An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

~ 130 — 131.

Condition for integration to be possible.

227

and this series likewise converges uniformly in the entire interval
from xo to x,.
The Theorems proved concerning the definite integral show that
these investigations can be extended to series representing functions
that are discontinuous or infinite in separate points, or again, to
the definite integral with an infinite limit, always on the hypothesis
that the series of the integral functions remains convergent.
The examples adduced in last ~ can be regarded inversely also as
examples for the integration of infinite series.
We cite the following, due to Darboux, as an example in which
the integration is not effected, although the series of the integral
functions converges:
F(x) =xe-x2 = Z(xe - n( - n + ixe-+1x2)
n-l
is a convergent series for all values of x and a continuous function,
although the series converges unequably in the neighbourhood of the
point x == 0. In fact Rn (x) = n x e-" and for x -= it becomes
Y-2n
-R e- 2
e/2
By integrating the individual terms between 0 and x we obtain:
x x
Jf xe —nx2dx -f(n + l)xe-n+ +Lx =- - e- + -e-n+l
u 0
The infinite series formed of these integral functions:
n= co
n=l
converges, it expresses the function - Ie-x2 for every finite value
of x, but for x= O its value is 0, it is therefore a discontinuous
function at this point and not in general equal to:
x x
J F(x) dIx =Sxc 2dx - (1 -- e-).
0 0
131. Applying these Theorems to a series ascending by powers
of any continuous function f(x):
ao, c+ a,f(x) + c(2 { f(x) } * + + a-n I f(X) } n +.,
we see:
First: Such a series is a continuous function of x within its
convergency; for, if we put f(x) = S, the absolutely convergent series:
aO + a1 z + a22 + * * * + an1 + -..
15*