Mathematics. — “The Remainder in the Binomial-series.” By 
Prof. Frep. Scaun. (Communicated by Prof. D. J. Korreweg.) 
(Communicated in the meeting of June 28, 1919). 
Dn 
é ; 7 . mM). fe = (m) 
1. We consider the binomial-series XZ u Dad Dl de ee 
a j 
gnd 
sijl 
=H (m—f). We suppose « real and m not zero and not a 
= k=0 
positive integral number (there otherwise the series is finite). 
a. “The series is*convergent if vel <1, if e=1,m > —1 and 
if «= —i1, m > 0, divergent in the other cases. 
3. If |w|<1 and if |e)=1, m >—1 we have hm ul” 10); 
n= @ 
as appears from N°. 2. 
4. According to Mac Laurin’s series we have: 
dt)" — LS ae 
j=0 / 
in which the remainder is given by: 
En (1—6)"—p n—1 
a es m—k)(p 0), 
TE ae (p > 9) 
hence: 
(m) 
u, 
Hi rr 5 
(1 + Ox)n—m ( ) 
Me irs =p, m—p (m—1) 5) 
en 1+ 0e GE A VERN 
p ee) Kga a 
5. The aim of this paper is to show that lim R„=—=0 in all 
ns 
cases in which the series perros, that is to demonstrate for those 
cases the validity of (1 + 2)" = = u exclusively from the remainder 
in. 
in the series (which is done incorrectly in a great number of hand- 
1) ‘The numbers 6 occurring in these expressions are generally unequal, 6 depend- 
ing on ” and on p. 
