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taylor_s_formula [2013/09/15 21:38] nikolaj |
taylor_s_formula [2013/09/16 21:36] nikolaj |
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===== Taylor's formula ===== | ===== Taylor's formula ===== | ||
==== Theorem ==== | ==== Theorem ==== | ||
- | | @#88DDEE: $f\in C^k(\mathbb R^n,\mathbb R)$ | | + | | $k,n\in \mathbb N,\ k>n$ | |
- | | @#88DDEE: $k\in \mathbb N,\ k>n$ | | + | | $f\in C^k(\mathbb R^n,\mathbb R)$ | |
| @#55EE55: $f(x) = \sum_{|\alpha|\le k} \frac{1}{\alpha !} f^{(\alpha)}(0)\ x^\alpha + R_k(x) $ | | | @#55EE55: $f(x) = \sum_{|\alpha|\le k} \frac{1}{\alpha !} f^{(\alpha)}(0)\ x^\alpha + R_k(x) $ | | ||
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| @#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 $ | | | @#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]]. | + | where we use multi-index notation for $\alpha \in \mathrm{FinSequence}(\mathbb N)$, see [[Multi-index power]]. |
==== Discussion ==== | ==== Discussion ==== |