( 851 } 
b b 
7. op 
| F (#v) da = sf Un (ve) dw. 
0 
a a 
We may not conclude to this last equation as soon as one of the 
two limits of the domain of integration is just excluded from the 
interval of the uniform convergence. 
Wellknown instances have shown, that in that case the rule of 
the integration term-by-term may hold or not. 
oo 
Instances. The series % Up (w) converges uniformly for Of <1. 
0 
an 
I. 3% uU,(x) converges for x=. 
0 
Un (©) = (n+ 1) (1 — 2) ©" — (n+ 2) (1 — x) wrth. 
l 1 
Rit (is Ik adn - ; Ef u ds == 2 : 
2 0 2 
v 
0 
Un (2) —=N (1 — x) e—"A—#)? — (n +1) (1 — x) er ea—#., 
1 1 
Pin==0, i EU) ie : % | Un (x) dx == — = ; 
0 
0 0 
3 
eo = ahr (1 — x) (n+1)?2 
un (= 7 EAD 1) +17 
‘ 
‘ 
S 
Ea) = 0; fe (eda ==0~ 5 if u” (x) dx diverges. 
0 0 
I: S Un (x) diverges for ~=—1. 
0 
Un (x) (— 1)" aw, 
1 
(x)= Ln Fw) da — log 3 fu (alde == tog 8 
v = 142 ’ : AH « = gs ’ pe n \e “ Og e 
0 0 
Un (©) — (n + 1) 4" — (n+ 2) wrt, 
1 1 
Pint 1, [ro del, if Un (x) dr — 0. 
0 
0 0 
Un (2) == (—1)" (n+ 1) 2". 
_ 
l 
1 i! 2 ; 
ENT fF « dx — =- ’ Da Un HF; 1 , ar a 
F (x) der ah (x) da 5 if (x) de diverges 
0 0 
