Could some one please help me find the multivariable fourth order Taylor series expansion for f(x,y).
$$ f(x,y)|_{x,y = 0,0} $$
I'm sorry, but I need someone to write the full approximation out. My professor wrote down the summation form of the formula, but never did anything past a second order approximation. So I am not able to make heads or tails of the formula.
Let $f:\mathbb R^n \to \mathbb R$ and $f\in C^{k+1}$. Then we have $$f(x+h) = \sum_{0\leq \lvert \alpha \rvert \leq k} \partial^{\alpha}f(x)\frac{h^{\alpha}}{\alpha!}+o(\lvert h \rvert^{k+1})$$ where $\alpha = (\alpha_1,\dots,\alpha_n)$ where $\alpha_j \in \mathbb N_{\geq 0}$ and $\lvert \alpha \rvert = \alpha_1+\dots+\alpha_n$. The notation $\partial^{\alpha}$ means $\partial_{1}^{\alpha_1}\dots\partial_n^{\alpha_n}$ and $\alpha! = \alpha_1!\dots\alpha_n!$ and $h^{\alpha} = h_1^{\alpha_1}\dots h_n^{\alpha_n}$. The sum is to be interpreted as $\sum_{0\leq \lvert \alpha \rvert \leq k}=\sum_{j = 0}^k\sum_{\lvert \alpha \rvert = j}$.
