Taylor Remainder

In practice, given a function and a desired accuracy , we would like know which degree to choose for the Taylor polynomial approximation, such that

In other words, the error made when replacing by should not exceed . Higher degree Taylor polynomials provide better approximations, but are more and more expensive to compute. Can we find the minimal order ( cheapest to compute ) for which the approximation error is small enough ?

The following formula from Calculus is exactly what we need to estimate the error.

Let have continuous derivatives for . Construct the Taylor polynomial that approximates around . The difference between the function value and the polynomial value is

where is an unknown number between and . is therefore the approximation error; it is called the Taylor remainder.

To prove the formula, we use repeated integrations by parts.

The remainders can then be brought from the integral form to the standard form using the mean value theorem

where is a number between and .

For example, let and . We want to find the Taylor polynomial which approximates within for . The problem can be formulated as follows: find such that

According to the Taylor remainder formula

Since for the following inequality (error bound) holds:

It is therefore enough to find such that

satisfies this and, therefore, is the desired approximation.

When computing functions, or other mathematical objects (e.g. integrals) we are forced to make certain algorithmic approximations. In our case, we cannot evaluate the infinite Taylor series; we need to truncate it, i.e. to stop the computations after a finite number of terms. The resulting error is a first example of truncation error in numerical computing. Truncation is a second source of numerical errors, after roundoff. While roundoff is due to inexact computer arithmetic, truncation errors are due to ``inexact mathematical formulas'', i.e. to the algorithmic approximations we make in order to keep the computations feasible.