#7 
We cannot say, however, that this proposition is a substitute for 
that of NierseN, because a series of factorials sometimes converges 
for Re) < w,. Take, for instance, the function of Werersrrass 
k 
VE eM Ry eh aria a oe? nef ee ee 
This function has its cirele of convergence as a natural limit, 
because all parts of the circumference are equivalent, as appears 
from the substitution 
rn ih 
k 
(Meee ger eee 
where k and 4 are arbitrary integral numbers. The order w, in the 
point t=1 cannot, therefore, differ from the order w along the 
whole circumference, and this order is equal to 1, the characteristic 
d’ of the coefficients of the power-series being zero. The quantity à 
is, however, also zero and the series of factorials converges for 
Rv) > 0 and is, for all these values of 2, equal to the integral (1) *). 
1) The convergence of the series of factorials is, in this case, absolute, in the 
strip of plane 0 < R(x) <1, owing to the large distance between the coefficients 
of p(t) which differ from zero. If, therefore, we spoke continually in the preceding 
investigations of conditional convergence for R(x)< A’ +1, we meant this in 
general. 
Proceedings Royal Acad. Amsterdam. Vol. XXII. 
