Truncation Errors

Truncation errors arise when approximate formulae are used instead of exact one. Usually this is because a function can be expressed as a infinite Taylor series. Using only a finite number of terms results in an error.
Example: If  
  is small (much less than 1 in magnitude), an approximate expression for  
\[sin \: x\]
\[sin \: x = x- \frac{x}{3!} + \frac{x^5}{5!} + \frac{x^7}{7!} +... = \sum_{3}^{\infty} \frac{(-1)^n x^{2n-1}}{(2n-1)!}\]

\[sin \: x \simeq x- \frac{x}{3!}\]

The truncation error is the true value minus the approximate value:
\[Truncation \: Error = \frac{x^5}{5!} + \frac{x^7}{7!} +... = \sum_{3}^{\infty} \frac{(-1)^n x^{2n-1}}{(2n-1)!}\]