Historical synopsis of the Taylor remainder

Abstract

In this paper we give an historical synopsis of various Taylor remainders and their di erent proofs (without being exhaustive). We overview the formulas and the proofs given by such names as Bernoulli, Taylor, MacLaurin, Lagrange, Lacroix, Cauchy, Schlomilch, Roche Cox, Turquan, Bourget, Koenig, Darboux, Amigues, Teixeira, Peano, Blumenthal, Wolfe and Gon calves. We end the paper with a new Taylor remainder which generalizes the well-known Lagrange remainder.