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What is the Lagrange remainder?

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The Lagrange remainder is a term that estimates the size of the error introduced by a partial sum of the Taylor series when it is used to compute an approximated value for the original function. The Lagrange remainder for the Taylor series is a tool not only for estimating errors but for giving a precise difference between the polynomial and the original function approximated by the polynomial.

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The Lagrange remainder may sometimes refer to the remainder when terms in the Taylor series are taken up the second last power. Lagrange's remainder is applicable to the binomial series studied by Jean d'Alembert, as it provides solutions to pertinent questions such as, determining what happens when X is equal to one. It also provides accurate answers as to the exact number of terms that must be taken when a series converges for the attainment of the desired accuracy.

It also clarifies on the degree of accuracy, which can be obtained once a series diverges. The Lagrange remainder is one of the various forms of the Taylor series remainder, while other explicit forms of the remainder include the mean-value forms and the Cauchy form. These forms are refinements of the Taylor theorem and are typically proven using the mean value theorem.

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