Taylor's formula

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

$f\in C^\infty(\mathbb R,\mathbb R)$, $a\in \mathbb R$

$f(x) = \sum_{n=0}^\infty f^{(n)}(a) \frac{1}{n!} (x-a)^n$