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# Lagrange Remainder

Given a Taylor series

 (1)

the error after terms is given by

 (2)

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

 (3)

for some (Abramowitz and Stegun 1972, p. 880).

Note that the Lagrange 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).

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

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