11 
12. It is convenient to observe on this occasion that by means of 
the proposition of N°. 10 we may infer from the first part of for- 
mula (43) that never convergence of the series of factorials (6) takes 
place for Riv) < 4, a truth already shewn in another way in N°.9 
of the precediug communication. 
For, if the upper limit of s„ for no is equivalent to „, then 
the function represented by the series 
oo 
D n8nt? 
0 
is, for t=1, at most of order 1: (1—7d)!+* and hence gp (A) is at 
most of order 1: (1—7)’, so that the number 2 is not greater than 0, 
whereas the limit between convergence and divergence of the series 
of factorials according to the lemma mentioned, is afforded by R(x) = 0. 
13. At the end of the first communication we said that, if a 
certain argument mentioned there and used by NrierseN were not 
erroneous, then another very general case of the possibility of expand- 
ing the integral 
1 
2 («) = (po agrar. Er ED 
‘0 
in a series of factorials would be proved by it, viz. the case when 
the series of factorials corresponding to that integral converges. We 
shall now prove that this case, though not established by 
NIELSEN, is, indeed, exact. In order to do this we shall make use 
of the following lemmas: 
l. If the series of factorials converges for a certain value x= c, 
then it converyes for any value of « whose real part R(x) is greater 
than that of c. 
Let 
mI! am 
m Pom = Sn, ly . . . 5 . . (44) 
n 
Siep we find on denn by parts 
ml! an net m! am Pe+m+1)_ 
Brey +1) oem P(c+m-+ 1) POS 
l-1 
Beene (el 1) 
M(@ +141) 
as Sn mn T(c 4+- m+ da) 
br P(e +m + 2) 
