  The aim of this paper is to derive a refined first-order expansion formula in
Rn, the goal being to get an optimal reduced remainder, compared to the one
obtained by usual Taylor's formula. For a given function, the formula we
derived is obtained by introducing a linear combination of the first
derivatives, computed at $n+1$ equally spaced points. We show how this formula
can be applied to two important applications: the interpolation error and the
finite elements error estimates. In both cases, we illustrate under which
conditions a significant improvement of the errors can be obtained, namely how
the use of the refined expansion can reduce the upper bound of error estimates.
