Q:

# What is the Lagrange remainder?

A:

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.

## Keep Learning

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.

Sources:

## Related Questions

• A:

Relative extrema is a term used in calculus to describe points on the graph of a function where there are minimums and maximums. It can be visualized as representing the peaks and valleys on a line graph.

Filed Under:
• A:

The definition of a limit in calculus is the value that a function gets close to but never surpasses as the input changes. Limits are one of the most important aspects of calculus, and they are used to determine continuity and the values of functions in a graphical sense.

Filed Under:
• A:

The integral of the secant function is the natural logarithm of the absolute value of the secant of x plus the tangent of x, plus a constant. Expressed in mathematical notation, it reads as the integral of sec x dx = ln |sec x + tan x| + C.