Lagrange remainder ($R_n(x)=f^{(n+1)}(\xi )(x-a{)}^{n+1}/(n+1)!$)

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