I am asking myself wether the following fact is true:
Let $f(x_1,\ldots,x_k) \colon U \subseteq \mathbb{R}^n \to \mathbb{R}$ be a multivariate function.
Do the multivariate Taylor series $$MT_{f}(x_1,\ldots,x_k) = \sum_{\alpha\in\mathbb{N}^k} \frac{\partial^\alpha f(0,\ldots,0)}{\alpha!~\partial x^\alpha} x^\alpha \qquad \text{where $\alpha$ is used as a multi-index}$$ and iterating univariate Taylor series expansion (i.e. treating the other variables as constants) $$ \begin{align*} T_{f}(x_1,\ldots, x_k) &= \sum_{\alpha\in\mathbb{N}} \frac{\partial^\alpha f(0,x_2,\ldots,x_k)}{\alpha!~\partial{x_1^\alpha}} x_1^\alpha\\ T_{T_{f}}(x_1,\ldots,x_k) &= \sum_{\alpha\in\mathbb{N}} \frac{\partial^\alpha T_{f}(x_1,0,\ldots,x_k)}{\alpha!~\partial{x_2^\alpha}} x_2^\alpha\\ &\;\vdots\\ IT_{f}(x_1, \ldots, x_k)&= \sum_{\alpha\in\mathbb{N}} \frac{\partial^\alpha T_{T_{\ddots_{T_{f}}}}(x_1,x_2,\ldots,0)}{\alpha!~\partial{x_k^\alpha}} x_k^\alpha \end{align*} $$ coincide? Specifically:
- if the multivariate Taylor series $MT$ does not exist, the iterated one $IT$ also does not exist
- if they both exist, they are equal and the iteration method is invariant under variable permutation, e.g., the sequences $x_1\to x_2\to x_3\to \ldots \to x_k$ and $x_2 \to x_1 \to x_k \to \ldots$ should result in the same Taylor series.
