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taylor_s_formula [2013/09/15 21:37] nikolaj created |
taylor_s_formula [2013/09/16 10:45] nikolaj |
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| ===== Taylor's formula ===== | ===== Taylor's formula ===== | ||
| ==== Theorem ==== | ==== Theorem ==== | ||
| - | | @#88DDEE: $X,Y$ ... Banach spaces with topology | | + | | $k,n\in \mathbb N,\ k>n$ | |
| - | | @#88DDEE: $f\in C^k(\mathbb R^n,\mathbb R)$ | | + | | $f\in C^k(\mathbb R^n,\mathbb R)$ | |
| - | | @#88DDEE: $x_0\in \mathbb R^n $ | | + | |
| - | | @#88DDEE: $k\in \mathbb N,\ k>n$ | | + | |
| | @#55EE55: $f(x) = \sum_{|\alpha|\le k} \frac{1}{\alpha !} f^{(\alpha)}(0)\ x^\alpha + R_k(x) $ | | | @#55EE55: $f(x) = \sum_{|\alpha|\le k} \frac{1}{\alpha !} f^{(\alpha)}(0)\ x^\alpha + R_k(x) $ | | ||
