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).