Lagrange Remainder

analysis

Definition

Lagrange Remainder

Let $f$ be $(n+1)$-times differentiable on an interval $I$ containing $x_0$ and $x$. The Lagrange remainder $R_n(x)$ is the error term in the Taylor expansion of $f$ of order $n$.

There exists a point $\xi$ between $x_0$ and $x$ such that:

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

This form of the remainder is useful for bounding the error of a Taylor polynomial approximation.

Error Bounding

If the $(n+1)$-th derivative is bounded on the interval, i.e., $|f^{(n+1)}(t)|\le M$ for all $t$ between $x_0$ and $x$, then the error is bounded by:

$$|R_n(x)|\le \frac{M}{(n+1)!}|x-x_0{|}^{n+1}$$