The Cauchy remainder is a different form of the remainder term than the Lagrange remainder. The Cauchy remainder after terms of the Taylor series
for a function
expanded about a point
is given by

where
(Hamilton 1952).

Note that the Cauchy remainder is also sometimes taken to refer to the remainder when terms
up to the st
power are taken in the Taylor series, and that a
notation in which ,
, and is sometimes used (Blumenthal 1926; Whittaker
and Watson 1990, pp. 95-96).

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.Hamilton, H. J. "Cauchy's
Form of
from the Iterated Integral Form." Amer. Math. Monthly59, 320,
1952.Whittaker, E. T. and Watson, G. N. "Forms of the
Remainder in Taylor's Series." §5.41 in A
Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University
Press, pp. 95-96, 1990.