| 
eee aay 
| U(x) 
increases with wv. The series of the integrals 
oo (—1)*-1 
I'(s) { ——— 
9 ns 
converges for s >0O. The theorem therefore holds and we find for 
all positive values of s 
pol ao (___] )n—1 
{2 genes. 
ey +-1 1 ns 
0 
In general we shall often be able to use this theorem when eva- 
luating an integral of the form 
1 
[fog eae 
Suppose it possible to replace f(x) by a power series Y @, v" such that 
0 
oo 
the series = anr”g(x) diverges for «=1, but that the series of 
0 
the integrals 
1 
mn 
= au fer g (a) de 
0 
0 
is still convergent. Then this series will certainly be equal to the 
integral, if only g(z) does not change its sign in the domain of 
integration, because then all conditions under which the theorem 
holds are satisfied. 
We shall likewise, if the development 
. 5 
alt= > a, at 
C 
holds for all finite values of x, be allowed to conclude by means 
of the theorem to the equation 
a 
Go 
fre da = = n! ay, 
0 
0 
if this last development converges. 
