[Wikipedia](https://en.wikipedia.org/wiki/Hessian_matrix#Use_in_optimization) 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.