o/ an infinite series over an infinite range 5 



rx m 

 Proof. By (7), if lim I S | w„ {x) \ dx exists, so does 



«-»■», X-*-ao J a n=X 

 rX m 



lim S w„ {x) dx. 



m-^-oo , X-*-<x> J an=l 

 CX m 



Also Sm, x=\ ^ \Un{x)\dx^ 8, 



J a n = l 



for all X and m. But Sm,x increases with X for each m. Hence 



rx m 



% \u„(x)\ 

 J an=l 



rX m 



lim I 2 I w„ (ic) I dx 

 X^oo J an=l 



exists for each m and is less than 8. 

 And therefore 



rX f rx m rX m-1 "j 



lim 1 I u„ (x) I dx = lim j I S 1 1*„ (x) \dx — I % \Un{x)\dx> , 

 X^-co J a X^i-co [J an=l J a n=l J 



exists for each m, i.e. 1 | ^^^ (/») | dx and therefore I ?/.„ (x) dx con- 



^ a J a 



verges for each m. This is (a) of (I). 



Again, 8m, x increases with m for each X. 



Hence lim S.m,,x exists and is finite for each X. So from 



(C 1) [X 00 



X \Un (x) I dx 

 Jan=l 



00 



is finite. Thus 2 | m„ (x) I is summable in (a, X). But 



mm 00 



71=1 n = l 71 = 1 



and so by (C 2) 



rx m rx CO 



lim I S Un (x) dx, I 2 w„ (a;) c?^, 



m-*-oo .' a n. = l ./a7i.= l 



exist and are finite and equal. This is (c) of (I). Now (b) of (I) 

 is satisfied by hypothesis. 



Thus all the conditions of (I) are satisfied and so both sides of 

 (1) exist and are equal. 



Deductions from the general theorems. 

 6. A. // u^{x) = (f>(x)f„{x), 



where 2 /„ (x) converges for x'^a, 



n = l 



V 



I ^ fn{x)\< G, for x'^a and all v, 



n = l 



