Introductio in analysin infinitorum

294 DE USU SERIERUM RECURRENTIUM LIB. I. $P, Q, R, S, T, \&c.$, termini Seriei recurrentis a principio jam longissime remoti, ita ut cum progressione geometrica confundantur; fitque $T = \alpha S + \beta R + \gamma Q + \delta P$, seu scala relationis $\alpha, +\beta, +\gamma+\delta$. Ponatur valor fractionis $\frac{Q}{P} = x$; erit $\frac{R}{P} = xx$; $\frac{S}{P} = x^3$ & $\frac{T}{P} = x^4$, qui in superiori aequatione substituti dabunt $x^4 = \alpha x^3 + \beta x^2 + \gamma x + \delta$. unde patet quotum $\frac{Q}{P}$ tandem præbere radicem unam æquationis inventæ. Hoc vero & præcedens methodus indicat, præterea vero docet fractionem $\frac{Q}{P}$ dare maximam æquationis radicem. 355. Potest quoque hæc methodus investigandarum radicum saepenumero utiliter adhiberi, si æquatio fit infinita. Ad quod ostendendum proposita fit æquatio $\frac{1}{2} = z - \frac{z^3}{6} + \frac{z^5}{120} - \frac{ [TRUNCATED]