45 
proposed exceeds the powers of computation more and more 
as y 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 | ; 
and would actually become infinite when g reaches this limit. 
The conclusion, therefore, that 1?—2? + 3?—...= 0 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 peut démontrer rigoureusement 
qu’on aura, pour toutes les valeurs de  inferieures a 7, 
5 = sine — dsin22+4sin32 — &c. 
Il semble qu’on pourrait conclure que la meme formule aurait 
lieu pour c= 7; mais cela donnerait 
2 
and as (1) is itself less than ] 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 1? — 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 ? 
