192 3Ir Jollife, On. Certain Trigonometrical Series which have 



A very trifling modification of the analysis which follows will 

 show that, so far as an interval which includes ^ = is concerned, 

 the same results hold for the series 



2a„+i (cos \n^ — cos \n+i^) cosec 1)9, 



where h is any fixed number. If either |(X„+i — A,„) or ^(Xn+i + X„) 

 is always an integral multiple of some fixed number b, then X„. 

 differs by a constant from an integral multiple of 26, and the series 

 is periodic with a period Tr/b. In this case the results which are 

 true for an interval which includes ^ = are true for any interval. 

 The particular series 2a«sinn^ corresponds to b = ^, X,,j=» +^. 



2. Since the sum of the series when ^ = is zero, it follows 

 that, for continuity at 6 = 0, the sum of the series, when 6 is 

 different from zero, must tend to zero as 6 tends to zero in any 

 manner. In particular, the sum when 6 = 7r/2\„, must tend to 

 zero, as n tends to infinity. 



When 6 = 7r/2\,j, let m be the integer such that 



Xm-i^ $ TT < \n6- 



It should be noticed that we may have m — l = n, and that 



When m—1 >n, cos \p-i6 — cos \p6 is positive, so long as p is 

 not greater than ?7i — 1, and consequently 



0{U^ + Uo+ ... +Um-i) 



> a-n (cos Xi6 — cos \n^) + «w-i (cos X„^ - cos A,„,_i 0). 

 Also, by Abel's lemma, 



6 (Um + U,n+i + ... + U,n+g) > "m (cOS X^O - 1) 



for all values of q. 



Hence the sum of the series is greater than 



[an COS \6 - {a,n-i - a„i) cos Xn-iO - am]IO, 



which, since a,„,_i ^ a^ and cos Xm-i ^ is negative, is greater than 



(an cos X^d- a„,)/6 = 2Xn («« - «,»,)/-"■ + bn, 



where bn denotes a„(l —cosXi6)/d and consequentl}^ tends to zero 

 as n tends to infinity. 



When m — 1 = ?i, we can divide the series up into 



(U^ + U2+ ... + Un) + (Um + Um+i +...), 



and, noticing that cos X^-i 6 = 0, we see that the sum is greater 

 than (an cos Xj^ — am)/ 6, as before. 



