For some [[Smooth Function]] $f:X\to Y$, that has a given [[Taylor Series]], for any $n$ you can take any taylor series into its $n-1$ term and end it to look something like $\huge \begin{align} f(x)= \sum_{k=0}^{n-1} \frac{1}{k!} \mathcal D^{k}\{f\} (a)(x-a)^{k} + \frac{1}{n!} \mathcal D^{n}\{f\}(\xi)(x-a)^{n} \end{align}$ $\huge \xi\in[x,a]$ What this is saying is that for a function with a taylor series, [[Existential Quantifier|There Exists]] some element $\xi$ such that the tangent of the [[Function]] $f$ has the same slope (derivative) as the [[Secant Line]] between $f(a)$ and $f(x)$. $\huge f(x) = f(a) + f'(\xi )(x-a) $ ![[Truncated Taylor Series .excalidraw.svg]] %%[[Truncated Taylor Series .excalidraw.md|🖋 Edit in Excalidraw]]%%