Overview of De summis serierum Arithmo-Geometricarum infinitarum
Leibniz wrote this text in January 1678, i.e. a period shortly after his departure from Paris when questions about the analysis of progressions were gaining in interest. The problem Leibniz wanted to solve was to find the sum of what he called arithmo-geometric series: $0^p.t + 1^p.t^2 + 2^p.t^3 + \dots + (n-1)^p.t^n + \dots$ The terms of the series are composed of two factors, one of which, $t^n$, follows a law of geometric progression, and the other $n^p$ follows an arithmetic law, raised to a certain power $p$.
This text has many interests, starting with that of underlining the very important role played by the difference triangles that Leibniz developed at the beginning of his stay in Paris. Indeed, at first sight this text deals with quadrature problems, which the introductory diagram seems to confirm. The presence of numerous equations suggests the use of algebra to solve geometric problems, which is not original. But the framework that Leibniz chooses to deal with the question is actually much closer to the combinatorial framework in which Leibniz develops difference triangles. The diagrammatic aspect of these triangles is subsumed in the reasoning and leads the philosopher to the construction of another table, equally combinatorial.
Leibniz even announces the opposite, namely that the method of differences is only a special case of the method he presents here ("Finding the series by means of differences is only a specific case of this method"). But the technique of decomposition is the same, and so one can also see this particular case as a canonical example that serves as a model.
Indeed, Leibniz calculates the difference between the given series $X$ and the series obtained by shifting the series by one term $Y$, to find series that are of a lower degree $p$. The process is a recursive one, i.e. these series can in turn be decomposed into series of lower degree up to the degree $p=0$ where the series is necessarily the following geometric series: $t + t^2 + t^3 + \dots + t^n + \dots = \frac{t}{1-t}$ Leibniz calls this series $S$. But the series $X$ is in fact the product of the given series $Y$ by the variable $t$. So $Y-X = \frac{1}{t}X-X= \frac{1-t}{t}X = \frac{X}{S}$. Therefore $X = S\times (Y-X)$. Thus, Leibniz progressively decomposes all series into sums and products of the $S$-series, hence a polynomial in $S$.
In this double equation, Leibniz realises that the $Y$ series is eliminated without even calculating it ("the technique consists in this of supposing mean terms and new unknowns"). The underlying philosophical reflection here concerns the form of deductive reasoning. Here, pure analysis is not enough. An element external to the problem must be attached for the deduction to be able to start. If it were necessary to obtain a complete resolution of this added element, it would then constitute a part of the problem and its study would represent a stage of the analysis. But the fact that we do not need to determine this addition completely confirms that this object is external to the problem, and that this reasoning contains a part of synthesis, or in other words, of combinatorics.