4 Mr Pollard, On the term by term integration 



00 



(b) 2 u^ (x) converges for X'^a, 



n=\ 



rX m rX CO 



(c) lim S Un (oc) dx, 2 ?/„ {x) dx, 



m-*-ao J a n=l J an=l 



exist and are equal for all X ; then both sides of {!) exist and are 

 equal. 



rX m 



Proof. Write S u^ {cc) dx = Sm, x, 



J a n=l 



and let lim Sm,x = S. 



m-*-'x>,X^'OD 



Since I w„ («) dx converges for all n 



J a 



rX m 

 lim % u^{x)dx, i.e. lim Sm,. 



X 



exists for all m. Hence 



lim (lim >S^,x) ='S^ (3)» 



Wl^-oo X-^cc 



by(«)- . 



In virtue of (b) and (c) 



rX m ^ 



lim 1 2 Un (^) dx 



m^-as J a n=l 

 rX 00 

 exists and is equal to 2 u^ (x) dx. 

 Jan=l 



Thus lira Sm, x exists for X ^ a. 



Hence lim (lim Sm,x) exists and is equal to S. 



X-^-CC 77l-*-00 



rX 00 

 Taking lim S^, x in the form 2 u^ {^) dx, we see that 



m-*-oo J an = l 



/-■» 00 



S Un (^) dx = S. (4) 



J an=l 



And (3) and (4) give us our theorem. 



00 



II. J/ 2 I w„ (x) I converges for x^a and the double limit 



rXm 



rXm 



lim 2 I w„ (a;) I dx 



■00, X->-<x) J an=\ 



exists and is finite; then without further condition both sides of (1) 

 exist and are finite and equal. 



