Proof of Binomial Theorem. 13 



perform the additions. But it is not necessary to take this 

 trouble, for any of these identities may be established by the 

 application of the forei^oing theorem. For they are known to 

 hold true for an infinite number of values of s and /, viz., for 

 all their positive integral values, since, the binomial theorem 

 having been proved for such values of the exponent, we know 

 that the factor-series are equal to ( 1 + •I'Y and {1 -\- xY re- 

 spectively, and the product- series to(l-|-,r)* + '. Hence the 

 coefficients of the first r powers of x in the product of the two 

 series 



s s s—1 t t t—1 



1 H X -\ . x'^ + etc., and I -\ x -\ . x'^ + etc., 



112 112 



are identical with the coefficients of the like powers of x in the 



series 



S'+t .s-^t s-\-t—l 



1 H X -\ . x' -\-.... 



1 1 2 



however great r may be. 



Theorem. — The limit of the sum of the series 



n 11 11 — 1 



1 + -.r + -. x^ +...., 



1 1 2 



in which n is any negative integer and x is numerically less 



than unity, is (1 + xY . 



For a series in this form, being multiplied by another series 



in the same form 



m m m — 1 



I + - X + -. X' +...., 



I 1 2 



in which m is any positive integer numerically greater than n, 

 will yield as a product the series 



in -\- 11 m -\- n m + n — 1 



1 H X -\ . X-' +...., 



1 1 2 



which, since m + « is a positive integer, is a series of a finite 



number of terms, and known to be equal to (1 + a;)"* ^ ". Now 



since in the given series 



II n n—\ 



1 H X -\ . x^ -\- etc., 



1 1 2 



the value of x is numerically less than one, this series is abso- 

 lutely convergent; but the series by which it is multiplied and 



