Example: If
\[x\]
is small (much less than 1 in magnitude), an approximate expression for \[sin \: x\]
is\[sin \: x = x- \frac{x}{3!} + \frac{x^5}{5!} + \frac{x^7}{7!} +... = \sum_{3}^{\infty} \frac{(-1)^n x^{2n-1}}{(2n-1)!}\]
\[sin \: x \simeq x- \frac{x}{3!}\]
The truncation error is the true value minus the approximate value:
\[Truncation \: Error = \frac{x^5}{5!} + \frac{x^7}{7!} +... = \sum_{3}^{\infty} \frac{(-1)^n x^{2n-1}}{(2n-1)!}\]