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I'm trying to rewrite Taylor's formula on $\mathbb{R}^n$.

Let $p\in \mathbb{N}$, $V \subset \mathbb{R}^n$ open and convex, $\mathbf{x, a} \in V$, and $f:V\to \mathbb{R}$ where $f \in C^p(V)$. Then $f(\mathbf{x}) = \sum_{k=0}^{p-1} \frac{1}{k!} D^{(k)} f(\mathbf{a}; \mathbf{x-a}) + \frac{1}{(p-1)!} \int_0^1 (1-t)^{p-1} D^{(p)} f(\mathbf{a} + t(\mathbf{x-a}); \mathbf{x-a}) dt.$

Surely, for $p=1$ the property evidently holds. I can then use an induction argument and integration by parts to prove this, is that correct? It was listed as a difficult exercise so I'm a bit cautious to waving my arms in the air too soon.

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Use Taylor's theorem in one dimension with remainder in integral form on the function $g(t) = f(a + t(x- a))$. If you write the Taylor expansion of $g(t)$ about $0$ and plug in $t = 1$ it translates into what you're trying to prove.

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    Ah yes, I never saw this theorem in one dimensional version, so employing it in a prove would be like cheating. But the proof for the 1-dim situation shown at wikipedia is completely analogue to my proposed method; induction to p where integration by partz seems to be the central trick. So I suppose it's solid.2011-03-11