10 Colorado College Studies. 



be convergent, the sum which it approaches is the product of 

 the sums approached by the two series which are its factors. 



Consider first the case in which all the terms of both series 

 are positive. If we denote by r some definite number, and sup- 

 pose each factor-series extended to and indefinitely beyond the 

 rth term, the first r terms in the product will contain all the 

 partial products in which the sum of the term-numbers does not 

 exceed r + 1. The aggregate of these partial products we will 

 denote by A. The complete product of the first r terms of the 

 first series by the first r terms of tlie second contains, in addi- 

 tion to the foregoing, those partial products in which, although 

 the sum of the two term-numbers exceeds r -f- 1, yet neither of 

 them separately exceeds r. The aggregate of these additional 

 partial products we will call B. And finally, the complete pro- 

 duct of the two series contains in addition to both A and B, 

 those partial products (the limit of whose sum we will call C) 

 in which the term-number of either factor exceeds r. Now let 

 us suppose the number r to be increased. Since the product is 

 a convergent series, the sum A of the first r terms approaches 

 the whole sum of the series. Hence B + C approaches zero. 

 But since both B and C consist of positive terms only, if 

 B + C approaches zero, B and C must each approach zero as r 

 is increased. Hence the sum A + B approaches the same sum 

 as A. But A + B contains the entire product of the first r 

 terms of the two series, and as r is increased the sum of the 

 first r terms approaches in each series the sum of that series. 

 Hence the sum which the product-series approaches is identi- 

 cal with the product of the sums of the separate series. 



Next, if either series or both contains negative terms, some 

 of the partial products will be negative, but if B approaches 

 zero when all the partial products which compose it are posi- 

 tive, it will certainly do so when some of them are negative, for 

 the sum of a number of quantities of two opposite signs must 

 always be nearer zero than the sum of the positive values of 

 the same quantities. Therefore, as before, the sum which A 

 and B approaches is the same as that which A approaches, and 

 the same conclusion follows as in the previous case. 



