8 Colorado College Studies. 



The form which the fallacy usually takes is the assumption 

 that when two infinite series are multiplied together, if the 

 factor series are both convergent, the product series will also 

 converge, and that to a sum which is the product of the 

 sums of the factor series. This proposition is not only 

 unproved but false; but though it is not explicitly stated, the 

 student is led to assume it, which of course he does without 

 hesitation. 



The distinction here required is that between merely con- 

 vergent and (so-called) absolutely convergent series, under- 

 standing by the latter such a series as remains convergent 

 when all of its negative terms are made positive without 

 change of numerical value, so that in the transformed series 

 all the terms are of like sign. Abel furnished a rigorous 

 proof of the binomial formula, by establishing that when 

 two absolutely convergent series are multiplied the product 

 is a convergent series; and that whenever the multiplication 

 of two convergent series yields a convergent result, the sum 

 which the product-series approaches is the product of the 

 sums approached by the factor-series. But Abel's proof has 

 generally been regarded as too abstruse for an elementary 

 work. 



The development of (j? + 2/) " ^hen n is of the form — m 



or {m, p, and q being positive integers), may be obtained 



a 



by the following method, which is believed to be both rigorous 

 and elementary, and involves the demonstration of a theorem 

 which forms a special case only of that of Abel. I assume that 

 the case in which n is a positive integer has been proved by 

 either of the usual rigorous methods {e. g., that of induction), 

 and that the ordinary definitions and theorems concerning con- 

 vergence have been given. From one of these, viz. : that " a 

 series whose successive terms diminish indefinitely is conver- 

 gent," (and I would add, absolutely convergent) "when the 

 ratio of the m th term to the term preceding, as m is indefin- 

 itely increased, approaches a constant value which is numer- 



