Function/R/Infinitely often differentiable/Taylor series/Definition

Taylor series

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

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

an infinitely often differentiable function, and ${}a\in I$. Then

$\sum _{k=0}^{\infty }{\frac {f^{(k)}(a)}{k!}}(x-a)^{k}$

is called the Taylor series of ${}f$ in the point ${}a$.