408 JACKSON. 



term in the braces, without the factor pn~'^-~, may be WTitten in the 

 form 



(38) gnrCOtiw/v) Q—mTigh'i — h'ii-\ — — + (w+«>.)5rC0t {-k/v) —{iio-\-in)Tri . 



The factor e'"""*, of course, is ec{ual to ( — 1)". The third factor 

 approaches a limit Hi—Hd as n becomes infinite. The second term 

 in the braces, without the power of pn, may be WTitten similarly as 

 ^rnt cot (ir/;-) gTiTTi niultiplied by a factor which approaches the limit 

 Hi + Hii. Consequently the whole expression in braces is equal to 

 ( — l)"pn~^~^ gWJT cot u/;-) multiplied by a quantity which approaches 

 2Hi. It follows that the absolute value of the expression is from 

 some point on greater than 



?_ pmrCOt{Tr/v) 



where Cg is a positive constant, provided that Hi 9^0. 



The condition that Hi be ec{ual to zero is that the imaginary part 

 of the limiting value approached by the exponent of the third factor 

 in (38), namely, the expression 



(q + 2)Ti 



V 



be an odd multiple of ^iri. There is no apparent reason why this 

 condition should not be satisfied, in particular cases But it can 

 surely not be satisfied for two successive values of q. We may accord- 

 ingly adopt the following rule, with the assurance that the indicated 

 choice of q is always possible: 



Let iV be any positive integer. Then q shall be the first integer 

 greater than or equal to N, for which Hi ^ 0. 



Assuming that q is so chosen, we infer from (37) that 



/„ 



/ (x) Vn (x) dx 

 



''^ ,9+2 



from some point on, and from this inequality and (36), that 



Clo 



an 



n' 



g+2' 



where Cg and Cjo are positive. 



The remainder of the proof that the series diverges need not be 

 elaborated. The details are suggested partly by the special discus- 



