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- Apr 14, 2013

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Let $U\subset \mathbb{R}^n$ be an open set and $f:U\rightarrow \mathbb{R}$ is a $k$-times continusouly differentiable function.

Let $x_0\in U$ be fixed.

The $k$-th Taylor polynomial of $f$ in $x_0$ is $$T_k(x_1,\ldots ,x_n)=\sum_{m=0}^k\frac{1}{m!}\sum_{i_1=1}^n \ldots \sum_{i_m=1}^n \frac{\partial}{\partial{x_{i_1}}}\ldots \frac{\partial}{\partial{x_{i_m}}}f(x_0)\cdot x_{i_1}\cdot \ldots x_{i_m}$$

Show that $T_k(x_1,\ldots ,x_n)$ is the only polynomial of degree $k$ such that $$T_k(0)=f(x_0) \\ \frac{\partial}{\partial{x_{i_1}}}\ldots \frac{\partial}{\partial{x_{i_m}}}T_k(0)=\frac{\partial}{\partial{x_{i_1}}}\ldots \frac{\partial}{\partial{x_{i_m}}}f(x_0)$$ for all $m\in\{1,\ldots , k\}$, $i_\ell\in \{1, \ldots ,n\}$ and $\ell \in \{1, \ldots , m\}$.

To show that $T_k$ satisfies these properties we just have to replace $x_i=0$ for all $i$ for the first property and for the second one we have to calculate these partial derivatives, right?

I am trying to show that $$\frac{\partial}{\partial{x_{i_1}}}\ldots \frac{\partial}{\partial{x_{i_m}}}T_k(0)=\frac{\partial}{\partial{x_{i_1}}}\ldots \frac{\partial}{\partial{x_{i_m}}}f(x_0)$$

Is it maybe as follows? \begin{align*}\frac{\partial}{\partial{x_{i_1}}}T_k(x_1,\ldots ,x_n)&=\frac{\partial}{\partial{x_{i_1}}}\left [\sum_{m=0}^k\frac{1}{m!}\sum_{i_1=1}^n \ldots \sum_{i_m=1}^n \frac{\partial}{\partial{x_{i_1}}}\ldots \frac{\partial}{\partial{x_{i_m}}}f(x_0)\cdot x_{i_1}\cdot \ldots x_{i_m}\right ]\\ & =\sum_{m=0}^k\frac{1}{m!}\sum_{i_1=1}^n \ldots \sum_{i_m=1}^n \frac{\partial}{\partial{x_{i_1}}}\ldots \frac{\partial}{\partial{x_{i_m}}}f(x_0)\cdot \frac{\partial}{\partial{x_{i_1}}}\left (x_{i_1}\cdot \ldots x_{i_m}\right )\\ & =\sum_{m=0}^k\frac{1}{m!}\sum_{i_1=1}^n \ldots \sum_{i_m=1}^n \frac{\partial}{\partial{x_{i_1}}}\ldots \frac{\partial}{\partial{x_{i_m}}}f(x_0)\cdot x_{i_2}\cdot \ldots x_{i_m}\end{align*} Or how do we calculate the partial derivatives?

To show the uniqueness, we have toconsider that there is also an other polynomial with the specific properties and then we have toshow that this polynomial must be $T_k$, right? But how do we do that exactly?