TITLE: What exactly are the "higher order terms" (H.O.T.) in Taylor series expansion in multivariable case?
QUESTION [8 upvotes]: I know the Taylor series expansion in single variable case:
$$ f(x) = f(x_0) + f'(x_0)(x - x_0) + \frac{1}{2}f''(x_0)(x-x_0)^2 + \frac{1}{3!}f^{(3)}(x_0)(x-x_0)^3 + \frac{1}{4!}f^{(4)}(x_0)(x-x_0)^4 + \frac{1}{5!}f^{(5)}(x_0)(x-x_0)^5 + \dots $$
Multivariable case is stated as:
$$ f(\bar{x}) = f(\bar{x_0}) + \bar\nabla f(\bar{x_0})^T(\bar{x} - \bar{x_0}) + \frac{1}{2}(\bar{x} - \bar{x_0})^T\bar\nabla^2 f(\bar{x_0})(\bar{x} - \bar{x_0}) + H.O.T. $$
I can find the expression above everywhere, but it is not possible to find the open expression for the H.O.T. (at least I wasn't able to find it). No source states the explicit expression for the higher order terms. Do people avoid them for some reason? For example, are those terms too complex or too long to write?
So, what are these higher order terms in the multivariable case? Do they involve a $\bar\nabla^3 f(\bar{x_0})$, $\bar\nabla^4 f(\bar{x_0})$, $\bar\nabla^5 f(\bar{x_0})$, ... terms; if yes, how are they defined? Please write a few terms from H.O.T. to make the pattern clear.

REPLY [2 votes]: In addition to below answers the explicit form for three variables are
$$f(x,y,z)=f(x_0,y_0,z_0)$$ 
$$+\frac{\partial f_0}{\partial x}(x-x_0)+\frac{\partial f_0}{\partial y}(y-y_0)+\frac{\partial f_0}{\partial z}(z-z_0)\quad \Rightarrow Order 1$$
$$+\frac{1}{2} \bigg(\frac{\partial^2 f_0}{\partial x^2}(x-x_0)^2+\frac{\partial^2 f_0}{\partial y^2}(y-y_0)^2+\frac{\partial^2 f_0}{\partial z^2}(z-z_0)^2+2\frac{\partial^2 f_0}{\partial x\partial y}(x-x_0)(y-y_0) $$
$$+2\frac{\partial^2 f_0}{\partial x\partial z}(x-x_0)(z-z_0)+2\frac{\partial^2 f_0}{\partial z\partial y}(z-z_0)(y-y_0)\bigg)\quad \Rightarrow Order 2$$
And it goes like this to higher orders