14 Colorado College Studies. 



the product-series are both finite. Hence the sum approached 

 by the given series, multiplied by the sum of the other factor- 

 series [which is (1 + x)^'] will be equal to the sum of the 

 product series, which is (1 + x)^ + ^. Therefore, the limit 

 required is (1 + x^' + '» -^ (1 + a?)"^ or (1 + a?)". 

 Theorem. — The limit of the sum of the series 

 n n n — 1 



1 1 2 



P 

 in which n has any fractional value — , (when p and q are in- 



tegers and q is positive), and in which x is numerically less than 



_p 

 unity, is a value of the radical (1 + a?) ^ or (1 + x)^ . 



From the form of this series, and the fact that in it x is 

 numerically less than unity, it is known that the series is abso- 

 lutely convergent. Let y represent its limit, and let the given 

 series be multiplied by itself. The resulting series is in the 



form 



n n n—1 



I + - X + -. a;' + 



1 1 2 



2p 

 {n having the value — ) and, in consideration of the value of x 



is known to be absolutely convergent. Hence its sum ap- 

 proaches the limit y' Let the operation of multiplying by the 

 given series be repeated until that series has been used q times 

 as a factor. It is proved as at the first multiplication that the 

 sum of each product-series approaches as a limit one of the suc- 

 cessive powers of y, the last being y'^ ; also that the last series 



is in the form 



n 11 n — 1 



1 H X ^ . x^ +. . • , 



1 1 2 



QP 

 where n has the value — or p. But by the previous theorem 



the limit of this series is (1 + a;)P. Hence y" = {1 + x) p ov 



P 

 y = {1 -\- x) — , as was to be shown. 



g. 



