Introductio in analysin infinitorum

FACTORUM TRINOMIALIUM. 123 161. Nunc, quoniam in omnibus Factoribus terminus con- CAP. IX. stans est $= 1$, ipsa Functio $e^{b+x} \pm e^{c-x}$ per ejusmodi constantem dividi debet, ut terminus constans fiat $= 1$, seu ut ejus valor, posito $x=0$, fiat $= 1$; talis Divisor erit $e^b \pm e^c$, & hanc ob rem expressio hæc $\frac{e^{b+x} \pm e^{c-x}}{e^b \pm e^c}$ per Factores numero infinitos exponi poterit. Erit ergo, si va- leat signum superius atque $m$ denotet numerum imparem, $\frac{e^{b+x} + e^{c-x}}{e^b + e^c} = (1 + \frac{4(b-c)x+4xx}{\pi\pi+(b-c)^2})(1 + \frac{4(b-c)x+4xx}{9\pi\pi+(b-c)^2})$ $(1 + \frac{4(b-c)x+4xx}{25\pi\pi+(b-c)^2})$ &c., fin autem signum inferius va- leat, atque ideo $m$ denotet numerum parem, cafuque $m=0$ radix Factoris quadrati ponatur, erit $\frac{e^{b+x} - e^{c-x}}{e^b - e^c} =$ $(1 + \frac{2x}{b-c})(1 + \frac{4(b-c)x+4xx}{4\pi\pi+(b-c)^2})(1 + \frac{4(b-c)x+4xx}{16\pi\pi+(b-c)^2})$ $(1 + \frac{4(b-c)x+4xx}{36\pi\pi+(b-c)^2})$ &c. 162. Ponatur $b=0$, quod fine detrimento universalitatis fieri potest, eritque $\frac{e^x + e^{c-x}}{1+e^c} = (1 - \frac{4cx+4xx}{\pi\pi+cc})$ $(1 - \frac{4cx+4xx}{9\pi\pi+cc})(1 - \frac{4cx+4xx}{25\pi\pi+cc})$ &c.; $\frac{e^x - e^{c-x}}{1-e^c}$ $= (1 - \frac{2x}{c})(1 - \frac{4cx+4xx}{4\pi\pi+cc})(1 - \frac{4cx+4xx}{16\pi\pi+cc})$ $(1 - \frac{4cx+4xx}{36\pi\pi+cc})$ &c.. Jam ponatur $c$ negativum, atque habebuntur hæ duæ æquationes: $\frac{e^x + e^{-c}e^{-x}}{1+e^{-c}} =$ Q 2 $(1+$