The application of a method of differences to the species of series whose sums are obtained by Mr. <i>Landen</i> , by the help of impossible quantities

Abstract The nature of this paper is such, as renders it absolutely incapable of abridgement. By way of introduction to it the author observes, that having some years back, when reading the learned Mr. Landen’s fifth memoir, discovered the manner of applying a method of differences to the species of series whose sums are there obtained by the help of impossible quantities, and having since extended that application, he now ventures to offer it to the consideration of others. The practice of this method, in most cases, appears, he says, extremely simple, and on that account he is almost induced to imagine that it has already been considered by mathematicians. And he acknowledges that, since the greatest part of the paper was written, he has, in Euler’s Institutiones Calculi Integrals, met with two simple series, which are in that work summed by multiplications similar to those employed in the investigation of the principal theorems contained in this paper. But whether that learned mathematician has pursued the method any further, he has not been able to ascertain.