Taylor formula with integral remainder in $\mathbb{R}^{d}$

Question

Let $V\in \mathcal{C}^{\infty}(\mathbb{R}^{d})$

How to write the Taylor formula with integral remainder of order $n$ for the gradient of $V$ in some element $x_0\in\mathbb{R}^d$?

Thanks

Answer 1

Apply the 1-dimensional version of Taylor's theorem to the real-variable function $g(t)=V(x_0 + th)$ at $0$, i.e.,

$$g(t)=\sum_{k=0}^{n-1} \frac{g^{(k)}(0)t^k}{k!} + o(t^n)$$ which implies $$V(x_0 + th)=\sum_{k=0}^{n-1} \frac{\nabla^{(k)}V(x_0)h^k}{k!} + o(t^n)$$ where $h^k$ = $(h, h, ..., h) \in (\mathbb{R}^d)^k$.