A Treatise on Summation & Interpolation of Infinite Series

Abstract

Just as curves are not determined by some given ordinates no matter how many, but by the general relation between the abscissae and the ordinates, so series are not determined by some given terms no matter how many, but by the relation between successive terms. For any quantities which are finite in number can form terms in different series: in fact the series is unique which has the same initial terms and the same law for forming the remaining terms up to infinity. Therefore in the first place the relations of the terms have to be investigated; then when these have been found they are to be specified by difference equations just as Des Cartes has defined curves by algebraic equations: when these things have been obtained, problems about summation and interpolation and other matters of that type concerning series will be solved by an analysis no less exact than common algebra is.