# 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 $s_1+\dots+s_n \le m-1$.

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Your last sum is no equation, just a sum. –  Pedro Tamaroff Aug 14 '13 at 20:36
Well, better start grouping terms that are the same, need to apply a Theorem that allows for swapping order of direction differentiation. You'll also need some combinatorics that can count the number of ways to rearrange for example a word like XXYYZZZ. –  Evan Aug 14 '13 at 23:17