I have a doubt about Taylor's formula in two or more variables:are these two scripts equivalent? if yes, why?
$$f(\vec{x}) = f(\vec{x}^0) + \sum\limits_{i=1}^{m-1} \frac{1}{i!} \left( \dfrac{\partial^i f}{\partial x_{j_1} \partial x_{j_2} \dots \partial x_{j_i}} \right)_{\vec{x}^0} (x_{j_1} -x^0_{j_1}) (x_{j_2} -x^0_{j_2}) \dots (x_{j_i} -x^0_{j_i}) + \sigma$$
and
$$f(\vec{x}) = f(\vec{x}^0) + \sum\limits_{i=1}^{m-1} \frac{1}{i!} \left[ \dfrac{\partial f}{\partial x_1} (x_1 - x^0_1) + \dfrac{\partial f}{\partial x_2} (x_2 - x^0_2) + \dots \dfrac{\partial f}{\partial x_n} (x_n - x^0_n) \right]^{i} + \sigma$$
Thank you in advance