For a function $f: \Bbb R^n \to R$, the $2$nd order Taylor expansion is:
$$f(\mathbf x+\mathbf h) \approx f(\mathbf x)+ Df(\mathbf x) \mathbf h + \frac{1}{2}\mathbf h^T H(f)(\mathbf x) \mathbf h$$
What's the next term (the last term in the $3$rd order Taylor expansion)?
I can see that $Df(\mathbf x)$ is a row vector, so to make it scalar, we need to right multiply a column vector. And similarly $Hf(\mathbf x)$ is an $n\times n$ matrix so to make it a scalar we need to left and right multiply by vectors (row and column, respectively). I can't think what the next term would have to be, though.
Best Answer
The next term is
$$ \frac{1}{3!}D^3f\big|_x(h,h,h) $$
Where
$$ D^3f\big|_x = \sum \frac{\partial^3 f}{\partial x_i \partial x_j \partial x_k} dx_i \otimes dx_j \otimes dx_k $$
In other words, the third term is
$$ \frac{1}{3!}\sum \frac{\partial^3 f}{\partial x_i \partial x_j \partial x_k} h_i h_j h_k $$