# Taylor Polynomial in Multivariable Case

I am doing Problem 9.30 in Rudin's Principles of Mathematical Analysis and have done part (a) and (b) but got stuck on part(c).

In part (b) I have finished the proof that: $$f(\mathbf a+\mathbf x)=\sum_{k=0}^{m-1}{1 \over{k!}}\sum (D_{i_1 \dots i_k}f)(\mathbf a)x_{i_1}\dots x_{i_k}+r(\mathbf x),(*)$$ where $$\lim_{x\to 0}{{r(\mathbf x)}\over{|\mathbf x|^{m-1}}}=0.$$ Now I need to show that the equation ($*$) is in fact equivalent to $$\sum{{(D_1^{s_1}\dots D_n^{s_n}f)(\mathbf a)}\over{s_1 !\dots s_n !}}x_1^{s_1}\dots x_n^{s_n},$$where the summation extends over all ordered $n$-tuples $(s_1,\dots,s_n)$ such that each $s_i$ is a nonnegative integer and $s_1+\dots+s_n \le m-1$.