Taylor's Expansion Revisited: A General Formula for the Remainder

Taylor’s polynomial is a central tool in any elementary course in mathematical analysis. Nowadays, its importance is centred on its applications, for instance, to asymptotic analysis or to obtain satisfactory numerical or integral inequalities see, e.g., 1–5 . The core of these results comes from manipulations on the explicit formula of the remainder, that is, the error estimation when considering the Taylor’s polynomial expansion instead of the function. In this paper, we provide a new explicit formula for the remainder that generalizes classic ones, namely, Schölomilch, Lebesgue, Cauchy, and Euler’s remainders. Inspired by the explicit expression for an arbitrary polynomial