Mathematics. — “Observations on the expansion of a function in 
a series of factorials.” Ul. By Dr. H. B. A. Bockwinxet. 
(Communicated by Prof. H. A. Lorenz). 
(Communicated in the meeting of May 3, 1919). 
10. We shall now prove that the theorem of NreLsEN is znexact. 
For this we use the following lemma: 
co 
Lf the series Ss: an diverges in such a way that the upper limit 
0 
of the sum 
n 
Sn = Sin am oe Shes kis pce ee (35) 
0 
for n= ts equivalent to n°, where @ is a certain positive number, 
then the series 
oo 
—~ An 
eS Mer nnn 
0 
converges or diverges according as a>>0 or a< 8. 
Summation by parts gives 
n n—1 
Amn Sn au 1 1 
SS SS n Sm ee > 5 ° 37 
~ mt ne zr >» sy (= <i) i) 
0 0 
For a > 0, the limit of the expression s/n is, by the hypothesis, 
zero for n= oo. Further the upper limit, for m= oo of the general 
term of the series in the right-hand member is equivalent to m—@1*), 
if dis a certain positive number, so that the series converges if n 
is made infinite. So the series in the left-hand member, too, converges 
fors soon 
Again, writing 
n 
hE Am 
Sa, n= INGE) . © . . 5 . . (36 ’) 
me 
0 
we find on summation by parts 
n n n—1 
=e as a Am 
) n= Nr me te = NI San — mM Som [(m +- Le — m*] (38) 
0 0 0 
From this equality we may infer that, for a< 6, the upper limit 
