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Taylor Remainder

The Taylor remainder R_n(x) is the difference between a function f(x) and its Taylor polynomial of degree n about a,

R_n(x)=f(x)-sum_(k=0)^n(f^((k))(a))/(k!)(x-a)^k.

Under the hypotheses of Taylor's theorem, the remainder can be expressed in forms including the Lagrange remainder, Cauchy remainder, and integral remainder. Bounds on R_n measure the error made by replacing f with its Taylor polynomial.

See also

Cauchy Remainder, Lagrange Remainder, Taylor Series, Taylor's Theorem

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References

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.

Cite this as:

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

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