Introductio in analysin infinitorum

PER SERIES INFINITAS. 53 $(a+2b)^m+(a+3b)^m + \text{&c.}$ exhibeat progressionem ordinis $m$; erit ob naturam Serierum recurrentium CAP.IV. $0 = a^m - \frac{n}{1} (a+b)^m + \frac{n(n-1)}{1 \cdot 2} (a+2b)^m - \frac{n(n-1)(n-2)}{1 \cdot 2 \cdot 3} (a+3b)^m + \dots \pm \frac{n}{1} (a+(n-1)b)^m \mp (a+nb)^m$; ubi signa superiora valent si $n$ fit numerus par, inferiora autem si $n$ fit numerus impar. Hæc ergo æquatio semper est vera si fuerit $n$ numerus integer major quam $m$. Hinc ergo intelligitur quam late pateat doctrina de Seriebus recurrentibus. 68. Si denominator fuerit potestas non binomii sed multinomii, natura Seriei quoque alio modo explicari potest. Sit nempe hæc fractio $\frac{1}{(1-\alpha z - \beta z^2 - \gamma z^3 - \delta z^4 - \text{&c.})^{m+1}}$ proposita, erit Series infinita hinc nata $1 + \frac{m+1 [TRUNCATED]