41 
already animadverted upon at page 35—that the series (7) is 
strictly the differential of the series (8) which involves the 
term 1.2.3...(u—1)¢", m 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 m terms, so also must 
st: if therefore a new term ¢ be prefixed to — sé, in order that 
t— st may commence with the same terms as the series (8), 
the series ¢ — st will contain n + 1 terms; that is, however 
great m may be, ¢ — s¢ will contain, besides the whole of the 
series (8), an additional term still more remote: so that if 
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] + 1.2—1.2.3 + .... = °596347362324 
1—1.2 + 1.2.8 —....... = °621449624236 
1—1.2.3 + 1.2.3.4.5..... = *343279002556 
&e. &e. 
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 tha Ne 
series. This will be manifest from what follows. 
VOL, III, E 
