Differentiable function/R/n-times/Taylor polynomial of degree n/Definition

Taylor polynomial

Let ${}I\subseteq \mathbb {R}$ denote an interval,

$f\colon I\longrightarrow \mathbb {R}$

an ${}n$-times differentiable function, and ${}a\in I$. Then

${}T_{a,n}(f)(x):=\sum _{k=0}^{n}{\frac {f^{(k)}(a)}{k!}}(x-a)^{k}\,$

is called the Taylor polynomial of degree ${}n$ for ${}f$ in the point ${}a$.