Introductio in analysin infinitorum

PER SERIES INFINITAS. 47 Functio foret integra; num vero per hujusmodi terminorum se- CAP. IV. riem infinitam exhiberi possit, si quis dubitet, hoc dubium per ipsam evolutionem cujusque Functionis tolletur. Quo au- tem hæc explicatio latius pateat, præter potestates ipsius $z$ ex- ponentes integros affirmativos habentes, admitti debent pote- states quæcunque. Sic dubium erit nullum quin omnis Functio ipsius $z$ in hujusmodi expressionem infinitam transmutari possit: $Az^{\alpha} + Bz^{\beta} + Cz^{\gamma} + Dz^{\delta} + \&c.$ denotantibus exponen- tibus $\alpha, \beta, \gamma, \delta, \&c.$ numeros quoscunque. 60. Per divisionem autem continuam intelligitur fractionem $\frac{a}{\alpha + \zeta z}$ resolvi in hanc seriem infinitam $\frac{a}{\alpha} - \frac{a \zeta z}{\alpha^2} + \frac{a \zeta^2 z^2}{\alpha^3} - \frac{a \zeta^3 z^3}{\alpha^4} + \frac{a \zeta^4 z^4}{\alpha^5} - \&c.$, quæ, cum quilibet terminus ad se- quentem habeat rationem constantem $1 : \frac{\zeta z}{\alpha}$, vocatur series geo- metrica. Potest vero quoque hæc series ita inveniri, ut ipsa initio pro incognita habeatur: ponatur enim $\frac{a}{\alpha + \zeta z} = A + Bz + Cz^2 + Dz^3 + Ez^4 + \&c.$ atque ad æqualitatem producendam quærantur coëfficientes $A, B, C, D, \&c.$ Erit ergo $a = (\alpha + \zeta z)(A + Bz + Cz^2 + Dz^3 + \&c.)$, & multiplica- tione actu peracta fiet $a = \alpha A + \alpha B z + \alpha C z^2 + \alpha D z^3 + \alpha E z^4 + \&c.$ $+ \zeta A z + \zeta B z^2 + \zeta C z^3 + \zeta D z^4 + \&c.$ Quamobrem esse debet $a = \alpha A$, ideoque $A = \frac{a}{\alpha}$, & coëf- ficientium uniuscujusque potestatis ipsius $z$ summa nihilo æqualis est ponenda: unde prodibunt hæ æquationes, $\alpha B + \zeta A = 0$ cognito ergo quovis coëfficiente $\alpha C + \zeta B = 0$ facile reperitur sequens; si enim $\alpha D + \zeta C = 0$ fuerit coëfficiens termini cujusque $= P$ $\alpha E + \zeta D = 0$ & sequens $= Q$ erit $\alpha Q + \zeta P = 0$ $\&c.$ sive $Q = - \frac{\zeta P}{\alpha}$. Cum