Cauchy’s_Remainder_Theorem

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Definition

Let $f$ be a function with derivatives of all orders on an interval containing $0$. For all $n \in \mathbb{N}$ and $x$ in this interval, there exists a point $c$ between $0$ and $x$ such that the error function $E_n(x) = f(x) - S_n(x)$ satisfies

$$ E_n(x) = \frac{f^{(n+1)}(c)}{n!}(x-c)^n x $$

where $S_n(x)$ is the $n$th partial sum of the Taylor series.