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

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

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

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

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

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