General Multivariable Taylor Series

Definition

• For function $f(x):{\mathbb{R}}^n\to \mathbb{R}$
• If $f$ is a Class Function $C^k$
• Then, the class function has an approximation $f(x)\sim {\sum }_{\alpha =1}^k\frac{\frac{{\partial }^{\alpha }f}{\partial x^{\alpha }}(x_0)}{\alpha !}{x}^{\alpha }$ Where:
• $x_0$ is a vector that satisfies $x\in {\mathbb{R}}^n,|x_0|=\alpha$