Apparently the second order multivariable Taylor expansion is: $$f(\mathbf x+\mathbf h)=f(\mathbf x)+ \partial_i f(\mathbf x) h_i + \frac 12 \partial_j \partial_i f(\mathbf x + t \mathbf h) h_i h_j$$ for some $t$, $0 \le t \le 1$, where some twice differentiable $f$ is defined in some neighborhood $N(x) \subset \Bbb R^n$ and $(\mathbf x + \mathbf h) \in N(x)$, and using the Einstein summation convention.
Am I right then in thinking that the entire (infinite order?) Taylor expansion is: $$f(\mathbf x+\mathbf h)=\sum_{i=0}^\infty \frac 1 {i!}[\partial]_i f(\mathbf x) [h]_i$$ where I'm using the notation (which I just invented) $[\partial]_k=\underbrace{\partial_i \partial_j \cdots}_{k\text{ of these}}$ and likewise for $[h]_i$? If so, what happened to the $+ t\mathbf h$ part? If not, what's the actual formula?
The correct taylor formula is $$f(x+h) = \sum_{k=0}^\infty \sum_{|\alpha| = k} \frac1{\alpha!} \partial^\alpha f(x) h^\alpha$$ Where for a multiindex $\alpha\in\mathbb N_0^n$ $$\alpha! = \prod_{j=1}^n \alpha_j!\\ x^\alpha = \prod_{j=1}^n x_j^{\alpha_j}\\ \partial^\alpha = \prod_{j=1}^n \frac{\partial^{\alpha_j}}{\partial x_j^{\alpha_j}}$$ Or even shorter $$f(x+h) = \sum_{\alpha\in\mathbb N_0^n} \frac1{\alpha!} \partial^\alpha f(x) h^\alpha$$
Unfortunately I can only provide a reference for the $n=1$ case ($f:\mathbb R^m \to\mathbb R^n$) and a wikipedia reference for the general case.