178 SERIES OF FUNCTIONS 
If thef H ... la are not integrable, we have 
f J P=2j/ ll ...... 
«Ai «Al 
Example. 
co 
^=2 
o (1 + wx 2 )(l + (ro + l)^ 2 ) 
does not converge uniformly at x— 0. Cf. 140, Ex. 8. 
1 
F„ = 1 - 
fl 
1 + nx 2 
for X =#= 0, 
for # = 0. 
Í(I) - to 
CfcIx = 1, 
«A 
,J dx 
_ 1 _ arctg Vn ^ 
■y/n 
Thus we can integrate F termwise although F does not converge 
uniformly in (0, 1). 
151. That uniform convergence of the series 
-^O) =/iO) +/ 2 0) + ••• (1 
with integrable terms, in the interval 2Í = (a < 5) is a sufficient 
condition for the validity of the relation 
J r»6 rb fb 
I Fdx= I f 1 dx+ I f^dx + ••• 
a *■' a a 
is well illustrated graphically, as Osgood has shown.* 
Since 1) converges uniformly in ?l by hypothesis, we have 
and 
for any x in 21. 
F n (x) = F(x) - F n (x) 
I F n (x) I < e n > m 
* Bulletin Amer. Math. Soc. (2), vol. 3, p. 59.