3T 



It is plain that, however far such a series as this be ex- 

 extended, a supplementary correction is always necessary to 

 complete the equation ; which correction must be infinite in 

 value if the series be infinitely extended : and the analytical 

 considerations offered above fully accord with this statement, 

 the contrary of which could never have been entertained had 

 not analysis seemed to justify the strange conclusion. All 

 that analysis really authorizes us in saying, in reference to the 

 extreme cases here considered, is — as the French analysts ex- 

 press it — that " la methode ordinaire est en defaute." 



Having mentioned the name of Abel in connexion with 

 this subject, it may not be out of place to notice here, that 

 that distinguished genius seemed inclined to trace the erro- 

 neous results above to another cause : " On applique aux 

 series infinies toutes les operations, comme si elles 6taient 

 finies ; mais cela est-il bien permis? Je crois que non. Ou 

 est il demontre qu'on obtient la differentielle d'un serie infinie 

 en en prenant la differentielle de chaque terme?" And he 

 then adduces the result, 



^ = cos X — cos 2x + cos 3x — &c. 



■which he pronounces to be " resultat tout faux."* 



But I submit that no such results of differentiation can 

 ever be absurd, unless the absurdity attaches to one or more 

 of the individual terms. 



In the formjer part of this paper the examination was re- 

 stricted to those classes of diverging series which arise from 

 the development of fractions into geometrical series, and from 

 the expansion of a binomial : but it is plain that the reasonings, 

 in reference to the former developments, equally apply to 



f(x) 

 those which arise from any fraction -~ ; and the reasoninor, 



(pix) ° 



in reference to the latter, equally applies to any root or power 

 of (x). And, in what is shewn above, we see how divergent 



* QSuvres, tome ii. p. 208. 



