# Multivariable Taylor series convergence

Let us have a Taylor series expansion for multuvariable function $$f(x_1,x_2,\ldots,x_N)=\sum_{n=0}^{\infty}{\sum_{j_1+j_2+\ldots +j_N=n}{\dfrac{1}{j_1!j_2!\ldots j_N!}\dfrac{\partial^{n}f}{\partial x_1^{j_1}\partial x_2^{j_2}\ldots \partial x_N^{j_N}}\Bigg|_{x=0}x_1^{j_1}x_2^{j_2}\ldots x_N^{j_N}}},$$ where in the second sum summation goes over all possible combinations of non-negative integer solutions of the equation $j_1+j_2+\ldots + j_N=n$. In particular I'm interested in the expansion of the function $$f(x_1,x_2,\ldots,x_N)=\exp\left\{\dfrac{1}{2}D_{st}{x_s}{x_t}\right\},$$ where $D_{st}x_sx_t$ means sum $\sum_{s=1}^{N}{\sum_{t=1}^{N}{D_{st}x_sx_t}}$, because that is a moment-generating function for multivariable normal distribution with a covariance matrix $D$.

I need to prove that in the case of such a function it's Taylor expansion converges to this function for every possible vector $(x_1,x_2,\ldots,x_N)\in \mathbb{R}^{N}$.

I know that for a function of one variable its Taylor series $$\sum_{i=0}^{n}{\dfrac{f^{(i)}(0)}{i!}x^{i}}+R_{n}(x)$$ converges to it if and only if its remainder term $R_n(x)$ has a limit $$\lim_{n\rightarrow \infty}{R_n(x)}=0,$$ and for any given interval $x\in (-R,R)$ this is true if, for example, $$\left|f^{(n)}(\xi)\right|\leq M$$ for any $n$ and $\xi\in(-R,R)$.

I'm interested if there some way to prove that for multivariable function in question its series converges to it on all $\mathbb{R}^N$ so that its series unambiguously defines it.

This question has stemmed out of that The theorem inverse to Isserlis' theorem