>[!Note] Theorem (Taylor's theorem for univariate real valued function, Cauchy remainder) >If $f:I \to \mathbb{R}$ is $n$ times differentiable on the open interval $I$ containing $x_{0}$ and $0\leq k\leq n-1$ then $\begin{align} f(x) =\; &f(x_{0}) + f'(x_{0})(x-x_{0}) +\dots \frac{f^{(n-1)}(x_{0})}{(n-1)!} (x-x_{0})^{n-1} \\ &+ \frac{f^{(n)}(t)}{(n-1)!} (x-x_{0})(x-t)^{n-1} \end{align}$for some point $t$ between $x$ and $x_{0}.$ ^c01cd3 *Proof*. Follows from Schlömilch remainder form of Taylor's theorem with $k=n-1.$