Lagrange remainder
= Lagrange remainder
{c}
{title2=$R_n(x)=f^{(n+1)}(\xi)(x-a)^{n+1}/(n+1)!$}
The <Taylor remainder> in this form evaluates one higher <derivative> at an intermediate point $\xi$ between the expansion point $a$ and $x$. If $|f^{(n+1)}|\leq M$ on that interval, then $|R_n(x)|\leq M|x-a|^{n+1}/(n+1)!$. This bounds truncation error without knowing the particular intermediate point. The differentiability hypotheses are those of the <Taylor theorem with Lagrange remainder>.