and latest at this point the motivation for B-trees and B-series begins to make sense.