# What is the more than 3rd order Taylor series approximation for a multi-variate function?

Suppose $f$ is a infinite continuously differentiable map: $R^n\to R$, and $x,x_0 \in R^n$, then we have the following second order Taylor expansion of $f(x)$ at $x_0$:

$$f(x)\approx f(x_0)+(x-x_0)^T\nabla f(x_0)+\dfrac{(x-x_0)^T \nabla^2f(x_0)(x-x_0)}{2}$$

What is the next iterm? Do I need to use tensor? what is a simple representation?

and how to conduct the multiplication between tensors and matrices and vectors?