Definition:Taylor Series/Remainder/Lagrange Form

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

• Definition:Cauchy Form of Remainder of Taylor Series

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: $20.2$
• 1969: J.C. Anderson, D.M. Hum, B.G. Neal and J.H. Whitelaw: Data and Formulae for Engineering Students (2nd ed.) ... (previous) ... (next): $4.$ Mathematics: $4.4$ Differential calculus: $\text {(iii)}$ Taylor's expansion
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Lagrange form of the remainder
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): remainder: 2.
• 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 22$: Taylor Series: Taylor Series for Functions of One Variable: $22.2.$

• Weisstein, Eric W. "Lagrange Remainder." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LagrangeRemainder.html