In numerical analysis and scientific computing, truncation error is an error caused by approximating a mathematical process. The term truncation comes from the fact that these simplifications often involve the truncation of an infinite series expansion so as to make the computation possible and practical.

A summation series for ${\displaystyle e^{x}}$ is given by an infinite series such as $${\displaystyle e^{x}=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+{\frac {x^{4}}{4!}}+\cdots }$$

In reality, we can only use a finite number of these terms as it would take an infinite amount of computational time to make use of all of them. So let's suppose we use only three terms of the series, then $${\displaystyle e^{x}\approx 1+x+{\frac {x^{2}}{2!}}}$$

In this case, the truncation error is ${\displaystyle {\frac {x^{3}}{3!}}+{\frac {x^{4}}{4!}}+\cdots }$

Example A:

Given the following infinite series, find the truncation error for x = 0.75 if only the first three terms of the series are used. $${\displaystyle S=1+x+x^{2}+x^{3}+\cdots ,\qquad \left|x\right|<1.}$$

Solution

Using only first three terms of the series gives $${\displaystyle {\begin{aligned}S_{3}&=\left(1+x+x^{2}\right)_{x=0.75}\\&=1+0.75+\left(0.75\right)^{2}\\&=2.3125\end{aligned}}}$$