Taylor's Inequality

Taylor's inequality is an estimate result for the value of the remainder term R_n(x) in any n-term finite Taylor series approximation.

Indeed, if f is any function which satisfies the hypotheses of Taylor's theorem and for which there exists a real number M satisfying |f^((n+1))(x)|<=M on some interval I=[a,b], the remainder R_n satisfies


on the same interval I.

This result is an immediate consequence of the Lagrange remainder of R_n and can also be deduced from the Cauchy remainder as well.

See also

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

This entry contributed by Christopher Stover

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Stover, Christopher. "Taylor's Inequality." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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