Rit (Vísindafélag Íslendinga) - 01.06.1946, Page 38

38
If -JVx Ms a very small quantity the summation of the
right hand side of (69) may be replaced by an integration
where ~'\ly<.t=-dz and therefore
b
'du0J
dx
2 T
TC^Jxt.
'dz
(69a)
which is a well known formula if b is infinitely great.
6. REMARKS ON DIVERGENCE
Probably somebody does not agree to my treatment of
the divergent series, and an explanatory note will, there-
fore, be needed:
I discern sharply between the infinite series which I
denote with the sign 'y anx" and the infinite series which
n = 0
00
is denoted as anxn. The common definition of the last
n — 0
mentioned series may be stated as follows:
Let
■ f (n,x) = a0 + atx -j- a2x -j- .... + anxn (70)
Then
00
V anxn = lim f(n,x) (71)
7z = 0 n —> oo
From (70) we conclude that
dn + rf(n,x)
dx
n + r
0, r= 1,2,3,...
02)
and as this is valid for all values of n it holds also when