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Schlömilch Remainder


A Taylor series remainder formula that gives after n terms of the series

 R_n=(f^((n+1))(x^*))/(n!p)(x-x^*)^(n+1-p)(x-x_0)^p

for x^* in (x_0,x) and any p>0 (Blumenthal 1926, Beesack 1966), which Blumenthal (1926) ascribes to Roche (1858). The choices p=n+1 and p=1 give the Lagrange and Cauchy remainders, respectively (Beesack 1966).


See also

Cauchy Remainder, Lagrange Remainder

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References

Beesack, P. R. "A General Form of the Remainder in Taylor's Theorem." Amer. Math. Monthly 73, 64-67, 1966.Blumenthal, L. M. "Concerning the Remainder Term in Taylor's Formula." Amer. Math. Monthly 33, 424-426, 1926.Maak, W. An Introduction to Modern Calculus. New York: Holt, Rinehart, and Winston, p. 99, 1963.Roche. Mem. de l'Acad. de Montpellier. 1858.Schlömilch, O. Kompendium der höheren Analysis. Braunschweig, Germany: Vieweg, 1923.

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Schlömilch Remainder

Cite this as:

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

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