Introductio in analysin infinitorum

FACTORUM TRINOMIALIUM. 119 CAP. IX. est enim $e^{-x} = 1 - \frac{x}{1} + \frac{x^2}{1.2} - \frac{x^3}{1.2.3} + \&c.$; quæ cum §. 151. comparata dat $n=i$, $a=1+\frac{x}{i}$ & $z=1-\frac{x}{i}$ unde hujus expressionis Factor erit $= aa - 2az.cof.\frac{2k}{n}\pi + zz = 2 + \frac{2xx}{ii} - 2(1-\frac{xx}{ii})cof.\frac{2k}{i}\pi + \frac{4kk\pi\pi xx}{i^4}$, ob $cof.\frac{2k}{i}\pi = 1 - \frac{2kk\pi\pi}{ii}$. Functio ergo $e^x - e^{-x}$ divisibilis erit per $1 + \frac{xx}{kk\pi\pi} - \frac{xx}{ii}$, ubi autem terminus $\frac{xx}{ii}$ tuto omittitur, quia etsi per $i$ multiplicetur, tamen manet infinite parvus. Præterea vero ut ante, si $k=0$, erit primus Factor $=x$. Quocirca, his Factoribus in ordinem redactis, erit [TRUNCATED]