Lagrange Remainder

Given a Taylor series


the error R_n after n terms is given by


Using the mean-value theorem, this can be rewritten as


for some x^* in (x_0,x) (Abramowitz and Stegun 1972, p. 880).

Note that the Lagrange 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

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

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Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972.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.Firey, W. J. "Remainder Formulae in Taylor's Theorem." Amer. Math. Monthly 67, 903-905, 1960.Fulks, W. Advanced Calculus: An Introduction to Analysis, 4th ed. New York: Wiley, p. 137, 1961.Nicholas, C. P. "Taylor's Theorem in a First Course." Amer. Math. Monthly 58, 559-562, 1951.Poffald, E. I. "The Remainder in Taylor's Formula." Amer. Math. Monthly 97, 205-213, 1990.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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Lagrange Remainder

Cite this as:

Weisstein, Eric W. "Lagrange Remainder." From MathWorld--A Wolfram Web Resource.

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