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Proposition: Taylor's Formula with Remainder Term of Lagrange
Let $I\subset\mathbb R$ be an interval and $f:I\to\mathbb R$ be a $(n+1)$ times continuously differentiable function. For any values $a,x\in I,$ there is a value $\xi$ between $a$ and $x$ such that $f(x)$ can be written as
$$f(x)=f(a)+\sum_{k=1}^n \frac{f^{\{k\}}(a)}{k !}(x-a)^k+\frac{f^{\{n+1\}}(\xi)}{(n+1) !}(x-a)^{n+1}.$$
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References
Bibliography
- Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983
