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 taylor_s_formula [2013/09/16 10:45]nikolaj taylor_s_formula [2014/03/21 11:11] (current) Both sides previous revision Previous revision 2013/09/16 21:36 nikolaj 2013/09/16 10:45 nikolaj 2013/09/15 21:38 nikolaj 2013/09/15 21:38 nikolaj 2013/09/15 21:37 nikolaj created Next revision Previous revision 2013/09/16 21:36 nikolaj 2013/09/16 10:45 nikolaj 2013/09/15 21:38 nikolaj 2013/09/15 21:38 nikolaj 2013/09/15 21:37 nikolaj created Line 4: Line 4: | $f\in C^k(\mathbb R^n,\mathbb R)$ | | $f\in C^k(\mathbb R^n,\mathbb R)$ | - | @#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 ==== Line 18: Line 18: === Reference === === Reference === ==== Parents ==== ==== Parents ==== - === Requirements ​=== + === Context ​=== [[Fréchet derivative]],​ [[Function integral]] [[Fréchet derivative]],​ [[Function integral]]