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

`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
`

`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
`

`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
`

`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.
`