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taylor_s_formula [2013/09/15 21:38]
nikolaj
taylor_s_formula [2014/03/21 11:11] (current)
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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: postulate ​  | @#55EE55: $f(x) = \sum_{|\alpha|\le k} \frac{1}{\alpha !} f^{(\alpha)}(0)\ x^\alpha + R_k(x) $ |
  
 with 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 $ |+| @#55EE55: postulate ​  | @#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 ====
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 === Reference === === Reference ===
 ==== Parents ==== ==== Parents ====
-=== Requirements ​===+=== Context ​===
 [[Fréchet derivative]],​ [[Function integral]] [[Fréchet derivative]],​ [[Function integral]]
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