Let $I\subset\mathbb R$ be an interval and $f:I\to\mathbb R$ be a $n$ times continuously differentiable function. For a given $a\in I$ and all $x\in I$ the value $f(x)$ can be written as
$$f(x)=f(a)+\sum_{k=1}^n \frac{f^{\{k\}}(a)}{k!}(x-a)^k+r(x)(x-a)^n$$ where $r(x)$ is a function converging to $0$ at the point $a$, i.e. with $\lim_{x\to a}r(x)=0.$
Proofs: 1