292 
books). At the same time the advantage of the expression (2) for the 
remainder in the binomial-series will be obvious. 
6. If «20, we find according to (1) if we choose n > m: 
Bn Sy wm) |, hence lim Rk, = 0 if the series converges. 
n= 
7. If 1e <0, its ensues, from; (2), -if we, take p=1: 
Al << A|maum |, in which A is the greatest of the 
numbers 1 and (4 + mt. This leads to lim KR, = 0. 
ns 
8. If v=—l, m > 0, it follows from (3) ‘by taking p= m: 
(Ott) 
Al ME a SM ee) 
In connexion with N°. 3 it follows that lun A, = 0 (the inequality 
NZL 
m—1 > —1 being satisfied), a result that cannot be obtained 
from LAGRANGE's or CavcHy’s form of the remainder if m < 1. 
It ean further be observed, that (3) leads to the identity 
_m(m—l) PTO Ne (ei), 
1 — mt ee + (— 1)" ———-—— af = 
„(mln —.2),. … (m — zo) 
== (== 15) —— Es ES ee =F 
1. 
the identity can also easily be demonstrated by mathematical induction, 
from which it appears that the identity also holds good for m <0. 
9. If we make use of Aper’s theorem about the continuity of 
power-series, the examination of the remainder in the case | wv | <1 
suffices. In order to demonstrate, without distinguishing various cases, 
that then im FH, == 0; in_(2) we take p= 4 (as in NO. 7): 
n= 
