I couldn't find a reference for this:
Let $f:\mathbb{R}^n\to\mathbb{R}^m$ $k$ times differentiable and $P_k$ its degree $k$ Taylor polynomial centered at $0$. Then, it is true that $$f(x) = P_k(x) + o(\|x\|^k)?$$ where $\frac{o(\|x\|^k)}{\|x\|^k}\to0$ when $\|x\|\to0$. What is exactly the form of $P_k(x)$?
I could only find references for this, but when $f$ is $C^k$ and with real values.