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taylor_s_formula [2013/09/16 10:45] nikolaj |
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| - | ===== Taylor's formula ===== | ||
| - | ==== Theorem ==== | ||
| - | | $k,n\in \mathbb N,\ k>n$ | | ||
| - | | $f\in C^k(\mathbb R^n,\mathbb R)$ | | ||
| - | | @#55EE55: $f(x) = \sum_{|\alpha|\le k} \frac{1}{\alpha !} f^{(\alpha)}(0)\ x^\alpha + R_k(x) $ | | ||
| - | |||
| - | with | ||
| - | |||
| - | | @#55EE55: $ R_k(x) = \sum_{|\alpha|=k+1} \frac{k+1}{\alpha !} \left( \int_0^1\ (1-s)^k\ F^{(\alpha)}(s\ x)\ \mathrm ds \right)\ x^\alpha $ | | ||
| - | |||
| - | where we use multi-index notation for $\alpha \in \mathrm{FinSequence}(\mathbb N)$, explained in [[Finite sequence]]. | ||
| - | |||
| - | ==== Discussion ==== | ||
| - | $f\in C^\infty(\mathbb R,\mathbb R)$, $a\in \mathbb R$ | ||
| - | |||
| - | ^ $ f(x) = \sum_{n=0}^\infty f^{(n)}(a) \frac{1}{n!} (x-a)^n$ ^ | ||
| - | === Reference === | ||
| - | ==== Parents ==== | ||
| - | === Requirements === | ||
| - | [[Fréchet derivative]], [[Function integral]] | ||
