TAYLOR'S THEOREM WITH REMAINDER: THE LEGACY OF LAGRANGE, AMPERE, AND CAUCHY

This dissertation undertakes an analysis of the various treatments of the remainder term in Taylor's formula, from its initial consideration by Lagrange to its use in convergence theorems by Cauchy. The main authors whose works are analyzed are Lagrange, Ampere and Cauchy, with passing reference to d'Alembert, Prony, Laplace, and Lacroix, among others.;Ampere's main achievement was to relate the Taylor formula (polynomial plus remainder term) to the Newton interpolatory formula (polynomial plus remainder term). At different times, Ampere found two different remainder formulas:;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI).;and.;Lagrange's main contributions are shown to be his important upper and lower estimates for the remainder, and secondly, two explicit formulas for the remainder: the intermediate-value formula and, in essence, the familiar integral formula. Basic to these results are his "monotonicity lemma" and a "telescoping identity" for a suitable derivative of the remainder. Much of his approach remains valuable, and eminently teachable today.;(x-a)('n)f(,n){a,a,...,a,x}.;Here f(,n)(x(,0),x(,1),...,x(,n)) stands for the nth divided difference of f; Ampere required this to be suitably defined even when there are coincidences among its arguments. By contrast, Ampere derived his first remainder in a self-contained way, with no reference to divided differences. It is shown that this derivation can be interpreted as a modification of the standard derivation of Newton's interpolatory formula, but with successive derivatives everywhere replacing successive divided differences. Ampere showed how his first remainder leads to Lagrange's estimates. The dissertation shows how this remainder also leads to the integral formula, and moreover clarifies the relation between Ampere's two remainders.;With Cauchy, one arrives at essentially modern exposition. Although Cauchy was in some ways indebted to Ampere (and of course, Lagrange), by the time he had finished his repeated attacks on expressions for the remainder, virtually all the devices now used in exposition were in place. Apart from certain lapses involving uniformity concepts, Cauchy's various remainder derivations involving single integrals, multiple integrals, integration by parts, ordinary mean value theorem, Cauchy mean value theorem, and the modern form of the telescoping identity were all presented in virtually impeccable form.