Experiment 1 — Maths

12 Jul, 2025

This is an experiment.

Given an integer $k \geq 1$ , and a function $f : D \to ℝ$ (where $D \subseteq ℝ$ ), we have that for any $x, x_{0} \in D$ , as long as all of its first $k + 1$ derivatives exist and all, along with $f$ itself, are continuous on an interval that includes both $x$ and $x_{0}$ ,

f (x) = {\hat{f}}_{k} (x) + e_{k + 1} (x, x_{0}),

where

{\hat{f}}_{k} (x) = \sum_{i = 0}^{k} \frac{f^{(i)} (x_{0})}{i!} (x - x_{0})^{i}

and

e_{k + 1} (x, x_{0}) = \frac{f^{(k + 1)} (ξ)}{(k + 1)!} (x - x_{0})^{k + 1}

for some $ξ \in (x, x_{0})$ .