Proof of Binomial Theorem. 11 



Corollary. If, instead of absolutely convergent series, any 

 of the three series mentioned in the foregoing theorem, as fac- 

 tors or product, should consist of a finite number of terms, the 

 theorem will still apply, for a finite series may be regarded as a 

 convergent one in which all the terms beyond a certain number 

 vanish. 



Theorem. ■ In an equation containing several literal quanti- 

 ties, when each member is reducible to a finite number of 

 rational terms, if the equation be true for an unlimited number 

 of values for each literal quantity, while the selection of partic- 

 ular \*alues for one or more of the latter does not afPect the pos- 

 sible values assignable to the others, the numbers of the equa- 

 tion must be identically equal.* 



For when we have chosen particular values of all the letters 

 but one, the equation, if it were not an identity, would deter- 

 mine the value of the remaining letter, so that it could not have 

 an indefinite number of values. By selecting successively sev- 

 eral different sets of values for all but one of the letters we may 

 obtain as many different equations as we please for the remain- 

 ing letter, say as many as there are different powers of that let- 

 ter in the equation, and this group of equations must by the 

 terms of the hypothesis be satisfied by any one of an indefinite 

 number of values of that letter, whereas if the original equa- 

 tion were not identical, such a group of equations would de- 

 termine linearly each power of the quantity to a single value. 



Definition. By "a series in the form 



n n n — 1 



1 + — iK + — . x'' +...." 



I 1 2 



is meant any series derived from this by assigning to n some 

 particular value, whether positive or negative, integral or frac- 

 tional. 



Theorem. If two series, each in the form 



II n n—\ 



1 H X ^ . .-c^ + 



1 1 2 



• Note that the hypothesis Is not that each quantity may take any value whatever, 

 ( which would make the theorem tautological ) but that it may take any one of an 

 indefinite number of values. 



