Truncation error

In numerical analysis and scientific computing, truncation error is the error caused by approximating a mathematical process.

In reality, we can only use a finite number of these terms as it would take an infinite amount of computational time to take use all of them. So let's suppose we use only three terms of the series, then

Given the following infinite series, find the truncation error for x=0.75 if only the first three terms of the series are used.

The definition of the exact first derivative of the function is given by

This implies that we are finding the area under the curve using infinite rectangles. However, if we are calculating the integral numerically, we can only use a finite number of rectangles. The error caused by choosing a finite number of rectangles as opposed to an infinite number of them is a truncation error in the mathematical process of integration.

find the truncation error if a two-segment left-hand Reimann sum is used with equal width of segments.

Using two rectangles of equal width to approximate the area (see Figure 2) under the curve, the approximate value of the integral

Occasionally, by mistake, round-off error (the consequence of using finite precision floating point numbers on computers), is also called truncation error, especially if the number is rounded by chopping. That is not the correct use of "truncation error"; however calling it truncating a number may be acceptable.