Lagrange error bounds for Taylor approximations

Understand the idea

The Lagrange remainder bounds Taylor approximation error using a derivative of one order higher than the polynomial degree. A valid bound controls that derivative throughout the interval between the center and the input being approximated.

Choose a number $\displaystyle M$ that bounds the absolute value of the required derivative on the whole interval. It need not be the smallest possible bound, but it must be valid. A derivative that happens to be zero at the center need not be zero elsewhere on the interval.

The error estimate uses $\displaystyle |x-a|^{n+1}$ divided by $\displaystyle (n+1)!$ . Increasing the degree changes both the derivative to bound and the factorial. The resulting number is an upper bound for absolute error, not a claim that the actual error equals that number.

|f(x)-T_n(x)|\le\frac{M|x-a|^{n+1}}{(n+1)!},\qquad |f^{(n+1)}(t)|\le M

A method you can apply

Identify the polynomial degree $\displaystyle n$ and the interval from the center to the evaluation input.
Bound $\displaystyle |f^{(n+1)}(t)|$ by a valid constant $\displaystyle M$ over that interval.
Substitute into the remainder inequality and compare the result with the requested tolerance.

Worked example

Bound the error when $\displaystyle T_2(x)=1+x+x^2/2$ approximates $\displaystyle e^{0.1}$ .

The center is $\displaystyle 0$ and $\displaystyle n=2$ , so the derivative to bound is the third derivative, $\displaystyle e^t$ .
For $\displaystyle 0\le t\le0.1$ , $\displaystyle e^t<2$ , so the convenient choice $\displaystyle M=2$ is valid.
The error is at most $\displaystyle 2(0.1)^3/3!=0.002/6$ .

The absolute error is at most

\displaystyle 1/3000

. The polynomial value is

\displaystyle 1.105

, with the true value guaranteed to lie within that error distance.

Try it yourself

Using $\displaystyle n=2$ and the bound $\displaystyle M=1$ , bound the error of $\displaystyle 1-x^2/2$ as an approximation to $\displaystyle \cos(0.2)$ .

Show solution and reasoning

The third derivative of $\displaystyle \cos x$ is $\displaystyle \sin x$ , whose absolute value is at most $\displaystyle 1$ . Therefore the error is at most $\displaystyle (0.2)^3/3!=1/750$ . This is a valid bound even though it is not the sharpest available.