Rn=f(n)(ξ)(x−a)nn! where a≤ξ≤x, ( Lagrangue's form )
Rn=f(n)(ξ)(x−ξ)n−1(x−a)(n−1)! where a≤ξ≤x, ( Cauch's form )
This result holds if f(x) has continuous derivatives of order n at last.
If limn→+∞Rn=0, the infinite series obtained is called Taylor series for f(x) about x=a.
If a=0 the series is often called a Maclaurin series.