Definition:Taylor Series/Remainder/Lagrange Form
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Definition
Let $f$ be a real function which is smooth on the open interval $\openint a b$.
Let $\xi \in \openint a b$.
Consider the remainder of the Taylor series at $x$:
- $\ds \map {R_n} x = \int_\xi^x \map {f^{\paren {n + 1} } } t \dfrac {\paren {x - t}^n} {n!} \rd t$
The Lagrange form of the remainder $R_n$ is given by:
- $R_n = \dfrac {\map {f^{\paren {n + 1} } } {x^*} } {\paren {n + 1}!} \paren {x - \xi}^{n + 1}$
where $x^* \in \openint \xi x$.
Also see
Source of Name
This entry was named for Joseph Louis Lagrange.
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 20$: Taylor Series for Functions of One Variable: $20.2$
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Entry: Lagrange form of the remainder
- Weisstein, Eric W. "Lagrange Remainder." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LagrangeRemainder.html