Let $f$ be a function, that has continuous partial derivatives of order $m+1$ in an open ball $B(a,r)$ in $R^n$. Show for $k=1, \dots, m$ that the equation is valid:
$$\frac1{k!}\sum_{j_1=1}^n \cdots \sum_{j_k=1}^n \frac{\partial^k f}{\partial x_{j_1} \cdots \partial x_{j_k}}(\mathbf a)x_{j_1}\cdots x_{j_k}=\sum_{|\mathbf{\alpha}|=k}\frac1{\alpha!}D_\alpha f(\mathbf a)\mathbf x^\alpha$$
where $\mathbf \alpha$ is an $n$-dimensional multi-index.