Truncation error

Remarks

Example: let $f$ have a continuous second derivative and take the forward difference $A(h) = (f(x+h) - f(x))/h$, with limit $A = f'(x)$. Taylor's theorem gives $A - A(h) = -\tfrac12 f''(\xi) h$ for some $\xi$ between $x$ and $x + h$, so the truncation error is of order $h$. Example: for $A_n = \sum_{k=0}^{n} 1/k!$ and $A = e$, the truncation error is $\sum_{k > n} 1/k!$, which lies between $1/(n+1)!$ and $2/(n+1)!$ for $n \ge 1$. Non-example: the difference between $1/3$ and its stored value $0.333$ in three-digit decimal arithmetic is a round-off error, since no limit process is cut off.