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Cauchy Remainder

The Cauchy remainder is a different form of the remainder term than the Lagrange remainder. The Cauchy remainder after n terms of the Taylor series for a function f(x) expanded about a point x_0 is given by

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

where x^* in [x_0,x] (Hamilton 1952).

Note that the Cauchy remainder R_n is also sometimes taken to refer to the remainder when terms up to the (n-1)st power are taken in the Taylor series, and that a notation in which h->x-x_0, x^*->a+thetah, and x-x^*->1-theta is sometimes used (Blumenthal 1926; Whittaker and Watson 1990, pp. 95-96).

See also

Lagrange Remainder, Schlömilch Remainder, Taylor's Inequality, Taylor Series

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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.Hamilton, H. J. "Cauchy's Form of R_n from the Iterated Integral Form." Amer. Math. Monthly 59, 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.

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Cauchy Remainder

Cite this as:

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

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