A Taylor series remainder formula that gives after
terms of the series
for
and any
(Blumenthal 1926, Beesack 1966), which Blumenthal (1926) ascribes to Roche (1858).
The choices
and
give the Lagrange and Cauchy
remainders, respectively (Beesack 1966).
Beesack, P. R. "A General Form of the Remainder in Taylor's Theorem." Amer. Math. Monthly73, 64-67, 1966.Blumenthal,
L. M. "Concerning the Remainder Term in Taylor's Formula." Amer.
Math. Monthly33, 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.
terms of the series
and any 
(Blumenthal 1926, Beesack 1966), which Blumenthal (1926) ascribes to Roche (1858).
The choices 
and 
give the Lagrange and Cauchy
remainders, respectively (Beesack 1966).
