relatim to Geometrical Series. 199 



•to 



It may be observed that this formula retains the same signs 

 whether r be positive or negative. 



(5.) If each side of theorem I. be multiplied by the quan- 

 tity 



1+r + r -f- ... — r +r , 



s n ;j we have 



1— r + r — ... — r -\-r 



Theorem II. 



1 +r2'» ^^2.2»*_^ ,.(''»-2)2'»_^_^(m-l)2n 



1-r + r- ,,m-2_^^m-l 



= {l+r + rV r— Vr-^'}X 



{l-r%r^-. ^(.-2)2^^.(-i)2| ^ &C.&C.&C 



X |^l_,.2%>2!:^c r^m-2)2''-]im-^)2''-^^^ tO 72 



factors of m terms each ; or, 



l+/V,-''-^+&C ^(m-2)!i'^^(m-l)2'' 



, , 2 o w— 2 m-1 



1— r + r — &c r -j-r 



— J -1 . .2,0 m—2 , w — 11 -. 



= |l+r + r+&c r +r ]• x 



2{ !_,%.«_ +,(™-')2}n-l . 



(6.) Cor. y^r'='\r^'''\ ,.(^-.)^^" 



- ^ , 2A- (m-\)k 



1 — -r 4-r — r^ 



Theorem III. 



(7.) Let ^, 5^, r be any numbers of which p y q and m an 

 odd number, then 



l +r^ + r^-^&c ,.(«-ii)i!%,.(^-l)ii'' 



X+/+r''-^\^C ,.(m-2)2t,.^(».-:)2' 



