Let $I \subseteq \mathbb{R}$ be an open interval, let $x_0 \in I$, and let $f: I \to \mathbb{R}$ be $n$ times [differentiable](/page/Derivative) at $x_0$. Then for $h$ such that $x_0 + h \in I$, \begin{align*} f(x_0 + h) = \sum_{k=0}^{n} \frac{h^k f^{(k)}(x_0)}{k!} + E_n, \end{align*} where $E_n = o(h^n)$ as $h \to 0$. If, moreover, $f^{(n+1)}$ exists on $(x_0, x_0 + h)$ and $f^{(n)}$ is [continuous](/page/Continuity) on $[x_0, x_0 + h]$, then $E_n = O(h^{n+1})$ as $h \to 0$. More precisely, there exists $\xi$ with $x_0 \leq \xi \leq x_0 + h$ such that \begin{align*} E_n = \frac{f^{(n+1)}(\xi)}{(n+1)!} h^{n+1}. \end{align*}