45 



proposed exceeds the powers of computation more and more 

 as g approaches to 1 ; involving at length terms infinitely 

 great, and thus tending to no finite limit. In other words, 

 however many terms of this series be summed, the results 

 would diverge more and more from zero as g approaches to 1 ; 

 and would actually become infinite when g reaches this limit. 

 The conclusion, therefore, that P — 2- -f- 3^— . . . = is, as 

 in the other instances discussed in this Essay, erroneous to an 

 infinite extent : and it thus affords one more example of the 

 truth of the doctrine here advanced. 



The general analytical principle announced above has 

 been misapplied, or improperly neglected, in many important 

 inquiries connected with series. It may not be uninstructive 

 to advert more particularly to some instances of this. 



At page 267 of the second volume of his works, Abel has 

 the following remark: " On pent demontrer rigoureusement 

 qu'on aura, pour toutes les valeurs de x inferieures a tt, 



^ z: sin .?; — ^ sin 1x ■\- ^ sin 3*' — &c. 



II semble qu'on pourrait conclure que la meme formule aurait 

 lieu pour x-=.ir; mais cela donnerait 



and as ( ) is itself less than 1 for every finite value of n, however 



great, it follows that g may approach so near to 1 as to postpone the point 

 of convergency beyond any finite limit ; which is tantamount to saying that 

 this point can never actually be reached. The series, therefore, cannot tend 

 to merge into zero as g approaches to 1 ; so that zero is not the limit to 

 which the series continuously approaches as g approaches continuously to 1 ; 

 and therefore the general principle stated in the text does not countenance 



the conclusion that l'^ — 2^ + 3^ — =0. 



I cannot help regarding the criterion of convergency proposed by Cauchy 

 (Cours d' Analyse, p. 152) as open to objection ; since, according to it, we 

 should pronounce a series to be convergent under circumstances in which 

 the point of convergency would be postponed beyond any finite limits : more- 

 over, what security have we that neutrality may not have place before diver- 

 gency commences ? 



