SERIERUM SINGULARIUM 
29 
sive 
W = 2(1 — r A )(1— 2 ) (1 — r —3 ) (l-f-r“ 4 ). . . (1— r ~* n+i ) . . [8] 
Porro quum sit r* n = — i, adeoqne 
l —(— t 2 = — r**~ 2 (l—r-****) 
1 + r“ 4 = — T**-* (l — r~* n+4 ) 
1-f r~ 6 = — r*" —6 (1 __ r —1"+8) etc. 
productumque e factoribus —r* n ~~ 2 , —r***“ 4 , — r * n ~' b etc. usque ad —r l fiat 
= (— r T* nn ~i n ^ aequatio praecedens ita quoque exhiberi potest 
W= 2(—l) in ~ 1 r™ nn ~ in (1—r _1 ) (1 — r~ 2 ) (1—r -3 ) (1—r“ 4 ). . . .(1 — r-* n+l ) 
Quum habeatur 
1 — r 1 = —t 1 (1—r~ n+l ) 
1—r~ 2 = — r~ 2 (1—r~ n+2 ) 
1—r -3 = — r~ 3 (1—r~ n+3 ) etc. 
erit 
(1—r _1 )(l —r _2 )(l — r“ 3 ). . .(1 — r~* n+1 ) 
— ( l)* nr ~ i r — r —in— 1) (1 r -4n-2j Jj_ r —t«-3j _ _ 
adeoque 
W — 2 ( I)!»“ 2 ij ^ r —J«-j j r -\n-3^ _ _ _ q «+!) 
Multiplicando hunc valorem ipsius W per prius inventum, adiungendoque utrim 
que factorem 1 — r~^ n , prodit 
w 2 = 4 (— 1 ) n ~~ 3 r~~* n {1 — 0(1 — r" 2 )(l—r- z ). . . .(1 — r - n+i ) 
Sed fit 
1 — r“*" = 2 
(--l) n “ 3 = —1 
r —in __ r \n 
(1 r _1 )(l—r _2 )(l —r“ 3 ). . . . (1 — r~ n+1 ) = n 
Unde tandem concluditur