On the Remainder in the Taylor Theorem

Abstract

This formula is the main tool for bounding the remainder of the Taylor expansion in calculus classes, especially when this subject is taught before integration. One would like to have some natural proof for it. In [3] it is suggested that induction seems suitable, since P'n is the Taylor polynomial of f of order n ? 1, so R'n(x) is given by induction. This approach fails, since one cannot integrate R'n(x) because the point ? = t=(x) depends on x. While we were teaching a first calculus course for chemistry and physics majors at Tel-Aviv University, we observed that this obstacle can be removed if we change the problem to finding a bound on the remainder. This is just as useful, since a bound is all that is needed to show that the Taylor series converges to the function. From our personal experience, we believe that this approach enables students to grasp the mate rial more easily. Furthermore, Lagrange's formula can be deduced from the bound, as we show at the end of this note.