Theorem
| $k,n\in \mathbb N,\ k>n$ |
| $f\in C^k(\mathbb R^n,\mathbb R)$ |
| postulate | $f(x) = \sum_{|\alpha|\le k} \frac{1}{\alpha !} f^{(\alpha)}(0)\ x^\alpha + R_k(x) $ |
with
| postulate | $ 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 $\alpha \in \mathrm{FinSequence}(\mathbb N)$, see Multi-index power.
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
Context