Mémoire Sur Les Suites

The theory of series is one of the most important objects of Analysis: all problems which reduce to some approximations, and consequently nearly all the applications of Mathematics to Nature, depend on this theory; thus we see that it has principally fixed the attention of the geometers; they have found a great number of beautiful theorems and ingenious methods, either in order to expand function into series, or in order to sum series exactly or for approximation; but they have attained them only by some indirect and particular ways, and we can not doubt that, in this branch of Analysis, as in all others, there is a general and simple manner to view it, from which the already known truths derive, and which lead to many new truths. The research of a similar method is the object of this Memoir; that to which I am come is founded on the consideration of that which I name generating functions: this is a new kind of calculus which we can name calculus of generating functions, and which has appeared to me to merit being cultivated by the geometers. I exhibit first some very simple results on these functions and I deduce from them a method to interpolate series, not only when the consecutive differences of the terms are convergent, that which is the sole case which we have considered until now, but yet when the proposed series converges towards a recurrent series, the final ratio of its terms being given by a linear equation in finite differences of which the coefficients are constants. Integration of this kind of equation is a corollary of this analysis. In passing next from the finite to the infinitely small, I give a general formula to interpolate the series of which the final ratio of the terms is represented by a linear equation in infinitely small differences, of which the coefficients are constants; whence I conclude the integration of these equations. By applying the same method to the transformation of series, there results from it a quite simple way to transform them into some others of which the terms follow a given law; finally the relationship of the generating functions to the corresponding variables leads me immediately to the singular analogy of the positive powers with the differences and of the negative powers with the integrals, an analogy observed first by Leibnitz, and since brought to greater light by Mr de la Grange (Mémoirs de Berlin, 1772); all the theorems of which the