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Let $f: \mathbb{R}^n \to \mathbb{R}$ and additionally suppose that $f \in \mathcal{C}^{\infty}(\mathbb{R}^n)$. Given a point $a \in \mathbb{R}^n$, let the $l$-th Taylor polynomial of $f$ about $a$, $T_{a,l} : \mathbb{R}^n \to \mathbb{R}$ be given by:

$$T_{a,l}(x) = f(a) + \sum_{|z|=1}^l \frac{1}{z!} \frac{ \partial^{z} f}{\partial x^{z}}(a)(x-a)^{z}$$ where $z$ is an $n$-dimensional multi-index. I would like to prove that

$$\lim_{h \to 0} \frac{f(a+h)-T_{a,l}(a+h)}{||h||^l}=0$$

Any hints?

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Fix $h$ and define $g(t):=f(a+th)$. Apply Taylor's formula for smooth functions of one variable to get a bound. – Davide Giraudo Oct 3 '12 at 12:46

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