Introductio in analysin infinitorum

PER SERIES INFINITAS. 49 tur, sicque reperietur Series infinita $A+Bz+Cz^2+Dz^3+\&c.$ CAP. IV. Functioni fractæ propositæ $\frac{a+bz}{\alpha+\beta z+\gamma z^2}$ æqualis. EXEMPLUM. Si fuerit proposita hæc fractio $\frac{1+2z}{1-z-z^2}$, huicque æqualis statuatur Series $A+Bz+Cz^2+Dz^3+\&c.$ ob $a=1$; $b=2$; $\alpha=1$; $\beta=-1$; $\gamma=-1$; erit $A=1$; $B=3$; tum vero erit $C=B+A$ quilibet ergo coëfficiens æqualis est sum- $D=C+B$ mæ duorum præcedentium; quare si co- $E=D+C$ gniti fuerint duo coëfficientes contigui $F=E+D$ $P$ & $Q$, erit sequens $R=P+Q$. &c. Cum igitur duo coëfficientes primi $A$ & $B$ sint cogniti, fractio proposita $\frac{1+2z}{1-z-z^2}$ in hanc Seriem infinitam transmutatur $1+3z+4z^2+7z^3+11z^4+18z^5+\&c.$, quæ nullo nego- tio quousque libuerit continuari potest. 62. Ex his jam satis intelligitur indoles Serierum infinitarum, in quas Functiones fractæ transmutantur; tenent enim ejusmodi legem, ut quilibet terminus ex aliquot præcedentibus deter- minari possit. Scilicet, si denominator fractionis propositæ fue- rit $\alpha+\beta z$, atque Series infinita statuatur $A+Bz+Cz^2+\dots+Pz^n+Qz^{n+1}+Rz^{n+2}+Sz^{n+3}+\&c.$; quilibet coëfficiens $Q$ ex præcedente $P$ solo ita definietur ut sit $\alpha Q+\beta P=0$. Sin denominator fuerit trinomium $\alpha+\beta z+$ $\gamma z^2$, quilibet coëfficiens Seriei $R$ ex duobus præcedenti- bus $Q$ & $P$ ita definietur ut sit $\alpha R+\beta Q+\gamma P=0$: simili modo si denominator fuerit quadrinomium, ut $\alpha+\beta z+\gamma z^2+$ $\delta z^3$, quilibet coëfficiens seriei $S$ ex tribus antecedentibus $R$, $Q$ & $P$ ita determinabitur, ut sit $\alpha S+\beta R+\gamma Q+\delta P=0$, Euleri *Introduct. in Anal. infin. parv.* G sicque