Pagina:Werkecarlf02gausrich.djvu/39

sive

$$W=2(1-r^{-1})(1+r^{-2})(1-r^{-3})(1+r^{-4})\ldots (1-r^{-{\frac {1}{2}}n+1})\dots \dots \dots [8]$$

Porro quum sit ${\textstyle r^{{\frac {1}{2}}n}=-1}$, adeoque

$${\begin{aligned}&1+r^{-2}=-r^{{\frac {1}{2}}n-2}(1-r^{-{\frac {1}{2}}n+2})\\&1+r^{-4}=-r^{{\frac {1}{2}}n-4}(1-r^{-{\frac {1}{2}}n+4})\\&1+r^{-6}=-r^{{\frac {1}{2}}n-6}(1-r^{-{\frac {1}{2}}n+6}){\text{ etc.}}\end{aligned}}$$

productumque e factoribus ${\textstyle -r^{{\frac {1}{2}}n-2}}$,${\textstyle -r^{{\frac {1}{2}}n-4}}$,${\textstyle -r^{{\frac {1}{2}}n-6}}$ etc. usque ad ${\textstyle -r^{2}}$ fiat ${\textstyle =(-1)^{{\frac {1}{4}}n-1}r^{{\frac {1}{16}}nn-{\frac {1}{4}}n}}$, aequatio praecedens ita quoque exhiberi potest

$$W=2(-1)^{{\frac {1}{4}}n-1}r^{{\frac {1}{16}}nn-{\frac {1}{4}}n}(1-r^{-1})(1-r^{-2})(1-r^{-3})(1-r^{-4})\ldots (1-r^{-{\frac {1}{2}}n+1})$$

Quum habeatur

$${\begin{aligned}&1-r^{-1}=-r^{-1}(1-r^{-n+1})\\&1-r^{-2}=-r^{-2}(1-r^{-n+2})\\&1-r^{-3}=-r^{-3}(1-r^{-n+3}){\text{ etc.}}\end{aligned}}$$

erit

$${\begin{aligned}(1-r^{-1})&(1-r^{-2})(1-r^{-3})\ldots (1-r^{-{\frac {1}{2}}n+1})\\=&(-1)^{{\frac {1}{2}}n-1}r^{-{\frac {1}{8}}nn+{\frac {1}{4}}n}(1-r^{-{\frac {1}{2}}n-1})(1-r^{-{\frac {1}{2}}n-2})(1-r^{-{\frac {1}{2}}n-3})\ldots (1-r^{-n+1})\end{aligned}}$$

adeoque

$$W=2(-1)^{{\frac {3}{4}}n-2}r^{-{\frac {1}{16}}nn}(1-r^{-{\frac {1}{2}}n-1})(1-r^{-{\frac {1}{2}}n-2})(1-r^{-{\frac {1}{2}}n-3})\ldots (1-r^{-n+1})$$

Multiplicando hunc valorem ipsius ${\textstyle W}$ per prius inventum, adiungendoque utrimque factorem ${\textstyle 1-r^{-{\frac {1}{2}}n}}$, prodit

$$(1-r^{-{\frac {1}{2}}n})W^{2}=4(-1)^{n-3}r^{-{\frac {1}{4}}n}(1-r^{-1})(1-r^{-2})(1-r^{-3})\ldots (1-r^{-n+1})$$

Sed fit

$${\begin{alignedat}{2}1-r^{-{\frac {1}{2}}n}&=2\\(-1)^{n-3}&=-1\\r^{-{\frac {1}{4}}n}&=-r^{{\frac {1}{4}}n}\\(1-r^{-1})(1-r^{-2})(1-r^{-3})&\ldots (1-r^{-n+1})=n\end{alignedat}}$$

Unde tandem concluditur