Introductio in analysin infinitorum

54 DE EXPLICATIONE FUNCTIONUM LIB. I. turæ Serierum recurrentium consentaneam. Illa vero lex non constans tantum locum habet si numerator fractionis fuerit unitas seu quantitas constans; si enim quoque aliquot potestates ipsius $z$ contineret, tum illa lex multo magis fieret complicata, id quod post tradita calculi differentialis principia facilius patebit. 69. Quoniam hactenus posuimus primum denominatoris terminum constantem non esse $= 0$, ejusque loco unitatem collocavimus; nunc videamus cujusmodi Series oriantur, si in denominatore terminus constans evanescat. His casibus ergo Functio fracta hujusmodi formam habebit $\frac{a + bz + cz^2 + \text{&c.}}{z(1 - \alpha z - \beta z^2 - \text{&c.})}$, convertatur ergo, neglecto denominatoris Factore $z$, reliqua fractio $\frac{a + bz + cz^2 + \text{&c.}}{1 - \alpha z - \beta z^2 - \text{&c.}}$ in Seriem recurrentem $A + B z + C z^2 + D z^3 + \text{&c.}$ atque manifestum est fore $\frac{a + bz + cz^2 + \text{&c.}}{z(1 - \alpha z - \beta z^2 - \gamma z^3 - \text{&c.})} = \frac{A}{z} + B + Cz + Dz^2 + Ez^3 + \text{&c.}$ Simili modo erit $\frac{a + bz + cz^2 + \text{&c.}}{z^2(1 - \alpha z - \beta z^2 - \text{&c.})} = \frac{A}{z^2} + \frac{B}{z} + C + Dz + Ez^2 + \text{&c.}$, atque generatim erit $\frac{a + bz + cz^2 + \text{&c.}}{z^m(1 - \alpha z - \beta z^2 - \gamma z^3 - \text{&c.})} = \frac{A}{z^m} + \frac{B}{z^{m-1}} + \frac{C}{z^{m-2}} + \frac{D}{z^{m-3}} + \text{&c.}$ quicunque numerus fuerit exponens $m$. 70. Quoniam per substitutionem loco $z$ alia variabilis $x$ in Functionem fractam introduci, hocque pacto Functio fracta quævis in innumerabiles formas diversas transmutari potest; hoc modo eadem Functio fracta infinitis modis per Series recurrentes explicari poterit. Sit scilicet proposita hæc fractio $y = \frac{1+z}{1-z-z^2}$ & per Seriem recurrentem $y = 1+2z+3z^2+5z^3+8z^4+\text{&c.}$: ponatur $z = \frac{1}{x}$ erit $y =$