41 



already animadverted upon at page 35 — that the series (7) is 

 strictly the diiferential of the series (8) which involves the 

 term 1.2.3 . . . (n — l)t'\ n being infinitely great, and for the 

 differential of which the calculus seems to make no provision. 

 But, waiving these objections, the deduction (9) is palpably 

 erroneous, and altogether fatal to the final conclusion. For 

 the series s is evidently coextensive with the series (8), and 

 so, of course, is st; that is, if (8) contain n terms, so also must 

 st : if therefore a new term t be prefixed to — st, in order that 

 t — st may commence with the same terms as the series (8), 

 the series t — st will contain n + I terms; that is, however 

 great n may be, t — st will contain, besides the whole of the 

 series (8), an additional term still more remote : so that if n 

 be infinite, and we assume, as above, that the two series are 

 equal, we commit an error infinitely great. And this is the 

 error, thus introduced, which will be found to vitiate all 

 Euler's processes for summing divergent series by definite in- 

 tegrals : an error which obviously has no existence for the 

 convergent cases of those series ; since the additional term, 

 noticed above, is, in such cases, not infinite, but zero. We 

 may safely infer, therefore, that the results so often quoted in 

 analysis, viz. 



1-1 4. 1,2-^1.2.3 + .... = -596347362324 



1 — 1.2 4. 1.2.3 - zz -621449624236 



1 — 1.2.3 + 1.2.3.4.5 = -343279002556 



&c. &c. 



all involve errors infinitely great ; and this, as it ought to be, 

 is quite consistent with the common-sense view of diverging 

 infinite series. 



V. — There is another method of investigation by which 

 these erroneous results appear to be established : the method 

 suggested by the well-known differential theorem. But, as 

 in the processes already considered, so here, that theorem will 

 be found upon examination to be applicable only to convergent 

 series. This will be manifest from what follows. 



VOL. III. E 



