DE MOIYUE. 179 



The general term of the expansion of (1 — r)~^ in powers of 



r is ^-^ '" r^ ; the sum of the first n terms of 



the expansion is equivalent to the following expression 



^ ^ 1.2 ^ ^ n—\ i^J— 1 



(l-r)" 



This may be easily shewn to be true when n= 1, and then, 

 by induction, it may be shewn to be generally true. For 



r«+i = r"|l-(l-r)}, 

 so that 



r-^^ + (^ + 1) r"^^ (1 - r) + ^'' "^ ^ ^l "^ ^^ r""^^ (1 _ r)^ + . . . 

 = r« |l _ (1 _,.)} +(«+!) r» (1 - r) |l - (1 - r)} 



n{n-\-V\ \n + p— 2 



= ^" + ^^" (1 -r) + ""^V ^'" (1 -0'+ • • • + ^ Ti 1 ''" (1 - ^')' 



^ ^ l.z ^ ^ ?i — Iw — 1 ^ 



P 



^ ,.« (1 _ ^)p. 



\n\j) — 1 



Thus the additional term obtained by changing oi into n + 1 



\n -\-p— 1 



is I — , r- r" as it should be ; so that if De Moivre's theorem is 



\np—\. ' 



true for any value of ?2, it is true when n is changed into ii-\-l. 



321. Another theorem may be noticed ; it is enunciated by 

 De Moivre on his page 229. Having given the scales of relation 

 of two Recurring Series, it is required to find the scale of relation 

 of the Series arising from the product of corresponding terms. 



For example, let w^^r" be the general term in the expansion 

 according to powers of r of a proper Algebraical fraction of which 

 the denominator is 1 —fr 4- gr'^ ; and let t'„a" be the general term 

 in the expansion according to powers of a of a proper Algebraical 



12—2 



