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Lagrange Remainder

Given a Taylor series

f(x)=f(x_0)+(x-x_0)f^'(x_0)+((x-x_0)^2)/(2!)f^('')(x_0)+... +((x-x_0)^n)/(n!)f^((n))(x_0)+R_n,
(1)

the error R_n after n terms is given by

R_n=int_(x_0)^xf^((n+1))(t)((x-t)^n)/(n!)dt.
(2)

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

R_n=(f^((n+1))(x^*))/((n+1)!)(x-x_0)^(n+1)
(3)

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 Remainder, Taylor Series, Taylor's Inequality

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References

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 Resource. https://mathworld.wolfram.com/LagrangeRemainder.html

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