Proof of Binomial Theohem. 9 



ically less than unity," may be deduced the conclusion that a 

 series in the form 



n n 11—1 n n—\ n—1 



1 + — a- 4- — . ,r^ + — . . .V +.... 



1 12 12 3 



is absolutely convergent without regard to the value oi n, pro- 

 vided X has a value numerically less than one. 



I require to establish the following definitions and the- 

 orems: 



Definition. — By the product of two series (whether finite or 

 convergent or divergent), is meant a new series formed by mul- 

 tiplying the several terms of the one series by those of the 

 other, and grouping the partial products as in the example 

 below; viz.: so that any term (the 7nt\\) of the product-series 

 consists of the sum of m- partial products (monomials), being 

 all of those partial products wherein the sum of the term-num- 

 bers of the two factors is m -\- 1. (By "term-number " I mean 

 the number of the term formed by the factor in question in the 

 series from which it is taken. Thus the third term of the mul- 

 tiplier has the term-number 3, and the second term of the mul- 

 tiplier has 2, and ths product of these two factors, as the sum of 

 the term-numbers is 5, will form a part of the 4th term of the 

 product. In the example the term-numbers appear as sub- 

 scripts). For example, multiply 



Ol + «2 + O3 -t- O4 + 



by h, + b, + b, -h \ + 



Qibi + a.Jji + 0361 + a^bi + 



+ a^b.^ + a^b^ + a^b.2 + 



+ ttibi + a^bs -f 



+ ai?>4 + 



1st, 2d, 3d, 4th, term of the product. 



As a second example, the first four terms of the product of 



the series a -\- b -\- c -\- d -\- . . . . by m -{- n -\- j) -{- q -\- 



are (am) + [an + bni) -\- (ap + hn + cm) -\- {aq -\- 6j)+ 

 en -f dm). 



Theorem. If the product of two absolutely converijent series 



