Question

What is Lagrange’s form for the remainder? Why is Lagrange’s form usually more useful for analyzing the remainder than the definition of the remainder or the integral provided by Taylor theorem?

Short answer

Lagrange form for the remainder states that if a function that can be differentiated (n+1) times in some open intervalI that contains the point $x_0.\mathrm{And}{R}_n\left(x\right)$ be the nth remainder for the function at $x=x_0$.

This form is useful for analyzing the remainder.

Step-by-step solution

Step 1. Given Information

The given term is Lagrange’s form for the remainder.

Step 2. Explanation

Lagrange form for the remainder states that consider a function that can be differentiated (n+1) times in some open intervalI that contains the point $x_0.\mathrm{And}{R}_n\left(x\right)$ be the nth remainder for the function at $x=x_0$.

Hence, for each point $x\in I$, there is at least one c between $x_{0}andx$such that,

$R_n\left(x\right)=\frac{f^{n+1}\left(c\right)}{(n+1)!}{(x-x_0)}^{n+1}$

It is more difficult to analyze the remainder by using integral provided by Taylor theorem. Hence, Lagrange form is usually useful for analyzing the remainder.